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<pre>
        Proceedings of the Workshop
            “Logic & Cognition”
            European Summer School in Logic, Language, and Information
                                Opole, Poland
                              13-17 August, 2012


         edited by Jakub Szymanik and Rineke Verbrugge

          Copyright ⓒ 2011 for the individual papers by the papers' authors.
Copying permitted only for private and academic purposes. This volume is published and
                               copyrighted by its editors.
                                        Preface



1      The workshop theme


The roots of logic go back to antiquity, where it was mostly used as a tool for analyzing
human argumentation. In the 19th century Gottlob Frege, one of the founders of modern
logic and analytic philosophy, introduced anti-psychologism in the philosophy of mathe-
matics. In the following years anti-psychologism, the view that the nature of mathematical
truth is independent of human ideas, was one of the philosophical driving forces behind
the success of mathematical logic. During the same period in the 19th century, also mod-
ern psychology (Helmholtz, Wundt) was born. However, the notion of anti-psychologism
often stood in the way of a potential merge of the disciplines and led to a significant sepa-
ration between logic and psychology research agendas and methods. Only since the 1960s,
together with the growth of cognitive science inspired by the ‘mind as computer’ meta-
phor, the two disciplines have started to interact more and more. Today, we finally ob-
serve an increase in the collaborative effort between logicians, computer scientists, lin-
guists, cognitive scientists, philosophers, and psychologists.

Topics of the workshop revolve around empirical research motivated by logical theories
as well as logics inspired by experimental studies reflecting an increasing collaboration
between logicians and cognitive scientists.


2      Invited talks at the workshop

In addition to the contributed talks, of which the articles are gathered in this volume, the
workshop also presents two invited talks:
    • Paul Egré and David Ripley Vagueness and hysteresis: a case study in color cat-
         egorization
    • Iris van Rooij Rationality, intractability and the prospects of “as if” explanations
The abstracts of both presentations may be found in this volume.
3      Best papers

The Programme Committee of the workshop decided to award the Best Paper Prize to:
Nina Gierasimczuk, Han van der Maas, and Maartje Raijmakers for the paper Logical and
Psychological Analysis of Deductive Mastermind.

The Best Student Paper Prize was awarded to Fabian Schlotterbeck and Oliver Bott for
the paper Easy Solutions for a Hard Problem? The Computational Complexity of Recipro-
cals with Quantificational Antecedents.



4      Acknowledgements

We would like to thank all the people who helped to bring about the workshop Logic and
Cognition. First of all, we thank all invited speakers and contributed speakers for ensuring
an interesting conference.

Special thanks are due to the members of the program committee for their professionalism
and their dedication to select papers of quality and to provide authors with useful, con-
structive feedback during the in-depth reviewing process:

Program Committee:

Leon de Bruin
Eve Clark
Robin Clark
Paul Egré
Fritz Hamm
Alice ter Meulen
Marcin Mostowski
Maartje Raijmakers
Iris van Rooij
Keith Stenning
Marcin Zajenkowski


In addition, Catarina Dutilh Novaes and Raphael van Riel shared their expertise to help
review the papers. Thanks! We would also like to thank the organizers of the European
Summer School in Logic, Language and Information, Opole, 2012, that which hosts our
workshop.
Finally, we would like to express our gratitude to NWO for largely financing this work-
shop through Vici grant NWO 227-80-00, Cognitive Systems in Interaction: Logical and
Computational Models of Higher-Order Social Cognition.

Jakub Szymanik
Rineke Verbrugge
Groningen, July 2012
Vagueness and hysteresis: a case study in color
               categorization

                        Paul Egré and David Ripley

                      Institut Jean-Nicod (CNRS), Paris



  Abstract. This paper presents the first results of an experimental study
  concerning the semantic status of borderline cases of vague predicates.
  Our focus is on the particular case of color predicates (such as ‘yellow’,
  ‘orange’, ‘blue’, ‘green’), and on the influence of context in the catego-
  rization of color shades at the border between two color categories. In an
  unpublished study, D. Raffman and colleagues found that subjects have
  no difficulty in categorizing the same color shade as ‘blue’ or ‘green’ de-
  pending on the direction of the transition between the two categories,
  suggesting a phenomenon of hysteresis or persistence of the initial cate-
  gory. Hysteresis is a particularly intriguing phenomenon for vagueness for
  two reasons: i) it seems to comport with the tolerance principle, which
  says that once applied, a category can be extended to cases that dif-
  fer very little ii) it seems to suggest that borderline cases are cases of
  overlap, rather than underlap between semantic categories (see Raffman
  2009, Egré 2011, Ripley 2012). In our first study, we probed for hysteresis
  in two different tasks: in the first, subjects had to perform a task of color
  matching, namely to decide of each shade in a series between yellow and
  orange (respectively blue and green) whether it was more similar to the
  most yellow or to the most orange kept on the display. In the second
  task, subjects had to decide which of the two color labels ‘yellow’ or
  ‘orange’ was the most suitable. Shades were presented in three different
  orders, random, ascending from yellow to orange, and descending. While
  we found no order effect in the perceptual matching task, we found an
  effect of negative hysteresis in the linguistic task in each color set, namely
  subjects switched category at a smaller position rather than at a later
  position depending on the order. In a second study, we used the same
  design but asked subjects to report agreement or disagreement with var-
  ious sentential descriptions of the shade (viz. ‘the shade is yellow/not
  yellow/yellow and not yellow’). No order effect was found in that task.
  These findings raise two particular issues concerning the boundaries of
  vague semantic categories, which we discuss in turn: the first concerns
  the interpretation of negative, as opposed to positive hysteresis. Another
  concerns the sensitivity of order effects to the task.

  This is a joint work with Vincent de Gardelle.
Rationality, intractability and the prospects of ‘as
                  if’ explanations

                                 Iris van Rooij

                       Radboud University Nijmegen
             Donders Institute for Brain, Cognition and Behavior



    Abstract. Proponents of a probabilistic (Bayesian) turn in the study of
    human cognition have used the intractability of (non-monotonic) logics
    to argue against the feasibility of logicist characterizations of human
    rationality. It is known, however, that probabilistic computations are
    generally intractable as well. Bayesians have argued that, in their own
    case, this is merely as pseudoproblem. Their argument is that humans do
    not really perform the probabilistic calculations prescribed by probability
    theory, but only act as if they do–much like the planets do not calculate
    their own orbits, and birds fly without any knowledge of the theory of
    aerodynamics.
    The prospects of such an ‘as if’ explanation dissolving the intractability
    problem depends inter alia on what is meant by ‘as if’. I analyze some of
    the most plausible meanings that are compatible with various statements
    in the literature, and argue that none of them circumvents the problem
    of intractability.
    The analysis will show that, even though the constraints imposed by
    tractability may prove pivotal for determining adequate characterizations
    of human rationality, these constraints do not directly favor one type of
    formalism over another. Cognitive science would be better off realizing
    this and putting efforts into dealing with the problem of intractability
    head-on, rather than playing a shell game.

    This is joint work with:
    Cory Wright (University of California, Long Beach, USA),
    Johan Kwisthout (Radboud University Nijmegen, The Netherlands),
    Todd Wareham (Memorial University of Newfoundland, Canada).
Logical and Psychological Analysis of Deductive
                 Mastermind

        Nina Gierasimczuk1 , Han van der Maas2 , and Maartje Raijmakers2
    1
        Institute for Logic, Language and Computation, University of Amsterdam,
                             Nina.Gierasimczuk@gmail.com,
                 2
                    Department of Psychology, University of Amsterdam



        Abstract. The paper proposes a way to analyze logical reasoning in a
        deductive version of the Mastermind game implemented within the Math
        Garden educational system. Our main goal is to derive predictions about
        the cognitive difficulty of game-plays, e.g., the number of steps needed for
        solving the logical tasks or the working memory load. Our model is based
        on the analytic tableaux method, known from proof theory. We associate
        the difficulty of the Deductive Mastermind game-items with the size of
        the corresponding logical tree derived by the tableau method. We dis-
        cuss possible empirical hypotheses based on this model, and preliminary
        results that prove the relevance of our theory.

        Keywords: Deductive Mastermind, Mastermind game, deductive rea-
        soning, analytic tableaux, Math Garden, educational tools


1   Introduction and Background

Computational and logical analysis has already proven useful for the investi-
gations into the cognitive difficulty of linguistic and communicative tasks (see
[1–5]). We follow this line of research by adapting formal logical tools to directly
analyze the difficulty of non-linguistic logical reasoning. Our object of study is
a deductive version of the Mastermind game. Although the game has been used
to investigate the acquisition of complex skills and strategies in the domain of
reasoning about others [6], as far as we know, it has not yet been studied for
the difficulty of the accompanying deductive reasoning. Our model of reasoning
is based on the proof-theoretical method of analytic tableaux for propositional
logic, and it gives predictions about the empirical difficulty of game-items. In
the remaining part of this section we give the background of our work: we ex-
plain the classical and static versions of the Mastermind game, and describe the
main principles of the online Math Garden system. In Section 2 we introduce
Deductive Mastermind as implemented within Math Garden. Section 3 gives a
logical analysis of Deductive Mastermind game-items using the tableau method.
Finally, Section 4 discusses some hypotheses drawn on the basis of our model,
and gives preliminary results. In the end we briefly discuss the directions for
further work.
2

1.1   Mastermind Game

Mastermind is a code-breaking game for two players. The modern game with
pegs was invented in 1970 by Mordecai Meirowitz, but the game resembles an
earlier pen and paper game called Bulls and Cows. The Mastermind game, as
known today, consists of a decoding board, code pegs of k colors, and feedback
pegs of black and white. There are two players, the code-maker, who chooses a
secret pattern of ` code pegs (color duplicates are allowed), and the code-breaker,
who guesses the pattern, in a given n rounds. Each round consists of code-breaker
making a guess by placing a row of ` code pegs, and of code-maker providing the
feedback of zero to ` key pegs: a black key for each code peg of correct color and
position, and a white key for each peg of correct color but wrong position. After
that another guess is made. Guesses and feedbacks continue to alternate until
either the code-breaker guesses correctly, or n incorrect guesses have been made.
The code-breaker wins if he obtains the solution within n rounds; the code-maker
wins otherwise. Mastermind is an inductive inquiry game that involves trials of
experimentation and evaluation. As such it leads to the interesting question of
the underlying logical reasoning and its difficulty. Existing mathematical results
on Mastermind do not provide any answer to this problem—most focus has
been directed at finding a strategy that allows winning the game in the smallest
number of rounds (see [7–10]).
    Static Mastermind [11] is a version of the Mastermind game in which the goal
is to find the minimum number of guesses the code-breaker can make all at once
at the beginning of the game (without waiting for the individual feedbacks), and
upon receiving them all at once completely determine the code in the next guess.
In the case of this game some strategy analysis has been conducted [12], but,
more importantly, Static Mastermind has been given a computational complex-
ity analysis [13]. The corresponding Static Mastermind Satisfiability Decision
Problem has been defined in the following way.
Definition 1 (Mastermind Satisfiability Decision Problem).

Input A set of guesses G and their corresponding scores.
Question Is there at least one valid solution?

Theorem 1. Mastermind Satisfiability Decision Problem is NP-complete.

This result gives an objective computational measure of the difficulty of the task.
NP-complete problems are believed to be cognitively hard [14–16], hence this
result has been claimed to indicate why Mastermind is an engaging and popular
game. It does not however give much insight into the difficulty of reasoning that
might take place while playing the game, and constructing the solution.


1.2   Math Garden

This work has been triggered by the idea of introducing a dedicated logical
reasoning training in primary schools through the online Math Garden system
                                                                                     3

(Rekentuin.nl or MathsGarden.com, see [17]). Math Garden is an adaptive train-
ing environment in which students can play various educational games especially
designed to facilitate the development of their abstract thinking. Currently, it
consists of 15 arithmetic games and 2 complex reasoning games. Students play
the game-items suited for their level. A difficultly level is appropriate for a stu-
dent if she is able to solve 70% items of this level correctly. The difficulty of tasks
and the level of the students’ play are being continuously estimated according
to the Elo rating system, which is used for calculating the relative skill levels
of players in two-player games such as chess [18]. Here, the relative skill level is
computed on the basis of student v. game-item opposition: students are rated by
playing, and items are rated by getting played. The rating depends on accuracy
and speed of problem solving [19]. The rating scales go from − infinity to +
infinity. If a child has the same rating as an item this means that the child can
solve the item with probability .5. In general, the absolute values of the ratings
have no straightforward meaning. Hence, one result of children playing within
Math Garden is a rating of all items, which gives item difficulty parameters. At
the same time every child gets a rating that reflects her or his reasoning ability.
One of the goals of this project is to understand the empirically established item
difficulty parameters by means of a logical analysis of the items. Figure 5 in the
Appendix depicts the educational and research context of Math Garden.


2    Deductive Mastermind

Now let us turn to Deductive Mastermind, the game that we designed for the
Math Garden system (its corresponding name within Math Garden is Flower-
code). Figure 1 shows an example item, revealing the basic setting of the game.
Each item consists of a decoding board (1), short feedback instruction (2), the
domain of flowers to choose from while constructing the solution (3) and the
timer in the form of disappearing coins (4). The goal of the game is to guess
the right sequence of flowers on the basis of the clues given on the decoding
board. Each row of flowers forms a conjecture that is accompanied by a small
board on the right side of it. The dots on the board code the feedback about
the conjecture: one green dot for each flower of correct color and position, one
orange dot for each flower of correct color but wrong position, and one red dot
for each flower that does not appear in the secret sequence at all. In order to
win, the player is supposed to pick the right flowers from the given domain (3),
place them in the right order on the decoding board, right under the clues, and
press the OK button. She must do so before the time is up, i.e., before all the
coins (4) disappear. If the guess is correct, the number of coins that were left as
she made the guess is added to her overall score. If her guess is wrong, the same
number is subtracted from it. In this way making fast guesses is punished, but
making a fast correct response is encouraged [19].
    Unlike the classical version of the Mastermind game, Deductive Mastermind
does not require the player to come up with the trial conjectures. Instead, Flower-
code gives the clues directly, ensuring that they allow exactly one correct solution.
4




             Fig. 1. Deductive Mastermind in the Flowercode setting




Hence, Deductive Mastermind reduces the complexity of classical Mastermind
by changing from an inductive inference game into a very basic logical-reasoning
game. On the other hand, when compared to Static Mastermind, Deductive Mas-
termind differs with respect to the goal. In fact, by guaranteeing the existence
of exactly one correct solution, Deductive Mastermind collapses the postulated
complexity from Theorem 1, since the question of the Static Mastermind Satis-
fiability Problem becomes void. It is fair to say that Deductive Mastermind is a
combination of the classical Mastermind game (the goal of the game is the same:
finding a secret code) and the Static Mastermind game (it does not involve the
trial-and-error inductive inference experimentation). Its very basic setting al-
lows access to atomic logical steps of non-linguistic logical reasoning. Moreover,
Deductive Mastermind is easily adaptable as a single-player game, and hence
suitable for the Math Garden system. The simple setting provides educational
justification, as the game trains very basic logical skills.

    The game has been running within the Math Garden system since Novem-
ber 2010. It includes 321 game-items, with conjectures of various lengths (1-5
flowers) and number of colors (from 2 to 5). By January 2012, 2,187,354 items
had been played by 28,247 primary school students (grades 1-6, age: 6-12 years)
in over 400 locations (schools and family homes). This extensive data-collecting
process allows analyzing various aspects of training, e.g., we can access the in-
dividual progress of individual players on a single game, or the most frequent
mistakes with respect to a game-item. Most importantly, due to the student-
item rating system mentioned in Section 1.2, we can infer the relative difficulty
of game-items. Within this hierarchy we observed that the present game-item
domain contains certain “gaps in difficulty”—it turns out that our initial diffi-
culty estimation in terms of non-logical aspects (e.g., number of flowers, number
of colors, number of lines, the rate of the hypotheses elimination, etc.) is not
precise, and hence the domain of items that we generated does not cover the
whole difficulty space. Providing a logical apparatus that predicts and explains
the difficulty of Deductive Mastermind game-items can help solving this problem
and hence also facilitate the training effect of Flowercode (see Appendix, Figure
7).
                                                                                          5

3     A Logical Analysis
Each Deductive Mastermind game-item consists of a sequence of conjectures.
Definition 2. A conjecture of length l over k colors is any sequence given by
a total assignment, h : {1, . . . , `} → {c1 , . . . , ck }. The goal sequence is a distin-
guished conjecture, goal : {1, . . . , `} → {c1 , . . . , ck }.
Every non-goal conjecture is accompanied by a feedback that indicates how
similar h is to the given goal assignment. The three feedback colors: green,
orange, and red, described in Section 2, will be represented by letters g, o, r.
Definition 3. Let h be a conjecture and let goal be the goal sequence, both of
length l over k colors. The feedback f for h with respect to goal is a sequence
                                  a          b         c
                               g . . . g o . . . o r . . . r = g a ob r c ,
                               z }| { z }| { z }| {

where a, b, c ∈ {0, 1, 2, 3, . . .} and a + b + c = `. The feedback consists of:
 – exactly one g for each i ∈ G, where G = {i ∈ {1, . . . `} | h(i) = goal(i)}.
 – exactly one o for every i ∈ O, where O = {i ∈ {1, . . . , `}\G | there is a j ∈
   {1, . . . , `}\G, such that i 6= jand h(i) = goal(j)}.
 – exactly one r for every i ∈ {1, . . . , `}\(G∪O).

3.1   The informational content of the feedback
How to logically express the information carried by each pair (h, f )? To shape
the intuitions let us first give a second-order logic formula that encodes any
feedback sequence g a ob rc for any h with respect to any goal:

          ∃G⊆{1, . . . `}(card(G)=a ∧ ∀i∈G h(i)=goal(i) ∧ ∀i∈G
                                                            / h(i)6=goal(i)
 ∧ ∃O⊆{1, . . . `}\G (card(O)=b ∧ ∀i∈O ∃j∈{1, . . . `}\G(j6=i ∧ h(i)=goal(j))
                             ∧ ∀i∈{1, . . . `}\(G∪O) ∀j∈{1, . . . `}\G h(i)6=goal(j))).

     Since the conjecture length, `, is fixed for any game-item, it seems sensible to
give a general method of providing a less engaging, propositional formula for any
instance of (h, f ). As literals of our Boolean formulae we take h(i) = goal(j),
where i, j ∈ {1, . . . `} (they might be viewed as propositional variables pi,j , for
i, j ∈ {1, . . . `}). With respect to sets G, O, and R that induce a partition of
{1, . . . `}, we define ϕgG , ϕoG,O , ϕrG,O , the propositional formulae that correspond
to different parts of feedback, in the following way:
  – ϕgG := i∈G h(i)=goal(i) ∧ j∈{1,...,`}\G h(j)6=goal(j),
              V                        V


 – ϕoG,O :=
              V        W
                  i∈O (    j∈{1,...,`}\G,i6=j h(i) = goal(j)),

 – ϕrG,O :=
              V
                  i∈{1,...`}\(G∪O),j∈{1,...`}\G,i6=j h(i) 6= goal(j).
6

Observe that there will be as many substitutions of each of the above schemes
of formulae, as there are ways to choose the corresponding sets G and O. To fix
the domain of those possibilities we set G := {G|G ⊆ {1, . . . , `} ∧ card(G)=a},
and, if G ⊆ {1, . . . , `}, then OG = {O|O ⊆ {1, . . . , `}\G ∧ card(O)=b}. Finally,
we can set Bt(h, f ), the Boolean translation of (h, f ), to be given by:
                                  _ g      _
                     Bt(h, f ) :=    (ϕG ∧     (ϕoG,O ∧ ϕrG,O )).
                                    G∈G         O∈OG

Example 1. Let us take ` = 2 and (h, f ) such that: h(1):=c1 , h(2):=c2 ; f :=or.
Then G={∅}, O{∅} ={{1}, {2}}. The corresponding formula, Bt(h, f ), is:
(goal(1)6=c1 ∧ goal(2)6=c2 ) ∧ ((goal(1)=c2 ∧ goal(2)6=c1 ) ∨ (goal(2)=c1 ∧ goal(1)6=c2 ))

   Each Deductive Mastermind game-item consists of a sequence of conjectures
together with their respective feedbacks. Let us define it formally.
Definition 4. A Deductive Mastermind game-item over ` positions, k colors
and n lines, DM (l, k, n), is a set {(h1 , f1 ), . . . , (hn , fn )} of pairs, each consist-
ing of a single conjecture together with its corresponding feedback. Respectively,
Bt(DM (l, k, n)) = Bt({(h1 , f1 ), . . . , (hn , fn )}) = {Bt(h1 , f1 ), . . . , Bt(hn , fn )}.
Hence, each Deductive Mastermind game-item can be viewed as a set of Boolean
formulae. Moreover, by the construction of the game-items we have that this set
is satisfiable, and that there is a unique valuation that satisfies it. Now let us
focus on a method of finding this valuation.

3.2     Analytic Tableaux for Deductive Mastermind
In proof theory, the analytic tableau is a decision procedure for propositional
logic [20–22]. The tableau method can determine the satisfiability of finite sets
of formulas of propositional logic by giving an adequate valuation. The method
builds a formulae-labeled tree rooted at the given set of formulae and unfolding
breaks these formulae into smaller formulae until contradictory pairs of literals
are produced or no further reduction is possible. The rules of analytic tableaux
for propositional logic, that are relevant for our considerations are as follows.3
                             ϕ∧ψ                     ϕ∨ψ
                                  ∧                     ∨
                             ϕ, ψ               ϕ              ψ

By our considerations from Section 3 we can now conclude that applying the
analytic tableaux method to the Boolean translation of a Deductive Mastermind
game-item will give the unique missing assignment goal. In the rest of the paper
we will focus on 2-pin Deductive Mastermind game-items (where ` = 2), in
particular, we will explain the tableau method in more detail on those simple
examples.
3
    We do not need the rule for negation because in our formulae only literals are negated.
                                                                                            7

2-pin Deductive Mastermind Game-items Since the possible feedbacks consist
of letters g (green), o (orange), and r (red), in principle for the 2-pin Deductive
Mastermind game-items we get six possible feedbacks: gg, oo, rr, go, gr, or. From
those: gg is excluded as non-applicable; go is excluded because there are only two
positions. Let us take a pair (h, f ), where h(1)=ci , h(2)=cj , then, depending on
the feedback, the corresponding boolean formulae are given in Figure 2. We can
compare the complexity of those feedbacks just by looking at their tree repre-
sentations created from the boolean translations via the tableau method. Those

     Feedback     Boolean translation
         oo       goal(1) 6= ci ∧ goal(2) 6= cj ∧ goal(1) = cj ∧ goal(2) = ci
         rr       goal(1) 6= ci ∧ goal(2) 6= cj ∧ goal(1) 6= cj ∧ goal(2) 6= ci
        gr        (goal(1) = ci ∧ goal(2) 6= cj ) ∨ (goal(2) = cj ∧ goal(1) 6= ci )
         or       (goal(1) 6= ci ∧ goal(2) 6= cj ) ∧
                  ((goal(1) = cj ∧ goal(2) 6= ci ) ∨ (goal(2) = ci ∧ goal(1) 6= cj ))


       ci , cj       ci , c j              ci , cj                     ci , c j
            oo             rr                                            or
                                            gr
    goal(1)6=ci   goal(1)6=ci                                goal(1)6=c1      goal(2)6=c2
    goal(2)6=cj   goal(2)6=cj                                goal(2)=c1       goal(1)=c2
    goal(1)=cj    goal(1)6=cj    goal(1)=ci goal(2)=cj       goal(1)6=c2      goal(2)6=c1
    goal(2)=ci    goal(2)6=ci    goal(2)6=cj goal(1)6=ci     goal(2)6=c2      goal(2)6=c1


     Fig. 2. Formulae and their trees for 2-pin Deductive Mastermind feedbacks

representations clearly indicate that the order of the tree-difficulty for the four
possible feedbacks is: oo<rr<gr<or. As the feedbacks oo, rr are conjunctions,
they do not require branching (the other two include disjunctions, and as such
demand reasoning by cases). Unlike rr, the feedback oo in fact gives the solution
immediately. Within the two remaining rules, gr requires less memory to store
the information in each branch.
    We will now briefly discuss the tableau method on an example. Let us con-
sider the following Deductive Mastermind game-item (Figure 3). The tree on
the left stands for the reasoning which corresponds to analyzing the conjectures
as given, from top to bottom. The first branching gives the result of applying
the gr feedback. In the next level of the tree we apply the oo feedback to the
second conjecture. We must first do so assuming the left branch of the first con-
jecture to be true. This leads to a contradiction—on this branch we get that
goal(2)=c1 and goal(2)6=c1 . Then we move to the right branch of the first con-
jecture. This assumption leads to discovering the right assignment, goal(1)=c2
and goal(2)=c1 , there is no contradiction on this branch. This tableau proce-
dure is required to build the whole tree for the game-item. That is not always
necessary. The right-most part of Figure 3 shows what happens if you chose to
start the analysis from the second conjecture. We first apply the feedback oo
to the second conjecture. We immediately get: goal(1)=c2 goal(2)=c1 . The full
8
                                                    Bt(h1 , f1 )                        Bt(h2 , f2 )
                                                    Bt(h2 , f2 )                        Bt(h1 , f1 )
                                                      gr                                    oo
                                             goal(1)=c1       goal(2)=c1        goal(1)=c2
                                             goal(2)6=c1      goal(1)6=c1       goal(2)=c1
                                             Bt(h2 , f2 )     Bt(h2 , f2 )      Bt(h1 , f1 )
                                                 oo               oo
                                             goal(1)=c2       goal(1)=c2
                                             goal(2)=c1       goal(2)=c1

Fig. 3. Comparison of two different trees for one item. The formalization is as follows:
c1 stands for sunflower, c2 for tulip; h1 (1)=c1 , h1 (2)=c1 , f1 =gr, etc. Green branch gives
the right valuation. The tree on the right hand-side analyzes feedback oo first.

unique assignment with no contradiction. We can stop the computation at this
point—if the other conjecture contradicted this assignment, then it would mean
that the two conjectures must be inconsistent and hence not satisfiable. This
contradicts the setting of our game.
   The tree might not always give us the complete valuation explicitly. In some
game-items it is required to use a flower that did not appear in the conjectures.
This is so in the example in Figure 4; the right-most branch does not give a
contradiction, it does assign color 1 to the second position of the goal conjecture.
In such case we draw the remaining color, c3 (a tulip in the picture), as the
missing value of the first position of the goal conjecture, i.e., goal(1)=c3 .
                                                       Bt(h1 , f1 )
                                                       Bt(h2 , f2 )
                                                         gr
                                               goal(1)=c1        goal(2)=c1
                                               goal(2)6=c1       goal(1)6=c1
                                               Bt(h2 , f2 )      Bt(h2 , f2 )
                                               gr                     gr
                                  goal(1)=c2 goal(2)=c1          goal(1)=c2 goal(2)=c1
                                  goal(2)6=c1 goal(1)6=c2        goal(2)6=c1 goal(1)6=c2

Fig. 4. Comparison of the trees of two different items, while processing the conjectures
from top to bottom. Here, c1 stands for marguerite, c2 for sunflower, c3 for tulip. This
item rating is 4.2 (see Figure 6).

4    Hypotheses and Preliminary Results

Normatively speaking, the full tree generated by the tableau method for the
set of formulae corresponding to a Deductive Mastermind game-item gives its
ideal reasoning scheme and thus the size of the tree can be thought of as an
abstract complexity measure. Obviously, the shape and the size of the tree for
each Deductive Mastermind item depends to some extent on the order in which
the formulae are analyzed (see Figure 3). Hence, using the tableau method it is
even possible to analyze whether and in what way the students apply reasoning
                                                                                   9

strategies, i.e., how they manipulate with the elements of the task in order to
optimize the size of the reasoning tree (i.e., the length of the computation). In
this way, items’ logical difficulty can be expressed via the size of their minimal
trees. The empirical data, resulting from children playing the Deductive Mas-
termind game in Math Garden, includes item ratings. In the first analysis of
this data we aimed at relating the item ratings to the parameters of the trees.
To this end we define a computational method based on the tableau method.
The computational method makes two assumptions. First, the formulae are not
processed from top to bottom, but instead the order depends on the length of
the rule that is associated with the feedback. That is, feedback is processed in
the following order: oo, rr, gr, or. Ties are solved by processing the top formula
first. Second, the computational method is stopped once a consistent solution is
found, assuming that there exists at least one solution. Based on these principles
and the tableau method we programmed a recursive algorithm for calculating
the type and number of steps until solution for each item is reached. We define
4 measures of theoretically derived item difficulties; the required number of oo,
rr, gr, and rr steps, which together might predict item difficulty. Below we will
describe the empirically derived item ratings of the 2-pin items and we will show
how these relate to the theoretically derived measures of item difficulties.

Method Participants were 28,247 students from grades 1-6, of age: 6-12 years.
Together, they played 2,187,354 items between November 2010 and January
2012. From the total of 321 items in the Master Mind game, 100 items have
two pins. From these 100 items, 10 items involved 2 colors, 30 items involved 3
colors, 30 items involved 4 colors and 30 items involved 5 colors.

Results To test the relation between empirically derived item ratings and the-
oretically derived measures of item difficulty we did a regression analysis that
includes the basic features of the items (number of colors and number of guesses)
and the required number of oo, rr, gr, and or analysis steps as predicted by
the tableau-based computational algorithm (F (6, 93)=33, p < .0001, R2=.66).
All these factors but one (i.e., number of gr evaluations) were significant in
predicting item difficulties: number of colours (β=1.07, p=.02), number of hy-
potheses (β=1.75, p<.01), number of oo feedbacks (β= − 5.1, p<.001), number
of rr feedbacks (β= − 3.19, p<.0001), number of gr evaluations (ns), number of
or evaluations (β=1.6, p<.0001). Note that among the rules, only the required
number of or steps increases item difficulty. A second aspect of the observed
item ratings to explain is the shape of its distribution. The distribution of the
item difficulties shows a remarkable property for the 2-pin items (see Appendix,
Figure 6). The distribution is bimodal, meaning that there is a cluster of more
difficult items (item ratings >0) and a cluster of easier items (item ratings <0).
In the cluster of easy items, there are items with 2, 3, 4, and 5 colors. The cluster
of difficult items consists of items with 3, 4 and 5 colors. The two clusters could
be expressed fully in terms of model-relevant features. It appeared that items
are easy in the following two cases: (1) no or feedback and no gr feedback; (2)
no or feedback, at least one gr feedback, and all colors are included in at least
10

one of the conjectures. Items are difficult otherwise. This shows that or steps
make the item relatively difficult, but only if or is required to solve the item. A
second aspect that makes an item difficult is the exclusion of one of the colors
in the solution from the hypotheses rows.


5    Conclusions and Future Work

In this paper we proposed a way to use a proof-theoretical concept to analyze
the cognitive difficulty of logical reasoning. The first hypotheses that we have
drawn from the model gave a reasonably good prediction of item-difficulties,
but the fit is not perfect. However, it must be noted that several non-logical
factors may play a role in the item difficulty as well. For example, the motor
requirements to give the answer also introduce some variation in item difficulty,
which depends not only on accuracy but also on speed. The minimal number of
clicks required to give an answer varies between items. We did not take these
aspects into account so far. In particular, the two difficulty clusters observed in
the empirical study can be explained with the use of our method.
    Our further work can be split into two main parts. We will continue this
study on the theoretical level by analyzing various complexity measures that
can be obtained on the basis of the tableau system. We also plan to compare
the fit with empirical data of the tableau-derived measures and other possible
logical formalizations of level difficulty (e.g., a resolution-based model). On the
empirical side we first would like to extend our analysis to game-items that con-
sist of conjectures of higher length. This will allow comparing difficulty of tasks
of different size and measure the trade-off between the size and the structure
of the tasks. We will also study this model in a broader context of other Math
Garden games. This would allow comparing individual abilities within Deductive
Mastermind and the abilities within other games that have been correlated for
instance with the working memory load. Finally, it would be interesting to see
whether the children are really using the proposed difficulty order of feedbacks in
finding an optimal tableau—this could be checked in an eye-tracking experiment
(see [23, 24]).


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20. Beth, E.W.: Semantic entailment and formal derivability. Mededelingen Van
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    MALLOW, CUER Workshop Proceedings (2010)
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12

Appendix

The appendix contains three figures. The first one illustrates the educational
and research context of the Math Garden system (Figure 5); the second shows
the distribution of the empirical difficulty of all items in Deductive Mastermind
(Figure 6); and the third gives the frequency of players and game-items with
respect to the overall rating (Figure 7). We refer to those figures in the proper
text of the article.


                                      Investigators
                                                                      instruction
                                                                       methods
                                                new items,
                                  data
                                                  tasks
                   game playing                              report
         Student                   Math Garden.com                       Teacher


                                       instruction


                Fig. 5. Math Garden educational and research context




Fig. 6. The distribution of the empirically established game-item difficulties for all 321
game-items in the Deductive Mastermind game. The item number (x-axis) is some
arbitrary number of the game-item. The y-axis shows the item ratings. For example,
the item presented in Figure 4 is number 23 and its rating is 4.2.
                                                                                      13




Fig. 7. The frequency of players (green, y-axis on the left) and game-items (blue, y-axis
on the left) with respect to the overall rating (x-axis).
                 Euclid’s Diagrammatic Logic and
                         Cognitive Science

                           Yacin Hamami1 and John Mumma2
    1
        Centre for Logic and Philosophy of Science, Vrije Universiteit Brussel, Brussels
                                 yacin.hamami@gmail.com
                  2
                    Max Planck Institute for the History of Science, Berlin
                                  john.mumma@gmail.com



          Abstract. For more than two millennia, Euclid’s Elements set the stan-
          dard for rigorous mathematical reasoning. The reasoning practice the
          text embodied is essentially diagrammatic, and this aspect of it has been
          captured formally in a logical system termed Eu [2, 3]. In this paper, we
          review empirical and theoretical works in mathematical cognition and
          the psychology of reasoning in the light of Eu. We argue that cognitive
          intuitions of Euclidean geometry might play a role in the interpretation
          of diagrams, and we show that neither the mental rules nor the mental
          models approaches to reasoning constitutes by itself a good candidate
          for investigating geometrical reasoning. We conclude that a cognitive
          framework for investigating geometrical reasoning empirically will have
          to account for both the interpretation of diagrams and the reasoning
          with diagrammatic information. The framework developed by Stenning
          and van Lambalgen [1] is a good candidate for this purpose.


1       Introduction
A distinctive feature of elementary Euclidean geometry is the natural and in-
tuitive character of its proofs. The prominent place of the subject within the
history and education of mathematics attests to this. It was taken to be the
foundational starting point for mathematics from the time of its birth in ancient
Greece up until the 19th century. And it remains within mathematics education
as a subject that serves to initiate students in the method of deductive proof.
No other species of mathematical reasoning seem as basic and transparent as
that which concerns the properties of figures in Euclidean space.
    One may not expect a formal analysis of the reasoning to shed much light
on this distinctive feature of it, as the formal and the intuitive are typically
thought to oppose one another. Recently, however, a formal analysis, termed
Eu, has been developed which challenges this opposition [2, 3]. Eu is a formal
proof system designed to show that a systematic method underlies the use of
diagrams in Euclid’s Elements, the representative text of the classical synthetic
tradition of geometry. As diagrams seem to be closely connected with the way
we call upon our intuition in the proofs of the tradition, Eu holds the promise
of contributing to our understanding of what exactly makes the proofs natural.
                                                                                 15

    In this paper, we explore the potential Eu has in this respect by confronting
it with empirical and theoretical works in the fields of mathematical cognition
and the psychology of reasoning. Our investigation is organized around the two
following issues:
 1. What are the interpretative processes on diagrams involved in the reasoning
    practice of Euclidean geometry and what are their possible cognitive roots?
 2. What would be an appropriate cognitive framework to represent and inves-
    tigate the constructive and deductive aspects of the reasoning practice of
    Euclidean geometry?
By providing a formal model of the reasoning practice of Euclidean geometry,
Eu provides us with a tool to address these two issues. We proceed as follows. To
address the first issue, we first state the interpretative capacities that according
to the norms fixed by Eu are necessary to extract, from diagrams, information for
geometrical reasoning. We then present empirical works on the cognitive bases
of Euclidean geometry, and suggest that cognitive intuitions might play a role
in the interpretative aspects of diagrams in geometrical reasoning. To address
the second issue, we compare the construction and inference rules of Eu with
two major frameworks in the psychology of reasoning—the mental rules and the
mental models theories. We argue that both have strengths and weaknesses as a
cognitive account of geometrical reasoning as analyzed by Eu, but that one will
need to go beyond them to provide a framework for investigating geometrical
reasoning empirically.
    The two main issues are of course intimately related. In a last section, we
argue that the framework developed by Stenning and van Lambalgen in [1],
which connects interpretative and deductive aspects of reasoning, might provide
the right cognitive framework for investigating the relation between them.


2   A Logical Analysis of the Reasoning Practice in
    Euclid’s Elements: The Formal System Eu
Eu is based on the seminal paper [4] by Ken Manders. In [4] Manders challenges
the received view that the Elements is flawed because its proofs sometimes call
upon geometric intuition rather than logic. What is left unexplained by the re-
ceived view is the extraordinary longevity of the Elements as a foundational
text within mathematics. For over two thousand years there were no serious
challenges to its foundational status. Mathematicians through the centuries un-
derstood it to display what the basic propositions of geometry are grounded on.
The deductive gaps that exist according to modern standards of logic story were
simply not seen.
    According to Manders, Euclid is not relying on geometric intuition illicitly in
his proofs; he is rather employing a systematic method of diagrammatic proof.
His analysis reveals that diagrams serve a principled, theoretical role in Euclid’s
mathematics. Only a restricted range of a diagram’s spatial properties are per-
mitted to justify inferences for Euclid, and these self-imposed restrictions can be
16

explained as serving the purpose of mathematical control. Eu [2, 3] was designed
to build on Manders’ insights, and precisely characterize the mathematical sig-
nificance of Euclid’s diagrams in a formal system of geometric proof.
    Eu has two symbol types: diagrammatic symbols ∆ and sentential symbols A.
The sentential symbols A are defined as they are in first-order logic. They express
relations between geometric magnitudes in a configuration. The diagrammatic
symbols are defined as n × n arrays for any n. Rules for a well-formed diagram
specify how points, lines and circles can be distinguished within such arrays.
The points, lines and circles of Euclid’s diagrams thus have formal analogues in
Eu diagrams. The positions the elements of Euclid’s diagrams can have to one
another are modeled directly by the position their formal analogues can have
within a Eu diagram.
    The content of a diagram within a derivation is fixed via a relation of dia-
gram equivalence. Roughly, two Eu diagrams ∆1 and ∆2 are equivalent if there
is a bijection between its elements which preserves their non-metric positional
relations.3 The equivalence relation is intended to capture what Manders terms
the co-exact properties of a Euclidean diagram. A close examination of the El-
ements shows that Euclid refrains from making any inferences that depend on
a diagram’s metric properties. At the same time, Euclid does rely on diagrams
for the non-metric positional relations they exhibit—or in Manders’ terms, the
co-exact relations they exhibit. Diagrams, it turns out, can serve as adequate
representations of such relations in proofs.
    Eu exhibits this by depicting geometric proof as running on two tracks:
a diagrammatic one, and a sentential one. The role of the sentential one is to
record metric information about the figure and provide a means for inferring this
kind of information. The role of the diagrammatic track is to record non-metric
positional information of the figure, and to provide a means for inferring this
kind of information about it. Rules for building and transforming diagrams in
derivations are sensitive only to properties invariant under diagram equivalence.
It is in this way that the relation of diagram equivalence fixes the content of
diagrams in derivations.
    What is derived in Eu are expressions of the form

                                 ∆1 , A1 −→ ∆2 , A2

where ∆1 and ∆2 are diagrams, and A1 and A2 are sentences. The geometric
claim this is stipulated to express is the following:

      ?Given a configuration satisfying the non-metric positional relations de-
      picted in ∆1 and the metric relations expressed in A1 , then one can
      obtain a configuration satisfying the positional relations depicted by ∆2
      and metric relations specified by A2 .

3
     For a more detailed discussion of Eu diagrams and diagram equivalence, we refer
     the reader respectively to sections A and B of the appendix.
                                                                                 17

3      Interpretative Aspects of Geometrical Reasoning with
       Diagrams
Diagrams in Euclid’s Elements are mere pictures on a piece of paper. How can a
visual experience triggered by looking at a picture lead to a cognitive representa-
tion that can play a role in geometrical reasoning? From a cognitive perspective,
taking seriously the use of diagrams in reasoning requires an account of the
way diagrams are interpreted in order to play a role in geometrical reasoning.
The formal system Eu provides a theoretical answer to this question. Here we
compare this theoretical account with recent works in mathematical cognition
probing the existence of cognitive intuitions of Euclidean geometry. We will ar-
gue that the Eu analysis of geometrical reasoning suggests a possible role for
these cognitive intuitions in the interpretation of diagrams.

3.1     Interpretation of Diagrams in Eu
As described in section 2, the key insight behind the Eu analysis of the diagram-
matic reasoning practice of Euclid’s Elements is the observation that appeals to
diagrams in proofs are highly controlled: proofs in Euclid’s Elements only make
use of the co-exact properties of diagrams. From a cognitive perspective, the
practice of extracting the co-exact properties from a visual diagram is far from a
trivial one. According to Eu, the use of diagrams presupposes the two following
cognitive abilities:
 1. The capacity to categorize elements of the diagram using normative concepts:
    points, linear elements and circles.
 2. The capacity to abstract away irrelevant information from the visual-perceptual
    experience of the diagram.
   The formal syntax of Eu, combined with the equivalence relation between
Eu diagrams, can be interpreted as specifying these capacities precisely. More
specifically:
    – The first capacity amounts to the ability to see in the diagram the elements
      p, l, c of a particular Eu diagram hn, p, l, ci, which respectively denote the
      sets of points, lines and circles.
    – The second capacity amounts to the ability to see in the particular diagram
      positional relations that are invariant under diagram equivalence.
    Thus, the interpretation of a visual diagram according to Eu results in a for-
mal object which consists in an equivalence class of Eu diagrams. It is precisely
on these formal objects, the interpreted diagrams, that inference rules operate
in the Eu formalization of reasoning in elementary geometry.

3.2     Intuitions of Euclidean Geometry in Human Cognition
Recently, several empirical studies in mathematical cognition [5–7] have directly
addressed the issue of the cognitive roots of Euclidean geometry. These studies
18

might be classified in two categories: one approach consists in providing empirical
evidence for the existence of abstract geometric concepts [5, 7], the other approach
consists in providing empirical evidence for the perception of abstract geometric
features [6]. Here we successively report some of the empirical findings provided
by these two approaches.
     The two studies [5] and [7] have been conducted on an Amazonian Indi-
gene Group called the Mundurucu, with no previous education in formal Eu-
clidean geometry. In the first study [5], the experiment consisted in identifying
the presence, or the absence, of several topological and geometric concepts in
Mundurucu participants. To this end, the experimenter proposed, for each con-
cept under investigation, six slides in which five of the images displayed the
given geometric concept (e.g., parallelism), while the last one lacked the consid-
ered property (non-parallel lines). The test shows that several basic geometrical
concepts are present in Mundurucu conceptual systems, such as the concepts of
straight line, parallel line, right angle, parallelogram, equilateral triangle, circle,
center of circle. Nevertheless, the study [5] does not address geometric concepts
that go beyond perceptual experience. Such concepts are the topic of another
study [7]. In [7], empirical evidence is provided for the capacity of Mundurucu
to reason about the properties of dots and infinite lines. In particular, most of
the Mundurucu participants consider that, given a straight line and a dot, we
can always position another straight line on the dot which will never cross the
initial line.
     The second approach for detecting the presence of intuitions of Euclidean
geometry is based on the framework of transformational geometry [8]. Accord-
ing to this framework, a geometric theory is identified with respect to the
transformations that preserve its theorems. Euclidean geometry is then iden-
tified by its four types of transformations—translation, rotation, symmetry and
homothecy—leaving then angle and length proportions as the main defining fea-
tures of figures in Euclidean geometry. The experimental approach based on this
framework consists in investigating the capacity of participants to perceive ab-
stract geometric features in configurations where irrelevant features are varied.
The experiments reported in [6] adopt such an approach. The empirical results
show that both children and adults are able to use angle and size to classify
shapes, but only adults are able to discriminate shapes with respect to sense,
i.e., the property distinguishing two figures that are mirror images of each other.
     The two approaches reported here to investigate cognitive intuitions of Eu-
clidean geometry bring out cognitive abilities which seem related to the abilities
for the interpretation of diagrams postulated by Eu. This observation suggests a
possible role for the cognitive intuitions of Euclidean geometry in the reasoning
practice of elementary geometry, as we will now see.

3.3   Cognitive Intuitions and the Interpretation of Diagrams
How might the cognitive intuitions of Euclidean geometry relate to the deductive
aspects of Euclid’s theory of geometry? In a correspondence in Science [9] on
the study reported in [5], Karl Wulff has challenged the claim that Dehaene et
                                                                                 19

al. address Euclidean geometry by denying a role for these cognitive intuitions in
the demonstrative aspects of Euclid’s theory of geometry. An opposing position
can be found in [10]:
    The axioms of geometry introduced by Euclid [. . . ] define concepts with
    spatial content, such that any theorem or demonstration of Euclidean ge-
    ometry can be realized in the construction of a figure. Just as intuitions
    about numerosity may have inspired the early mathematicians to de-
    velop mathematical theories of number and arithmetic, Euclid may have
    appealed to universal intuitions of space when constructing his theory of
    geometry. [10, p. 320]
    Interestingly, stating precisely the postulated cognitive abilities involved in
the interpretation of diagrams according to Eu might suggest a role for the
cognitive intuitions of Euclidean geometry in the reasoning practice of Euclid’s
Elements.
    Firstly, we have noticed that one of the key abilities in the interpretation
of diagrams, according to Eu, is the capacity to categorize or type the different
figures of the visual diagram using the normative concepts of geometry. This
ability seems to be universal, according to the empirical data that we reported
in the previous section, as the Mundurucu people, without previous formal ed-
ucation in Euclidean geometry, seem to possess an abstract conceptual systems
which contains the key normative concepts of elementary geometry, and are able
to use it to categorize elements of visual diagrams.
    Secondly, interpretation of diagrams in Eu requires an important capacity
of abstraction which is formally represented by an equivalence relation between
diagrams. This equivalence relation aims to capture the idea that some features
of the diagram are not relevant for reasoning: for instance, the same diagram
rotated, translated or widened, would play exactly the same role in a geometrical
proof of the Elements. This capacity of abstraction seems to connect with the
cognitive ability of perceiving abstract geometric features, such as angle and
length proportions, while abstracting away irrelevant information from the point
of view of Euclidean geometry.
    We now turn to the deductive and constructive aspects of geometrical reason-
ing. In section 5, we argue that a plausible cognitive framework for an empirical
investigation of geometrical reasoning will have to bring together the interpreta-
tive, deductive and constructive aspects of geometrical reasoning to be faithful
to the mathematical practice of Euclidean geometry.

4   Deductive and Constructive Aspects of Geometrical
    Reasoning with Diagrams
In this section, we begin by presenting the Eu formalization of constructive
and deductive steps in Elementary geometry, and then discuss the capacity of
the mental rules and the mental models theories to represent these reasoning
steps. We conclude that an adequate framework for an empirical investigation
of geometrical reasoning will have to go beyond these two theories.
20

4.1     Eu Construction and Inference Rules

The Eu proof rules specify how to derive ∆1 , A1 −→ ∆2 , A2 expressions. They
specify, in particular, the operations that can be performed on the pair ∆1 , A1
to produce a new pair ∆2 , A2 . Given the intended interpretation of ∆1 , A1 −→
∆2 , A2 given by ?, the fundamental restriction on these rules is that they be
geometrically sound. In other words, if ∆2 , A2 is derivable from ∆, A1 , then with
any configuration satisfying the geometric conditions represented by ∆1 A1 one
must either have a configuration satisfying the conditions represented by ∆2 , A2
(if ∆1 = ∆2 ), or be able to construct a configuration satisfying the conditions
represented by ∆2 , A2 (if ∆2 contains objects not in ∆1 ).
    The formal process whereby ∆2 , A2 is derived has two stages: a construction
and demonstration. Hence Eu has two kinds of proof rules: construction rules
and inference rules. The construction stage models the application of a proof’s
construction on a given diagram. The demonstration stage models the inferences
made from the assumed metric relations and the particular diagram produced
by the construction. The soundness restriction thus applies only to the inference
rules. The construction rules together codify a method for producing a represen-
tation that can serve as a vehicle of inference. The demonstration rules codify
the inferences that can be made from such a vehicle.4


4.2     Mental Rules Theory

The mental rules theory [11, 12] represents reasoning as the application of formal
rules, rules which are akin to those of natural deduction. According to this theory,
reasoning is conceived as a syntactic process whereby sentences are transformed.
These transformations are made according to specific rules defined precisely in
terms of the syntactic structures of sentences. Deduction of one sentence from
a set of other sentences (premisses) is seen as the search for a mental proof,
which consists precisely in the production of the conclusion from the premisses
by application of the rules a finite number of times. Consequently, the mental
rules theory represents reasoning as consisting in syntactic operations on the
logical forms of sentences.
    The formalization of geometrical reasoning provided by Eu shares one impor-
tant feature with the mental rules theory: Eu represents geometrical reasoning
in terms of syntactic rules of inference. This is made possible by considering di-
agrams as a kind of syntax, and then by stipulating rules that control inferences
that are drawn from diagrams. Thus, Eu diagrams might be seen as representing
something like the logical form of concrete visual diagrams. One could perhaps
say that Eu diagrams represent their geometric form, and that Eu inference
rules operate on these forms. From this point of view, the formalization of ge-
ometrical reasoning provided by Eu appears to be in direct line with the the
mental rules theory of reasoning.
4
     For the formal details, see sections 1.4.1 and 1.4.2 of [2], available online at
     www.johnmumma.org.
                                                                                 21

    Nevertheless, the mental rules theory seems to run into troubles when we
consider the construction operations on diagrams, which are central to the rea-
soning practice of Euclidean geometry, and which are formalized by Eu’s con-
struction rules. One might still argue that Eu construction rules are syntactic
operations on Eu diagrams, since diagrams are included its syntax, and so Eu
construction rules fit the framework of mental rules theory. However, this does
not seem correct as the mental rules are considered to be deduction rules; the
soundness restriction applies to them. They are thereby of a very different nature
than Eu’s construction rules. Consequently, even though the mental rules theory
could suitably represent inferential steps in geometrical reasoning, it seems that
the theory lacks the necessary resources to account for the construction opera-
tions on diagrams, an aspect fundamental to the reasoning practice in Euclidean
geometry.

4.3   Mental Models Theory
The mental models theory [13, 14] postulates that reasoning depends on envisag-
ing possibilities. When given a set of premisses, an individual constructs mental
models which correspond to the possibilities elicited by the premisses. Different
reasoning strategies are then available to extract information from the mental
models: one may represent several possibilities in a diagram and draw a conclu-
sion from the diagram, one may make an inferential step from a single possibility,
or one can use a possibility as a counter-example for falsifying a conclusion. One
interesting feature of the theory is that mental models are supposed to be iconic:
the structure of a mental model is supposed to reflect the structure of the possi-
bility that it represents. This feature is nicely illustrated when the mental models
theory is applied to account for reasoning with visual-spatial information [15].
    Contrary to the mental rules theory, the mental models theory seems suited
to provide an account of the representation of the different construction opera-
tions on diagrams: visual diagrams used in geometrical proofs can be conceived
as mental models entertained by the reasoner. Mental models associated to di-
agrams would then be of a visual-spatial nature, reflecting the spatial relations
between the different elements of the diagram. This just seems to be a descrip-
tion of Eu diagrams, which encode the information that can be legitimately used
in geometrical reasoning.
    Accordingly, if Eu diagrams are conceived as mental models, the construction
operations on diagrams would be explained in terms of the ways mental models
are constructed. Nevertheless, one might worry about the capacity of the mental
models theory to account for the specific use of diagrams in geometrical rea-
soning. One of the main points of Eu is to exhibit precisely that diagrammatic
information enters legitimate mathematical inferences in a very controlled way.
In order to account for the use of diagrams in geometrical reasoning, the mental
models theory would have to be supplemented with a regimentation of the in-
formation that can be legitimately extracted from diagrams conceived as mental
models. The formal system Eu shows that this can be done using syntactic rules
that operate on diagrams represented as syntactic objects.
22

4.4     Beyond the Mental Rules vs Mental Models Debate
Geometrical reasoning, as practiced in Euclid’s Elements, constitutes an inter-
esting test for current theories in the psychology of reasoning. Our previous
comparison predicts that the mental rules and mental models theories present
both advantages and disadvantages as an account of the reasoning practice of
geometrical reasoning with diagrams. Indeed, it seems that these two theories
are actually complementary in their capacity to account for geometrical reason-
ing as described by Eu: the mental rules theory seems adequate to represent
inferences in geometrical reasoning, while the mental models theory seems ade-
quate to represent the diagrammatic constructions that support such inferences.
Thus, it seems that to provide a framework for an empirical investigation of
geometrical reasoning with diagrams one will need to go beyond these two theo-
ries. Such a move is also suggested, although for different reasons, in [16, 17]. In
this respect, Eu constitutes a possible starting point for developing a cognitive
framework for geometrical reasoning which would be faithful to both deductive
and constructive aspects of the mathematical practice of Euclidean geometry.

5     Interpretation and Reasoning in Elementary Geometry
According to the Eu analysis of geometrical reasoning, interpretation and rea-
soning with diagrams are intimately related. This observation, originating in the
study of the reasoning practice in Euclid’s Elements [4], is directly in line with
a recent approach to the psychology of reasoning developed by Stenning and
van Lambalgen [1] which attributes a central role to interpretative processes in
human reasoning:
      We [. . . ] view reasoning as consisting of two stages: first one has to
      establish the domain about which one reasons and its formal properties
      (what we will call ‘reasoning to an interpretation’) and only after this
      initial step has been taken can ones reasoning be guided by formal laws
      (what we will call ‘reasoning from an interpretation’). [1, p. 28]
    Geometrical reasoning with diagrams, as formalized by Eu, precisely fits
within this framework: reasoning to an interpretation corresponds to the pro-
cess of interpreting a visual diagram along with an associated metric assertion,
resulting in Eu into a pair ∆, A; reasoning from an interpretation is then repre-
sented as the application of the formal rules of Eu to prove geometric claims.
    This perspective unifies the two main issues addressed in this paper. Our main
conclusions can then be restated as follows: (i) intuitions of Euclidean geometry
as studied by mathematical cognition are likely to play a role in reasoning to an
interpretation of a diagrams, (ii) the mental rules and mental models theories
of reasoning are inadequate for representing reasoning from an interpretation,
as none of them is able to account for both deductive and constructive aspects
of geometrical reasoning. In the perspective of Stenning and van Lambalgen [1],
Eu appears as a good candidate for representing the process of reasoning from
an interpretation in elementary geometry.
                                                                                     23

6    Conclusion
The empirical investigation of the cognitive bases of Euclidean geometry is a
multi-disciplinary enterprise involving both mathematical cognition and the psy-
chology of reasoning, and which shall benefit from works in formal logic and the
nature of mathematical practice. In this paper, we used the formal system Eu to
review existing empirical and theoretical works in cognitive science with respect
to this enterprise. Our investigation shows: (i) the necessity of dealing jointly
with interpretation and reasoning, (ii) the relevance of works on the cognitive
bases of Euclidean geometry for the interpretation of diagrams and (iii) the ne-
cessity to go beyond the mental rules vs mental models distinction for accounting
for both constructive and deductive aspects of geometrical reasoning.


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 1. Stenning, K., Van Lambalgen, M.: Human Reasoning and Cognitive Science. The
    MIT Press, Cambridge, MA (2008)
 2. Mumma, J.: Intuition Formalized: Ancient and Modern Methods of Proof in Ele-
    mentary Geometry. PhD thesis, Carnegie Mellon University (2006)
 3. Mumma, J.: Proofs, pictures, and Euclid. Synthese 175(2) (2010) 255–287
 4. Manders, K.: The Euclidean diagram. In Mancosu, P., ed.: Philosophy of Mathe-
    matical Practice. Clarendon Press, Oxford (2008)
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    etry in an Amazonian indigene group. Proceedings of the National Academy of
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 8. Klein, F.: A comparative review of recent researches in geometry. Bulletin of the
    New York Mathematical Society 2 (1893) 215–249
 9. Wulff, K.: Examining knowledge of geometry. Science 312(5778) (2006) 1309
10. Izard, V., Pica, P., Dehaene, S., Hinchey, D., Spelke, E.: Geometry as a universal
    mental construction. In Brannon, E., Dehaene, S., eds.: Space, Time and Number
    in the Brain: Searching for the Foundations of Mathematical Thought. Number
    XXIV in Attention & Performance. Oxford University Press, Oxford (2011) 319–
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11. Rips, L.: Cognitive processes in propositional reasoning. Psychological Review
    90(1) (1983) 38–71
12. Rips, L.: The Psychology of Proof: Deductive Reasoning in Human Thinking. The
    MIT Press, Cambridge, MA (1994)
13. Johnson-Laird, P.: Mental Models. Cambridge University Press, Cambridge (1983)
14. Johnson-Laird, P.: Mental models and human reasoning. Proceedings of the Na-
    tional Academy of Sciences 107(43) (2010) 18243–18250
15. Knauff, M.: A neuro-cognitive theory of deductive relational reasoning with mental
    models and visual images. Spatial Cognition & Computation 9(2) (2009) 109–137
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17. Goel, V.: Anatomy of deductive reasoning. Trends in Cognitive Sciences 11(10)
    (2007) 435–441
24

Appendix

A    Eu diagrams

The syntactic structure of diagrams in Eu has no natural analogue in stan-
dard logic. Their underlying form is a square array of dots of arbitrary finite
dimension.




The arrays provide the planar background for an Eu diagram. Within them
geometric elements—points, linear elements, and circles— are distinguished. A
point is simply a single array entry. An example of a diagram with a single point
in it is




Linear elements are subsets of array entries defined by linear equations expressed
in terms of the array entries. (The equation can be bounded. If it is bounded
on one side, the geometric element is a ray. If it is bounded on two sides, the
geometric element is a segment.) An example of a diagram with a point and
linear element is
                                                                                  25

Finally, a circle is a convex polygon within the array, along with a point inside it
distinguished as its center. An example of a diagram with a point, linear element
and a circle is




    The size of a diagram’s underlying array and the geometric elements distin-
guished within it, comprise a diagram’s identity. Accordingly, a diagram in Eu
is a tuple
                                   hn, p, l, ci
where n, a natural number, is the length of the underlying array’s sides, and
p, l and c are the sets of points, linear elements and circles of the diagram,
respectively.
     Like the relation symbols which comprise metric assertions, the diagrams
have slots for variables. A diagram in which the slots are filled is termed a labeled
diagram. The slots a diagram has depends on the geometric elements constituting
it. In particular, there is a place for a variable beside a point, beside the end of
a linear element (which can be an endpoint or endarrow), and beside a circle.
One possible labeling for the above diagram is thus



                                 A        E      D



                                      B                C




Having labeled diagrams within Eu is essential, for otherwise it would be im-
possible for diagrams and metric assertions to interact in the course of a proof.
We can notate any labeled diagram as

                                  hn, p, l, ci[A, R]

where A denotes a sequence of variables and R a rule matching each variable to
each slot in the diagram.
26

B    Diagram equivalence

The definition of diagram equivalence is based on the notions of a diagram’s
completion and that of co-exact map. The completion ∆0 of a diagram ∆ is simply
the diagram obtained by adding to ∆ intersection points to the intersections
present in it. (See page 38-40 of [2]) Two diagrams are then equivalent if there
is a co-exact map between their completions.
    The key idea behind the definition of diagram equivalence, then, is that of
of a co-exact map. A necessary condition for their to be such a map between
diagrams ∆1 and ∆2 is that both diagrams contain the same number of points,
linear elements and circles, and these objects are labeled by the same variables
and variable pairs.
    Suppose ∆1 and ∆2 are two such diagrams, and φx is the bijection that maps
an element in ∆1 to the element in ∆2 with the same label. In virtue of their
position with respect to the underlying array of ∆1 , the elements of ∆1 satisfy
various geometric relations. The bijection φx is a co-exact map if and only if it
preserves a certain sub-set of these relations. Precisely φx is a co-exact map if
and only if it satisfies the following nine conditions, where P and Q are points
of ∆1 , N and M are linear elements of ∆1 (i.e. segments, rays, or lines) and C1
and C2 are circles of ∆1 .
 – P lies on M in ∆1 ←→ φx (P ) lies on φx (M ) in ∆2 .
 – P and Q lie on a given side M in ∆1 ←→ φx (P ) and φx (Q) lie on the same
   side of φx (M ) in ∆2 . (The same side of φx (M ) in is determined via the
   orientation specified by the two variable label of M and φx (M ).)
 – P lies inside/on/outside circle C1 in ∆1 ←→ φx (P ) lies inside/on/outside
   φx (C1 ) in ∆2 .
 – M intersects N at a point/along a segment in ∆1 ←→ φx (M ) intersects
   φx (N ) at a point/along a segment in ∆2 .
 – M cuts in one point/cuts in two points/is tangent at one point to/is tangent
   along a segment to C1 in ∆1 ←→ φx (M ) cuts in one point/cuts in two
   points/is tangent at one point to/is tangent along a segment to φx (C1 ) in
   ∆2 .
 – C1 has the same intersection signature with respect to C2 in ∆1 as φx (C1 ) has
   with respect to φx (C2 ) in ∆2 . (The intersection signature classifies the way
   C1 and C2 intersect each other as convex polygons. For its precise definition,
   see appendix A in the thesis.)
 – Circle C1 lies outside/ is contained by circle C2 in ∆1 ←→ φx (C1 ) lies
   outside/is contained by circle φx (C2 ) in ∆2 .
 – Circle C1 lies within a given side of M in ∆1 ←→ φx (C1 ) lies within the
   same side of φx (M ) in ∆2 .
 – End-arrow P is on a given side of M in ∆1 ←→ φx (P ) is on the same side
   of φx (M ) in ∆2 .
The Double Disjunction Task as a Coordination Problem

                                          Justine Jacot

                         University of Lund, Department of Philosophy
                                 justine.jacot@fil.lu.se



        Abstract. In this paper I present the double disjunction task as introduced by
        Johnson-Laird. This experiment is meant to show how mental model theory ex-
        plains the discrepancy between logical competence and logical performance of
        individuals in deductive reasoning. I review the results of the task and identify
        three problems in the way the task is designed, that all fall under a lack of coor-
        dination between the subject and the experimenter, and an insufficient represen-
        tation of the semantic/pragmatic interface. I then propose a reformulation of the
        task, that makes explicit the underlying semantic reasoning and emphasizes the
        difference of interpretation of the ddt between the experimenter and the subjects.


1     Introductory Remarks
1.1    The Double Disjunction Task
The double disjunction task (ddt), introduced by Johnson-Laird et al. (1992), is an
inferential task where subjects are asked to say “What, if anything, follows” from a set
of two premises, typically of the form:
            Alice is in India or Barbara is in Pakistan [or both/but not bot]                 (P1 )
       Barbara is in Pakistan or Cheryl is in Afghanistan [or both/but not bot]               (P2 )
    The optional parameters (in square brackets) are always covariant in premises. In
the terminology introduced by Johnson-Laird et al. (1992), the authors determine an
inclusive and an exclusive variant, when (resp.) the first and second values are chosen.
These variants are called affirmative, as opposed to those obtained when (P2 ) is substi-
tuted with:
      Barbara is in Bangladesh or Cheryl is in Afghanistan [or both/but not bot]              (P02 )
which are called negative, because of the incompatibility between (P1 ) and (P02 ), that
gives rise to a contradiction. With subscript for inclusive or exclusive, and superscripts
for positive and negative, one obtains four variants, ddt+i , ddt+e , ddt−i and ddt−e .
    The task was aimed at enforcing some predictions by the mental model theory (see
below) according to which the more models an agent has to represent the less she is able
to make correct inferences from the premises. Inclusive disjunctions should be therefore
more difficult to cope with than exclusive ones, as well as disjunctions involving a con-
tradiction in the premises. For matters of clarity, here is the truth table for the inclusive
(noted ∨) and exclusive (noted ∨) disjunctions:
28

                                        P Q P ∨ Q P∨Q
                                        TT T       F
                                        T F T      T
                                        F T T      T
                                        F F F      F

                   Fig. 1. Truth table for inclusive and exclusive disjunctions


1.2    The Mental Model Theory
The mental model theory (mmt) has been developed in reaction against the idea of a
‘mental logic’ according to which individuals have logical rules in their mind and draw
logical inferences in line with these rules. According to the mmt, people make logical in-
ferences by following neither the rules of natural deduction nor those of truth tables, but
by constructing mental models of the premises from which they derive a model of the
conclusion. Deductive reasoning does not depend on syntactic rules but on a three-steps
semantic process. First, semantic information is mentally construed as a representation
of situations given by the premises. Second, an attempt to draw a conclusion from the
mental models of the premises is made. Third, the validity of the conclusion is checked
by ensuring that no model of the premises renders it false. For instance, a disjunction
such as:
      There is a club or there is a spade
calls for two models:
      ♣
           ♠
      When the following premise is added:
      There is no club
   the model representing the club is discarded and the informational content of the
new premise is added to the remaining model:
      ¬♣       ♠
    The conclusion ‘There is a spade’ can be drawn, since there is no model of the
premises that makes this conclusion false.
    Relying on the assumption that individuals are fundamentally rational but make er-
rors in practice, Johnson-Laird identifies three principles for deductions made by ‘logi-
cally untutored individuals’:
      [They] eschew conclusions that contain less semantic information than premises.
      [...] Similarly, they seek conclusions that are more parsimonious than the premises.
      [...] They try to draw conclusions that make explicit some information only im-
      plicit in the premises. In short, to deduce is to maintain semantic information,
      to simplify, and to reach a new conclusion. (Johnson-Laird et al. 1992, p. 419)
                                                                                         29

    These constraints on deductive reasoning are close to Grice’s maxims of conversa-
tion (Grice 1989). These maxims ask for agents involved in a communicative situation
to be as informative as is required (maxim of quantity), to try to make their contribution
true (maxim of quality), to be relevant (maxim of relevance), and to be perspicuous
(maxim of manner). Although Grice is mentioned only in passing by Johnson-Laird
et al. (1992, p. 419), their analysis of deduction is recovered in these maxims: main-
taining semantic information pertains to the maxim of quantity, non redundancy and
simplification to the maxim of manner, and reaching a new conclusion to the maxim of
relevance, while deduction guarantees the quality of the conclusion, conditional on the
quality of the premises.
    Grice stresses that conversations where the maxims apply are a special case of coop-
erative situations, and that these maxims are instantiations of a more general Coopera-
tion Principle (cp). An experiment is, after all, a cooperative situation, and therefore the
effect of cp should be evaluated at the beginning of the task, because deductive reason-
ing of lay people is constrained by semantic principles as well as pragmatic principles.
    The remainder of this paper is organized as follows. The next section firstly dis-
cusses the results of the ddt, secondly identifies the main issues at stake in the inter-
pretation of the results and the way the task is formulated. Section 3 shows that most
of these problems arise because of a lack of coordination between the subject and the
experimenter in the interpretation of the task. Section 4 suggests some possible reformu-
lations that could improve the base rate of originally expected answer, by eliciting both
the adequate modal representation, and the selection of relevant information, through
pragmatic principles.


2     Results and Interpretation of the ddt

2.1   Results

Johnson-Laird et al. (1992) discuss only the selection made for the ‘positive’ ddt, con-
sisting in the following two statements:

 Alice is in India and Cheryl is in Afghanistan, or Barbara is in Pakistan, or both (C+i )
 Alice is in India and Cheryl is in Afghanistan, or Barbara is in Pakistan, but not both
                                                                                     (C+e )

    The valid conclusion for the negative disjunctions were predicted to be more dif-
ficult to make because they suppose the ability to first detect the contradiction in the
premises. The percentages of valid conclusions made by the subjects for the four kind
of disjunctions are displayed in Fig. 2.


                                           Affirmative Negative
                               Exclusive      21%        8%
                               Inclusive       6%        2%

                      Fig. 2. Percentage of valid conclusions for the ddt
30

     As predicted by the mmt, the percentage of success decreases when the number of
models to build increases, coupled with a floor effect: when a deduction needs three or
more models, it is almost impossible for the subjects to make a correct inference, or
even to decide whether a conclusion follows from the premises.
     But while Johnson-Laird is interested in the link between one particular case of
error with one particular subset of the models of the premises (Johnson-Laird et al.
1992, p. 433-434), and although the prediction of mmt for relative rates of success
are confirmed, the low base rates remain only imperfectly explained. Besides, a non
negligible number of subjects shows an ability to perform disjunctive reasoning. The
results reported by Toplak and Stanovich (2002) show a higher success rate for the ddt
(37%), because the authors collapsed in one ‘super-category’ all the conclusions that
displayed a “disjunctive pattern” of reasoning.1
     The two categories collapsed into the disjunctive responses were what Toplak and
Stanovich (2002) call ‘the partial contingency’ response and the ‘Alice-Cheryl response’,
which were both given by 12% of the subjects. The partial contingency response, for
instance, ‘If Alice is in India and Cheryl is in Afghanistan, Barbara is not in Pakistan’,
is given when people draw “the implications of one of the disjunctive path but not the
other.” (Toplak and Stanovich 2002, p. 203). According to Toplak and Stanovich (2002),
it seems in this case that subjects are able to assign hypothetic truth values to each dis-
junct in the premises but unable to represent situations where all states of affairs hold,
comprising those where Alice is not in India and Cheryl is not in Afghanistan.

2.2     Identifying the Problems in the Design of the ddt
mmt explains failure in terms of the number of mental models to be construed in order to
solve the task. Reasoning being semantic, the semantic content affects the resolution of
the task. The more models individuals have to process, the less they succeed in making
correct inferences, due to the difficulty to store in the working memory the totality of
the models of the premises needed to draw the correct inferences, particularly when
different situations have to be processed in parallel.
     While mmt assumes that “naive individuals have a modicum of deductive compe-
tence” (Johnson-Laird 1999, Johnson-Laird and Byrne 1991) but that they (sometimes)
fail at deductive performance, things have to be viewed the other way round. The the-
ory predicts that, faced with a deductive inference, individuals apply some filtering
constraints depending on the kind of inference they have to make. I suggest that they
should be viewed as applying some relevance constraints because of the form of the
reasoning they have to make, and then draw the conclusions that seem relevant.
     Changing the perspective preserves some insights from mmt (i.e. why ‘trivial’ con-
clusions are not even considered), but sheds light on other aspects left unexplained by
the theory. For instance, answers in disjunctive form are more common than in condi-
tional form (see Toplak and Stanovich 2002), which seems to indicate that people tend
to preserve the syntactic form of the premises.
 1
     What Toplak and Stanovich (2002) call ‘disjunctive reasoning’ is the construction of mental
     models for the premises and at least a partial combination of them in the conclusion. The
     normative answer (fully disjunctive) was given by only 13% of the subjects, which is lower
     than in Johnson-Laird et al. (1992).
                                                                                               31

    One may then identify three problematic categories in the formulation of the ddt.
The first category of problems concerns the logical form of the premises. First, the con-
junction of the two premises is implicit, but this conjunction must become explicit in
order for the intended (correct) conclusion to be drawn. Second, it is not clear whether
the disjunct ‘Barbara is in Pakistan’ should be at the same place in the two premises, or
at the first place in the first premise and at the second place in the second premise. In-
deed, it is not sure that not logically trained individuals always interpret the disjunction
as commutative in the ddt. But at the same time, it seems that people want to preserve
the syntactic form of the premises: answers in disjunctive form are more common than
in conditional form. Although Johnson-Laird et al. (1992, p. 433) explained subjects the
difference between an inclusive and an exclusive disjunction, guaranteeing a common
interpretation of “. . . or . . . or both” and “. . . or . . . but not both” (between participants
and experimenter), they give little detail about this explanation, save for the fact that it
did not rely on truth-tables or natural deduction rules.
    The second category pertains to the linguistic context opened by the ddt. First, it
seems that, in designing the task, Johnson-Laird and his colleagues wanted to avoid
the problem of an abstract task by giving the subjects a thematic task. But it is unclear
whether the difference would negatively affect the results. Indeed, the premises in the
ddt are rather abstract since they present names and places out of the blue. No context
or situation supports the story. Second, nothing in the task says that ‘Barbara’ in both
premises is the same individual, for the reasons mentioned above.
    The third category of problems relates to the way the ddt is framed in general,
and particularly the way the question is asked: ‘what, if anything, follows from these
premises?’ Either classical logic is assumed and an infinite number of valid conclusions
follows from the premises, and then subjects have no time or no space to give the full
answer; or the answer is intended to be the conclusion that preserves the most semantic
information from the premises and ‘if anything’ can be dropped from the question.
    In what follows I will show that these issues can be addressed if we reconsider
the ddt as a coordination problem between subject and experimenter. I will focus on
the second and third category of problems, the first one being answered in the new
formulation of the task I propose next.


3     A Coordination Problem

3.1   Narrowing Down the Context

Experimenters take often for granted that only one possible interpretation of the logi-
cal vocabulary is correct, and think that departure from this interpretation is an error.
But, as shown by Stenning and van Lambalgen (2008, chap. 3), the problem is that
experimenters also leave under-determined this interpretation for the subjects, to such
an extent that subjects sometimes complete the under-determined semantic content in
an unintended way. If mmt is correct and logical reasoning is indeed semantic, then the
ddt should be able to provide tests such that either the semantic content is determined
enough to yield a single interpretation, or the normative answer allows for different
interpretations.
32

    As noted by Shafir (1994), one problem with the ddt is that it demands subjects to
“think through” the disjuncts, and to draw a conclusion from both the premises. In order
to do so, subjects must process all the premises at the same time, but they appear on
paper as two separate sentences, and the conjunction of the premises remains implicit.
While the disjunctive form may trigger some ability to reason by cases (even though the
performance’s rate is low), what in fact is needed is to reason by cases together with the
ability to keep track of the different possibilities, as e.g. answers to different questions
regarding the premises.
    Moreover, as put forward by relevance theory (Sperber et al. 1995, Sperber and
Wilson 1995, Wilson and Sperber 2004) there is a discrepancy between the laboratory
task and the real life situations in which the subjects might have to perform these kind
of inferences. The lack of relevance of the task at stake, or the lack of communicative
interaction can yield some pragmatic ‘disabilities.’ On a linguistic level, some linguistic
forms produce some kind of relevance ‘filters’ content- and context- dependent which,
in turn, induce a preference ordering over possible interpretations (Girotto et al. 2001,
p. 70). On one hand, in the type of task presented to the subjects, logically relevant
and logically irrelevant content are mixed, whereas it is asked to the individuals to pay
attention only to the logically relevant content. On the other hand, in order to achieve the
task, individuals have to interpret it correctly. They must therefore pay attention to the
logically relevant and irrelevant content. But the logically irrelevant content sometimes
prompts individuals to a certain interpretation, which is not always the interpretation
the experimenter has in mind.2
    Further discussion of the relevance theory explanation of the success and failure of
logical reasoning is not necessary, since my argument needs no other agreement than
retaining its central insight. The ddt is content- and context-dependent and it seems
reasonable to think that subjects interpret it as such, in a way that intermingles semantic
and pragmatic inferences in reasoning.


3.2     Semantics vs. Pragmatics and the Principle of Cooperation

The way the question is asked in the ddt is disputable, for reasons that are admitted
by Johnson-Laird himself, but that he does not seem to factor in the formulation of the
question. Indeed, Johnson-Laird acknowledges that “infinitely many valid conclusions
follow from any premises” (Johnson-Laird 1999, p. 113) but, by saying ‘if anything’,
nevertheless implies that it could be the case that nothing follows from them. This is
obviously triggered by the three principles of deductive reasoning already mentioned
in section 1.2. Assuming these facts, it is highly improbable that subjects will draw
conclusions such as any conjunction of the premises.
    If it is true that logically untrained individuals do not draw ‘trivial’ conclusions from
a set of premises, there is no need to try to lure them in thinking that there might be
 2
     Sperber et al. (1995, p. 44): “Past discussions of subjects’ performance have tended to focus on
     the task as already interpreted (by the experimenter). [...] But interpreting the task is part and
     parcel of performing it, and obeys criteria of rationality in its own right. The study of ‘content
     effects’ is the study of sound cognitive processes that are by no means out of place in subjects’
     performance.”
                                                                                         33

some. Besides, the formulation of the question must indicate that there is some relevant
information that can be inferred from the premises. Indeed, the Gricean cp would ask for
a speaker’s intentions to be recognized by the hearer in order for the speaker’s utterance
to be correctly interpreted. Why not assuming such a process in the ddt? As previously
mentioned, a psychological experiment involves a certain level of cooperation between
subjects and the experimenter. Certainly cp governs inferences to intentions and goals
of others, and reconstructing them in ddt inevitably involves speculation. But it is not
impossible to reconstruct a possible route to C+i and C+e that takes cp more seriously
than the original theory.


4     Two Versions of the ddt
4.1    The Intended Interpretation
The theory predicts that subjects will: (a) compute a representation of three (two) men-
tal scenarios, or ‘mental models’, for each inclusive (resp.: exclusive) premise; (b) com-
bine the representations, eliminating those incompatible with the information; and: (c)
‘read off’ some appropriate conclusion and answer the question.
     In this general setting, the intended interpretation requires an abstraction step. In
step (a), subjects are supposed to replace the premises with a variable X ∈ {A, B, C}
where A, B, C stand for each of the individual in the disjunct, to which a value is as-
signed, equivalent to ‘I know/I do not know where is X’. Step (b), for each variant of
the ddt, is tantamount to assigning a value to each variable, compatible with the con-
straints embodied in the models of the premises.
     Comparatively, subjects should be expected to have greater difficulties to solve the
‘negative’ tasks. Johnson-Laird et al. explain this difficulty by the necessity of obtaining
an additional (implicit) premise, which follows from subjects background knowledge:
          Of course, if [Barbara] is in [Bangladesh], then [she] is not in [Pakistan].
      According to the model theory, affirmative deductions should be easier than
      such negative deductions because the latter call for the detection of the incon-
      sistency between the contrary constituents. (Johnson-Laird et al. 1992, p. 433)
    The detection of inconsistency is an additional (inferential) step, left unanalyzed by
Johnson-Laird et al. This step is equivalent to impose additional constraints to compute
the composition of the representations associated with (P1 ) and (P02 ), using background
knowledge. The additional complexity of tracking combinations of two values for pa-
rameter B in both ddt−i ddt−e , explains the cost of “detection of inconsistency” by the
stress imposed on computational abilities. In addition, ddt−e outputs an additional men-
tal model, as compared to ddt+e , inducing a heavier load on working memory.
    So far, this reconstruction of steps (a) and (b) of ddt agrees with Johnson-Laird
et al. However, reconstruction of step (c) raises serious issues with both the design of
the experiment, and the interpretation of its results. Johnson-Laird et al. discuss only
the selection made for the ‘positive’ ddt. Their justification for this selection is the
“support” mental models yield to these conclusions.
    Indeed, treating A, B and C as propositions, and substituting known (asserted) and
unknown (excluded) locations with 1 and 0, respectively, one obtains distributions of
34

truth-values that satisfy (C+i ) and (C+e ). However, this explanation is at odds with the
claim that reasoning based on mental models does not proceed by building and reading
equivalent of truth tables, since “truth tables [. . . ] are too bulky to be mentally manipu-
lated” (Johnson-Laird et al. 1992, p. 421).
     Considering the representation Johnson-Laird et al. gives for the set of final mental
models in ddt+i and ddt+e , reproduced Fig. 3, (C+e ) is the most natural of the answers
considered. If the mental construction is actually carried as hypothesized by Johnson-
Laird et al., (C+e ) can be ‘read off’ neglecting excluded locations, and using ‘or. . . but
not both’ to express the existence of (two) alternative scenarios. This method does not
rely on explicit substitution with truth-values, but has no straightforward equivalent in
ddt+i that would read (C+i ) off.


                      Alice   Barbara    Cheryl Alice   Barbara    Cheryl
                       [I]                [A]    [I]      [P]       [A]
                                 [P]             [I]      [P]
                                                 [I]                    [A]
                                                           [P]          [A]
                                                           [P]

                            Fig. 3. Mental Models for ddt+e and ddt+i


    One could hypothesize that subjects approximate ddt+i by addressing it as if it were
    +
ddte , and then simply switch the proviso from “but not both” to “or both.” However, this
rationale for the choice of (C+i ), is not consistent with the data, in particular the relative
difference between the rate of success in ddt+e and ddt+i . Johnson-Laird et al. may have
selected (C+i ) by the above reasoning, or simply by judging (C+e ) intuitively natural
enough, and, switching the proviso, assuming that (C+i ) was natural, too. However they
give no hint has to what their selection is based on, and rest content with the relative
frequencies, that validate the predictions of mmt.
     In the absence of an analysis of the ‘reading off’ method that outputs (C+i ), it seems
difficult to argue whether the answer corresponds to some natural method. Likewise,
the absence of discussion of answers expected to be given to ddt−i and ddt−e makes the
assessment of the task more complicated than Johnson-Laird et al. seem to assume.

4.2   The Abstraction Problem
The naturalness of (C+e ) depends essentially on the representation of the problem that
neglects specific locations, and in which one checks only whether a location is known,
or not. If one assumes with Johnson-Laird et al. that reasoners somehow automatically
select representations that are as implicit as possible, the inescapable conclusion is that
the underlying automatisms are not very good at selecting the level of abstraction that
makes the task solvable.
    Yet, abstraction in variants of the ddt may not be as simple as assumed by Johnson-
Laird et al. Reading (or hearing) (P1 ), (P2 ) and (P02 ), in either their inclusive or exclusive
variants, may elicit a very different representation, if the range of values is taken to be
                                                                                                       35

the set of locations. Then, upon reading (hearing) (P1 ), a subject may consider that the
set of relevant values is:3
                                    v(P1 ) = {i, p, ip}                            (v(P1 ) )
(where ip corresponds to any other possible location than India and Pakistan). However,
reading (P2 ) produces a new set of values for that premise, as well as an update of (v(??) ),
as follows:

                                         v0(P1 ) = {a, i, p, aip}                                (v0(P1 ) )
                                         v(P2 ) = {a, i, p, aip}                                 (v(P2 ) )

    The effect is even more striking with (P02 ), where the number of seemingly relevant
values (and therefore of their combinations) increases even more, with:

                                       v00(P1 ) = {a, b, i, p, abip}                             (v00(P1 ) )
                                       v(P02 ) = {a, b, i, p, abip}                              (v(P02 ) )

    Given the complexity of the task, one will typically not complete step (a) before
the time the answer has to be given. Again, when “unable to construct a set of mod-
els or [. . . ] to discern what holds over all of them” subject will respond that no valid
conclusion follows.


5      Conclusion

DDT as it is designed in psychology settings, is not a bad test for reasoning skills
in the sense that it would mistake the competence to test. The problem comes from
the way experimenters interpret the task itself. The difference in interpretation between
experimenters and subjects is what leads to a misinterpretation of the results, and shows
the role of relevance and computational complexity in the answers of the subjects. Re-
considering the ddt as a coordination situation brings to light the reasoning process of
the subjects in drawing inferences from a double disjunction. This process, as shown
in the sketch of the formal reformulation of the task above, is a sequential procedure
which starts from a step-by-step construction of representations task, and outputs a
global conclusion. Our reformulation emphasizes the discrepancy between the intended
interpretation of the ddt by the experimenters, and the task as it may be understood by
the subjects.
    What remains to do in a further research is, from a theoretical perspective, building a
complete formal representation of the ddt displaying the intended interpretation by the
experimenter and the possible interpretation(s) by the subjects. This reformulation of
the task relies on the difference made by Stenning and van Lambalgen (2008) between
reasoning to an interpretation and reasoning from an interpretation: the authors argue
that subjects must always first reason to the experimenter’s (intended) interpretation
before they can reason from it to a solution.
 3
     A symmetric argument can be given for the case where (P2 ) or (P02 ) are presented first.
36

    From an empirical perspective, we need to design an experiment based on the ex-
planatory hypothesis argued for in this paper, especially concerning the level of ab-
straction required from the subjects in order to solve the task. To this effect, it seems
desirable to handle both sides of the problem by testing subjects’ performance in a
fairly abstract task where the disjuncts are displayed as propositional variables such as
A, B, C, and compare the results with a ddt where the premises are contextually fleshed
out by a story to fulfill the need for a thematic task.
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         Conditionals, Inference, and Evidentiality

      Karolina Krzyżanowska1 , Sylvia Wenmackers1 , Igor Douven1 , and Sara
                                   Verbrugge2
1
      Faculty of Philosophy, University of Groningen, Oude Boteringestraat 52, 9712 GL
                               Groningen, The Netherlands
    2
       Department of Psychology, University of Leuven, Tiensestraat 102, 3000 Leuven,
                                           Belgium



         Abstract. At least many conditionals seem to convey the existence of
         a link between their antecedent and consequent. We draw on a recently
         proposed typology of conditionals to revive an old philosophical idea
         according to which the link is inferential in nature. We show that the
         proposal has explanatory force by presenting empirical results on two
         Dutch linguistic markers.


1      Introduction

Although various theories of conditional sentences have been proposed, none of
them seems to account for all the empirical data concerning how people use
and interpret such sentences.3 It is almost universally agreed that no theory of
conditionals counts as empirically adequate if it validates sentences like these:

(1) a. If badgers are cute, then 2 + 2 = 4.
    b. If weasels are vegetables, then unicorns hold Frege in particularly high
       esteem.

It is easy to understand why people are reluctant to accept conditionals like
(1a) and (1b): the antecedents of those conditionals have nothing to do with
their consequents. And it seems that using a conditional construction is meant
to convey the existence of some sort of link between the content of the if-clause
and the content of the main clause.
    But what kind of link might this be? According to an old philosophical idea,
the link is inferential in nature. If this idea is currently unpopular, at least in
philosophy, that may be due to the fact that theorists have failed to recognize
that the inferential connection need not be of one and the same type. We draw
on a recently proposed typology of conditionals to reinstate the old idea, at
least for a large class of conditionals that linguists have aptly termed “inferen-
tial conditionals.” We buttress our proposal by presenting experimental results
concerning two Dutch linguistic markers.
3
     For a survey of various accounts of conditionals and problems they are facing see,
     for instance, Edgington [1] or Bennett [2].
                                                                                     39

2   Inferential Conditionals
Probably the most general distinction to be made when it comes to conditionals
is that between indicative and subjunctive conditionals. In this paper, we will
only be concerned with indicative conditionals.4 For many theorists this is only
the beginning of a typology, though there is little unanimity as to what the ty-
pology should further look like. What has become common in linguistics, but is
rarely encountered in the philosophical literature, is to classify indicative condi-
tionals as inferential and content conditionals.5 The class of content conditionals
is not particularly well defined—its members are sometimes loosely said to de-
scribe relations between states of affairs or events as they happen in reality—but
those will not concern us here. We are going to limit our attention to inferen-
tial conditionals, that is conditional expressing a reasoning process, having the
conditional’s antecedent as a premise and its consequent as the conclusion, for
example: “If she has not had much sleep recently, she will perform poorly on her
exams” or “If he owns a Bentley, he must be rich.” Arguably, these constitute a
common, if not the most common, type among the conditionals we encounter in
natural language.
    That a conditional sentence can be considered as a kind of “condensed argu-
ment” [8, p. 15] is not altogether new to philosophy; it can be traced back at
least to Chrysippus, a stoic logician from the third century BC. He is believed
to have held the view that a conditional is true if it corresponds to a valid ar-
gument [9]. Obviously, if we limit our understanding of a valid argument only
to the classical deductive inference, we can easily find counterexamples to the
above claim. Yet deduction is not the only type of reasoning people employ, and
a plausible theory of inferential conditionals should not neglect this fact.
    Although linguists have proposed various finer-grained typologies of infer-
ential conditionals (see, e.g., Declerck and Reed [6]), most of these stem from
grammatical distinctions. However, we are interested in a specific typology re-
cently presented in Douven and Verbrugge [10] who acknowledge the variety of
inferential relations between a conditional’s antecedent and its consequent. A
first distinction these authors make is that between certain and uncertain infer-
ences, where certain inferences guarantee the truth of the conclusion given the
truth of the premises while uncertain inferences only tend to make the truth of
the conclusion likely given the truth of the premises; the former are standardly
referred to as “deductive inferences.” Douven and Verbrugge further follow stan-
dard philosophical usage in grouping uncertain inferences into abductive and
inductive ones, where the former are inferences based on explanatory considera-
tions and the latter are inferences based on statistical information. More exactly,
in an abductive inference we infer a conclusion from a set of premises because
4
  The distinction between indicative and subjunctive conditionals is not as clear-cut as
  one might wish, but the conditionals we used in our materials were all uncontroversial
  cases of indicative conditionals. Henceforth, indicative conditionals will be referred
  to simply as “conditionals.”
5
  See, among others, Dancygier [3,4], Dancygier and Sweetser [5], Declerck and Reed
  [6], and Haegeman [7].
40

the conclusion provides the best explanation for those premises; for example, we
may infer that Sally failed her exam from the premises that Sally had an exam
this morning and that she was just seen crying and apparently deeply unhappy:
that she failed the exam is the best explanation for her apparent unhappiness.
Inductive inferences rely on information about frequencies (which may be more
or less precisely specified); for instance, we infer that Jim is rich from the premise
that he owns a Bentley because we know that by far the most Bentley owners
are rich. The validity of both abductive and inductive inferences is a matter of
controversy and ongoing debate. It is largely uncontested, however, that people
do engage in these types of inferences on a routine basis.
    Douven and Verbrugge’s typology of inferential conditionals follows the afore-
said typology of inference. That is to say, they distinguish between certain (or
deductive) and uncertain inferential conditionals, and then divide the latter class
further into abductive and inductive inferential conditionals. More specifically,
they propose the following:
Definition 1. “If p, then q” is a deductive inferential (DI, for short) / inductive
inferential (I I) / abductive inferential (AI) conditional if and only if q is a de-
ductive / inductive / abductive consequence of p.
(Various formalizations of the inductive and abductive consequence relations
have been offered in the literature, though, like Douven and Verbrugge in their
paper, we refrain from committing to any in particular.) Douven and Verbrugge
note also that, often, the inference may rely on the antecedent p together with
background assumptions that are salient in the context in which the conditional
is asserted or evaluated. Such conditionals are called contextual DI, AI, or I I
conditionals, depending on the type of inference involved.6
Definition 2. “If p, then q” is a contextual DI / I I / AI conditional if and only
if q is a deductive / inductive / abductive consequence of {p, p1 , . . . , pn }, with
p1 , . . . , pn being background premises salient in the context in which “If p, then
q” is asserted or evaluated.
    Douven and Verbrugge do not claim that their typology is correct and the
ones that so far have been propounded by other theorists are incorrect. Indeed,
it might even be odd to think that there are natural kinds of conditionals that
a typology should try to chart. What they do claim is that their typology is
exceedingly simple and that it is non-ad hoc in that it relies on a time-tested
distinction between types of inference. More importantly still, they show in their
2010 paper that the typology has considerable explanatory force by recruiting it
in service of testing a thesis, first proposed by Adams [11] and championed by
many since, according to which the acceptability of a conditional is measured by
the probability of its consequent conditional on its antecedent.
6
     As Douven and Verbrugge [10, p. 304] note, in contextual AI conditionals, the con-
     sequent need not always be the best explanation of the antecedent. It may also be
     that the consequent is, in light of the antecedent, the best explanation of one of the
     background assumptions.
                                                                                       41

    We expand here on the main experiment of Douven and Verbrugge’s paper,
because it served as the starting point for our own empirical work. In their exper-
iment, Douven and Verbrugge divided the participants into two groups, asking
one group to judge the acceptability of ten DI, ten AI, and ten I I conditionals
and the other group to judge the corresponding conditional probabilities. This
is an example of a question asking for the acceptability of an AI conditional:7

     Context: Judy is waiting for a train. She is looking for her iPod. It is not in
     her coat. She suddenly sees that the zipper of her bag is open. She cannot
     remember having opened it. It is announced that there are pickpockets
     active in the train station.
     Conditional: If Judy’s iPod is not in her bag, then someone has stolen it.
     Indicate how acceptable you find this conditional in the given context:
       Highly unacceptable     1    2   3    4   5    6   7   Highly acceptable

The corresponding question asking for the conditional probability is this:

     Context: Judy is waiting for a train. She is looking for her iPod. It is
     not ain her coat. She suddenly sees that the zipper of her bag is open.
     She cannot remember having opened it. It is announced that there are
     pickpockets active in the train station. Suppose that Judy’s iPod is not in
     her bag.
     Sentence: Someone has stolen Judy’s iPod.
     Indicate how probable you find this sentence in the given context:
         Highly improbable     1    2   3    4   5    6   7   Highly probable

The results obtained in this experiment show that Adams’ thesis holds for DI
conditionals at best, and that for AI conditionals the most that can be said
is that acceptability and conditional probability are highly correlated; for I I
conditionals not even that much is true.
    This already shows that the typology of inferential conditionals proposed
by Douven and Verbrugge has considerable explanatory force. Here, we aim
to extend the case for this typology by relating it to two putative evidential
markers. Before we move on to report our experimental results on these markers
(in section 4), we briefly discuss the concept of an evidential marker.


3     Evidential Markers

In some languages, it is common for speakers to communicate information about
the evidential grounds for the contents of their assertions. Some languages also
possess a rich arsenal of prefixes, suffixes, particles, and other linguistic items for
7
    See Appendix A of Douven and Verbrugge [10] for the full materials used in this
    experiment.
42

this purpose.8 European languages do not encode evidentiality grammatically,
but this is not to say that speakers of European languages do not have the
resources to indicate evidential grounds, or that they never use those resources.
Sentences like:

(2) Adam works hard.

do not indicate the speaker’s source of information at all—the speaker can actu-
ally share an office with Adam and see him working hard on an everyday basis,
but she could also have inferred (2) from Adam’s work output or simply heard
someone else say so.
    But other sentences do suggest the speaker’s evidential grounds. For instance,
when we are wondering about the translation of a phrase in Latin and we know
that Susan studied classical languages for a number of years, we might say

(3) Susan should be able to translate this phrase.

This assertion would seem odd if we knew that (say) the phrase is from a text
which Susan recently published in English translation. Similarly, when an English
speaker tries to call her friend but does not get an answer, she may infer that
her friend is out and express the resulting belief by saying:

(4) She must be out.

Were she to see her friend walking on the street, her assertion of (4) would again
seem odd or even inappropriate.
    Some authors have argued that the modal auxiliary verb “must” makes the
assertion weaker than the one without it.9 But this is not generally true. For
sometimes “must” seems to indicate the necessity of what has been asserted: if
one knows that Mary has put a bottle of wine either in the fridge or in the
cupboard, and one has checked that it is not in the cupboard, it seems natural
for one to conclude:

(5) It must be in the fridge.

As noticed by von Fintel and Gillies [17,18], what (4) and (5) have in common
in the first place is that they signal the presence of an inference. Specifically,
the verb “must” indicates that the speakers’ grounds for their assertions are
inferential, and hence indirect, but as they also argue, this need not mean that
8
  On the basis of data from 32 languages, Willett [12] proposed a taxonomy of markers
  encoding main types of sources of information. The main distinction is between di-
  rect (perceptual) and indirect access, and the latter can be further divided into other
  speaker’s reports and inference. Since then, various aspects of evidentiality in lan-
  guage have been investigated, like for instance a developmental study by Papafragou
  and colleagues [13] on Korean speaking children’s learning of evidential morphology
  and their ability to reason about sources of information.
9
  See e.g. Karttunen [14, p. 12], Groenendijk and Stokhof [15, p. 69] or Kratzer [16,
  p. 645]
                                                                                     43

these grounds are weak or inconclusive. Again, some confusion could have been
avoided if the variety of inference relations had been attended to.
    Of course, sometimes we convey information about our evidential grounds
in more direct ways, as when we say that we saw that John crossed the street,
or that it seems to us that Harriet is worried, or by the use of such words
as “probably,” “presumably,” “possibly,” “apparently,” “allegedly,” “putatively,”
and so on. But in this paper we focus on “must” and “should” in their roles
as evidential markers, or rather, we focus on their Dutch counterparts “moet
wel” (“must” 10 ) and “zal wel” (“will,” “should”). Evidential markers can serve
a number of purposes. For instance, they may be used to indicate the source
of the speaker’s evidence: whether it is perceptual evidence, or evidence from
testimony, or evidence from some third type of source still. They may also be
used to indicate the quality of the evidence (e.g., indicate how reliable the source
was). We will be mainly interested in the question of whether “moet wel” and “zal
wel” play any distinctive role in signaling the kind of inference that is involved in
making whatever evidence the speaker has bear on the content of her assertion.
Can anything systematic be said about whether these markers go better with
some type or types of inference than with others?
    To clarify this question, note that the inference underlying the assertion of
(4) in our example is most plausibly thought of as being abductive, that is, as an
inference to the best explanation: that the friend is out is the best explanation
for the evidence that the speaker has, to wit, that her friend does not answer the
phone. In the example of Susan, it rather seems to be some form of inductive
reasoning that warrants the assertion of (3): the people we met in our lives
who had studied classical languages for a number of years were typically able
to translate Latin phrases; given that Susan studied classical languages for a
number of years, we expect her to be able to translate the designated phrase.
Naturally, the inferential connection between evidence and grounds for assertion
may also be deductive, as in the case of (5). The question we are interested in is
whether the use of “moet wel” and “zal wel” gives us any indication as to what
kind of inference (if any at all) led the speaker to feel warranted in making the
assertion she did on the basis of the evidence she had.


4      Experiment: Linguistic Markers
Our experiment makes use of the typology of inferential conditionals discussed
above. We look at a number of instances of the various types whose degrees of
acceptability have been ascertained in previous research and we look whether
these degrees are affected by inserting “moet wel” or “zal wel” into the sentences.
We encountered the putative English counterparts of these markers already in
our examples involving (4) and (3). “Must” and “zal wel” have also been de-
scribed as inferential markers in the literature. As for the latter, Verbrugge [20]
established a close connection between “zal wel” and inferential conditionals.
10
     “Wel” is a positive polar marker which has no counterpart in English; see Nuyts and
     Vonk [19, p. 701]. In German, “bestimmt” comes close.
44

Specifically, in an elicitation task in which they were requested to complete con-
ditionals whose antecedents were given, participants tended to come up with
a significantly higher number of inferential conditionals (as opposed to content
conditionals) when they were in addition requested to use “zal wel” in the conse-
quents than when they were not. As for “must,” Dietz [21, p. 246] notes that in
“It must be raining,” the auxiliary indicates that the speaker only has (what he
calls) “inferential evidence,” and no direct observational evidence, that it is rain-
ing; see also the papers by von Fintel and Gillies cited earlier as well as Anderson
[22], Papafragou [23], and Nuyts and Vonk [19]. However, so far researchers have
not considered differentiating between the various types of inference by means of
the said markers. Might not one marker go better with the expression of one type
of inference and the other with the expression of another type of inference? Even
more fundamentally, is “must” really an inferential marker? We are not aware
of any empirical evidence that warrants a positive answer. Philosophers may be
convinced that “must” can serve as an evidential marker, but philosophers were
also convinced that the acceptability of a conditional is equal to the correspond-
ing conditional probability, and—as Douven and Verbrugge showed—empirical
evidence gives the lie to that thought.
    To investigate the aforementioned questions, we used the materials of the ex-
periment described in the previous section. We inserted “moet wel” and “zal wel”
into the consequents of the conditionals used in Douven and Verbrugge’s main
experiment and checked whether this made a difference to acceptability ratings
and their correlations with probability ratings. We were particularly interested
in the effect the presence of the auxiliaries has on these correlations for different
types of conditionals.

4.1   Method
Participants
Fifty seven students of the University of Leuven took part in the experiment.
Design
The type of conditional (DI / AI / I I) was manipulated within subjects. The dif-
ferent lexical markers were manipulated between subjects.
Materials and Procedure
All materials were in Dutch, the participants’ mother tongue. Thirty items were
presented in a booklet. Every participant had to evaluate ten abductive, ten
inductive, and ten deductive items. Items were randomized per booklet and the
booklets were randomly distributed in the lecture hall. The items consisted of the
same context–sentence pairs that were presented to the participants in the main
experiment of Douven and Verbrugge [2010] who were asked to judge the accept-
ability of conditionals, except that the conditionals now contained the markers
“moet wel” and, respectively, “zal wel.” For instance, to the AI conditional “Als
Judy’s iPod niet in haar tas zit, dan heeft iemand die gestolen” (“If Judy’s iPod
is not in her bag, then someone has stolen it”), whose acceptability participants
in Douven and Verbrugge’s experiment had been asked to grade, corresponded
                                                                                 45

in our experiment the AI conditionals “Als Judy’s iPod niet in haar tas zit, dan
moet iemand die wel gestolen hebben” (“If Judy’s iPod is not in her bag, then
someone must have stolen it”) and “Als Judy’s iPod niet in haar tas zit, dan zal
iemand die wel gestolen” (“If Judy’s iPod is not in her bag, then someone will
have stolen it”).
    Participants (N = 30) in the condition “moet wel” were asked to judge the
acceptability of the conditionals containing “moet wel.” Participants (N = 27) in
the condition “zal wel” were asked to judge the acceptability of the conditionals
containing “zal wel.” The instructions were the same as the ones used in the
acceptability condition of Douven and Verbrugge’s experiment.

Results
Comparisons with the condition investigating the probability were set up. We
computed the mean per sentence (ten abductive sentences, ten inductive sen-
tences, and ten deductive sentences) over the participants. We thus obtained
a mean for each of the thirty sentences. Then we computed the correlations
between the condition with marker (“zal wel” / “moet wel”) and the mean prob-
abilities per item obtained on the basis of Douven and Verbrugge’s experiment.
    For the thirty sentences in the experiment, probabilities as obtained in Dou-
ven and Verbrugge’s experiment and acceptability of the sentences with “zal wel”
were highly correlated: N = 30, Spearman R = .837712, t(N − 2) = 8.116914,
p < .0001. For the thirty sentences in the experiment, probabilities as ob-
tained in Douven and Verbrugge’s experiment and acceptability of the sen-
tences with “moet wel” were highly correlated: N = 30, Spearman R = .855871,
t(N − 2) = 8.756641, p < .0001.
    We next considered different types of conditionals. For the abductive sen-
tences, probability and “moet wel” were highly correlated: N = 10, Spearman
R = .993902, t(N − 2) = 25.49522, p < .00001. We obtained similar results for
“zal wel”: N = 10, Spearman R = .936175, t(N − 2) = 7.532386, p = .0001. For
the inductive sentences, correlations did not reach significance level; for “moet
wel” and “zal wel”: N = 10, Spearman R = .612121, t(N − 2) = 2.189453,
p = .059972. For the deductive sentences, probability was highly correlated with
“moet wel”: N = 10, Spearman R = .814593, t(N − 2) = 3.972223, p < .01.
For probability and “zal wel,” the correlation did not reach significance level:
N = 10, Spearman R = .613985, t(N − 2) = 2.200141, p = .058981.
    Comparison with the results from Douven and Verbrugge’s experiment showed
that, overall, the markers had little effect on the perceived acceptability of the
conditionals as well as, correspondingly, on the correlation between acceptabil-
ity and probability (see Table 1). However, splitting the results for the various
types of conditionals was more revealing. It appeared that, while for I I condition-
als, adding either of our two markers had virtually no effect on the correlation
between acceptability and probability, adding them to the AI conditionals did
increase the said correlation, even to the extent of yielding a near-to-perfect cor-
relation for the AI conditionals containing “moet wel” (for no marker, R = .8997;
for “zal wel,” R = .936175; and for “moet wel,” R = .993902). For DI conditionals,
46

Table 1. Correlations with and without markers (* = marginally significant; for all
other results p < .01)

                              All         DI          II          AI
             No marker      .851102    .818182     .620064*     .899700
               “zal wel”    .837712    .613985     .612121*     .936175
             “moet wel”     .855871    .814593     .612121*     .993902



adding “moet wel” had almost no effect, but adding “zal wel” led to a considerable
decrease of the correlation between acceptability and probability.


4.2   Discussion

These results confirm that “moet wel” and “zal wel” are inferential markers, given
that (i) inserting either of them in the AI conditionals has the effect of increasing
the correlation between acceptability and probability, and (ii) inserting “zal wel”
in the DI conditionals has the effect of decreasing that correlation. It is no
surprise that inserting “moet wel” in the DI conditionals does not have a similar
effect: like “must” in English, “moet wel” can also serve as an alethic modality,
and may thus be naturally interpreted in a DI conditional as underlining the
necessity of the inference.


5     Concluding Remarks

The typology proposed in Douven and Verbrugge [10] helps to explain why
adding inferential markers to conditionals makes the systematic kind of differ-
ence that we found in our experiment. That is further support for the thought
that this typology is of theoretical significance. The experimental work reported
here is part of a larger project. Experiments concerning the English “must,”
“should,” and “will” are currently being undertaken, and the results are to be
compared to the results of the experiment reported above. A further avenue
for future research concerns applications of these markers. For instance, if some
markers can be used as a kind of litmus test for distinguishing between various
types of conditionals, then testing for the effect that adding these markers to
conditionals has may help us in classifying conditionals whose type is controver-
sial. Perhaps the most important part of the project is to see whether the current
typology of conditionals can serve to ground a new semantics and/or pragmatics
of conditionals. Once it is recognized that various types of inferential connection
may be involved, it becomes quite plausible to claim that at least many condi-
tionals require for their acceptability the existence of an inferential link between
antecedent and consequent. Whether this is then to be taken as a brute prag-
matic fact, or whether it has a deeper explanation in terms of truth conditions,
is the question we ultimately hope to answer. The investigations reported here
are meant as a first step toward that answer.
                                                                                        47

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 Using Quantified Epistemic Logic as a Modeling
       Tool in Cognitive Neuropsychology
                                 Rasmus K. Rendsvig
                  Philosophy and Science Studies, Roskilde University


1     Introduction
The classic modus operandi for model construction in cognitive neuropsychology
(CN) is by use of box-and-arrow diagrams1 to capture the functional architecture
of the mental information-processing system under consideration. In such dia-
grams, of which Figure 1 is an example, boxes represent particular components
of the system and arrows represent pathways of information flow. Along with a
suitable interpretation, box-and-arrow diagrams represent typical CN theories,
consisting of statements regarding what specific modules are included in the sys-
tem as well as statements regarding how information may flow between these
components.
    Recently, it has been argued that this methodology should be augmented
by the use of computational models, allowing for the realization of CN theories
in the form of executable computer programs, structurally isomorphic to the
box-and-arrow version the theory, [3, p. 166]:
     This way of doing cognitive psychology is called computational cognitive psy-
     chology, and its virtues are sufficiently extensive that one might argue that all
     theorizing in cognitive psychology should be accompanied by computational
     modeling.

Though epistemic logics are not by themselves executable programs, as a mod-
eling tool they do however possess many of the features and virtues required
by [3]. It is the purpose of this paper to discuss the viability of using epistemic
logics as a modeling tool in cognitive neuropsychology.
Benefits of epistemic logic. There are three fields that could benefit from
epistemic logical modeling of theories from cognitive neuropsychology. First, a
motivation for constructing formal logical models aimed at cognitive neuropsy-
chology is that precision and logical entailment can provide explanatory force
and working hypotheses. Further, epistemic logics can be used to express higher
cognitive functions such as knowledge and belief in straightforward languages.
This further allows the formulation of derived principles, such as object recogni-
tion. That epistemic logics can straightforwardly express such functions further
makes it easy to read off predictions from logical theories, allowing for simple
comparisons with empirical observations.
    Second, logicians interested in modeling information flow and communication
acts could gain more realistic models of the internal parts of these processes, if
they took to modeling the functional architecture underlying our abilities to
perform such actions.
1
    The “universal notation in modern cognitive neuropsychology”, according to [4].
                                                                                49

    Finally, if logicians and cognitive neuropsychologists merged formal tools and
empirical insight, philosophers would stand to gain. Having empirical theories
couched in flexible modal logical frameworks would allow precise analyses of
philosophical problems using tools with which many philosophers are already
familiar. To exemplify, the model introduced below has been used in [14] to
give novel analyses of Frege’s Puzzle about Identity, based on formal notions of
semantic competence.
Plan of attack. As argued in [3], a proper modeling of a CN theory should
be true to that theory in an isomorphic sense: the formal model should include
exactly the modules and pathways of the CN theory, while maintaining their
separate and joint functions. However, while many CN theories include a seman-
tic system module, it is far from clear where such a collection of concepts can
be found in, e.g., a quantified S5 logic. To overcome this difficulty, we here take
the perspective that a logical model of a CN theory is composed of both a for-
mal logic as well as semantics for this logic. More specifically, then the modules
of a functional architecture is represented by model-theoretic structures over
which agents’ capabilities can then be expressed using formal logical syntax. A
complete logic for the model-theoretic structure may then be seen as a theory
detailing the capabilities of an agent with such a mental makeup.
    The present approach differs from the computational cognitive neuropsy-
chological (CCN) way outlined in [3] in an important aspect. The CCN approach
there discussed construct a model of normal behavior, which can the be ‘lesioned’
to simulate brain damage, and thereby compare to experimental observations.
In the present, we reverse this approach. Instead of constructing first a stronger
logic for normal behavior from which we can then remove, we construct a weaker
logic for the completely damaged, to which abilities can then be added. As such,
the logic is generic: in order to produce a subject-specific logic, further assump-
tions regarding specific knowledge and abilities must be made.
Evaluating logical models. Given that a motivation for logical modeling of
CN theories was a promise of working hypotheses, one would expect such hy-
potheses to be empirically testable. As the observables in CN tests are subjects’
performance based on given input, and these abilities are described by formal
logical statements over the model-theoretic structure modeling the functional
architecture, it is thus natural to take such hypotheses as constituted by the
formulas satisfied in the actual world of the semantic model. As will be shown
below, this is a feasible way of comparing formal model with empirical observa-
tions. However, there is no hope that any normal epistemic logical model can
produce hypotheses all of which are consistent with made observations. Qua the
problem of logical omniscience, any modeled agent will always know all logical
consequences of her knowledge [7], which no subject can do. This results in a
problem for the present approach in relation to evaluating, rejecting and refining
the formal models, for every hypothesis may now be wrong for two reasons: it
may be the case that the modeled functional architecture does not represent
reality, or it may be that the chosen hypothesis requires reasoning skills beyond
normal, human capabilities. In the latter case, this will result in a non-answer,
50

and the danger is that the correct functional architecture is rejected. We offer
no solution here, but note that this is a problem for which one needs to control
when performing theory evaluation.
Structure. The structure of the paper is as follows. In the ensuing section,
we introduce a simplified version of a CN theory of the structure of lexical,
semantic competence (SLC) from [12]. This is to act as a toy CN theory, which
will be modeled using a two-sorted quantified epistemic logic (QEL) introduced
in section 3. Most weight is on section 4, where the connections between the
QEL and the SLC are drawn. It is first argued that the modules of the SLC are
represented in the two-sorted model-theory. Secondly, it is argued that the three
distinct competence types of the SLC are expressible in the formal language,
and that their dissociations are preserved in the two-sorted logic.2 Finally, we
consider studies which show the shortcomings of the present model, and thereby
falsify the presented model. We then conclude.

2    The Structure of Lexical Competence
In [12], a box-and-arrow theory of the structure of semantic, l exical competence
(SLC) is constructed on the basis of studies from cognitive neuropsychology.3
The elements of the SLC consist of three competence types defined over four on-
tologies, two of these being mental modules, see Figure 1. The three competence
types are inferential competence and two types of referential competence, being
naming and application. The four ontologies include one of external objects, one
of external words (e.g., spoken or written words) and two mental modules: a
word lexicon and the semantic system. This structure is illustrated in Figure 1.
                                                           Application


                                    Inference


                         Naming

                Words        Word Lexicon       Semantic System          Objects

Fig. 1. A simplified illustration of the SLC. Elements in the WL are not connected,
only elements in the SS are. Inferential competence requires connecting two items
from the word lexicon through the semantic lexicon.
Word Lexicon, Semantic Lexicon and Inferential Competence. Infer-
ential competence lies between three ontologies: external word, word lexicon and
the semantic system. On input, the external word is first analyzed and related
to an mental representation from the word lexicon (WL). In [12], two word lexica
are included for different input, a phonetic and a graphical. Here, attention is
restricted to a simplified structure with only one arbitrary such, consisting only
2
  Due to space restrictions, proof theory will not be considered, but a complete ax-
  iomatization can easily be constructed based on the general completeness result for
  many-sorted modal logics from [13].
3
  For the review of these studies, arguments for the structure and references to relevant
  literature, the reader is referred to [12]. The presentation here differs slightly, but
  in-essentially, from that.
                                                                                    51

of proper names,4 as illustrated in Figure 1. Using a graphical lexicon as an
example, the word lexicon consists of the words an agent is able to recognize in
writing. Secondly, the mental representation of the word is related to a mental
concept in the semantic system (SS). The SS is a collection of non-linguistic,
mental concepts possessed by the agent, distinct from the WL. The semantic sys-
tem reflects the agent’s mental model of the world, and the items in this system
stand in various relations to one another.5 In contrast, in the WL connections
between the various items do not exist. Such only exist via the SS. The third
step is exactly a connection between two entries in the SS. Finally, the latter
of these are connected to an entry in the WL and output can be performed.6
Inferential competence is the ability to correctly connect lexical items via the
SS, e.g., connecting ‘dog’ to ‘animal’. This ability underlies performance such as
stating definitions, paraphrasing and finding synonyms.

Referential Competence and External Objects. Referential competence is
“the ability to map lexical items onto the world” [12, p. 60]. This is an ability
involving all four ontologies, the last being external objects. It consists of two
distinct subsystems. The first is naming. This is the act of retrieving a lexical
item from the WL when presented with an object. It is a two-step process, where
first the external object is connected to an suitable concept in the SS, which
is then connected to a WL item for output. The second subsystem is that of
application. Application is the act of identifying an object when presented with
a word. Again, this is a two-stage process, where first the WL item is connected
to an SS item, which is then connected to an external object. A naming or
application deficit can occur if either stage is affected: if, e.g., either an object
is not mapped to a suitable concept due to lack of recognition, or a suitable
concept is not mapped to the correct (or any) word, then a naming procedure
will not be successfully completed.

Empirical Backing for Multiple Modules and Competence Types. The
SLC may seem overly complex. It may be questioned why one should distin-
guish between word and semantic type modules, or why referential competence
is composed of two separate competence types, instead of one bi-directional. The
reasons for these distinctions are based on empirical studies from cognitive neu-
ropsychology where reviews of subjects with various brain-injuries indicate that
these modules of human cognition are separate (see [12] for references).
    The distinction between WL and SS is further supported in [8] by cases where
patients are able to recognize various objects, but are unable to name them (they
cannot access the WL from the SS). In the opposite direction, cases are reported
where patients are able to reason about objects and their relations when shown
4
  To only include proper names is technically motivated, as the modeling would oth-
  erwise require second-order expressivity. This is returned to below.
5
  Marconi uses the term ‘semantic lexicon’, but to keep this presentation in line with
  standard cognitive neuropsychological terminology, ‘semantic system’ is used instead.
6
  For simplicity, a distinction between input and output lexica will not be made. See
  [15] for discussion.
52

objects, yet unable to do the same when prompted by their names (i.e., the
patients cannot access the SS from the WL). The latter indicates that reasoning
is done with elements from the SS, rather than with items the WL.
    Regarding competence types, it is stressed in [12] that inferential and refer-
ential competence are distinct abilities. Specifically, it is argued that the ability
to name objects does not imply inferential competence with the used name, and,
vice versa, that inferential knowledge about a name does not imply the ability
to use it in naming tasks. No conclusions are drawn with respect to the rela-
tionship between inferential competence and application. Further, application is
dissociated from naming, in the sense that application can be preserved while
naming is lost. No evidence is presented for the opposite dissociation, i.e. that
application can be lost, but naming maintained.
    In the following section, a model will be constructed which include the men-
tioned ontologies over which the competence types can be defined, and in which
these are appropriately dissociated.

3      Modeling the Structure of Lexical Competence
To construct a toy model of the SLC, a two-sorted first-order epistemic logic will
be used. A very limited syntax is used, though the syntax and semantics could
easily be extended to include more agents, sorts, function- and relation symbols,
cf. [13].
     A two-sorted language is used to ensure that the model respects the disso-
ciation of word lexicon and semantic system. The first sort, σOBJ , is used to
represent external objects and the semantic system entries. As such, these are
non-linguistic in nature. The second sort, σLEX , is used to represent the exter-
nal words from the agent’s language and entries in the word lexicon. Had terms
been used to represent both simultaneously, the model would be in contradiction
with empirical evidence.
     The choice of quantified epistemic logic (QEL) fits well with the SLC, if one as-
sumes the competence types to be (perhaps implicitly) knowledge-based.7 The
notions of object identification required for application is well-understood as
modeled in the quantified S5 framework, cf. [5]. The ‘knowing who/what’ con-
structions using de re-constructions in QEL from [10] captures nicely the knowl-
edge required for object identification by the subjects reviewed in [12]. This is
returned to in the following section.

Syntax. Define language L to include two sorts, σOBJ and σLEX . For sort
σOBJ , include 1) a countable set of object constant symbols, OBJ = {a, b, c, ...},
and 2) a countably infinite set of object variables V AR = {x1 , x2 , ...}. The set
of terms of sort σOBJ is T EROBJ = OBJ ∪ V AR. For sort σLEX , include 1) a
countable set of name constant symbols, LEX = {n1 , n2 , ...}, and 2) a countably
infinite set of name variables, V ARLEX = {ẋ1 , ẋ2 , ...}. The set of terms of sort
σLEX is T ERLEX = LEX ∪ V ARLEX .
7
     In [3], visual recognition tasks are explicitly referred to in terms of knowledge (see,
     e.g., p. 149). Which type of knowledge is however not discussed.
                                                                                       53

     Include further in L a unary function symbol, µ, of arity T ERLEX −→
T EROBJ . The set of all terms, T ER, of L are OBJ ∪V AR ∪LEX ∪V ARLEX ∪
{µ(t)}, for all t ∈ LEX ∪ V ARLEX . Finally, include the binary relation symbol
for identity, =. The well-formed formulas of L are given by
                       ϕ ::= (t1 = t2 ) | ¬ϕ | ϕ ∧ ψ | ∀xϕ | Ki ϕ
The definitions of the remaining boolean connectives, the dual operator of Ki ,
K̂i , the existential quantifier and free/bound variables and sentences are all
defined as usual. Though a mono-agent system, the operators are indexed by i
to allow third-person reference to agent i.

Semantics. Define a 2QEL model to be a quadruple M = hW, ∼, Dom, Ii where

 1. W = {w, w1 , w2 , ...} is a set of epistemic alternatives to actual world w.
 2. ∼ is an indistinguishability (equivalence) relation on W × W .
 3. Dom = Obj ∪ N am is the (constant) domain of quantification, where Obj =
    {d1 , d2 , ...} is a non-empty set of objects, and N am = {ṅ1 , ṅ2 , ..., ṅk } is a
    finite, non-empty set of names.
 4. I is an interpretation function such that
       I : OBJ × W −→ Obj | I : LEX −→ N am | I : {µ} × W −→ Obj N am

To assign values to variables, define a valuation function, v, by
                   v : V AR −→ Obj | v : V ARLEX −→ N am
and a x-variant of v as a valuation v 0 such that v 0 (y) = v(y) for all y ∈
V AR(LEX) /{x}.
   Truth conditions for formulas of L are now defined as follows:
                                                 
                                                  v (ti )  if ti ∈ V AR ∪ V ARLEX
    M, w |=v (t1 = t2 ) iff d1 = d2 , where di = I (w, ti ) if ti ∈ OBJ
                                                   I (ti )  if ti ∈ LEX
                                                 
    M, w |=v ϕ ∧ ψ      iff M, w |=v ϕ and M, w |=v ψ
    M, w |=v ¬ϕ         iff not M, w |=v ϕ
    M, w |=v Ki ϕ       iff for all w0 such that w ∼ w0 , M, w0 |=v ϕ
    M, w |=v ∀xϕ (x) iff for all x-variants v 0 of v, M, w |=v0 ϕ (x)

Comments on the semantics are postponed to the ensuing section.

Logic. A sound and complete two-sorted logic for the presented semantics can
be found in [13]. The logic is here denoted QS5(σLEX ,σOBJ ) .

4    Model Validation
As mentioned above, the modules of the functional architecture of the SLC is
represented by model-theoretic structures, over which the agent’s capabilities
can then be expressed using the formal logical syntax. So far, the structures
introduced bear little resemblance to the SLC, and it will be a first task to extract
this hidden structure. Secondly, it is shown that the logical model can express
the three competence types and that the dissociation properties are preserved
in the logic.
54

4.1 Ontologies
The two sets of external objects and external words are easy to identify in the
model-theoretic structure. The external objects constitute the sub-domain Obj,
and are denoted in the syntax by the terms T EROBJ , when these occur outside
the scope of an operator. External words (proper names) constitute the sub-
domain N am denoted by the terms T ERLEX , when occurring outside the scope
of an operator.
    The word lexicon and the semantic system are harder to identify. The strategy
is to extract a suitable notion from the already defined semantic structure. To
bite the bullet, we commence with the more complicated semantic system.
Semantic System. In order to include a befitting, albeit very simple notion,
define an object indistinguishability relation ∼aw ⊆ Obj × Obj:
                d ∼aw d0 iff ∃w0 ∼ w : I (a, w) = d and I (a, w0 ) = d0 .
                                                                                a
and from this define the agent’s individual concept class for a at w by Cw        (d) =
   0       a,w 0
{d : d ∼i d }. The semantic system of agent i may then be defined as the
                                                         a        a
collection of non-empty concept classes: SSi = {Cw         (d) : Cw (d) 6= ∅}.
                a
     The set Cw (d) consists of the objects indistinguishable to the agent by a ∈
OBJ from object d ∈ Obj in the part of the given model connected to w by ∼.
As an example, consider a scenario with two cups (d and d0 from Obj) upside
down on a table, where one cup conceals a ball. Let a denote the cup containing
the ball, say d, so I (a, w) = d. If the agent is not informed of which of the two
cups contains the ball, i.e. which is a, there will be an alternative w0 to w such
that I (a, w0 ) = d0 . Hence, d ∼aw d0 so d0 ∈ Cw a
                                                    (d). The interpretation is that the
                                   0
agent cannot tell cups d and d apart with respect to which conceals the ball.8
     Important properties of individual concepts can be expressed in L (see [13]).
For present purposes, most importantly we have that
                           a
                        |Cw  (d)| = 1 iff M, w |=v ∃xKi (x = a),                    (1)
i.e. the agents has a singleton concept of a in w iff it is the case that the agent
knows which object a is, in the readings of [9,5]. The intuition behind this reading
is that the satisfaction of ∃xKi (x = a) requires that the interpretation of a is
constant across i’s epistemic alternatives. Hence, there is no uncertainty for
i with respect to which object possesses feature a – i knows which object a
is. Using a contingent identity system for objects, i.e. giving these a non-rigid
interpretation as done in the semantics above, results in the invalidity of both
(a = b) → Ki (a = b) and (a = b) → ∃xKi (x = b). Hence, agent i does not by
default know whether objects are identical when identified by different features,
and neither is the agent able to identify objects by default – as in the example
above. This is a good example of how the present is a weak, generic model:
subject specific abilities regarding identificatory abilities needs to be made as
further assumptions on a per subject basis.
Word Lexicon. A suitable representation of the word lexicon is simpler to
extract than for the SS. This is due to the non-world relative interpretation of
8
     Though the agent may be able to tell them apart with respect to other features, like
     their color or position.
                                                                                55

name constants n ∈ LEX, which so far has gone without comment. The inter-
pretation function I of the name constants is defined constant in order ensure
that the agent is syntactically competent. From the definition of I, it follows
that (n1 = n2 ) → Ki (n1 = n2 ) is valid on this class of models. This corresponds
formally to the incontingent identity system used in [11]. The interpretation is
that whenever the agent is presented by two name tokens of the same type of
name, the agent knows that these are tokens of the same name type. The as-
sumptions is adopted as the patients reviewed in [12] are able to recognize the
words utilized.
    Notice that identity statements such as (n1 = n2 ) do not convey any infor-
mation regarding the meaning of the names. Rather, they express identity of the
two signs. Hence, the identity ‘London = London’ is true, where as the identity
‘London = Londres’ is false – as the two first occurrences of ‘London’ are two
tokens (e.g. n1 , n2 ∈ LEX) of the same type (the type being ṅ ∈ N am), whereas
the ‘London’ and ‘Londres’ are occurrences of two different name types, albeit
with the same meaning.
    Due to the simpler definition of I for name constants, we can define i’s name
class for n directly. Where ṅ ∈ N am and n ∈ LEX this is the set Cin (ṅ) =
{ṅ0 : I (n) = ṅ0 }. The word lexicon of i is then the collection of such sets:
WLi = {Cin (ṅ) : n ∈ LEX}. Each name class is a singleton equivalence class,
and WLi is a partition of N am. Further, (1) holds for name classes if suitably
modified, and the construction of WLi therefore fits nicely fit the assumption of
syntactic competence.

4.2 Interlude: Word Meanings
In order to model knowledge of the meanings of word tokens n ∈ LEX, these
must first be assigned a meaning. In the clinical trials reviewed in [12], applying
a name to it’s meaning is done by extension identification. Therefore, a simple
purely extensional theory of meaning have been embedded in the framework:
the function symbol µ of arity T ERLEX −→ T EROBJ . A meaning function
rather than a relation is used as only proper names are included in the agent’s
language, and for these to have unambiguous meanings, the function requirement
is natural. Given it’s defined arity, µ assigns an element of T EROBJ to each
name in T ERLEX . From the viewpoints of the agents, µ hence assigns an object
(meaning) to each name.
    On the semantic level, M, w |=v (µ(n) = a) is taken to state that the meaning
of name n is the object a in the actual world w. The reference map µ is defined
world relatively, i.e. the value µ (n) for n ∈ LEX, can change from world to world.
This is the result of the world relative interpretation of µ given in the semantics
above. Hence, names are assigned values relative to epistemic alternatives.
   w1                w2
        n1       a        n1       a     Fig. 2. The meaning function µ is defined
             µ
                               µ         world relatively, so meaning of a name
        n2       b        n2        b    may shift across epistemic alternatives.
The motivation for a world relative meaning function is the generic nature of
the model. No agents will by default have knowledge of the meaning of words
56

from their language, but further assumptions to that effect can be assumed in
specific cases.
   The simplifying assumption that the WL should include only proper names
was technically motivated by the inclusion of µ. Had verbs been introduced in
the agent’s language, then µ should have assigned them relation symbols as
meanings, and a second-order logic would be required.
4.3 Competence Types
Inferential Competence. Due to the restriction to proper names, the model
is extremely limited in the features expressible regarding inferential competence.
The expressible instances of inferential competence are limited to knowing re-
lations between referring names and not inferential knowledge regarding names
and verbs. As an example, one cannot express that the agent knows the true
sentence ‘name is verb’ as the word lexicon does not contain a ‘verb’ entry. We
are however able to express knowledge of co-reference:
                                 Ki (µ(n) = µ(n0 ))                              (2)
(2) states that i knows that n and n0 co-refer, i.e. knows the two names to be
synonyms. Based on (2), we may define that agent i is generally inferentially
competent with respect to n by
                M, w |=v ∀ẋ((µ(n) = µ(ẋ)) → Ki (µ(n) = µ(ẋ)))                 (3)
where ẋ ∈ V ARLEX . If (3) is satisfied for all n, agent i will have full ‘encyclo-
pedic’ knowledge of the singular terms of her language. This may however be
‘Chinese Room style’ knowledge, as it does not imply that any names can be
applied nor that any objects can be named.
Referential Competence. Referential competence compromises two distinct
relations between names and objects, relating these through the semantic system,
namely application and naming. An agent can apply a name if when presented
with a name, the agent can identify the appropriate referent. This ability can be
expressed of agent i with respect to name n in w by
                             M, w |=v ∃xKi (µ(n) = x)                         (4)
i.e. there is an object which the agent can identify as being the referent of n.
Given the assumption of syntactical competence, there is no uncertainty regard-
ing which name is presented. Since the existential quantifier has scope over the
knowledge operator, the interpretation of µ(n) is fixed across epistemic alterna-
tives, and i thus knows which object n refers to.
     To be able to name an object, the agent is required to be able to produce
a correct name when presented with an object, say a. For this purpose, the de
re formula ∃ẋKi (µ(ẋ) = a) is insufficient as µ(ẋ) and a may simply co-vary
across states. This means that i will be guessing about which object is to be
named, and may therefore answer incorrectly. Since there may in this way be
uncertainty regarding presented objects, naming must include a requirement that
i can identify a, as well as know a name for a. This is captured by
                     M, w |=v ∃x∃ẋKi ((x = a) ∧ (µ(ẋ) = a)).                (5)
Here, the quantification and first conjunction ensures that i can identify the
presented object a and the second conjunct ensures that the name refers to a in
all epistemic alternatives.
                                                                                   57

Dissociations. As mentioned, inferential competence and naming are dissoci-
ated. This is preserved in the model in that neither (2) nor (3) alone imply (5).
Nor does (5) alone imply either of the two. The dissociation of application from
naming is also preserved, as (4) does not alone entail (5). That application does
not imply naming is illustrated in Figure 3.
w1                   w2
     µ(n)       d1        µ(n)      Fig. 3. Application and naming are not correla-
                                     d1
            I
                                 I  ted. In actual world w1 , n refers to a and i can
   a         d2    a          d2
                                    correctly apply n, but cannot name a using n:
  w1 |=v (µ(n) = a) ∧ ∃xKi (µ(n) = x), but w1 |=v ¬∃x∃ẋKi ((x = a) ∧ (µ(ẋ) = a)).
Here, i cannot name a due to an ambiguous concept. a may be either of d1 or d2 ,
and can therefore not be identified precisely enough to ensure a correct answer.

    Whether application entails inferential competence, and whether naming en-
tails application is not discussed in [12]. In the present model, however, these are
modeled as dissociated in the sense that (4) does not entail, nor is entailed by,
either (3) or (5). However, the modeled dissociations are single instances of the
various abilities. Once more instances are regarded simultaneously, implicational
relationships arise, as will be discussed below.

5     Hypotheses and Explanations
A motivation for constructing formal logical models is that precision and logical
entailment can provide explanatory force and working hypotheses. One testable
hypothesis of the present model predicts lack of dissociation between multiple
application instantiations and inferential competence. Specifically, the model
entails that subjects capable of applying two co-referring names will be knowl-
edgeable of their co-reference:
                          If M, w |=v (µ(n) = µ(n0 )) then
                                                                                 (6)
        M, w |=v ∃xKi (µ(n) = x) ∧ ∃yKi (µ(n0 ) = y) → Ki (µ(n) = µ(n0 ))
From this, an explanation why none of the studies reviewed in [12] show dissoci-
ation between application and inferential competence can be conjectured: simple
inferential competence can come about as a bi-product of application, memory
and deductive skill and may therefore require much damage before being severely
impaired.
    The formalizations of application and naming also suggests a reason why no
cases where naming was intact, but application broken, was reported in [12]: the
ability to name is very close to entailing the ability to apply a name. In fact,
once (5) is instantiated with a specific name, it implies (4) for the same name.
For application, the chosen object is identified by the subject via the mental
representation ‘the referent of n’, whereas for naming, the presented object must
first be identified by some other feature, e.g., a visual trademark. In case the
mental representation in the SS of this feature is then identical to that of ‘the
referent of n’, then the subject will be able to name a. Hence, one of the necessary
conditions for naming almost implies the necessary and sufficient condition for
application, why the latter will be observed accompanying the former.
58

    Implicational relationships as the mentioned should allow for the refutation
of the model. If subjects are found who possess abilities represented by the
antecedents, but lack those of the consequents, the model can be regarded as
refuted, though exactly what the problem is may not be obvious, qua the previ-
ously mentioned issue with logical omniscience.
    As mentioned above, both the difference between orthographic and and phono-
logical lexica as well as the difference between input and output lexica was
ignored in this presentation. Since the logical model therefore is based on an ar-
guably wrong functional architecture, it should be possible to find inconsistencies
between model and observed subject behavior. This is indeed the case. For, if
the presented model was correct, then the word lexicon entries should play both
orthographic and phonological roles for words the agent knows in both speech
and writing. Given such a word, an agent able to name with the word should
always be able to do so both orally and in writing. That is, the hypothesis that
agent i is able to name a, i.e. ∃x∃ẋKi ((x = a) ∧ (µ(ẋ) = a)) ((5) from above),
requires that the subject can produce name(s) ẋ both orally and in writing.
This, however, is not the case, as is illustrated, e.g., by the case of RCM, an
82-year old woman, reported on in [8, p. 191]. When prompted with a picture of
a turtle, RCM was able to correctly name it orally using ‘turtle’, but named it
incorrectly in writing, using ‘snake’. As RCM repeatedly made similar mistakes
with respect to written word tasks but not with oral naming tasks, this case can
be taken to show that damage to the orthographic (output) lexicon does not
imply damage to the phonological (output) lexicon. This is not possible in the
presented model, why the model is refuted.9

6      Conclusions and Further Perspectives
In the present paper, we have looked at the possibilities of using epistemic logic
as a modeling tool for cognitive neuropsychological theories by a toy model con-
struction. It has been shown how the functional architecture can be represented
using a combination of model theory and formal syntax. It was further shown
that the constructed model respected important dissociations, but also how the
model could be refuted by suitable empirical evidence contradicting an hypoth-
esis of the model. In conclusion, though the model is incorrect and simplistic, a
serious epistemic logical approach to modeling functional architecture theories
from cognitive science could possibly be of value. An attempt at making a proper
model of a full CN theory would be an obvious next step.
    A clear limitation of the presented model is that the formal semantic system
lacks content. The limitation to objects only should be lifted as to include various
properties and relations as well, and moreover, the representation of objects are
black boxed behind constants without a precise interpretation. Before a clear
picture such concepts’ role can be given, an explicit theory of object recognition
must be incorporated. It would be interesting to see the effects of formalizing,
e.g., geon theory of [2] and ‘plug it in’ in the place of the object constants.
9
     This can easily be remedied by distinguishing between phonological and orthographic
     word terms, i.e. by the addition of a further word sort.
                                                                                   59

More structure could also be provided by attempting to incorporate elements
from conceptual spaces theory [6]. Finally, knowledge operators are too strong in
some cases – RCM from above being a case in point. In situations where subjects
answer consistently but wrongly, belief operators would be better suited. A range
of competence levels could be captured using the various operators from [1].
    The style of modeling semantic competence presented here differs from the
way the conceptual theory of [12] and other cited literature tend to regard these
matters. Here, competence types was defined relative to specific words, and
competence judged on a case-by-case basis. Many studies from cognitive neu-
ropsychology base their conclusions on percent-wise correct performance over
one or more test batteries and therefore focus on impaired connections between
modules. In order to facilitate comparison of formal models and empirical re-
search, the case-by-case methodology must be reconciled with this more general
approach.

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    New Directions in Logic, Language, and Computation. Springer, 2012.
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    chology Press, 2001.
   Easy Solutions For a Hard Problem? The
 Computational Complexity of Reciprocals with
        Quantificational Antecedents?

                        Fabian Schlotterbeck and Oliver Bott

              Collaborative Research Center 833, University of Tübingen
              {fabian.schlotterbeck,oliver.bott}@uni-tuebingen.de



        Abstract. The PTIME-Cognition Hypothesis states that cognitive ca-
        pacities are limited to those functions that are computationally tractable
        (Frixione 2001). We applied this hypothesis to semantic processing and
        investigated whether computational complexity affects interpretation pref-
        erences of reciprocal sentences with quantificational antecedents, that is
        sentences of the form Q dots are (directly) connected to each other. De-
        pending on the quantifier, some of their interpretations are computation-
        ally tractable whereas others are not. We conducted a picture completion
        experiment and two picture verification experiments and tested whether
        comprehenders shift from an intractable meaning to a tractable interpre-
        tation which would have been dispreferred otherwise. The results suggest
        that intractable readings are possible, but their verification rapidly ex-
        ceeds cognitive capacities if it cannot be solved using simple heuristics.


1     Introduction

In natural language semantics sentence meaning is commonly modeled by means
of formal logics. Recently, semanticists have started to ask whether their models
are cognitively plausible (e.g. Pietroski, Lidz, Hunter & Halberda (2009)). Ob-
viously, for a semantic theory to be cognitively realistic the assumed meanings
have to be bounded in computational complexity since they have to be computed
in real time by processors with limited resources. In line with this consideration,
Frixione (2001) has proposed the PTIME-Cognition Hypothesis (PCH) as a gen-
eral research heuristic for cognitive science: cognitive functions have to be limited
to those functions that are computationally tractable (see also van Rooij 2008).
?
    This research was funded by the German Research Foundation in the Project B1
    of SFB 833. The authors would like to thank Fritz Hamm, Janina Radó, Wolfgang
    Sternefeld, Jakub Szymanik and three anonymous reviewers for valuable comments
    and discussions on earlier drafts of this paper. We would also like to thank our
    reviewers and audience at the International Workshop on Computational Semantics
    9 and the International Conference on Linguistic Evidence 2012 where we have
    presented parts of the present paper. For their assistance in conducting the reported
    experiments we would like to give special thanks to Anna Pryslopska and Aysenur
    Sarcan.
                                                                                 61

However, computational complexity has hardly received any attention in formal
semantics. In this paper we apply the PCH to semantic processing by looking at
a particularly interesting test case that has been discussed by Szymanik (2010).
    We investigated the interpretation of reciprocal sentences with quantifica-
tional antecedents of the form Q (some N, all N, most N, ...) stand in relation
R to each other. What makes these sentences particularly interesting for our
purposes is that, depending on the quantificational antecedent, the evaluation of
one of their readings is NP-hard whereas other antecedents stay within the realm
of PTIME verifiable meanings. We tested whether interpretations are limited to
those for which verification is in PTIME (i.e. computationally tractable) as pro-
posed in Szymanik (2010). In particular, we investigated whether comprehenders
shift from the preferred interpretation to other – under normal circumstances
dispreferred, but computationally tractable – readings in order to avoid having
to deal with an intractable verification problem1 . Consider (1).

(1)     a.   All of the students know each other.
        b.   All of the students followed each other into the room.

Reciprocals like (1) are notoriously ambiguous (Dalrymple, Kanazawa, Kim,
McHombo & Peters 1998). The intuitively preferred reading of (1-a) is that any
two members of the restriction (= the students) have to participate in the re-
ciprocal relation (= knowing one another ). By contrast, (1-b) states that the
students entered the room one after the other, i.e. they can be ordered on a
path. Quantified reciprocals like (1) may also exhibit a third reading, i.e. for any
member of the restriction there is some other member and these two partici-
pate in the reciprocal relation. We dub the first interpretation a complete graph
reading, the second a path reading and the third a pair reading. Note the logi-
cal dependencies. In fact, the complete graph reading is the logically strongest
interpretation entailing the others.
    To account for interpretation preferences of sentences like (1) Dalrymple
et al. (1998) have put forward the Strongest Meaning Hypothesis which proposes
that sentences like (1) receive the strongest interpretation consistent with the
reciprocal relation. This hypothesis is not undisputed, however, and Kerem,
Friedmann & Winter (2010) have presented empirical evidence that it does not
hold in general, but that interpreters will choose the most typical interpretation,
instead. They refined the original hypothesis accordingly and formulated their
Maximal Typicality Hypothesis.
    The strongest meaning of (1-a), which is presumably also the most typical
one, is PTIME verifiable. Interestingly, once we replace the aristotelian quantifier
all with a proportional quantifier like most or a cardinal quantifier like exactly
k, verification of the strong meaning becomes NP-hard (Szymanik 2010) since
it involves solving the CLIQUE problem. In contrast to the Strongest Meaning
Hypothesis, the PCH therefore predicts that complete graph readings should
not be possible for reciprocals with proportional or counting quantifiers as their
1
    A clarification may be in order: When we speak of intractable problems we always
    refer to NP-hard problems and assume silently that P 6= NP.
62

antecedents. As illustrated in this example, combining the Strongest Meaning
Hypothesis and the PCH yields specific predictions (and similar predictions can
be derived by combining the Maximal Typicality Hypothesis with the PCH). We
can think of computational complexity as a filter acting on the possible mean-
ings of reciprocal sentences. The effect of this filter should be that the logically
strongest meanings is preferred, as long as it is computationally tractable.
    The structure of the paper is as follows. In the next section we present a
picture completion experiment which was conducted to elicit the preferred in-
terpretation of reciprocals with three different quantificational antecedents. The
results show that, in line with our predictions, the interpretation preferences
clearly differed between aristotelian and proportional or cardinal quantifier an-
tecedents: we observed only a small proportion of complete graph interpretations
in the latter two quantifier types, whereas reciprocals with an aristotelian an-
tecedent were ambiguous. We then present two picture verification experiments
that tested whether an intractable complete graph reading was a viable inter-
pretation for proportional and cardinal quantifiers, at all. The results show that
potentially intractable complete graph readings are possible as long as they are
tested in graphs of limited size. We conclude with a discussion of whether the
obtained results require a parameterized version of the PCH (cf. van Rooij &
Wareham (2008)) or whether the findings could also be explained if we assume
that intractable meanings were approximated using simple heuristics that fail
under certain conditions.


2     Picture Completion Experiment

We measured interpretation preferences in a picture completion experiment in
which participants had to complete dot pictures corresponding to their preferred
interpretation of sentences like (2). As outlined above, the Strongest Meaning
Hypothesis predicts sentences like (2) to be preferably interpreted with their
complete graph meaning (3).

(2)    All/Most/Four of the dots are connected to each other.
(3)    ∃X ⊆ DOTS [Q (DOTS, X) ∧ ∀x, y ∈ X (x 6= y ← CONNECT (x, y))], where
       Q is ALL, MOST or FOUR.

In combination, the PCH and the SMH predict interpretation differences between
the three quantifiers. While the complete graph meaning of reciprocal all can
be verified in polynomial time, verifying the complete graph interpretation of
reciprocal most and reciprocal k (here: k = 4) is NP-hard. By contrast, the weaker
readings are computable in polynomial time for all three types of quantifiers. It
is thus expected that the choice of the quantifier should affect the preference
for complete graph vs. path/pair interpretations: reciprocal all should preferably
receive a complete graph reading, but reciprocal most/k should receive a path or
pair reading. The Maximal Typicality Hypothesis is hard to apply to these cases
because it is unclear what the most typical scenario for to be connected should
                                                                                63

look like. Most probably, the different readings shouldn’t differ in typicality and
should hence show a balanced distribution.


2.1   Method

23 German native speakers (mean age 24.3 years; 10 female) received a series of
sentences, each paired with a picture of (yet) unconnected dots. Their task was to
connect the dots in a way that the resulting picture matched their interpretation
of the sentence. We tested German sentences in the following three conditions
(all vs. most vs. four).

(4)    Alle (die meisten/vier) Punkte sind miteinander verbunden.
       All (most/four)         dots   are with-one-other connected.
       All (most/four) dots are connected with each other.

All-sentences were always paired with a picture consisting of four dots, whereas
most and four had pictures with seven dots. In addition to the fifteen experi-
mental trials (five per condition), we included 48 filler sentences. These were of
two types. The first half clearly required a complete graph. The second type was
only consistent with a path. We constructed four pseudorandomized lists, making
sure that two adjacent items were separated by at least two fillers and that each
condition was as often preceded by a complete graph filler as it was by a path
filler. This was done to prevent participants from getting biased towards either
complete graph or path interpretations in any of the conditions. The completed
pictures were labeled with respect to the chosen interpretation. We considered a
picture to show a complete graph meaning if it satisfied the truth conditions in
(3). A picture was taken to be a path reading if a sufficiently large subgraph was
connected by a continuous path, but there was no complete graph connecting
the nodes. A picture was taken to be a pair reading if the required number of
nodes were interconnected, but there was no path connecting them all. Since we
didn’t find any pair readings, we will just consider the complete graph and path
readings in the analysis. Cases that fulfilled neither interpretation were counted
as mistakes.


2.2   Results and Discussion

The proportions of complete graph meanings were analyzed using a logit mixed
effects model (cf. Jäger (2008)) with quantifier as fixed effect and random in-
tercepts of participants and items. Furthermore, we computed three pairwise
comparisons: one between all and most, one between all and four and one be-
tween most and four.
    In the all-condition, participants chose complete graph meanings 47.0% of the
time. By contrast, in the most-condition there were only 22.9% complete graph
interpretations among the correct pictures. The number of complete graphs
was even lower in the four-condition with only 17.4%. The statistical analy-
sis revealed a significant difference between all and the other two quantifiers
64

(estimate = −1.87; z = 4.14; p < .01). The pairwise comparisons revealed a sig-
nificant difference between all and most (estimate = −1.82; z = −3.99; p < .01),
a significant difference between all and four (estimate = −3.16; z = −5.51;
p < .01), but only a marginally significant difference between four and most
(estimate = .80; z = 1.65; p = .10).
    The error rates differed between conditions. Participants were 100% correct
in the all-condition. They made slightly more errors in the four-condition which
had a mean of 94.8% correct drawings. In the most-condition the proportion
of correct pictures dropped down to 83.5%. To statistically analyze error rates
we computed two pairwise comparisons using Fisher’s exact test. The analysis
revealed a significant difference between all and four (p < .05) and a significant
difference between four and most (p < .01).
    The observed preference for path interpretations, and in particular the very
low proportions of complete graph readings for most and four, matched the
predictions of the PCH . Both, most and four reciprocals, constitute intractable
problems and their strong interpretation shouldn’t hence be possible. The error
rates provide further support for the PCH. Most and four led to more errors than
all. This can be accounted for if we assume that participants were sometimes
trying to compute a complete graph interpretation, but due to the complexity
of the task did not succeed.
    An open question is whether the strong readings of reciprocal most and re-
ciprocal four are just dispreferred or completely unavailable. This was addressed
in a picture verification experiment.


3    Picture Verification Experiment 1

The second experiment tested reciprocals which were presented together with
pictures that disambiguated graph from path readings. To achieve clear disam-
biguation, we had to use different quantifiers than in the previous experiment.
This is because the quantifiers were all upward entailing and therefore complete
graphs are also consistent with a path reading. In the present experiment, we
used reciprocals with all but one, most and exactly k, as in (5). All but one and
exactly k are clearly non-monotone and hence none of the readings entails the
others. For most it is possible to construct complete graph diagrams in a way
that the other readings are ruled out pragmatically if we take its scalar impli-
cature (= most, but not all ) into account. Crucially, although intuitively more
complex than simple all, the complete graph reading of all but one is PTIME com-
putable. A brute force algorithm to verify reciprocals with all but one requires
approximately n-times as many steps as an algorithm to verify all reciprocals.
In order to verify a model of size n, at most the n subsets of size n − 1 have to be
considered. By contrast, verifying the strong meaning of (5-b,c) is intractable.
In particular, exactly k is intractable because k is not a constant, but a variable.
In the experiment we kept all but one constant and presented exactly k with the
values exactly three and exactly five.
                                                                                   65

(5)    (a)Alle bis auf einen / (b)Die meisten / (c)Genau drei (/fünf)
       (a)All but      one / (b)The most      / (c)Exactly three (/five)
       Punkte sind miteinander verbunden.
       dots    are with-one-other connected.
       (a)All but one/ (b)Most / (c)Exactly three (/five) dots are connected
       with each other.

We paired these sentences with diagrams disambiguating towards the complete
graph or the path reading. Sample diagrams are depicted in the appendix A in
Figure 4(a)/(e) and 4(b)/(f), respectively. As for complete graph pictures, the
PCH lets us expect lower acceptance rates for (5-b,c) than for (5-a). In order to be
able to find out whether the complete graph readings of (5-b/c) are possible at all
we paired them with false diagrams which served as baseline controls (see Figure
4(c)/(g)). The controls differed from the complete graph pictures in that a single
line was removed from the completely connected subset. If the complete graph
reading is possible for (5-b/c), we should observe more “yes, true” judgments in
the complete graph condition than in the false controls. As an upper bound we
included ambiguous conditions compatible both with complete graph and path
conditions (cf. Figure 4 (d)/(i)).
    It is possible that people can verify the strong meaning of (5-b,c) given small
graphs, but fail to do so for larger ones. Therefore, besides having the three
types of quantifiers, another crucial manipulation consisted in the size of the
graphs, i.e. the number of vertices. Small graphs always contained four dots (see
Fig. 4(a)-(d)) and large graphs consisted of six dots (see Fig. 4(e)-(i)). This way,
we also were able to keep the quantifier all but one constant and compare it to
exactly k with variable k. Exactly k was instantiated as exactly three and exactly
five, respectively. In total, this yielded 24 conditions according to a 3 (quantifier)
× 4 (picture type) × 2 (graph size) factorial design.


3.1   Method

We constructed nine items in the 24 conditions and used a latin square design
with three lists to make sure that each picture only appeared once for each
subject, but that the same picture appeared with all three quantifier types.
Each participant provided three judgments per condition resulting in a total of
72 experimental trials. We added 66 filler trials.
    36 German native speakers (mean age 26.9 years; 23 female) read reciprocal
quantified sentences on a computer screen. After reading the sentence, they had
to press a button, the sentence disappeared and a dot picture was presented for
which they had to provide a truth value judgment.


3.2   Results

The mean judgments are presented in Figure 1. The proportions of “yes, true”
judgments were subjected to two analyses. The lower bound analysis compared
66




(a) upper bound analysis                   (b) lower bound analysis

Fig. 1: Mean judgments in Picture Verification Experiment 1 (low : pictures with
4 dots; high: pictures with 6 dots)


the complete graph conditions with the false baseline controls in order to deter-
mine whether complete graph pictures were more often accepted than the latter.
The fixed effects of quantifier (three levels), graph size (two levels) and picture
type (two levels) were analyzed in a logit mixed effects model with random in-
tercepts of participants and items. Accordingly, upper bound analyses compared
the path conditions with the ambiguous conditions.
    Upper bound analysis: There was an across the board preference (7.3%
on average) of ambiguous pictures over pictures disambiguating towards a path
interpretation (estimate = −2.37; z = −2.88; p < .01). No other effects were
reliable.
    Lower bound analyses: Quantifiers differed with respect to how many
positive judgments they received. Most received reliably more positive judgments
than exactly k and all but one which led to a significant effect of quantifier
(estimate = 3.31; z = 8.10; p < .01). There was also a marginally significant
effect of truth (estimate = 0.72; z = 1.77; p = .07) which was due to slightly
higher (7.9%) acceptance of the complete graph pictures as compared to the
false baseline controls. No other effects were reliable.

3.3   Discussion
Most behaved rather unexpectedly. Surprisingly, it was rather often accepted in
the false baseline controls. The high acceptance rates in these two conditions
indicate that participants were canceling its scalar implicature (= most, but not
all ) and interpreted it equivalently to the upward monotone more than half. This
also explains the high acceptance rates of most in the complete graph conditions
which were, without the implicature, compatible with a path interpretation. Not
being able to tell path interpretations apart from complete graph interpretations,
we exclude most from the following discussion.
     Overall, the path reading was strongly preferred and the complete graph
reading was hardly available for any quantifier type. However, both the upper
                                                                                     67

bound and the lower bound analyses provide evidence that the complete graph
reading, even though strongly dispreferred, is available to some degree. Both
analyses revealed an effect of picture type. Path diagrams were accepted some-
what less often than the ambiguous diagrams and complete graph diagrams were
somewhat more often accepted than false diagrams. To our surprise, we didn’t
find any differences between quantifiers and graph sizes. However, we have to
be careful in interpreting these effects because judgments were very close to the
floor in the complete graph conditions.
    Why did we observe only so few complete graph interpretations even for
tractable all but one reciprocals? One possible explanation is that readers shifted
to path interpretations in all three quantifier conditions because the complete
graph readings may have been too complex. In the following we will show that
this is not the case, but that the low acceptability of complete graph interpre-
tations was for another reason. In particular, it is due to the reciprocal relation
be connected to each other. When reconsidering this relation we were not sure
whether it had the desired logical properties. It is possible that to be connected
is preferably interpreted transitively. This could have led to the low number of
positive judgments for the complete graph pictures, because, strictly speaking,
the complete graph reading would then be false in the complete graph pictures.
Note that there was a path connecting all the dots. Thus, assuming transitivity,
any dot was pairwise connected to all the others, violating the complete graph
reading of all but one and exactly k.


4     The Reciprocal Relation

To find out whether the reciprocal relation be connected with each other allows for
a transitive interpretation we conducted a picture verification experiment with
non-quantificational reciprocal sentences. We presented 80 participants with the
picture in Figure 2 and presented one of the following sentences to each of four
subgroups of 20 participants. Each participant provided only one true/false-
judgment. This was done to make sure that the data are unbiased.




    Fig. 2: Diagram used to test for (in)transitivity of the reciprocal relations.




(6)     a.   A und D sind miteinander verbunden.
             A and D are connected to each other.
68

       b.   A und D sind nicht miteinander verbunden.
            A and D are not connected to each other.
       c.   A und D sind direkt miteinander verbunden.
            A and D are directly connected with each other.
       d.   A und D sind nicht direkt miteinander verbunden.
            A and D are not directly connected with each other.

    The proportion of yes, true judgments of sentence (6-a) provides us with an
estimate of how easily be connected with each other can be interpreted transi-
tively. By contrast, judgments of the negated sentence in (6-b) inform us about
how easily the relation be connected with each other can be understood intran-
sitively. As a pretest for the next picture verification experiment, we included
another relation, be directly connected with each other, which, according to our
intuitions, should only be interpretable intransitively.


4.1   Results and Discussion

The distribution of judgments was as follows. (6-a) was accepted in 80% of the
cases, (6-b) was accepted in 42%, (6-c) in 10% and (6-d) in 80% of the cases.
Fisher’s exact test revealed that the proportions of “yes, true” judgments in
the conditions (6-a), (6-b) and (6-d) were significantly different from 0% (p <
.01), whereas condition (6-c) didn’t differ significantly from 0% (p = .49). With
respect to to be connected the results show that the relation is in fact ambiguous
although there was a preference for the transitive reading as indicated by the
higher proportion of “yes, true” judgments of sentence (6-a) than (6-b). The
transitive preference might have confined the acceptability of complete graphs in
the previous picture verification experiment. Regarding to be directly connected
there is a strong preference to interpret the reciprocal relation intransitively as
revealed by almost no acceptance of (6-c).


5     Picture Verification Experiment 2

In the first picture verification experiment the complete graph interpretation
seemed to be possible across quantifiers. If this was really the case it would pro-
vide evidence against the PCH. In the present experiment we thus investigated
whether this tentative finding can be replicated with a reciprocal relation like be
directly connected which is fully consistent with complete graphs. If the PCH,
in its original form, were correct, it would predict complete graphs to be accept-
able in tractable all but one reciprocals. By contrast, exactly k should lead to an
interpretive shift and reveal path interpretations, only. If, on the other hand, the
findings of the first picture verification experiment could in fact be generalized
to intransitive relations, we would expect to find complete graph interpretations
of exactly k, contra the PCH. Extending this line of reasoning, we might expect
the availability of a complete graph reading of exactly k reciprocals to be influ-
enced by graph size. In small sized graphs this reading may be fully acceptable,
                                                                                69

whereas with increasing size it may well exceed processing capacity. As for all but
one, both theoretical alternatives predict complete graphs to constitute possible
interpretations irrespective of the size of the model.

(7)    (a) Alle bis auf einen / (b) Genau drei (/fünf) Punkte sind
       (a) All but      one / (b) Exactly three (/five) dots    are
       miteinander verbunden.
       with-one-other connected.
       (a) All but one/ (b) Exactly three (/five) dots are connected with each
       other.

    To test these predictions we combined the sentences in (7) with the pictures
in Figure 5. The picture conditions were mostly identical to those of the first
picture verification experiment. The only difference was that the ambiguous
pictures were replaced by two other conditions (see Figure 5(d)/(h)). These were
false baseline controls included to find out whether path readings are available
with a clearly intransitive relation. The false controls depicted paths that were
one edge too short. Altogether this yielded 16 conditions in a 2(quantifier) ×
2(reading) × 2(truth) × 2(graph size) within subjects design.
    Thirty-four native German speakers (mean age: 27.5y, 20 female) took part
in the study. Except for the mentioned changes everything was kept identical to
the first picture verification experiment.


5.1   Results and Discussion

Mean acceptance rates are depicted in Figure 3. The path and complete graph
conditions were analyzed separately using logit mixed effect model analyses. The
path reading was generally accepted (true conditions: mean acceptance of 67.7%)
and led to almost no errors (false conditions: mean acceptance 4.2%) with both
quantifiers and graph sizes. The statistical analysis revealed that only the main
effect of truth was significant (estimate = 5.70; z = 8.70; p < .01).
    The true complete graph diagrams were also accepted across the board
(66.3%). Further, true complete graph diagrams were accepted significantly more
often than false ones (31%: estimate = 1.94; z = 4.08; p < .01). In the false com-
plete graph conditions there were, however, clear differences between diagrams
with four and six dots. In the former conditions participants made relatively few
errors (9.9%), whereas error rates increased drastically in the latter conditions
(52%). This led to a reliable effect of graph size (estimate = −6.46; z = −3.99;
p < .01) and a significant interaction of truth and graph size (estimate = 5.23;
z = 3.18; p < .01). Furthermore, the increase in error rates was greater for
exactly k than for all but one. For exactly k we observed an increase of 47.0%,
whereas there was only a 37.2% increase for all but one. Analyzing the false con-
ditions separately we found a significant interaction of graph size and quantifier
(estimate = 18.6; z = 2.04; p < .05).
    The relation to be directly connected clearly allowed for complete graph read-
ings. As opposed to the predictions of the PCH, this was the case for both all
70




(a) continuous paths                       (b) complete graphs

Fig. 3: Mean judgments in Picture Verification Experiment 2 (low : pictures with
4 dots; high: pictures with 6 dots)


but one and exactly k. In the false conditions we did, however, find clear effects
of semantic complexity. In these conditions errors drastically increased with the
number of vertices. Especially, for the intractable exactly k reciprocals error rates
increased with the number of dots.
    Why did semantic complexity only affect the false conditions? We think that
the reason for this may lie in the specific verification strategies participants were
able to use in the relevant conditions. In the true complete graph conditions the
complete subgraph was visually salient and could be immediately identified. In
combination with the fact that the CLIQUE problem is in NP, the true complete
graph conditions may, thus, in fact have posed a tractable problem to the par-
ticipants. As a consequence, only in the false conditions with one edge removed
from the graph participants really had to solve an intractable problem. Appar-
ently, this was still possible when the relevant subgraph consisted of only three
dots but already started to exceed cognitive capacities when it consisted of five
dots.


5.2   Conclusions

We started off with hypotheses that made too strong predictions. Firstly, the
linguistic work on reciprocal sentences by Dalrymple et al. (1998) let us ex-
pect that reciprocals should be interpreted in their logically strongest meaning.
When the antecedent is upward entailing a complete graph interpretation should
thus be chosen. Secondly, Szymanik (2010), building upon the PCH, predicted
interpretation shifts if reciprocals are intractable due to their quantificational
antecedents. Surprisingly, neither of these predictions was fully confirmed. In
contrast to the first prediction, complete graph readings were not the default
even for tractable reciprocals with an upward entailing antecedent. In the pic-
                                                                                    71

ture completion study path readings were equally often chosen for reciprocals
with all as the stronger complete graph reading. At first sight the predictions
of the PCH were borne out by the picture completion study. In this experiment
the quantificational antecedent affected interpretation preferences according to
the predictions of the PCH. When the complete graph reading was intractable it
was only rarely chosen. However, in picture verification intractable readings were
clearly available to our participants. This provides evidence against the PCH in
its most general form.
    Still, we did find effects of semantic complexity. Participants had problems
to correctly reject pictures not satisfying the complete graph reading with one
missing connection. This difficulty increased with the number of dots, especially
for intractable exactly k. How can these effects be explained? Firstly, it is possible
that participants approximated the intractable meaning of exactly k, thereby
effectively computing tractable functions. These tractable approximations may
have worked well in the true complete graph conditions but were inappropriate
for the false complete graph controls. We think of specific graphical properties
present in the true conditions, e.g. salience of the relevant subgraph which may
have simplified the task. Secondly, it is possible that participants were able to
compute intractable functions, but only within certain limits, e.g. up to a certain
number of elements or in pictures where the relevant subgraph was graphically
salient. No matter which of these explanations is correct our data provide an
interesting challenge for the PCH. A promising perspective in this respect may
be a parameterized version of the PTIME Cognition Hypothesis (cf., for instance,
van Rooij & Wareham 2008) which allows us to take into consideration the exact
instantiation of the problem. The presence or absence of a perceptually salient
group of objects may be a crucial factor for identifying a clique and should,
therefore, influence error rates. We plan to explore this in future research.


References
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    ters (1998), Reciprocal expressions and the concept of reciprocity, Linguistics and
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Frixione, Marcello (2001), Tractable competence, Minds and Machines, 11:379–397.
Jäger, Florian T. (2008), Categorical data analysis: Away from anovas (transformation
    or not) and towards logit mixed models, Journal of Memory and Language., 59:434–
    446.
Kerem, Nir, Naama Friedmann, & Yoad Winter (2010), Typicality effects and the logic
    of reciprocity, in Proceedings of SALT XIX.
Pietroski, Paul, Jeffrey Lidz, Tim Hunter, & Justin Halberda (2009), The meaning of
    ‘most’: semantics, numerosity, and psychology, Mind & Language, 24(5):554–585.
Szymanik, J. (2010), Computational complexity of polyadic lifts of generalized quan-
    tifiers in natural language, Linguistics and Philosophy, forthcoming.
van Rooij, Iris (2008), The tractable cognition hypothesis, Cognitive Science, 32:939–
    984.
van Rooij, Iris & Todd Wareham (2008), Parametized complexity in cognitive modeling:
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72

A    Sample diagrams


                                       n=4




         (a) complete    (b) continuous    (c) false base-   (d) ambigu-
         subgraph        path              line for com-     ous: complete
                                           plete sub-        subgraph &
                                           graphs            continous path
                                       n=6




 (e) complete    (f) continuous   (g) false base-   (h) false base-   (i) ambigu-
 subgraph        path             line for com-     line for com-     ous: complete
                                  plete sub-        plete sub-        subgraph &
                                  graphs            graphs (most)     continous path

Fig. 4: Sample diagrams presented in Picture Verification Experiment 1. The
upper row represents graphs with four dots. Graphs with six dots are represented
in the bottom row. The false baseline controls for complete graphs with six dots
were slightly different for all but one and exactly five (g) than for most (h). In
diagrams like (h), all dots were connected by a path, but in contrast to diagrams
like (g) there was no subset containing four or more elements forming a complete
graph. This way, for all three quantifiers falsity was due to a small number of
missing connections.

                                       n=4




         (a) complete    (b) continuous    (c) false base-   (d) false base-
         subgraph        path              line for com-     line for contin-
                                           plete sub-        uous paths
                                           graphs
                                       n=6




         (e) complete    (f) continuous    (g) false base-   (h) false base-
         subgraph        path              line for com-     line for contin-
                                           plete sub-        uous paths
                                           graphs

Fig. 5: Sample diagrams presented in the Picture Verification Experiment 2. The
upper row represents graphs with four dots. Graphs with six dots are represented
in the bottom row.
           Superlative quantifiers and epistemic
               interpretation of disjunction

                                    Maria Spychalska

                  Ruhr-University Bochum, Department of Philosophy



        Abstract. We discuss semantics of superlative quantifiers at most n and
        at least n. We argue that the meaning of a quantifier is a pair specifying a
        verification and a falsification condition for sentences with this quantifier.
        We further propose that the verification condition of superlative quanti-
        fiers should be interpreted in an epistemic way, that is as a conjunctive
        list of possibilities. We also present results of a reasoning experiment in
        which we analyze the acceptance rate of inferences with superlative and
        comparative quantifiers in German. We discuss the results in the light of
        our proposal.


1     Introduction

There is an ongoing debate (Look inter alia: (Geurts & Nouwen, 2007), (Koster-
Moeller et al, 2008), (Geurts et al., 2010), (Cummins & Katsos, 2010), (Nouwen,
2010), (Cohen & Krifka, 2011)) concerning the right semantical interpretation
of so-called superlative quantifiers, such as at most n and at least n, where
n represents a bare numeral, e.g two. Generalized Quantifier Theory (referred
here as a “standard account”) defines superlative quantifiers as equivalent to
respective comparative quantifiers: fewer than n and more than n, that is:

                     at most n (A, B) ⇐⇒ f ewer than n + 1(A, B)1                        (1)

                      at least n (A, B) ⇐⇒ more than n − 1(A, B)                         (2)


    It has been observed that in natural languages those equivalences (1) and (2)
might not hold, or at least they might not be accepted by language users based
on pragmatical grounds. There are numerous differences between comparative
and superlative quantifiers involving their linguistic use, acquisition, processing
and the inference patterns in which they occur. First of all, it seems that superla-
tive and comparative quantifiers are not freely exchangeable in same linguistic
contexts. Geurts & Nouwen (2007) provide, among many others, such examples:

     (a) I will invite at most two people, namely Jack and Jill.
     (b) I will invite fewer than three people, namely Jack and Jill.
1
    Q(A,B) means Q A’s are B, where Q is a h1, 1i generalized quantifier
74

     where (a) is considered a good sentence, while (b) is less felicitous. The con-
trast between (a) and (b) suggest that while embedding an indefinite expression
(two) in a superlative quantifier licenses a specific construal (namely Jack and
Jill ), the same is not licensed in the case of a comparative modifier.
     Secondly, it has been demonstrated that superlative quantifiers are mastered
later than the comparative ones during language development (Musolino, 2004),
(Geurts et al., 2010). Furthermore, there is ample data concerning processing of
those quantifiers. It has been for instance shown that verification of sentences
with superlative quantifiers requires a longer time than verification of sentences
with respective comparative quantifiers (Koster-Moeller et al, 2008),(Geurts et
al., 2010). Moreover, the processing of quantifiers is influenced by their mono-
tonicity. A quantifier Q(A, B) is upward monotone in its first argument A if it
licences inferences from subsets to supersets, that is if Q(A, B) and A ⊆ A0 , then
Q(A0 , B). A quantifier Q(A, B) is downward monotone in its first argument A if
it licences inferences from supersets to subsets, that is if Q(A, B) and A0 ⊆ A,
then Q(A0 , B). Understood as h1, 1i generalized quantifiers, at least n and more
than n are upward monotone in both their arguments, while at most n and fewer
than n are downward monotone. It has been shown that although the downward
monotone at most n and fewer than n take a longer time to be verified than the
upward monotone at least n and more than n, they are actually falsified faster
(Koster-Moeller et al, 2008).
     Finally, important arguments against the semantical equivalence between the
comparative and superlative quantifiers come from the analysis of people’s accep-
tance of inferences with those quantifiers. Empirical data show that a majority
of responders usually reject inferences from at most n to at most n+, (where n+
denotes any natural number greater than n), although they accept presumably
equivalent inferences with comparative quantifiers (Geurts et al., 2010), (Cum-
mins & Katsos, 2010). To illustrate it with an example: while people are unlikely
to accept that if at most 5 kids are playing in this room, then at most 6 kids are
playing in this room (2-14% in Cummins’ and Geurts’ experiments for this infer-
ence scheme) they are more likely to accept that if fewer than 5 kids are playing
in this room, then fewer than 6 kids are playing in this room (between 60-70%
in Cummins’and Geurts’). Such data seem to directly contradict the standard
account.


2    Modal semantics or clausal implicature?

Several theories have been developed to explain why seemingly logically equiv-
alent quantifiers show such big differences in the way people use them in the
language. Geurts (2007), (2010) proposes modal semantics for superlative quan-
tifiers and rejects the assumption that equivalences (1) and (2) hold in natural
languages. According to this proposal (referred here as a “modal account”), while
more than n and less than n have a conventional meaning defined in terms of
inequality relation, at least n and at most n have a modal component, namely:
                                                                                      75

    at least n A’s are B means that: a speaker is certain that there are n elements
    which are both A and B, and considers it possible that there are more than n.

    at most n A’s are B means that: a speaker is certain that there is no more
    than n elements that are both A and B, and considers it possible that there
    are n elements.

    According to this proposal, as semantically richer, superlative quantifiers are
expected to be harder to process than the respective comparative quantifiers.
Finally, defined as above, at most n A are B does not imply at most n+ A are
B : The latter implies that it is possible that there are (exactly) n+ A that are
B, which is contradicted by the semantics of at most n in the premise.
    There are strong arguments against the modal account. For instance this ac-
count seems unsatisfactory with regard to superlative quantifiers embedded in
conditional and various other contexts. Authors (Geurts & Nouwen, 2007),(Geurts
et al., 2010) realize themselves this problem and illustrate it with the following
example:

    If Berta had at most three drinks, she is fit to drive. Berta had at most two
    drinks. Conclusion: Berta is fit to drive.

    Such inferences, which are indeed licensed by the inference from at most n
(2) to at most n+ (3), are commonly accepted by speakers (over 96% in Geurts’s
experiment).
    Furthermore, while inferences from at most n to at most n+(1) are rejected
by majority of people (ca 84% in Geurts’ experiment) there are subjects who
do accept them (14% in Geurts’ and even more in our experiment — ca. 23%).
If at most n logically implies possible that n and not possible that more than
n, then the inference from at most n to at most n+ should be inaccessible
(except for cases of random mistakes) for any language users, due to the apparent
contradiction between the premise and the conclusion. Last but not least, to say
that possible that n is a part of the semantics of at most n, implies that at most
n cannot be paraphrased by not more than n. However such a paraphrase seems
totally eligible.
    A slightly different account was proposed by Cummins & Katsos (2010), who
observe that the considered linguistical phenomena could be better explained on
pragmatical grounds. The authors show that people do not evaluate at most n
and exactly n-1 as equally semantically incoherent as cases of obvious logical
incoherence, e.g. at most n and exactly n+1 or more than n and exactly n-1.
While sentence pairs, such as:

    Jean has at most n houses. Specifically she has exactly n+1 houses.

    get average coherence judgments very low, i.e. −4, in the scale from −5
(incoherent) to +5 (coherent), sentence pairs:

    Jean has at most n houses. Specifically she has n-1 houses
76

    get already +1.9. This result speaks against the “modal account”, whose
direct consequence is semantical incompatibility of at most n and exactly n-.
    Consequently, Cummins et al. agree with Geurts that at most n and at least
n both imply possible that n, but they claim that this is a pragmatical rather
than a logical inference, namely a co-called clausal implicature (Levinson, 1983).
Clausal implicature is a quantity implicature inferred due to use of epistemically
weak statement. Since the expressed statement with a superlative quantifier e.g.
at most n A are B, as equivalent to a disjunctive statement there are exactly n
or fewer than n elements that are both A and B, does not imply the truth of its
subordinate proposition p = there are exactly n elements that are A and B, the
possibility that p might or might not be true is inferred.
    Although we agree with the intuitions concerning a modal component in the
reasoning with superlative quantifiers, we reject the assumption by Geurts et al.
that this component is a part of their meaning. Furthermore, although we agree
with Cummins et al. that the mechanism that results with the observed inference
patterns is more of a pragmatical nature, we are not satisfied with the “causal
implicature” account. What we lack is a deeper insight into the source of this
kind of a pragmatical inference and how it interacts with the logical meaning of
those quantifiers in different reasoning contexts.
    Our motivation to further experimentally investigate reasoning with superla-
tive quantifiers is based inter alia on: (i) the lack of data concerning whether
people accept inferences: at least n → at least n -, (ii) the lack of satisfactory data
about how people accept inference with logically equivalent forms of superlative
quantifiers: such as not more than n/not fewer than n or n or fewer than n/n
or more than n, as well as how they accept mutual equivalences between these
forms, (iii) finally, the lack of data concerning people’s acceptance of logically
incorrect inferences with the quantifiers considered here.


3    Two semantic conditions for at most n

Krifka (1999) points out that semantic interpretation of a sentence is usually a
pair that specifies when the sentence is true and when it is false. Following Krifka,
we propose to define meaning of a quantifier as a pair hCF , CV i, where CV is a
verification condition (specifies how to verify sentences with this quantifier) and
CF is a falsification condition (specifies how to falsify sentences with this quanti-
fier). Furthermore, we propose that the interpretation of a quantifier depends on
a semantic context in which this quantifier is used, namely whether the context
requires the use of the verification condition or the falsification condition. Ver-
ification and falsification conditions are to be understood algorithmically, with
the “else” part of the conditional instruction being empty - thus, they verify (or
falsify) the formulas only if their conditional test is satisfied. From a perspective
of classical logic, these conditions should be dual, namely if C is a CV condition
for sentence φ, then C is a CF condition for sentence ¬φ, and vice versa. We
further, however, observe that in the case of superlative quantifiers, there is a
                                                                                          77

split between these two conditions. We suggest, that this split is a result of a
pragmatic focus on the expressed borderline n.
    One can think of the meaning of logical operators, thus also quantifiers, in
terms of algorithms, that have to be performed in order to verify (or falsify)
sentences with those operators. (See also (Szymanik, 2009), (Szymanik & Za-
jenowski, 2010), (Szymanik & Zajenowski, 2009)) Krifka (1999) observes, that
a sentence at most n x: φ(x)2 says only that more than n x: φ(x) is false, and
leaves a truth condition underspecified. In other words, the meaning of at most
n provides an algorithm for falsifying sentences with this quantifier, but not
(immediately) for verifying them. This corresponds with the experimental data
showing that it is easier to falsify sentences with at most than to verify them
(Koster-Moeller et al, 2008). Consequently, the primal semantical condition of at
most n x: φ(x) could be understood as an algorithm: “falsify when the number
of x that are φ exceeds n”, and would constitute what we understand by the
falsification condition.
Definition 1 ( falsification condition for at most)

                CF (at most x : φ(x)) := If ∃>n x(φ(x)), then f alsif y
But how can we know when a sentence with at most n is true? From the point
of view of an algorithm it is a so-called “otherwise” condition that defines in
this case the truth-condition. However a negation of a falsification condition is
in sense informationally empty: it does not describe any concrete situation in
which the given sentence can be verified. As a result, in those contexts that
require to directly verify a sentence, we refer to a verification condition, which
is specified independently. As expressing a positive condition, at most n may be
understood as a disjunction n or fewer than n (“disjunctive at most”).
Definition 2 ( verification condition for at most)
     CV (at most n : xφ(x)) := If (∃=n xφ(x) ∨ ∃<n xφ(x)), then verif y
                                                             Wn
  The disjunction Win 2 could be further broken down to: i=1 ∃=i xφ(x) ∨
                     n
¬∃xφ(x), in short: i=0 ∃=i xφ(x), where the disjunct ∃=o xφ(x) means that
        3
¬∃xφ(x).
  And ∃=n xφ(x) means precisely n x are φ, that is:

                                n
                                ^                ^                      n
                                                                        ^
∃=n xφ(x) ⇐⇒ ∃x1 ...∃xn [            φ(xi )∧          (xi 6= xj )∧∀y(       y 6= xi → ¬φ(y))]
                               i=1             1≤i<j≤n               i=1
                                                                                          (3)
2
    at most n x are φ
3
    Let us observe that ∃<n xφ(x) is a short notation that can be misleading, since it is
    not an existential sentence. As existential, fewer than n would imply that there has
    to be at least one (though less than n) such x that is φ. However we would like a
    sentence less than n x: φ(x) to be also true if no x’s are φ. Therefore, inVnfact, such a
    downward
    W          entailing sentence is a disguised universal sentence: ∀x1 ..xn ( i=1 φ(xi ) →

      1≤i<j≤n (xi = xj ))
78

3.1      Epistemic interpretation of disjunction

Following Zimmermann (2000) we adopt the view that disjunctive sentences
in natural language are likely to get so-called epistemic reading, that is they
are interpreted as conjunctive lists of epistemic possibilities. According to the
proposed solution a disjunction P1 or...or Pn is interpreted as an answer to a
question: Q: What might be the case? and, thus, is paraphrased as a (closed) list
L:

      L: P1 (might be the case) [and]... Pn (might be the case) [and (closure) nothing
      else might be the case].

      This results in the following reading of a disjunctive sentence:

Definition 3 (Zimmermann, 2000)

                           P1 ∨ ... ∨ Pn ⇐⇒ P1 ∧ ... ∧ Pn
      and (closure):

                       ∀P [P → [P ∩ P1 = ∅ ∨ ... ∨ P ∩ Pn = ∅]]

    The character of the closure requires a bit of our attention. Zimmermann
(2000) observes, that disjunctive sentences in natural languages could be under-
stood as closed (exhaustive lists of possibilities) or open (when other possibilities
are not excluded), which in the spoken language is usually marked by intona-
tion. Closure in Definition 3 indicates that the list is exhaustive. There are good
reasons to treat NL disjunctions as generally closed, and it would make sense
obviously also in the case of superlative quantifiers. In classical logic the truth
of ∃=n xφ(x) ∨ ∃<n xφ(x) semantically excludes the option that ∃>n xφ(x), as
contradictory. In our analysis, however, we want to treat closure as a merely
optional condition. This results from regarding the verification and falsification
conditions as independent from each other.


3.2      Verification condition for at most

If we assume that disjunctions in natural language are likely to be interpreted
as conjunctions of epistemic possibilities, then we get the following verification
condition for at most:

Definition 4 ( epistemic interpretation of the verification condition for at most)4
4
     A detailed description of the modal predicate logic needed for providing semantics
     of this kind of sentences is beyond the scope of this paper. For our present purposes
     it is just enough to assume that, for each possible world, we have a different domain
     of objects over which we quantify. We assume also standard semantics for modal
     operators, and we restrict to reflexive and transitive Kripke models.
                                                                                 79




     CW (at most n x : φ(x)) := If (∃=n xφ(x) ∧ ∃<n xφ(x)), then verif y

   and (closure)

                          If  ∃>n xφ(x), then f alsif y

Where:
                                                    n
                                                    ^
                 (∃=n xφ(x) ∧ ∃<n xφ(x)) ⇐⇒             ∃=i xφ(x)            (4)
                                                    i=o

     Now we can see the asymmetry between the falsification and verification con-
dition of at most n. While the falsification condition specifies precisely when the
sentence has to be rejected as false, the verification condition (in the epistemic
reading) provides a conjunctive list of epistemic possibilities that should all be
the case in order to verify it.
     The epistemic reading of the verification condition is also what differentiates
superlative quantifiers from the comparative ones. We propose that the disjunc-
tive form of the verification condition in the case of superlative quantifiers is a
result of the focus on the borderline n. The n mentioned in the superlative quan-
tifier constitutes the borderline of the truth-conditions, however the numeral n
that occurs in a comparative quantifier is not a part of the truth-conditions. The
borderline of truth-conditions (that is n-1 for fewer than n, and n+1 for more
than n) remains silent. Consequently, the disjunctive form of comparative quan-
tifiers (and hence also the epistemic reading), though logically possible, is not
pragmatically justified. In principle, a comparative quantifier can be as well in-
terpreted in the epistemic way (as a conjunction of epistemic possibilities). Such
a reading is, however, not equally likely to occur as in the case of a superlative
quantifier, since the borderline is not explicitly expressed.
     Let us observe that closure of the verification condition is stronger than
the falsification condition. Intuitively, if ¬  ψ (equivalently ¬ψ), then as well
¬ψ (here: ψ = ∃>n xφ(x)), i.e. ξ            ξ, however the theorem holds only in
reflexive Kripke frames. Furthermore, the optional character of the closure bases
on our assumption that the falsification and verification conditions are in a sense
independent and only as a pair constitute the full semantic interpretation. Since
the falsification condition, as defined in 2, is sufficient to account for the right
semantical criterion of when the sentence with at most n is false, the closure
of the verification condition is redundant and might or might not be considered
in the reasoning process. The optional character of closure turns out crucial in
evaluating validity of inferences with at most n.
     When at most n is interpreted as in Definition 4, then the inference from at
most n to at most n+1 is not valid. Namely, from ∃=n φ(x) ∧ ∃<n xφ(x) one
cannot infer ∃=n+1 xφ(x) ∧ ∃<n+1 xφ(x). It is easy to observe that the conjunct
∃=n+1 xφ(x) cannot be proven based on the premise, though it can be excluded
only if the closure of the premise is applied.
80

    On the other hand, the inference: at most n → at most n-1, which is invalid
in the standard account, in the epistemic interpretation is blocked only due
to closure of the conclusion. That is the ∃=n xφ(x) implied by the premise is
contradicted by the closure of the conclusion, i.e. ¬∃>n−1 xφ(x). It is important
to notice that without the closure the implication holds (if the epistemic reading
of the verification condition is applied).
    In some aspects our proposal might seem similar to Geurts’ “modal” ap-
proach, namely we define the verification condition of at most n and at least
n (see below) in modal terms. The main difference is that, in our account, it
is only the verification condition that is defined modally, while the falsification
condition remains standard. This results in a specific split or ambiguity in the
meaning of superlative quantifiers.


4    At least and bare numerals
The quantifier at least n might seem perhaps less interesting than at most n, as
it seems, as is also evident from our results (see Section (5)), less problematic
from the point of view of reasoning. As an upward monotone quantifier, at least
n appears to provide a clear verification algorithm: “verify when n x (that are
φ) are found”. Such a semantical interpretation would not, however, account for
the linguistical differences between at least n and more than n-1.
    Let us start with defining a falsification condition for at least n as follows:
Definition 5 ( falsification condition for at least)

               CF (at least n : xφ(x)) := If ∃<n xφ(x), then f alsif y

    What we expect from the verification condition is that it express the whole
range of epistemic possibilities in which the sentence with at least n can be true.
Understood as in formula (2), thus as equivalent to more than n-1, a sentence
with at least n can be expressed as an existential sentence that merely says that
there are n (x that are φ), but does not exclude the possibility that there are
more:

                                                        n
                                                        ^                  ^
 ∃≥n xφ(x) ⇐⇒ ∃>n−1 xφ(x) ⇐⇒ ∃x1 ...∃xn [                     φ(xi ) ∧             (xi 6= xj )] (5)
                                                        i=1              1≤i<j≤n

    This formula however, which would be a right verification condition for the
comparative more than n-1, does not outline the borderline n, which is empha-
sized in the superlative at least n. Therefore we further break down formula (5)
into a disjunctive formula: (exactly) n or more than n x are φ, which constitutes
our verification condition for at least n.

Definition 6 ( verification condition for at least n)

       CV (at least n : xφ(x)) := If (∃=n xφ(x) ∨ ∃>n xφ(x)), then verif y
                                                                                        81

     The latter can be handled as a conjunctive list of possibilities.

Definition 7 ( epistemic interpretation of the verification condition for at least n)

       CF (at least n : xφ(x)) := If (∃=n xφ(x) ∧ ∃>n xφ(x)), then verif y

     and (closure)

                                n−1
                                _
                           If (         ∃=i xφ(x)), then f alsif y
                                  i=0

    Having introduced the semantical conditions for at least n, we further ana-
lyze how they affect inferences with this quantifier, in particular we mean here
the inference (at least n→ at least n-1 ), as well as its presumably equivalent
disjunctive form: (n or more than n → n-1 or more than n-1 ). We propose
that the way people handle these inferences depends on how they interpret bare
numerals, such as n.
    From a logical perspective, a bare numeral n (e.g. “two”) can be interpreted
as denoting any set of at least n elements, or a set of exactly n elements.5 Thus
ψ = nx : φ(x) can simply mean that there are n x that are φ, without any further
constraints on whether there are more. Then, ψ gets the reading as in formula
(5). On the other hand, ψ could be understood with a kind of closure, that is
that there are exactly n x’s are φ, thus as in formula (3).
    It has been a matter of a wide debate in formal semantics and pragmatics
what is the right approach to interpreting bare numerals in natural languages. It
has been proposed that: (i) the literal meaning of n is at least n, while the condi-
tion exactly comes as a scalar implicture (Horn, 1972), (ii) the basic meaning of
n is exactly n, while both at least and at most readings would be context-based
(Breheny, 2008) (iii) n is ambiguous between at least n and exactly n (Geurts,
2006), (iv) n is underspecified and can receive at least, at most or exactly readings
depending on the context (Carston, 1998).
    Let us now show how the interpretation of n interacts with the validity of
inferences (n or more than n)→ (n-1 or more than n-1 ), given the epistemic
interpretation of disjunction. Suppose now that n is interpreted with a closure:
exactly n. It is easy to observe that, in such a case, possible that n and possible
that more than n does not imply possible that n-1 or possible that more than n-
1. The premise which V  is interpreted as in Definition 7 does not imply ∃=n−1 ∧
   >n−1                   n−2
∃       (with closure i=0 ¬  ∃=i xφ(x)) While ∃>n−1 follows from both ∃>n
and ∃=n , the problematic element is ∃=n−1 , which is directly contradicted by
the closure of the premise. But suppose that n does not get the “exact” reading,
but it is interpreted barely as there are n, so as (5). Then from possible that n
we can infer possible that n-1, since the latter does not exclude the possibility
that there is a bigger set of elements.
5
    or a set of at most n elements, however this interpretation seems to be counterintu-
    itive and allowed only in special contexts.
82

5      Experiment

We conducted a pilot reasoning experiment (in German) to check how people
reason with superlative quantifiers: at least n (mindestens n)6 and at most n
(höchstens n), as well as with logically equivalent but linguistically different
forms of those quantifiers: a comparative negative form and a disjunctive form.
These were quantifiers: not more than n (nicht mehr als n) and n or fewer than
n (n oder weniger als n) (as logically equivalent to at most n), and not fewer
than n (nicht weniger als n) and n or more than n (n oder mehr als n) (as
equivalent to at least n).
    We were particularly interested in comparing subjects’ acceptance of infer-
ences from at most n to at most n+1 (type B: see Table 1) with their acceptance
of logically equivalent forms of this inference: type D and type F. We were also
interested in people’s willingness to infer at most n from not more than n, and
vice versa (type G). Furthermore, we checked the respective inferences with at
least n, that is: at least n → at least n-1 (type A), and its equivalent forms:
type C and type E. Finally, we checked the inferences between not fewer than
n and at least n (type H). Last but not least we tested the respective incorrect
inferences with the considered quantifiers, in all forms (see Table 2).
    In the premises of our inferences we used four different quantifiers: at least
three, at most three, at least four, at most four, and their equivalent forms. All
the numbers were throughout spelled out in words according to the requirements
of German grammar. There was only one example for each distinct inference
relation (i.e. two per inference type). Every sentence content was different. Ad-
ditionally, to introduce more variation, sentences which had at most 4/at least
4 (or their equivalent forms) in the premise had a quantifier in the subject po-
sition (e.g. At least four computers are broken in the lab), sentences which had
at least 3 /at most 3 in the premise had a quantifier in the object position (e.g.
Arthur has at least three cars.)
    As fillers we used inferences with so-called bare numerals (e.g. four ): those
whose correctness depends on the “at least” reading of bare numerals, i.e. n →
n− (e.g. 4 → 3); and those that are logically incorrect independently of the
presumed reading, such as n → n+ (e.g. 4 → 5).


5.1     Procedure

The experiment was conducted on German native speakers, mainly students
of philosophy, psychology, neuroscience and computer sciences. There were 17
subjects (7 male). Subjects were asked to respond “yes” or “no” to the question
whether the second sentence (below the line) has to be true, assumed that they
know that the first sentence (above the line) is true. For a better understanding
of the task two examples were given: one of a valid inference, that should be
given a “yes” response:
6
     We give in brackets the German translation used in the experiment
                                                                                       83


                               Inga has done three exercises.
                           Inga has done more than two exercises.

      and one of an invalid inference, that requires a “no” response:

                                 Eva has done three exercises.
                            Eva has done fewer than two exercises.

    Note that the examples were selected in such a way that their validity did
not depend on the understanding of any of the tested inference relations, that
is the examples served as an instruction of what is an inference in general, but
not how to evaluate the inferences, that were tested in the experiment.
    After reading the instruction subjects saw 40 randomly ordered reasoning
tasks: one task at a time, displayed on a computer screen. There was no time
limit in the test.
    At the end of the experiment two additional control questions were asked, in
which the inference from at most n to at most n+1 was embedded in a deontic
context. Note that the logically correct response to first question is “yes”, while
to the second is “no”.
      (1) Erika promised to drink at most six beers. She drank at most four. Did she
      keep her promise?

      (2) Thomas is allowed to eat at most three cookies. He ate at most two. Did
      he break the rule?

      Tables: 1 and 2 summarize the inferences as well as the results.

5.2     Results
Our first observation is that people accepted the logically correct inferences much
more frequently than the logically incorrect ones. The incorrect inferences (apart
from disjunctive inferences: E’ and F’ which turned out specially problematic)
were mostly rejected and their acceptance rate was low enough (1 − 9%) to
be considered as a result of random mistakes (Table 2). On the other hand all
correct inferences were accepted on the level of at least 20%7 , with high variance
depending on the form of an inference, in this: inferences of type B and F seemed
the most problematic.
    The important result is that nearly 100% of responders did accept inference
of type G and H, that is (at most n → not more than n) as well as (not more
than n → at most n), and respective inferences between at least n and not
fewer than n, which suggests that they do see those expressions as equivalent.
Furthermore, while inferences from type B, namely the problematic (at most n
→ at most n+1 ) were accepted only by 23% of responders, the inferences of
type D (not more than n → not more than n+1 ), were already accepted by
44%. The difference was statistically significant: z = −2, 333, p = .02, r = −.4 8
Thus, it seems that paraphrasing at most n to the negative form not more than
n facilitates the inference.
84

                            Table 1. Logically correct inferences

                              Premise         Conclusion     Correct Responses Percentage
A At least                   at least 4        at least 3            13           79%
                             at least 3        at least 2            14
B At most                    at most 4         at most 5             3            23%
                             at most 3         at most 4             5
C Not fewer than         not fewer than 4 not fewer than 3           8            59%
                         not fewer than 3 not fewer than 2           12
D Not more than           not more than 4 not more than 5            7            44%
                          not more than 3 not more than 4            8
E N or more than n       4 or more than 4 3 or more than 3           10           67%
                         3 or more than 3 2 or more than 2           13
F N or fewer than n      4 or fewer than 4 5 or fewer than 5         6            23%
                         3 or fewer than 3 4 or fewer than 4         2
G “At most” Equivalence not more than 3        at most 3             16           98%
                             at most 3      not more than 3          17
                          not more than 4      at most 4             17
                             at most 4      not more than 4          17
H “At least” Equivalence not fewer than 3      at least 3            17           93%
                             at least 3    not fewer than 3          17
                         not fewer than 4      at least 4            16
                             at least 4    not fewer than 4          13
K Numerical                       5                 4                9            65%
                                  6                 2                11
                                  7                 6                13
                                  8                 5                11
  embedded                   at most 4         at most 6             17          100%
L                            at most 2         at most 3             16           94%



                           Table 2. Logically incorrect inferences

                               Premise         Conclusion     Correct Responses Percentage
     A’ At least              at least 4        at least 5            1             6%
                              at least 3        at least 4            1
     B’ At most               at most 4         at most 3             0             6%
                              at most 3         at most 2             2
     C’ Not fewer than    not fewer than 4 not fewer than 5           1             9%
                          not fewer than 3 not fewer than 4           2
     D’ Not more than      not more than 4 not more than 3            1             9%
                           not more than 3 not more than 2            2
     E’ N or more than n 4 or more than 4 5 or more than 5            3            20%
                          3 or more than 3 4 or more than 4           4
     F’ N or fewer than n 4 or fewer than 4 3 or fewer than 3         8            50%
                          3 or fewer than 3 2 or fewer than 2         9
     K’ Numerical                  4                 5                0            1,5%
                                   2                 6                1
                                   6                 7                0
                                   3                 8                0




   The inferences of type A, that is at least n→ at least n-1, turned out relatively
unproblematic for subjects, who accepted them in ca. 80% of cases. Interestingly
a paraphrase to the negative comparative form not fewer than n, made the
task more difficult (59% accepted; means comparison: z = −2.07, p = .038, r =
−.355). However, it is worth to note that inferences of type A were still rejected
7
     We give an overall result for a given type of an inference
8
     In all the cases we used Wilcoxon Signed Ranks test to compare means.
                                                                                 85

by ca. 20%, which suggests that there is some, at least pragmatic, mechanism
suppressing this inference.
    The results for the disjunctive inferences (E and F) are especially interest-
ing. First of all the response pattern for disjunctive counterpart of at least cor-
responds with the predictions of classical logic: While logically valid inferences
(E) were accepted on a relatively high level of 67% (which is lower, though not
significantly lower, compared to the acceptance of the basic form (type A)), the
invalid inferences (E’) were mostly rejected (only 20% accept). The opposite
effect, however, we got for the disjunctive form of at most. The logically valid in-
ferences (F) were mostly rejected (only 23% accept), whilst invalid inferences (F’)
were accepted in exactly 50% of cases. Interestingly the acceptance rate of (F)
inferences was similar to the acceptance rate of the basic form of inferences with
at most (B). In both cases the differences between acceptance rate of correct and
incorrect forms were statistically significant, and were as follows: The differences
between correct and incorrect inferences with “disjunctive at most” (F and F’):
z = −2.491, p = .013, r = −.43 and correct and incorrect “disjunctive at least”
(E and E’): z = −2.165, p = .030, r = −.37. The differences between disjunctive
at most and at least: incorrect (E’ and F’) z = −2.057, p = .040, r = −.36: and
correct (E and F) :z = −2.697, p = .007, r = −.46.
    The “correct” inferences with bare numerals were accepted in ca. 65% of
cases. There was only one mistake in the incorrect inferences with bare numerals.
Finally, both embedded at most inferences got nearly 100% correctness rate (one
mistake only for question 2).


6   Discussion

Although our results cannot be treated as a final evidence of our theory, our
experiment certainly provides various important observations that support the
plausibility of our proposal.
    First of all, all the implications between the negative comparative and su-
perlative forms of considered quantifiers were almost without exceptions ac-
cepted by our subjects. This result supports the assumption that those are se-
mantically equivalent forms in natural language.
    Secondly, the inferences (at least n → at least n-), although accepted by a
majority of responders, were not as obvious as the standard theory would pre-
dict, and 20% of subjects rejected them (A, Table 1). This suggests existence of
some, at least pragmatic, mechanism interfering in subject’s reasoning with at
least. What is also worth reminding, valid inferences with “disjunctive at least”
were rejected even more often (F, Table 1). We consider that this effect can be
explained in terms of the epistemic interpretation of at least n and its interaction
with the reading of bare numerals. Let us notice that our results provide a weak
evidence for the interplay between the reading of bare numerals and the treat-
ment of “disjunctive at least” inferences. In our experiment inferences of type
K: n → n−, which base on the “at least” reading of numerals were accepted
in ca. 65% of cases, thus our responders in 35% cases integrated the “exact”
86

reading of bare numerals, which resulted in their rejection of considered infer-
ences. However, “disjunctive at least” inferences (type E) ware rejected also in
ca. 33%. We suggest that rejection of type E inferences was as well a result of
an exact reading of a bare numeral n, as we have explained above. Although a
correlation between subject’s acceptance of type E and type K inferences failed
to reach significance, it was close to significant (Spearman’s rho= .426, p = .08)
and we expect that with a bigger sample it could reach the significance level.
    A similar effect is presumably the reason why 20% of subjects rejected type
A inferences: (at least n→ at least n-1 ). Namely, the application of the epis-
temic verification condition of at least n together with an exact reading of bare
numerals results in rejection of such inferences. This effect might be however
weaker and less likely to occur than in the case when the disjunctive form is
given explicitly.
    Thirdly, the surprising result that subjects accepted the invalid inferences
with “disjunctive at most” more frequently than the valid ones can be explained
by our proposal. As we have proposed above, closure in the verification condition
is optional, since the falsification condition is sufficient to account for the right
semantics. However, if context enforces applying one of the semantical conditions
(verification or falsification), then the other one tends to be ignored. While,
from the perspective of classical logic it should be enough to use only one of
the conditions (since the other can be defined via the first one), in the case of
superlative quantifiers the epistemic reading of the verification condition creates
the bifurcation in the meaning. This results in different inferential patterns in
which those quantifiers occur, depending on what the context primarily enforced:
the verification or falsification condition.
    In what follows, and as we have explained above, when the verification con-
dition is used, then n or fewer then n does not imply n+ or fewer then n+
due to the epistemic interpretation. Though, it also does not exclude it if no
closure is applied. However, n or fewer then n does imply n- or fewer then n- if
the verification condition is used but no closure is applied. Now we can explain
why inferences (F’, Table 2): (n or fewer than n) → (n-1 to fewer than n-1 )
got a 50% rate of acceptance, although they are invalid both in the standard ac-
count and in the epistemic account. Based on Definition 4, ∃=n xφ(x)∨∃<n xφ(x)
is interpreted as ∃=n xφ(x) ∧ ∃<n xφ(x) (with closure: ¬  ∃>n xφ(x)). But
∃<n xφ(x) can be broken down to: γ = (∃n−1 xφ(x) ∧ ∃<n−1 xφ(x)) Now
γ implies ∃=n−1 xφ(x) ∧ ∃<n−1 xφ(x) (here we use the assumption that the
world accessibility relation is transitive). In such a case it might happen that
the closure of the conclusion, that is: ¬  ∃>n−1 xφ(x) which contradicts the as-
sumption that ∃=n xφ(x) is ignored by subjects, which results in the high logical
mistake ratio.


7    Conclusions

We have argued that the meaning of a quantifier can be defined as a pair
hCF , CV i, in which the verification (CV ) and the falsification (CF ) condition
                                                                                               87

for sentences with this quantifier are specified separately. Though from the log-
ical point of view those conditions should be dual, in the case of superlative
quantifiers they are not. Namely, pragmatic focus on the borderline n in both
at least n and at most n enforces a disjunctive verification condition, which is
further interpreted as a conjunctive list of epistemic possibilities.
    Finally, we would like to say few words about why we want to understand the
verification and falsification conditions in terms of algorithms. Let us make an
observation that semantical equivalence and procedural identity of algorithms
are different things. Let us consider algorithms A1 and A2 :
    A1 : Count all x that are φ. If the number m of x that are φ is smaller than n − 1, then
    verif y.

    A2 : Count all x that are φ. If the number m of x that are φ equals n or is smaller than
    n, then verif y.


    A2 and A3 are semantically equivalent, namely they verify logically equiv-
alent formulas, e.g. ψ1 , ψ2 and ψ3 , however in a sense of procedures that are
executed they are not identical.

   ψ1 ⇐⇒ ¬∃>n xφ(x)
   ψ2 ⇐⇒ ∃<n+1 φ(x)
   ψ3 ⇐⇒ ∃=n xφ(x) ∨ ∃<n xφ(x)
   ψ1 ⇐⇒ ψ2 ⇐⇒ ψ3

   Furthermore, A3
    A3 : When the number m of x that are φ is bigger then n, then f alsif y.


    is dual to both A1 and A2 , namely adding an otherwise verify condition to
A3 and otherwise falsify condition to A1 and A2 would make A3 semantically
equivalent to both A1 and A2 . However, without the “otherwise” condition, A3
does not allow to verify any of the given sentences, while A2 or A1 do not allow
to falsify them. Then, hA1 , A3 i, or hA2 , A3 i could be considered pairs of partial
algorithms. Each pair could constitute a full semantical interpretation of each of
the sentences ψ1 , ψ2 , ψ3 .
    We consider that logically equivalent, but linguistically different, natural lan-
guage sentences may trigger different kinds of such partial algorithms, or pairs of
partial algorithms. First of all two equivalent sentences that differ in the linguis-
tical form can trigger as primary only one of the algorithmic conditions: verifi-
cation or falsification, while the complement condition is ignored. Secondly, they
might trigger non-identical verification/falsification procedures. Consequently, it
might happen that the executed procedures differ in complexity. Additionally,
if we take into account that some extra mechanisms, e.g. the above-discussed
epistemic interpretation of disjunctive conditions, might play a role, we not
only obtain different algorithms in the procedural sense but also semantically
non-equivalent algorithms for logically equivalent or even same sentences. For
instance, A2 would be replaced by:
88

     A02 : if [it is possible that there are exactly n x that are φ AND it is possible that there are
     fewer than n x that are φ], then verif y.


   While A2 is dual to A3 , A02 is not anymore. However a pair hA3 , A02 i could
constitute a semantical interpretation of a natural language sentence at most n :
xφ(x), which would explain the non-semantically coherent (from the point of
view of classical semantics) inference patterns in which this quantifier occurs.
                                                                                                  89

     Appendix: The complete list of pairs of sentences used in the
      experiment (premise, conclusion), translated from German.
At least four Anna’s dress are red. At least three Anna’s dresses are red.
Arthur has at least three cars. Arthur has at least two cars.
At most four books were stolen from the library. At most five books were stolen from the library.
Markus ate at most three pieces of cake. Markus ate at most four pieces of cake.
Not fewer than four cards are missing in the deck. Not fewer than three cards are missing in the
deck.
A child has painted not fewer than three pictures. A chid has painted not fewer than two pictures.
Not more than four students came today to the philosophy seminar. Not more than five students
came today to the philosophy seminar.
Sabine got not more than three presents. Sabine got not more than four presents.
Four or more than four students were sick this week. Three or more than three students were sick
this week.
Christopher speaks three or more than three languages. Christopher speaks two or more than two
languages.
Four or fewer than four students have passed the course. Five or fewer than five students have passed
the course.
Beate has three or fewer than three children. Beate has four or fewer than four children.
Five people came to the party. Four person came to the party.
Monika invited six guests to her birthday party. Monika invited two guests to her birthday party.
Seven fruits in the basket have spoilt. Six fruits in the basket have spoilt.
Alicia bought eight bottles of beer. Alicia bought five bottles of beer.
At least four of Carolina’s scarfs are blue. At least five of Carolina’s scarfs are blue.
Thomas has read at least three books. Thomas has read at least four books.
At most four computers in the lab are broken. At most three computers in the lab are broken.
Andrea baked at most three pizzas. Andrea baked at most two pizzas.
Not fewer than four professors attended the meeting. Not fewer than five professors attended the
meeting.
Hans had not fewer than three glasses of wine. Hans had not fewer than four glasses of wine.
Not more than four people have applied for this job. Not more than three people have applied for
this job.
Natalie wrote not more than three exercises. Natalie wrote not more than two exercises.
Four or more than four girls in the class are good in arts. Five or more than five girls in the class
are good in arts.
Christina’s cat gave birth to three or more than three kittens. Christina’s cat gave birth to four or
more than four kittens.
Four or fewer than four students failed in the exam. Three or more than three students failed in the
exam.
Tanja trains three or fewer than three times a week. Tanja trains two or fewer than two times a
week.
Four new students joined the chess club this week. Five new students joined the chess club this week.
Stephanie baked two cakes for her birthday. Stephanie baked six cakes for her birthday.
Six members of the library club came to the meeting. Seven member of the library club came to the
meeting.
Frank gave his mother three roses. Frank gave his mother eight roses.
Not more than three children have done their homework for today. At most three children have done
their homework for today.
At most three girls took part in the maths competition. Not more than three girls took part in the
maths competition.
Erika has not more than four necklaces. Erika has at most four necklaces.
Lena takes at most four courses at the university. Lena takes not more than four courses at the
university.
Not fewer than three new animals were born in the city zoo. At least three new animals were born
in the city zoo.
At least three exotic three have died in our botanic garden. Not fewer than three exotic threes have
died in our botanic garden.
Daniel plays not fewer than four times a week football. Daniel plays at least four times a week
football.
Albert has at least four exams this semester. Albert has not fewer than four exams this semester.
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       Quantifiers and visual cognition: the processing of
    proportional and superlative most in Bulgarian and Polish

                                      Barbara Tomaszewicz

                                 University of Southern California

                                       btomasze@usc.edu




       Abstract. I provide experimental evidence that quantifier semantics guides visual
       verification processes (Lidz et al. 2011). I tested the processing of two majority
       quantifiers in Bulgarian and Polish: the proportional Most1, the counterpart of Eng-
       lish most, and the superlative/relative Most2. Three obtained notable results have
       been obtained: (i) Most1 is verified by a Subtraction strategy, directly replicating the
       findings of Lidz et al. for Slavic; (ii) Most2 is verified by a Selection strategy in ac-
       cordance with its lexical semantics; (iii) each verification strategy is consistently
       used even in cases where either strategy would yield the correct truth value.


1      Introduction


   Lidz et al. (2011) argue that the verification of truth/falsity a declarative sentence is bi-
ased towards those procedures that are transparently associated with the semantic repre-
sentation of that sentence. They show that sentences containing the quantifier most such as
(1) are uniquely associated with truth conditions and a verification procedure involving
subtraction (2), despite the availability of other semantically equivalent specifications
(e.g. 3).

1. Most of the dots are yellow.
2. |Dot(x) & Yellow(x)| > |Dot(x)| – |Dot(x) & Yellow(x)|
3. |Dot(x) & Yellow(x)| > |Dot(x) & Red(x)| + |Dot(x) & Blue(x)| + |…|

   I provide further experimental evidence that quantifier semantics guides the verification
process. My evidence is based on the comparison of the verification patterns of two min-
imally distinct quantifiers and suggests that the properties of the linguistic input directly
influence the unconscious visual processes.
   The meaning of most intuitively refers to a comparison of quantities, where one of the
quantities is greater than others. For countable objects what is compared are cardinalities.
Visual perception of numerical information has been studied extensively and it is known
in psychology that the visual selection of a target "can be influenced by expectations and
strategies" (Trick, 2008). Manipulating a linguistic stimulus affects the patterns of the
visual search for obtaining a cardinality (or its estimation), however, at the same time the
choice of the visual verification strategy is also constrained by the psychophysics of visual
cognition. We can both formulate hypotheses and interpret the visual response pattern on
the basis of the findings about human perception in visual numerical judgment tasks. Un-
der time pressure, when precise counting becomes impossible, people switch to the Ap-
proximate Number System (ANS) that generates a representation of magnitude and is
governed by Weber’s law, i.e. the greater the distance between two numbers the better
discriminability (Dehaene, 1997). Numbers can be represented as 'noisy magnitudes' even
for the purposes of basic arithmetic operations like addition and subtraction. Quantifiers
can be verified against a visual display even when counting is blocked.1 Psychophysical
constraints, however, affect the accuracy of judgment.
   Lidz et al. (2011) hypothesize that the procedure in (3) (selection of each individual
color set in order to obtain the cardinality of the non-yellow set) should be computational-
ly costly if the verification involves more than one non-yellow set, because of the evi-
dence from Halberda et al. (2006) that on a 500ms display, multiple color sets can be
enumerated in parallel, but only for the total set of dots and two color subsets. The proce-
dure in (3) involves selection of each individual color set in order to obtain the cardinality
of the non-yellow set. The subtraction procedure in (2), on the other hand, is independent
of the number of color sets and is thus more suitable as a general verification strategy for
the quantifier most.
   I argue, however, that the choice of the procedure (2) over (3) for the verification of the
English quantifier most is not forced by psychophysics. My evidence suggests that (3) is
psychologically available as a procedure for visual verification, because a computationally
similar procedure is employed by the speakers of Bulgarian (Bg) and Polish (Pl) when
verifying the superlative majority quantifiers naj-mnogo (Bg) and najwięcej (Pl), cf. (4).

4. |Dot(x) & Yellow(x)| > |Dot(x) & Red(x)|,
   & |Dot(x) & Yellow (x)| > |Dot(x) & Blue(x)|,
   & |Dot(x) & Yellow (x)| > |Dot(x) & Green(x)|, & …

   I tested the processing of two majority quantifiers in Bulgarian and Polish: the counter-
part of English proportional majority quantifier most (povečeto in Bg, większość in Pl,
henceforth Most1) and a superlative/relative majority quantifier (naj-mnogo in Bg,
najwięcej in Pl, henceforth Most2). I obtained three notable results:

1
    Also Halberda, Taing and Lidz (2008) have shown that children who have not yet learned to count
     are perfectly able to understand sentences containing most.
• Most1 is verified by a Subtraction strategy as in (2) and not (3), directly replicating the
  findings of Lidz et al. for Slavic;
• Most2 is verified by a Selection strategy as in (4) in accordance with its lexical seman-
  tics;
• Each verification strategy remains to be used even in cases where either strategy would
  yield the correct truth-value.

   The results have some immediate implications for the semantics of quantifiers and the
interface of semantics with visual cognition. We can argue for the contribution of the
individual morphemes not only to the meaning of Most1 vs Most2 but also to the interface
with the visual cognition. The combined Bulgarian and Polish results further strengthen
the conclusions I presented in Tomaszewicz (2011) that quantifier semantics provides a
set of instructions to visual verification processes, since each of the two Polish Most1 and
Most2 biases a distinct verification strategy.
   .


2      Previous research

   Pietroski et al. (2008), Lidz et al. (2011) devised experimental paradigms to look “be-
yond” the truth conditions of (1) to see how the meaning of a sentence containing most
constrains the way people verify it against a visual scene. The two semantic specifications
in (5) are truth conditionally equivalent, but they differ in how the cardinality of the non-
yellow set of dots is arrived at. (5a) expresses a comparative relation between the cardi-
nalities of two sets, while (5b) is a one-to-one correspondence function that maps an or-
dered pair of sets (X, Y) to a truth value.

5. (a) |Dot(x)&Yellow(x)|>|Dot(x)&~Yellow(x)|
   (b) OneToOnePlus({Dot(x)&Yellow(x)},{Dot(x)&~Yellow(x)})

   Pietroski et al. (2008) obtained evidence that even when the arrangement of dots favors
the verification by strategy in (5b) (i.e. paired vs. unpaired arrangements of dots in two
colors), this strategy is not used. Using the same experimental paradigm requiring visual
verification under time pressure (screens displayed for 150 ms), Lidz et al. (2011) investi-
gated how the cardinality of the non-yellow set in (5a) is estimated when this set contains
dots in 1-4 different colors. They tested which of the two specifications in (6-7) most is
verified with.

6. Selection strategy
  |{Dot(x)&Red(x)} ∪{Dot(x)&Blue(x)} ∪ {Dot(x)&Green(x)} ∪…|
7. Subtraction strategy
   |Dot(x)| – |Dot(x)&Yellow(x)|
    Their proposed Subtraction strategy is based on the psychological evidence that a het-
erogeneous set is not automatically selectable (i.e. red, green, blue dots are not automati-
cally processed as one set unless it is the total set of dots), as well as on the findings of
Halberda et al. (2006) that humans can automatically (i.e. without a prompt) compute the
total number of dots and two color subsets but no more. Thus, Subtraction in (7) does not
depend on the number of colors of dots, while Selection in (6) should.
   In the experiment of Lidz et al. (2011) screens with dots in up to 5 colors in varying ra-
tios (yellow to non-yellow dots) were flashed for 150ms. Twelve participants evaluated
whether the sentence in (1) was true on each screen and the patterns of accuracy of their
responses were analyzed. No difference in accuracy was found as the function of the
number of colors of dots, but only as the function of the ratio (in adherence to Weber’s
law). This indicates that Subtraction was always used for the judgment of (1). Crucially,
on the screens with just 2 colors, Selection would be computationally less costly (it would
involve less steps than Selection as shown in Table 1) and thus more accurate.

     Table 1. Steps involved in Selection as opposed to Subtraction on two-color screens.
               Subtraction                                                 Selection
        (irrespective of no. of colors)                                   two colors
1. Estimate the total.                                    1. Estimate the target set.
2. Estimate the target set.                               2. Estimate the distractor set.
3. Subtract the target set from the total.                3. Compare with the target set.
4. Compare the difference with the target set.

    Yet, even on the two-color condition Subtraction appeared to be used, since the accu-
racy was not higher than on the multi-color screens, i.e. the verification procedure failed
to make use of the automatically obtained information, the cardinality of the two subsets
that could be compared directly (Halberda et al. 2006). Thus, Lidz et al. conclude that
Subtraction is the default procedure for verifying most under time pressure. On the basis
of this finding they formulate the Interface Transparency Thesis: “the verification proce-
dures that speakers employ, when evaluating sentences they understand, are biased to-
wards algorithms that directly compute the relations and operations encoded by the rele-
vant logical forms” (Pietroski et al. 2011).2


2
    What is crucial when comparing different strategies is evidence that a more advantageous strategy
     is failed to be used in favor of one that can be directly linked to semantics. Pietroski et al. (2011)
     “take it as given that speakers use various strategies in various situations. For us, the question is
     whether available procedures are neglected—in circumstances where they could be used to good
     effect—in favor of a strategy that reflects a candidate logical form for the sentence being evalu-
     ated.” A strategy may be abandoned in favor of one that is unrelated to a semantic representation,
     but that cannot be taken as evidence against a particular semantic specification.
3       Most1 and Most2 in Bulgarian and Polish

   Bulgarian and Polish have “two” versions of the English majority quantifier most.
Most1 in both languages has the same proportional reading as the English most has, so
that (8a) and (9a) are equivalent to (1).

 8.     (a) Povečeto točki sa žəәlti.                       Bulgarian
            Most1     dots are yellow
            'Most dots are yellow.'
        (b) Naj-mnogo točki sa žəәlti.
            Most2          dots are yellow
            'Yellow dots form the largest subset.’
 9.     (a) Większość kropek jest żółta.                    Polish
            Most1          dots is yellow
            'Most dots are yellow.'
        (b) Najwięcej jest kropek żółtych.
             Most2    is dots yellow
            'Yellow dots form the largest subset.'

   Most2 in Bulgarian (8b) and Polish (9b) contains superlative morphology in contrast to
Most1 as illustrated in (11). In accordance with the standard meaning of the superlative
morpheme (the relative reading), Most2 modifying a noun says that what the noun denotes
is the most numerous thing among other things of the same type, in our case, the set of
yellow dots is more numerous than any other color set.

    Table 2. The morphological make-up of Bulgarian and Polish Most1 and Most2.

                                                               	
  
                     Most1                                                Most2
               “more than half”                                       “largest subset”
                                            Bulgarian
                  po-veče-to                                            naj-mnogo
                  er+many+the                                           est+many
                                              Polish
                 więk-sz-ość                                           naj-więc-ej
                 many+er+nominalizer                                   est+many+er

   I predicted that Most1, being equivalent to the English most, should be compatible with
the Subtraction strategy. Thus, the number of color sets was expected to not affect the
accuracy of judgments with Most1 (it should only be affected by the ratio of yellow to
non-yellow dots). Since the semantics of Most2 can be specified as in (10), which I call
Stepwise Selection, I expected to find both an effect of ratio and of number of colors in
contrast to Most1.
10. Stepwise Selection strategy
    |Dot(x) & Yellow(x)| > |Dot(x) & Red(x)|, &
    |Dot(x) & Yellow (x)| > |Dot(x) & Blue(x)|, &
    |Dot(x) & Yellow (x)| > |Dot(x) & Green(x)|, & …

   Both of the predictions were met. The results of the Experiment 1 (on Bulgarian) and
of the Experiment 2 (on Polish) contain exactly the same main effects in the two condi-
tions.


3.1       Experiments 1 and 2: Materials and methods

   I conducted two on-line visual-display verification studies designed along the lines of
the experiment of Lidz et al. (2011). A group of 39 native speakers of Bulgarian partici-
pated in Experiment 1 and 20 native speakers of Polish participated in Experiment 2. The
Polish experiment is reported in Tomaszewicz (2011).
   The procedure was identical in both experiments. The participants evaluated the truth
of the sentences in (8-9) by pressing Yes or No buttons while viewing displays of arrays
of colored dots on a black background, flashed on a computer screen for 200ms. I manipu-
lated the ratio of the yellow target to the rest (1:2, 2:3, 5:6, i.e. 3 levels of the ratio varia-
ble) and the number of color sets (1, 2 or 3 other distractor colors, i.e. 3 levels of the dis-
tractor variable). The numbers of colors in each bin are presented in Table 5 in the Ap-
pendix.
    As the schema in Fig. 8 in the Appendix shows, 360 displays were presented in 2
blocks (180 for Most1 and 180 Most2, half of each requiring a yes response and half a no
response). Participants had 380ms to indicate their response by a button press. The exper-
iment was performed using Presentation® software (Version 14.2, www.neurobs.com).


3.2       Experiments 1 and 2: Results

   For Most1 accuracy rates were significantly affected only by ratio, and not by number
of color sets (Table 3, rows (a),(c)). For Most2 accuracy rates were significantly affected
both by ratio and by number of color sets (Table 3, rows (b),(d)). I analyzed each quanti-
fier with a 3x3x2 Repeated Measures ANOVA crossing the 3 levels of the ratio variable, 3
levels of distractor, and the truth/falsity of screens:

      Table 3. Accuracy rates.
                                            ratio                           color sets
                                    1:2      2:3    5:6               2       3        4
(a)     Bg    Most1 (povečeto)    .858      .778    .643   p<.001    .764    .748   .767   p.=.321
(b)           Most2 (najmnogo)    .827      .742    .617   p<.001    .807    .731   .648   p<.001
(c)   Pl         Most1 (większość)                   .871    .785     .673    p<.001            .797         .763   .769   p.=.215
(d)              Most2 (najwięcej)                   .866    .76      .63     p<.001            .801         .767   .688   p<.001

   The accuracy rates with Most1 in Bulgarian and Polish are significantly affected only
by ratio3 and not by number of color sets. These results are the same as for the English
most in Lidz et al. (2011) and are entirely consistent with the prediction that Most1 is
verified by Subtraction. The graphs in Fig. 2 and Fig. 3 clearly show the lack of a main
effect of number (Bulgarian: F(2, 76) = 1.153, p = .321; Polish: F(1.47, 27.98) = 1.637, p
= .2154).

        ‘Yes’	
  on	
  true	
  screens	
  	
                                 ‘No’	
  on	
  false	
  screens	
  




                                                       Fig. 2. Most1 in Bulgarian.


        	
  ‘Yes’	
  on	
  true	
  screens	
  	
                             ‘No’	
  on	
  false	
  screens	
  




3
  Bulgarian: F(2, 76) = 171.791, p < .001, Polish: F(2, 38) = 76.072, p < .001. Post hoc tests using
   the Bonferroni correction revealed significant differences (p < .001) between all levels of the ra-
   tio variable.
4
  Because of the violations of sphericity (p = .019), we are reading the Greenhouse-Geisser correct-
   ed value. Whether or not we use this correction, there is still no significance: F(2, 38) = 1.64, p =
   .208.
                                                        Fig. 3. Most1 in Polish.



   The results for Most2 are entirely compatible with the view that it is verified by Selec-
tion. In both Bulgarian (Fig. 4) and Polish (Fig. 5) the accuracy rates are significantly
affected both by ratio (Bulgarian: F(2, 76) = 182.449, p < .001, Polish: F(2, 38) = 124.77,
p < .001) and number of color sets (Bulgarian: F(2, 76) = 72.612, p < .001, Polish: F(2,
38) = 17.34, p < .001).5

          	
  ‘Yes’	
  on	
  true	
  screens	
  	
                          ‘No’	
  on	
  false	
  screens	
  




                                                       Fig. 4. Most2 in Bulgarian.

          	
  ‘Yes’	
  on	
  true	
  screens	
  	
                          ‘No’	
  on	
  false	
  screens	
  


5
    Pair-wise comparisons for the main effect of ratio and the main in effect of distractor in Bulgarian
     (using a Bonferroni correction) revealed significant differences (p < .001) between all levels. For
     Polish the differences between all levels of the ratio variable were significant (p < .001). The dif-
     ferences between 1-3 and 2-3 distractors were significant (p < .001 and p = .001 respectively),
     while the difference between 1-2 distractors was not (p = .316). Note that the Polish sample
     (N=20) is much smaller than the Bulgarian sample (N=39).
                                              Fig. 5. Most2 in Polish.

   It is also evident in the graphs in Fig. 5 and Fig. 6 that the accuracy with Most2 is af-
fected by the truth/falsity of screens. The present design does not allow us to determine
the reason for this, however, with Selection correct estimation of both the target set and
each color set is expected to be affected by a higher number of factors than Subtraction.
   Crucially, the significant effect of number of colors in addition to the effect of ratio in-
dicate that both the yellow set and the other color sets are selected for the verification of
Most2 in conformity with its semantics.6
   Importantly, on screens with 2 color sets (identical for both quantifiers) both Bulgarian
and Polish participants were significantly less accurate and slower confirming the truth of
Most1 than of Most2. This indicates that Subtraction continues to be used with Most1 and
Selection with Most2 even on the condition, where switching between the two procedures
would provide more accurate results.
   Participants could have used whichever strategy is computationally less costly/more
accurate under time pressure, since both strategies are otherwise used by the speakers of
Bulgarian and Polish. If the semantic representation guides verification, then with Most2
the non-yellow set should be selected directly – the accuracy should be greater than with
Most1 where the non-yellow set is computed (cf. Lidz et al. 2011), which is exactly what
we find on true screens.

             	
  	
  
	
                                           	
  




6
       Note that successful selection and comparison of 3-4 color sets in 200ms is not inconsistent with
        the findings of Halberda et al. (2006). The three set limit is on the automatically obtained infor-
        mation without a stimulus that creates expectations and directs attention to some specific aspect
        of the display. The superlative morphology clearly contributes an expectation that multiple sets
        should be compared.
       ‘Yes’	
  on	
  true	
  screens	
  	
                                ‘No’	
  on	
  false	
  screens	
  




                                    Fig. 6. Two color condition: Most1 vs. Most2 in Bulgarian.

       	
  ‘Yes’	
  on	
  true	
  screens	
  	
                            ‘No’	
  on	
  false	
  screens	
  




                                        Fig. 7. Two color condition: Most1 vs. Most2 in Polish.

    Both Bulgarian and Polish participants were significantly better with Most2 than
Most1 on true screens (Bulgarian: (F(1, 38) = 32.970, p < .001, Polish: F(1, 19) = 10.49, p
= .004). On false screens Most1 is significantly better than Most2 (Bulgarian: (F(1, 38) =
4.892, p = .033, Polish: F(1, 19) = 11.122, p =.003).
   Notably, the two languages also behave exactly the same with respect to the reaction
times. The accuracy is higher despite faster RTs and lower despite slower RTs (Table 4).
On true screens Most2 is faster (Bulgarian: F(1, 38) = .587, p = .448, Polish: F(1, 19) =
5.173, p = .035). On false screens Most1 is faster (Bulgarian: F(1, 38) = 9.884, p = .003,
Polish: F(1, 19) = .351, p = .561). See Table 6 in the Appendix for mean RTs. The RT
data shows that it is not the case that people are more prone to errors as they make judg-
ments faster. Instead, we can see that the procedure with Most2 on true screens is easier
(faster, more accurate judgments) which is expected if the two color sets are selected di-
rectly. On false screens Most1 is judged faster and more accurately, which does not seem
to follow from Subtraction vs. Selection difference. However, the correct disconfirmation
probably involves more factors that cannot be identified on the present design.
    Crucially, the accuracy patterns together with RTs consistent in both languages indicate
that participants do not switch to the more advantageous strategy, e.g. they don’t use Se-
lection to more accurately confirm the truth of Most1. This is the more interesting given
the findings of Halberda et al. (2006) that the cardinality of two color sets is automatically
computed. Yet the semantics of Most1 apparently precludes the use of this automatically
available information.
    Different behavior with each quantifier on the very same screens indicates that partici-
pants do not switch between the procedures and that the way those procedures differ is
specified by the semantics. Computation for both Most1 and Most2 involves the compari-
son between the yellow and the non-yellow set. The components provided by the visual
system are exactly the same: yellow set, non-yellow set, superset. However, the algo-
rithms must be different. To verify Most2 one has to (i) estimate target, (ii) estimate com-
petitor, (iii) compare. To verify Most1 one needs to (i) estimate target, (ii) estimate total,
(iii) subtract target from total. The lexical meaning of the functional morphemes that
build up Most1 and Most2 and their logical syntax are interfacing with the visual system
during the verification process.


4         Conclusions

     In conclusion, our experiments indicate that semantics provides a direct set of instruc-
tions to the visual cognition processes, and that these instructions are followed even when
computationally more advantageous strategies are available.
    We have met the prediction that Bulgarian and Polish proportional majority quantifier
Most1, just like English most, is verified using Subtraction strategy (we found a main
effect of ratio and no effect of number of colors). The superlative/relative majority quanti-
fier Most2 requires the Stepwise Selection strategy (as evidenced by the effect of ratio
together with the effect of number of colors)7. Importantly, in a within-subject design the
same group of participants behaves differently depending on the quantifier. The overall
patters of accuracy are exactly the same in Bulgarian and Polish.

7
    As one of the reviewers observes, my evidence for the different verification processes for Most1
     and Most2 is based on the use of the ANS representation of magnitude for the comparisons
     required by the semantics. If the superlative Most2 incurs a larger processing cost, it would be in-
     teresting to see if we find evidence for it also in experiments where counting is not precluded.
     Note, however, we cannot just “switch off” ANS, e.g. the effects of ratio-dependency character-
     istic of ANS are present also with judgments involving Arabic numerals,s although the quantities
     evoked by Arabic numerals may be more precise than those evoked by sets of dots (Dehaene
     1997).
   On two color screens (where Most1 and Most2 are either both true or both false) the
verification procedure depends on the lexical item used. The patterns of accuracy for
Most1 and Most2 were conspicuously different (but had the same direction in both Bul-
garian and Polish) indicating that computationally Most1 and Most2 are different.
    My results confirm and extend the findings of Pietroski et al. (2008) and Lidz et al.
(2011) and indicate that semantics provides inviolable instructions to visual cognition
processes.


References:
 1. Dehaene, S. (1997) The number sense: How the mind creates mathematics. New York: Oxford
    University Press.
 2. Halberda, J., Sires, S.F. & Feigenson, L. (2006) Multiple spatially overlapping sets can be
    enumerated in parallel. Psychological Science, 17.
 3. Halberda, J., Taing, L. & Lidz, J. (2008) The development of “most” comprehension and its po-
    tential dependence on counting-ability in preschoolers. Language Learning and Development,
    4(2).
 4. Lidz, J., P. Pietroski, T. Hunter & J. Halberda (2011) Interface Transparency and the Psycho-
    semantics of most. Natural Language Semantics 19.
 5. Pietroski, P., J. Lidz, T. Hunter & J. Halberda. (2008) The meaning of most: Semantics, numer-
    osity and psychology. Mind and Language, 24(5).
 6. Pietroski, P., Lidz, J., Hunter, T.,Odic, D. and Halberda, J. (2011) Seeing what you mean, most-
    ly. In J. Runner (ed.), Experiments at the Interfaces, Syntax & Semantics 37. Bingley, UK:
    Emerald Publications.
 7. Tomaszewicz, B. M., (2011) “Verification Strategies for Two Majority Quantifiers in Polish”,
    In Reich, Ingo et al. (eds.), Proceedings of Sinn & Bedeutung 15, Saarland Unversity Press:
    Saarbrücken, Germany.
 8. Trick, L. M. (2008) More than superstition: Differential effects of featural heterogeneity and
    change on subitizing and counting. Perception and Psychophysics, 70.



Acknowledgements:
    I would like to thank Roumi Pancheva as well as Elsi Kaiser, Barry Schein and Toby Mintz, for
    their invaluable comments and/or help with the experiments. Thanks are also due to Hagit Bor-
    er, Jeff Lidz, Martin Hackl, Victor Ferreira, Jon Gajewski, Robin Clark, Tom Buscher, Katy
    McKinney-Bock, Mary Byram, Ed Holsinger, Krzysztof Migdalski, Dorota Klimek-Jankowska,
    Grzegorz Jakubiszyn, Ewa Tomaszewicz, Marcin Suszczyński, Patrycja Jabłońska, Petya Ose-
    nova, Halina Krystewa, Petya Bambova, Ewa Panewa, Bilian Marinov, Marieta and the stu-
    dents of IFA at Wroclaw University and WSF Wroclaw and Polish Institute in Sofia.
    My work was partially supported by an Advancing Scholarship in the Humanities and Social
    Sciences Grant from USC awarded to R. Pancheva.
 Appendix:
       Table 5. The numbers of dots in each bin.
Most1 - total number of screens: 180                                   Most2 - total number of screens: 180

 no. of            ratio    dis-   no. of   no. of non- total no. of    no. of            ratio    dis-   no. of   no. of dots total no. of
screens                    trac-   yellow   yellow dots    dots        screens                    trac-   yellow    in closest    dots
                            tors    dots                                                           tors    dots    competitor

 30   10   true    1:2      1      8 - 12      4-6         12 - 18      30   10   true    1:2      1      8 - 12      4-6         12 - 18
      10                    2      8 - 12      4-6         12 - 18           10                    2      8 - 12      4-6         13 - 23
      10                    3      8 - 12      4-6         12 - 18           10                    3      8 - 12      4-6         14 - 27
 30   10   true    2:3      1      8 - 12      5-8         13 - 20      30   10   true    2:3      1      8 - 12      5-8         13 - 20
      10                    2      8 - 12      5-8         13 - 20           10                    2      8 - 12      5-8         15 - 27
      10                    3      8 - 12      5-8         13 - 20           10                    3      8 - 12      5-8         16 - 33
 30   10   true    5:6      1      8 - 12     7 - 10       15 - 22      30   10   true    5:6      1      8 - 12     7 - 10       15 - 22
      10                    2      8 - 12     7 - 10       15 - 22           10                    2      8 - 12     7 - 10       19 - 31
      10                    3      8 - 12     7 - 10       15 - 22           10                    3      8 - 12     7 - 10       22 - 29
 30   10   false   1:2      1      5-9        10 - 18      15 - 27      30   10   false   1:2      1      5-9        10 - 18      15 - 27
      10                    2      5-9        10 - 18      15 - 27           10                    2      5-9        10 - 18      18 - 35
      10                    3      5-9        10 - 18      15 - 27           10                    3      5-9        10 - 18      19 - 42
 30   10   false   2:3      1      5-9        8 - 14       13 - 23      30   10   false   2:3      1      5-9        8 - 14       13 - 23
      10                    2      5-9        8 - 14       13 - 23           10                    2      5-9        8 - 14       16 - 31
      10                    3      5-9        8 - 14       13 - 23           10                    3      5-9        8 - 14       17 - 38
 30   10   false   5:6      1      5-9        6 - 11       11 - 20      30   10   false   5:6      1      5-9        6 - 11       11 - 20
      10                    2      5-9        6 - 11       11 - 20           10                    2      5-9        6 - 11       14 - 28
      10                    3      5-9        6 - 11       11 - 20           10                    3      5-9        6 - 11       15 - 32




      Table 6. Reaction Times (RTs, in tenth of millisecond).
                                            Most2                                   Most1
                               true screens      false screens         true screens     false screens
      1:2 ratio       BG           8784               9475                 9066               9114
                      PL           8434               9922                 8978               9706
      2:3 ratio       BG          `9724              10316                 9411               9423
                      PL           8797              10498                 9892               9789
      5:6 ratio       BG           9884              10793                10461               9730
                      PL           9569              10204                10217              10660
                                Most2 slower on false screens,          Most1: no sig. difference be-
                                  BG: p = .002, PL: p <.001           tween true and false screens
                                                   Most2 faster on true screens:
                                        BG: p = .45 (ratio*quantifier p = .027), PL: p <.001
                                                  Most2 slower on false screens:
                                BG: p = .003, PL: p = .56 (ratio*quantifier p = .003, ratio p = .065)
Fig. 8. A schema of the experimental procedure.
The experimental investigation of the updated traditional
       interpretation of the conditional statement

                                      András Veszelka

              Pellea Human Research & Development Ltd., Budapest, Hungary
                           andras.veszelka@gmail.com



       Abstract. It is well known that the interpretation of the conditional statement in
       "everyday life" deviates from the official logical approach. It is conceivable,
       however, that the ancient logicians who first demonstrated the official approach
       erroneously characterised the "if P or R then Q" relationship in place of the "if P
       then Q" statement. When fixing this error, it turns out that the equivalent inter-
       pretation of the conditional statement, which is traditionally seen as one of the
       most common everyday fallacies, is in fact exactly the correct interpretation.
       Since classical logic has not been built on mathematical grounds but rather on
       philosophical argumentations and insights, its findings can be tested with the
       tools of today's human sciences, among others, with empirical experiments. The
       main experimental tools support this updated logical approach, and show that
       everyday thinking can be made compatible with logic. These results are sum-
       marized in this study.

       Keywords. scholastic logic, psychology of reasoning, human experimentation,
       Wason selection task


1      Introduction

The conditional statement is a glaring example of how the abstractions of logic and
the everyday use of the logical connectives deviate from each other. Many interpret
this as the difference between formal and natural languages (e.g. [1]). This differentia-
tion can be traced back to the beginning of the 20th century, where, for example, Fre-
ge [2] argued that the difference between the interpretations of the conditional state-
ment as prescribed in logic and as used in "everyday life" reveals linguistic or psycho-
logical components. This is where the search for the linguistic or psychological com-
ponents deemed different from logic began, first in the philosophy of language, then
in linguistics and finally in psychology. However, the so-called everyday interpreta-
tion of the conditional statement does not merely deviate from such formal languages
as propositional logic, which was born in Frege's era, but also from the classical inter-
pretation of the conditional statement, which is basically the same as that of proposi-
tional logic, but which has been clearly created not in a mathematical, but in a linguis-
tic, philosophical and psychological environment. Thus, the discrepancy between the
everyday interpretation and the classical abstraction is within the same system.
                                                                                        107


1.1    The abstraction error in classical logic

That said, when looking back to the classical interpretation, it can be seen that it is
erroneous. Instead of the "if P then Q" statement, classical logicians have erroneously
abstracted the "if P or R then Q" statement. Let's take an example from Jevons [3] (p.
70), a late scholastic logician:

If the snow is mixed with salt, it melts

As is well known, in this if P then Q statement from the snow mixed with salt (P)
antecedent, it is correct to infer the snow melts (Q) consequent (modus ponens, MP),
and from the snow not melting (not-Q) it is correct to infer that it has not been mixed
with salt (not-P) (modus tollens, MT). However, the denial of the antecedent (DA, if
the snow is not mixed with salt (not-P), it does not melt (not-Q)) and the affirmation
of the consequent (AC, if the snow melts (Q), it was mixed with salt (P)) are incor-
rect. Jevons has argued that, for example, from the snow melting (Q), it does not fol-
low that it has been mixed with salt (P) because it can melt by other means as well. It
is impossible to find any other explanation, even going back several hundred years, as
to why these two latter inferences are incorrect. On the contrary, this interpretation
can be traced back even to Aristotle, who wrote that:

The refutation, which depends upon the consequent, arises because people suppose
that the relation of consequence is convertible. For whenever, suppose A is, B neces-
sarily is, they then suppose also that if B is, A necessarily is. This is also the source of
the deceptions that attend opinions based on sense perception. For people often sup-
pose bile to be honey because honey is attended by a yellow colour: also, since after
rain the ground is wet in consequence, we suppose that if the ground is wet, it has
been raining; whereas that does not necessarily follow ([4], 167b1ff.).

To complete this argument, the inference that it has been raining does not necessarily
follow because there are other possible means to make the ground wet. However, if
we refer to additional possible causes, that is, to additional possible antecedents dur-
ing the abstractions, these have to be denoted. In logic, it is fundamental that "we
restricted ourselves to explicitly stated premises" ([5], p 6.). With their denotation,
however, it can be seen that with the above explanations we characterized the “if P or
R then Q” statement. Jevons' example was therefore the "If the snow is mixed with
salt or, for example, the sun is shining, the snow melts" statement that he erroneously
characterised in terms of P and Q only. These non-abstracted alternative antecedents
can be found in every example provided for the interpretation of the conditional
statement in the history of logic.
108


1.2    The correct abstraction of the conditional statement


The question arises therefore, of what the correct abstraction of the conditional state-
ment can then be, that is, the abstraction of the relationship in which there is no wedg-
ing “or R” component. I believe the correct inference pattern is the equivalence, in
which all the MP, MT, AC, DA inferences are valid. For example, we endorse the
equivalent AC and DA inferences for the “if-then” connective, even in the case of the
“if P or R then Q” statement. For instance from Q, we endorse the affirmation of the
consequent (AC) inference, and we deduce to “P or R”. As the traditional interpreta-
tion goes, within this we do not infer exclusively to P because it can be R as well.
This can be demonstrated the same way in the case of all three other classical infer-
ences as well. On the other hand, classical equivalent statements such as, for instance,
“if the sun is in the sky then it is day” are equivalent because the context of these
statements does not allow one to wedge any alternative antecedents. There can be day
only if the sun is in the sky. Even propositional logic refers to the alternative anteced-
ents, as when it differentiates the equivalence (to use another term, the biconditional)
with the artificial expression “If P then and only then Q” from the “if P then Q” con-
ditional, in the latter case of which, by parity of argument, several antecedents can
lead to Q. All of this is illustrated in further detail by Veszelka [6]. When explaining
the interpretation of propositional logic within the if-then statement, Geis and Zwicky
[7] reinvented and employed the aforementioned scholastic interpretation, and by
mentioning alternative antecedents they managed to block one of the most common
fallacies that people commit in the case of the conditional statement, the equiva-
lent/biconditional inferences. This approach has subsequently been implemented in
psychology, and its mechanism has to an extent been experimentally verified. Byrne
[8] has demonstrated that if the second antecedent is connected to the initial anteced-
ent with an “and” connective, then in terms of P and Q only, the MP and MT infer-
ences would be invalid and the DA and AC inferences would remain valid. This is the
case, for instance, in the example of the “If the snow is mixed with salt and it is not
extremely cold, it melts” (If P and R then Q) statement. These are very interesting
relationships, however, and as a consequence of the historical reasons illustrated in
the beginning of this study, this phenomenon is interpreted in linguistics and in psy-
chology as a linguistic, pragmatic effect, which is contrary to logic. It was neverthe-
less demonstrated above that this phenomenon is actually the update of the classical
logical interpretation of the conditional statement. It is the exact definition of what
differentiates between the two well-known inference patterns on the conditional
statement, the traditionally accepted conditional inference pattern, which allows only
MP and MT, and the equivalence. In antiquity, the rule of thumb used was that since
the conditional statement can evoke both conditional and equivalent inferences, one
should label only those inferences that are prescribed by both of them as valid, that is,
the MP and the MT [9]. Obviously, the new definition is more accurate. However,
many important psychological experiments are in conflict with this approach.
                                                                                      109


2      The experimental investigation of the conditional statement

2.1    The demonstration of the biconditional inferences

The most important task of this type, the “single most investigated problem in the
literature on deductive reasoning” ([10], p. 224) is Wason's abstract selection task.
Consequently, Byrne, who introduced the study of the alternative antecedents into
psychology, has rejected the basic biconditional interpretation of the conditional
statement [8]. In this task, participants are shown four schematic cards having a letter
on one side and a number on the other. Participants are then asked what card or cards
they would turn over in order to decide whether, for example, the “If there is a letter E
on one side there is a number 4 on the other side” conditional statement is true. On the
cards, the “E” (P), “K” (not-P), “4” (Q) and “7” (not-Q) can be seen. In this task,
abstract letters and numbers are used in order to assure that the context and the con-
tent have no influence on the results and so they accurately display how people inter-
pret the if-then statement itself. The traditional conditional interpretation would be
selecting the cards P and not-Q, since these could have falsifying instances on their
other side, while the biconditional interpretation would be the selection of all four
cards. The customary response is, however, merely the P and Q value. In the psycho-
logical field on logical reasoning, the logical negation is expressed in three different
ways. It can be implicit (e.g. “A”, and its negation = “K”), explicit (“A” and its nega-
tion “not-A”) and dichotom, which is the same as the implicit, but in which the task
instruction states that only two possible values can be found (e.g. “A” and “K”). The
result is P and Q with all three negatives [11]. This result constitutes an important
basis for many theories in the field. There are three additional main tasks:

─ Truth-table evaluation task, in which the given co-occurrences of the truth table of
  propositional logic, for instance the co-occurrence of P and Q, must be evaluated in
  terms of whether it verifies or falsifies the conditional statement, or is irrelevant to
  it.
─ Inference task, in which on the basis of the provided conditional statement, people
  must decide if the given conclusions follow from the minor premises or not, for
  example whether or not from not-P, not-Q follows.
─ Inference production task, in which participants themselves write down what fol-
  lows from the minor premises.

The available results from the combination of the four tasks and the three types of
negatives are shown in Table 1.
110


Table 1. Results of the main task types with the tree types of negatives
______________________________________________________________
                        Implicit neg.    Explicit neg.    Dichotom neg.
______________________________________________________________
Selection task          P&Q              P&Q              P&Q
Truth table task        Def. table [12]1 Def. table [12]1 ≡ (83%) [13]2
Inference task          ≡ (48%) [14]2    ?                ≡ (60%) [14]2
Inf product.task        ?                ?                ≡ (92%) [13]2
______________________________________________________________
1
  Defective truth table
2
  Biconditional

As can be seen in Table 1, although there are biconditional solutions, the results are
generally inconsistent and there are missing data. For this reason, I have retested all of
the tasks [15] except for the abstract selection task, which has robust results for all
three types of negatives. Consequently, for the selection task, I tested two thematic
tasks that have an evidently biconditional context in order to check if the results of
these tasks deviate from the results of the abstract selection task, or if they also evoke
the preference of the P and Q values, as was already observed by some researchers.
My results are shown in Table 2.

Table 2. My results of the main task types with the tree types of negatives [15]
______________________________________________________________
                     Implicit neg.    Explicit neg.    Dichotom neg.
______________________________________________________________
Selection task       P&Q              P&Q              P&Q
Truth table task     Def. truth table Def. truth table Bicon (50%)
Inference task       Def. truth table Def. truth table Bicon (42%)
Inf product.task     Bicon (73.3%) Bicon (52%)         Bicon (67%)
______________________________________________________________


The reasoning contained in the defective truth tables1 require further analysis, alt-
hough there are several explanations for this phenomenon that are compatible with the
updated scholastic approach. It is still apparent that the predominant response is the
biconditional. With regard to the selection task, instead of the biconditional responses,
both tested biconditional problems evoked the P and Q preference, the characteristic
response of the abstract selection tasks. According to my hypothesis [15], which has
been also formulated and partially tested by Wagner-Egger [14] one month prior to
my study, people avoid the selection of all the cards in the selection task. They be-
lieve that selecting all four cards would be contrary to the task instruction, which in

1
    In the defective truth table the co-occurrence of “P and Q” verifies the conditional state-
    ment, the co-occurrence of “P and not-Q” falsifies it, and the “not-P and Q” and the “not-P
    and not-Q” co-occurrences are irrelevant to it.
                                                                                     111


fact requires them to select from among the cards. This is fairly apparent in the case of
the following task, which was one of the tasks involving biconditional context that I
have tested:

On one side of each card, there is the name of a city and on the other side there is a
mode of transportation. Let us suppose that when someone goes to Budapest, he al-
ways goes by car, and when he goes to Szeged, he always goes by train. Likewise,
when he travels by car, he always goes to Budapest, and when he travels by train, he
always goes to Szeged. Mark the card or cards that must be turned over in order to
decide whether this is true.

The following statements were printed on the cards: “going to Budapest”, “going to
Szeged”, “going by train” and “going by car” ([15], Experiment 3).

In this task, which was tested on 2x20 participants, everyone produced the bicondi-
tional answer in the inference production task, but only 10% did so in the selection
task. However, as it can be seen, the task was in fact a pseudo-problem, because it
contained a clear description of what follows from what, or what value has to figure
on the other side of the cards. I obtained the same result on another clearly bicondi-
tional problem, the so-called “ball-light” problem [16]. This problem is commonly
accepted in the literature as a biconditional task which, being tested on 2x30 partici-
pants, has produced biconditional answers in 96% of inference production tasks, but
only in 23% of selection tasks [15]. Since participants do not find a better solution
than the avoided biconditional response, they finally select those instances that are
named in the conditional statement, the P and Q values. Thus, the main experimental
tasks altogether support the biconditional approach.


2.2    The demonstration of the response traditionally deemed correct

One half of the updated classical interpretation of the conditional statement, the basic
equivalent interpretation, can be therefore experimentally demonstrated. Another
empirical obstacle to this approach is to trigger the P and not-Q answer, the tradition-
ally expected response in logic. The elicitation of the “correct” answers has so far
been studied almost exclusively with selection tasks, and in so-called thematic selec-
tion tasks researchers obtained the allegedly correct P and not-Q response several
decades ago. The most cited task of this type is the drinking-age task [17], in which
participants have to imagine that they are on-duty police officers who must check if
everyone observes the rule that “If someone is drinking beer, he must be older than 18
years”. “Drinking beer”, "Drinking soft drink”, “21 years old,” and “17 years old”
appear on the cards. A large proportion of participants select “Drinking beer” and the
“17 years old” cards in this task—that is, the P and not-Q cards. This result is inter-
preted to arise from various effects, such as from pragmatic reasoning schemas [18],
from relevance [19], from deontic context [20], from precautions [21], from cheater
detection [22], from altruist context [23], from perspective switching [24], or from
benefits or costs [5]. However, these are not normatively valid explanations, because
112


in classical logic or propositional logic, where the abstraction itself has been defined,
such components were clearly not present. This can be easily seen in the examples
mentioned at the beginning of this study as well. It can be observed, however, that
there is a wedging of information in the easy-to-resolve selection tasks as well, which
mainly correspond with the effect of the alternative antecedents in the updated inter-
pretation of classical logic: In the above task everyone knows that people above 18
years can drink both alcohol and other beverages, although this is not explicitly com-
municated in the task instruction. One of the experiments of Hoch and Tschirgi [25]
can be seen as a means to test this additional information, in which they used in an
abstract task, with the appropriate substitutions, the statement that “Cards with a P on
the front may only have Q on the back, but cards with not-P on the front may have
either Q or not-Q on the back” ([25], p 203). Although this cue facilitation produced
56% correct results in the experiment of Hoch and Tschirgi [25], in the replication of
the experimental condition [26] the rate was only 36% in the usual experimental pop-
ulation, and participants with knowledge of logic were not filtered out; this could
evidently improve the result. With the usual experimental population, only a modest
improvement was received with this type of facilitation [27]. This task has so far been
tested only in selection tasks. In an unpublished experiment (with 21 participants), I
also received correct answers only in 14% of selection tasks, but the rate was 76%
when the very same task was presented in the inference-production task Pearson Chi-
Square (1, 42) = 16.243, p < .0001, Cramer’s V = .622. Perhaps in this case once
again, people in the selection task would test the complete relationship, and test, for
instance, that both P and not-P can figure on the card with a Q on its other side. This
would again require turning over all four cards, and as such the distorting effect men-
tioned above could reappear. Similarly, it can be observed that, contrary to this facili-
tation attempt, in the above easy-to-resolve drinking-age task the relationship that
people above 18 years can also drink soft drinks is from outside of the task, it is not
included in the investigated conditional statement. As a result, it must not be part of
the examination. To test this assumption in an abstract selection task, in an un-
published experiment I used the following task:

Imagine that four cards are lying in front of you on the table. On one side of the card
there is either the number 4 or the number 6; on the other side, there is either „divisi-
ble by two” or „divisible by three”. Your task is to check whether the four cards on
the table each conforms with the reality, namely, with the rule that

If the number is 4, then it is divisible by only two

Which card or cards would you turn over to check this?

In the control task I replaced 6 with 3 in the instruction and on the second card, and in
order to assure better text comprehension, I removed the word “only” from the if-then
statement. According to my interpretation, therefore, the two tasks evoke two differ-
ent relationships as shown in Figure 1.
                                                                                            113




Fig. 1. Cards and evoked relations in the experiment on the “easy to resolve” abstract selection
task

With number 6, the task produces the conditional inference pattern, and with number
3 the biconditional pattern. Indeed, with 21x22 participants I obtained P and not-Q
responses on the conditional task in 41% of cases. The rate of the “P and Q” and “all”
responses being in conformity with the biconditional was only 13.5%, whereas in the
control task the rate of the “P and not-Q” answers was merely 4%, and the “P and Q”
and “all” comprised 86% of the results. The difference is obvious with Pearson’s Chi-
Square (3, 43) = 20.157, p < .0001, with Cramer’s V = .685, and because of the fact
that the minimal differences between the tasks explain themselves. However, at a
different university, where participants were given twice as much time to resolve the
task, I failed to reproduce these results. It was then raised by a colleague that IQ
scores have a similar difference between the two universities, and that IQ could pos-
sibly also play a role in the way the tasks are resolved. For this reason, with Anikó
Kecse Nagy, we tested the task in the summer camp of Mensa HungarIQa. This or-
ganization collects Hungarians older than 17 years of age and who obtained a score
on the Raven Advanced Matrices IQ test higher than 98% of the general Hungarian
population. In this experiment, performed with 20x16 participants, people with and
without knowledge of logic participated equally. The results altogether were 69% vs.
30% “P and not-Q” answers, Pearson Chi-Square (1, 36) = 5.355, p < .021, Cramer’s
V = .0386 for the conditional task, versus the biconditional task. The difference
among the “all” biconditional answers was also significant in the opposite direction
114


(25% vs. 0%), Pearson Chi-Square (1, 36) = 4.654, p < .031, Cramer’s V = .359. De-
constructing the results further, 55% of the participants with no knowledge of logic (9
subjects) gave “P and not-Q” answers to the conditional task, while 25% of them gave
the same answers to the biconditional task. Even participants with knowledge of logic
produced significantly more P and not-Q answers to the conditional task (86% vs.
33%), Pearson Chi-Square (1, 19) = 4.866, p < .027, Cramer’s V = .506, and more
“all” responses to the biconditional task (0 vs. 33%), Pearson Chi-Square (1, 19) =
2.956, p < .086, Cramer’s V = .394). Although the 55% rate of correct responses of
the Mensa members not familiar with logic is still below the 70-75% rate of easy-to-
resolve thematic tasks, Fiddick and Erlich ([28], Exp 1) have received P and not-Q
selections in only 54% of cases even when the participants were explicitly instructed
to search for the falsifying co-occurrence of P and not-Q in the abstract selection task.
It is therefore conceivable that this is the maximum one could obtain from this task.


3      Conclusion

In general, the functioning of the updated classical logical interpretation of the condi-
tional statement can be demonstrated by the main experimental tasks used in experi-
mental psychology. People basically interpret the conditional statement as an equiva-
lent relation, and with the effect of the alternative antecedents this modifies into the
relationship known as the conditional. This approach can be defended from the point
of view of the history of logic [6], and is normatively valid. Many researchers assume
that the description of human inferences necessitates the introduction of non-
monotonic logics, or that the everyday interpretation of the conditional statement is
not truth-functional [30]. Still, the results presented here could be well described with
a merely slightly updated classical logic. In addition, this approach can also describe
the everyday interpretation of syllogisms [29]. Non-monotonic logics (e.g. default
logic [31], defeasible logic [32]) are introduced with reference to the effect of a cer-
tain type of context, without, however, denoting this context. To reiterate, this seems
to be a mistake, as in logic “we restricted ourselves to explicitly stated premises” ([5],
p 6). Leaving the context undenoted, or for example the traditional interpretation of
logical necessity and logical truth is probably the heritage of a classical logic that, in
consequence of the erroneous interpretation of the conditional statement, was rigid
and unable to develop, and did not allow to describe the effect of the context. When
fixing this error, however, the basic effect of the context can be seen even when the
equivalent relationship transforms into a conditional relationship—and this can be
quite precisely described. The purpose of non-monotonic logics is also to describe
such belief revisions. A similar example of the basic context is the otherwise mathe-
matical content, which can be observed at the end of this study in the easy-to-resolve
abstract selection task. The conditional statement itself is basically the same in the
two experimental conditions “If the number is 4, then it is divisible by (only) two”,
the underlining relationships (3 or 6) are, however, different. Still, these underlying
relationships can be precisely described, they do not require to introduce a specific
apparatus just because the conditional statement in one of the cases evokes equivalent,
                                                                                        115


and in the other case a conditional relationship, and with the addition of further con-
textual components, could behave again in quite a different way. I believe that more
complex contexts and even the concepts themselves behave in accordance with the
same principle. Naturally, in a more complex case, we cannot predict the exact con-
text or conceptual network in someone’s mind, but without precise information we
cannot predict which numbers someone is adding in his mind either. We could make
only vague or probabilistic predictions, just as happens in the case of the vague or
probabilistic approaches of the conditional statement. However, addition and subtrac-
tion written down on paper are still very useful tools.
          In another logical approach of the field, Stenning and van Lambalgen [33],
[34] worked precisely on defining the components behind the differences of such
individual inferences. According to them, participants in the abstract selection task
have first to define the parameters, and the differently chosen parameters produce the
many different answers, all of which are correct within the given parameters. The
authors themselves note that the parameters discussed by them are difficult to demon-
strate experimentally, and they assume that further parameters could be discovered. In
this respect, this study also defines such parameters, with markedly significant results,
such as, for example, the basic equivalent inference, the avoidance of the selection of
all cards and the effect of the alternative antecedent. And, of course, the whole litera-
ture investigating the relationship between logic and everyday inferences can be in-
terpreted as the search for and testing of such parameters—components that influence
how people resolve the tasks. According to this study, however, the many different
answers appearing in the abstract selection task are merely artefacts resulting from the
avoidance of the equivalent inferences. The same many different answers making
altogether the preference of P and Q cards also appear in the evidently biconditional
ball-light selection task already mentioned in this study [15]. It is, however, evident
that the equivalent inference is the only correct solution in this thematic task. So, in
the selection task, the search for the parameters that follows the rejection of the cor-
rect equivalent responses does not necessary reveal much about the basics of the in-
ferential processes. Still, they can provide important information on how people try to
resolve a situation that was made logically ambiguous. It is true that in the verbal
reports presented by Stenning and van Lambalgen participants do not speak about
avoiding the equivalent response. However, if logic has been unsure about the inter-
pretation of the conditional statement for 2,400 years, layman participants cannot be
expected to formulate a clear picture about this in the 5-10 minutes that they are given
to resolve the tasks. They particularly cannot be expected to be so sure about their
interpretation that, on the basis of this, they question the hidden instruction in the task,
going against the equivalent responses. As a matter of fact, even the good perfor-
mance on the easy-to-resolve drinking-age thematic task already mentioned in this
study drops back to half (75% to 35%) by presenting only two P and two not-Q cards
to the subjects, hence requiring the turning over of each of them [35].
          In this study, instead of analysing the individual responses I intended to de-
fine the overall correct responses and to demonstrate empirically that people generally
adhere to them. According to this approach, the greater the extent to which a task can
be resolved in the same way, the more it appears easy and evident to the experimental
116


participants. As the rate of characteristic response drops from 100% to just 20-30%,
so the task becomes more and more obscure to the participants. The more the task
become obscure, the more contextual effects activate in their mind in a great varia-
tion—giving a wider variety of parameters. The most characteristic solution for a task
is a sort of vote on what people believe is the correct solution in that task. This study
demonstrates that this voting/belief can be equated with some logical rules, which are
very simple and hence can probably also be easily programmed into a machine.

Acknowledgements. I am grateful to Professor László Bernáth for all his support
during this research, for Péter Bencsik for creating Figure 1 and for the comments of
three anonymous reviewers.

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