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. 1
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 mathematics. 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 modern 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 separation 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’ metaphor, the two disciplines have started to interact more and more. Today, we finally observe an increase in the collaborative effort between logicians, computer scientists, linguists, 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
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 categorization • Iris van Rooij Rationality, intractability and the prospects of “as if” explanations The abstracts of both presentations may be found in this volume.
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 Reciprocals with Quantificational Antecedents. 4
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, constructive feedback during the in-depth reviewing process: Program Committee:
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 workshop through Vici grant NWO 227-80-00, Cognitive Systems in Interaction: Logical and Computational Models of Higher-Order Social Cognition.
Vagueness and hysteresis: a case study in color categorization
Paul Egré and David Ripley
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 categorization 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’ depending on the direction of the transition between the two categories, suggesting a phenomenon of hysteresis or persistence of the initial category. 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 differ 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 various 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.
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 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 di culty 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 di culty of the Deductive Mastermind game-items with the size of the corresponding logical tree derived by the tableau method. We discuss possible empirical hypotheses based on this model, and preliminary results that prove the relevance of our theory. 1
Introduction and Background Computational and logical analysis has already proven useful for the investigations into the cognitive di culty of linguistic and communicative tasks (see [1{5]). We follow this line of research by adapting formal logical tools to directly analyze the di culty 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 di culty 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 di culty of game-items. In the remaining part of this section we give the background of our work: we explain 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 brie y discuss the directions for further work. 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 di culty. Existing mathematical results on Mastermind do not provide any answer to this problem|most focus has been directed at nding 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 nd 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 complexity analysis [13]. The corresponding Static Mastermind Satis ability Decision Problem has been de ned in the following way.
De nition 1 (Mastermind Satis ability Decision Problem). Input A set of guesses G and their corresponding scores.
Question Is there at least one valid solution? Theorem 1. Mastermind Satis ability Decision Problem is NP-complete. This result gives an objective computational measure of the di culty 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 di culty 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 (Rekentuin.nl or MathsGarden.com, see [17]). Math Garden is an adaptive training 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 di cultly level is appropriate for a student if she is able to solve 70% items of this level correctly. The di culty 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 in nity to + in nity. 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 di culty parameters. At the same time every child gets a rating that re ects her or his reasoning ability. One of the goals of this project is to understand the empirically established item di culty 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
Flowercode). Figure 1 shows an example item, revealing the basic setting of the game.
Each item consists of a decoding board (
Unlike the classical version of the Mastermind game, Deductive Mastermind does not require the player to come up with the trial conjectures. Instead, Flowercode gives the clues directly, ensuring that they allow exactly one correct solution. 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 Mastermind di ers 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 Satisability 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: nding a secret code) and the Static Mastermind game (it does not involve the trial-and-error inductive inference experimentation). Its very basic setting allows 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 justi cation, as the game trains very basic logical skills.
The game has been running within the Math Garden system since November 2010. It includes 321 game-items, with conjectures of various lengths (1-5 owers) 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 individual progress of individual players on a single game, or the most frequent mistakes with respect to a game-item. Most importantly, due to the studentitem rating system mentioned in Section 1.2, we can infer the relative di culty of game-items. Within this hierarchy we observed that the present game-item domain contains certain \gaps in di culty"|it turns out that our initial di culty estimation in terms of non-logical aspects (e.g., number of owers, 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 di culty space. Providing a logical apparatus that predicts and explains the di culty of Deductive Mastermind game-items can help solving this problem and hence also facilitate the training e ect of Flowercode (see Appendix, Figure 7).
A Logical Analysis Each Deductive Mastermind game-item consists of a sequence of conjectures. De nition 2. A conjecture of length l over k colors is any sequence given by a total assignment, h : f1; : : : ; `g ! fc1; : : : ; ckg. The goal sequence is a distinguished conjecture, goal : f1; : : : ; `g ! fc1; : : : ; ckg.
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. De nition 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 zg :}:|: g{ oz :}:|: o{ zr :}:|: r{ = gaobrc; where a; b; c 2 f0; 1; 2; 3; : : :g and a + b + c = `. The feedback consists of: { exactly one g for each i 2 G, where G = fi 2 f1; : : : `g j h(i) = goal(i)g. { exactly one o for every i 2 O, where O = fi 2 f1; : : : ; `gnG j there is a j 2 f1; : : : ; `gnG; such that i 6= jand h(i) = goal(j)g.
{ exactly one r for every i 2 f1; : : : ; `gn(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 rst give a second-order logic formula that encodes any feedback sequence gaobrc for any h with respect to any goal: ^ 9O 9G f1; : : : `g(card(G)=a ^ 8i2G h(i)=goal(i) ^ 8i2=G h(i)6=goal(i) f1; : : : `gnG (card(O)=b ^ 8i2O 9j2f1; : : : `gnG(j6=i ^ h(i)=goal(j)) ^ 8i2f1; : : : `gn(G[O) 8j2f1; : : : `gnG h(i)6=goal(j))):
Since the conjecture length, `, is xed 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 2 f1; : : : `g (they might be viewed as propositional variables pi;j , for i; j 2 f1; : : : `g). With respect to sets G, O, and R that induce a partition of g f1; : : : `g, we de ne 'G; 'oG;O; 'rG;O, the propositional formulae that correspond to di erent parts of feedback, in the following way: { 'gG := Vi2G h(i)=goal(i) ^ Vj2f1;:::;`gnG h(j)6=goal(j), { 'oG;O := Vi2O(Wj2f1;:::;`gnG;i6=j h(i) = goal(j)), { 'rG;O := Vi2f1;:::`gn(G[O);j2f1;:::`gnG;i6=j h(i) 6= goal(j). 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 x the domain of those possibilities we set G := fGjG f1; : : : ; `g ^ card(G)=ag, and, if G f1; : : : ; `g, then OG = fOjO f1; : : : ; `gnG ^ card(O)=bg. Finally, we can set Bt(h; f ), the Boolean translation of (h; f ), to be given by: Bt(h; f ) := _ ('gG ^
_ ('oG;O ^ 'rG;O)): G2G
O2OG
Example 1. Let us take ` = 2 and (h; f ) such that: h(
Each Deductive Mastermind game-item consists of a sequence of conjectures together with their respective feedbacks. Let us de ne it formally.
De nition 4. A Deductive Mastermind game-item over ` positions, k colors and n lines, DM (l; k; n), is a set f(h1; f1); : : : ; (hn; fn)g of pairs, each consisting of a single conjecture together with its corresponding feedback. Respectively, Bt(DM (l; k; n)) = Bt(f(h1; f1); : : : ; (hn; fn)g) = fBt(h1; f1); : : : ; Bt(hn; fn)g. 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 satis able, and that there is a unique valuation that satis es it. Now let us focus on a method of nding 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 satis ability of nite 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
' ^
',
^
'
' _
_
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.
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(
((goal(
rr
goal(
We will now brie y discuss the tableau method on an example. Let us
consider 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 rst 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 rst do so assuming the left branch of the rst
conjecture to be true. This leads to a contradiction|on this branch we get that
goal(
oo 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 satis able. 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 ower 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 rst position of the goal conjecture, i.e., goal(
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 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 di culty can be expressed via the size of their minimal trees. The empirical data, resulting from children playing the Deductive Mastermind game in Math Garden, includes item ratings. In the rst analysis of this data we aimed at relating the item ratings to the parameters of the trees. To this end we de ne 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 rst. 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 de ne 4 measures of theoretically derived item di culties; the required number of oo, rr, gr, and rr steps, which together might predict item di culty. 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 di culties. 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
theoretically derived measures of item di culty 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 signi cant in
predicting item di culties: number of colours ( =1:07, p=:02), number of
hypotheses ( =1:75, p<:01), number of oo feedbacks ( = 5:1, p<:001), number
of rr feedbacks ( = 3:19; p<:0001), number of gr evaluations (ns), number of
or evaluations ( =1:6, p<:0001). Note that among the rules, only the required
number of or steps increases item di culty. A second aspect of the observed
item ratings to explain is the shape of its distribution. The distribution of the
item di culties 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
di cult 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 di cult 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: (
Conclusions and Future Work In this paper we proposed a way to use a proof-theoretical concept to analyze the cognitive di culty of logical reasoning. The rst hypotheses that we have drawn from the model gave a reasonably good prediction of item-di culties, but the t is not perfect. However, it must be noted that several non-logical factors may play a role in the item di culty as well. For example, the motor requirements to give the answer also introduce some variation in item di culty, 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 di culty 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 t with empirical data of the tableau-derived measures and other possible logical formalizations of level di culty (e.g., a resolution-based model). On the empirical side we rst would like to extend our analysis to game-items that consist of conjectures of higher length. This will allow comparing di culty of tasks of di erent size and measure the trade-o 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 di culty order of feedbacks in nding an optimal tableau|this could be checked in an eye-tracking experiment (see [23, 24]). Appendix The appendix contains three gures. The rst one illustrates the educational and research context of the Math Garden system (Figure 5); the second shows the distribution of the empirical di culty 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 gures in the proper text of the article.
Student game playing
Investigators data Math Garden.com instruction new items, tasks report instruction methods
Teacher
Euclid's Diagrammatic Logic and
Cognitive Science
Yacin Hamami1 and John Mumma2 Abstract. For more than two millennia, Euclid's Elements set the standard 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 intuitive 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 gures 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.
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 (
(a) Povečeto točki sa žəәlti.
Most1 dots are yellow 'Most dots are yellow.' (b) Naj-mnogo točki sa žəәlti.
Most2 dots are yellow 'Yellow dots form the largest subset.’ (a) Większość kropek jestżółta.
Most1 dots is yellow 'Most dots are yellow.' (b) Najwięcej jestkropek żółtych.
Most2 is dots yellow 'Yellow dots form the largest subset.' Bulgarian Polish
Most2 in Bulgarian (8b) and Polish (9b) contains superlative morphology in contrast to
Most1 as illustrated in (
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 (
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 conditions.
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 participated 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 (
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 experiment 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 quantifier with a 3x3x2 Repeated Measures ANOVA crossing the 3 levels of the ratio variable, 3 levels of distractor, and the truth/falsity of screens: (a) (b)
Bg
Most1 (povečeto) Most2 (najmnogo)
1:2 .858 .827 ratio 2:3 .778 .742 5:6 .643 .617 .871 .866 .785 .76 .673 .63 .797 .801
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(
Fig. 2. Most1 in Bulgarian.
‘Yes’ on true screens
3 Bulgarian: F(
Fig. 3. Most1 in Polish.
The results for Most2 are entirely compatible with the view that it is verified by
Selection. In both Bulgarian (Fig. 4) and Polish (Fig. 5) the accuracy rates are significantly
affected both by ratio (Bulgarian: F(
Fig. 4. Most2 in Bulgarian. ‘Yes’ on true 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 differences 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 affected 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 indicate 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 information 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.
Fig. 6. Two color condition: Most1 vs. Most2 in Bulgarian.
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(
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(
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 Selection 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 participants 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 comparison 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 algorithms must be different. To verify Most2 one has to (i) estimate target, (ii) estimate competitor, (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 instructions 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 quantifier 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 interesting 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 characteristic 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 Bulgarian 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:
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 Borer, 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 Osenova, Halina Krystewa, Petya Bambova, Ewa Panewa, Bilian Marinov, Marieta and the students 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. no. of screens ratio distractors 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 no. of yellow dots 8 - 12 8 - 12 8 - 12 8 - 12 8 - 12 8 - 12 8 - 12 8 - 12 8 - 12 5 - 9 5 - 9 5 - 9 5 - 9 5 - 9 5 - 9 5 - 9 5 - 9 5 - 9
Most2 - total number of screens: 180 no. of non- total no. of yellow dots dots no. of screens ratio distractors no. of dots total no. of in closest dots competitor
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 interpretation 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 summarized in this study.
Keywords. scholastic logic, psychology of reasoning, human experimentation,
Wason selection task 1 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 differentiation can be traced back to the beginning of the 20th century, where, for example, Frege [2] argued that the difference between the interpretations of the conditional statement as prescribed in logic and as used in "everyday life" reveals linguistic or psychological components. This is where the search for the linguistic or psychological components deemed different from logic began, first in the philosophy of language, then in linguistics and finally in psychology. However, the so-called everyday interpretation 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 interpretation of the conditional statement, which is basically the same as that of propositional logic, but which has been clearly created not in a mathematical, but in a linguistic, philosophical and psychological environment. Thus, the discrepancy between the everyday interpretation and the classical abstraction is within the same system. 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 incorrect. Jevons has argued that, for example, from the snow melting (Q), it does not follow 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 necessarily 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 suppose 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 during 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. 1.2
The correct abstraction of the conditional statement The question arises therefore, of what the correct abstraction of the conditional statement can then be, that is, the abstraction of the relationship in which there is no wedging “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 interpretation 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 inferences 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 antecedents, 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” conditional, 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 equivalent/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 antecedent with an “and” connective, then in terms of P and Q only, the MP and MT inferences 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 psychology as a linguistic, pragmatic effect, which is contrary to logic. It was nevertheless 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. 2.1
The experimental investigation of the conditional statement 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 content have no influence on the results and so they accurately display how people interpret 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 psychological 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 negation “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 follows 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. 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. The reasoning contained in the defective truth tables1 require further analysis, although 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 believe 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 statement, 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. 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 always 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 biconditional 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 biconditional 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 participants, 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.
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
traditionally 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
selection 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
interpreted 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
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
interpretation 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
communicated 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
population, 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
ChiSquare (
Conclusion In general, the functioning of the updated classical logical interpretation of the conditional statement can be demonstrated by the main experimental tasks used in experimental psychology. People basically interpret the conditional statement as an equivalent 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 nonmonotonic 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 certain 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 mathematical 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, and in the other case a conditional relationship, and with the addition of further contextual 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 context 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 subtraction 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 demonstrate 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 literature investigating the relationship between logic and everyday inferences can be interpreted 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 correct equivalent responses does not necessary reveal much about the basics of the inferential 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 interpretation 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 performance 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 define 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 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 variation—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.
References