=Paper=
{{Paper
|id=Vol-2483/AIC19_paper4
|storemode=property
|title=Pink panthers and toothless tigers: three problems in classification
|pdfUrl=https://ceur-ws.org/Vol-2483/paper4.pdf
|volume=Vol-2483
|authors= Guendalina Righetti,Daniele Porello,Oliver Kutz, Nicolas Troquard,Claudio Masolo
|dblpUrl=https://dblp.org/rec/conf/aic/RighettiPKTM19
}}
==Pink panthers and toothless tigers: three problems in classification==
https://ceur-ws.org/Vol-2483/paper4.pdf
Pink Panthers and Toothless Tigers:
Three Problems in Classification
Guendalina Righetti2 , Daniele Porello1 , Oliver Kutz2 ,
Nicolas Troquard2 , and Claudio Masolo1
1
Laboratory for Applied Ontology, ISTC-CNR, Trento, Italy
2
Conceptual and Cognitive Modelling Group CORE,
KRDB Research Centre for Knowledge and Data,
Free University of Bozen-Bolzano, Italy
Abstract. Many aspects of how humans form and combine concepts
are notoriously difficult to capture formally. In this paper, we focus on
the representation of three particular such aspects, namely overexten-
sion, underextension, and dominance. Inspired in part by the work of
Hampton, we consider concepts as given through a prototype view, and
by considering the interdependencies between the attributes that define
a concept. To approach this formally, we employ a recently introduced
family of operators that enrich Description Logic languages. These oper-
ators aim to characterise complex concepts by collecting those instances
that apply, in a finely controlled way, to ‘enough’ of the concept’s defin-
ing attributes. Here, the meaning of ‘enough’ is technically realised by
accumulating weights of satisfied attributes and comparing with a given
threshold that needs to be met.
1 Introduction
A green chair is a chair that is also green, it is an untroubled combination of
attributes that gives us a conjunctive concept GuC. It is untroubled because the
specific colour of a chair is an attribute that does not affect the ‘chairness’ of the
chair: changing or losing the colour green will not make the chair a non-chair.
However, many combinations of concepts are not easily understood as simple
conjunctions, because some defining features of the combined concepts can enter
into a subtle conversation: this paper discusses and formally analyses some of
those phenomena studied within psychology and cognitive science, where the
combination of two (or more) concepts has a more subtle semantics than inter-
section of sets.
In the classical view of concepts, the extensions of concepts are precisely
defined: for each object it would be possible to specify whether it falls within
the definition of the concept or not. If the object fits all the necessary and
sufficient conditions in the definition of the concept, it will be categorised as
an instance of the concept, otherwise it will not. Moreover the combination of
concepts is defined in a compositional way. This means that the meaning of a
complex concept functionally depends on the meaning of the primitive concepts
Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution
4.0 International (CC BY 4.0).
(or predicates) that compose it. In other words, the definition of a complex
concept is exclusively determined by its basic components, and this is often seen
to explain in part the ease and prolific ability of humans in creating meaningful
phrases. This perspective, that can be traced back to Frege [1891], became a
cornerstone of Classical Logic, and moved from there to be also a paradigm in
Description Logic. In spite of this, experimental cognitive sciences have shown
that this view has great limitations and have proposed alternative frameworks.
Cognitive models of concepts and classification are usually grouped into the
prototype view, the exemplar view, and the knowledge view also called theory-
theory (see Margolis and Laurence [1999]; Murphy [2002]), but also Gärdenfors’s
theory of conceptual spaces Gärdenfors [2000] and Barsalou’s theory of frames
Barsalou [1992] enter this category.
The crisp view of concepts in the traditional logic-based paradigm excludes
indeterminacy, graded membership and typicality effects from the process of
categorisation, while the merely extensional compositionality of concepts does
not allow for the representation of a number of cognitive phenomena linked to
concept combination.
Overextension in conjunction [Hampton, 2017], situational effects [Barsalou,
1992], dominance effects [Hampton, 1988a], and attributes emergence [Hampton,
1987] are just a subset of the phenomena that hardly reconcile with composition-
ality of the extensions: the sole sets of entitites that fall under two concepts do
not determine the set of entities that fall under the combined concept. How to
integrate these phenomena in a logic-based framework is still an open challenge,
and the topic of this paper.
We begin with some examples to set the scene. A toothless tiger is a tiger
that has no teeth.1 Analysing the literal, rather than the metaphorical, mean-
ing of this conjunctive concept, we include all those tigers which are, although
tigers, lacking teeth, thereby losing an important, although not essential prop-
erty. Similar to the green chair , this can be understood as a simple conjunctive
concept when the concept of ‘tiger’ is given in such a way that it allows for grades
of typicality (e.g., wrt. a prototype). However, a definition of tiger in terms of
classical logic will exclude toothless tigers as non-tigers.
Now consider a pink panther as compared to the green chair . Whereas a
green chair might be quite like a red chair , just disagreeing in the colour, a pink
panther is quite unlike a black panther . Whilst it is extremely rare for a panther
to be black, it is still an optional colour quality attached to a panther.2 A pink
panther , on the other hand, assumes a colour which, by definition of the natural
1
The phrase comes from the Chinese cultural sphere, and a ‘toothless tiger’ in English
is essentially synonymous to a ‘paper tiger’ (a tiger made from paper, Chinese phrase
‘zhilaohu’): both are metaphorical concepts refering to “something or someone that
claims or appears to be powerful and/or threatening, but is actually ineffectual
and unable to withstand challenge” (see https://en.wikipedia.org/wiki/Paper_
tiger, accessed May 2019).
2
In 2018, a black panther was spotted in Kenya, the first scientifically confirmed sight-
ing in almost a century (1909). The location was very close to the fictional kingdom
2
concept of panther , makes it a non-panther (compare Gärdenfors [2000][p. 114]
for the related discussion of a pink elephant).
This is an example for the first of our three classification problems, overex-
tension, where we include more instances that belong to (the intersection of)
the defining concepts. To see the effect in a more straightforward example, take
the concepts of domestic worker and home appliance: clearly, these do not in-
tersect? Now consider a robotic vacuum cleaner : It is an autonomous domestic
robot with a floor cleaning system. It is not a domestic worker in the usual sense
of worker , since it is not a person, but often it would be categorised by people
as a domestic worker which is also a home appliance.
If we consider the category of the Tools that are also Weapons we could
consider different examples: hammers, chainsaws, razors, scissors, screwdrivers,
pliers and so on. But maybe we are interested in considering only the most
typical ones (say: hammer, razor and scissor).
This is an example for the second of our three classification problems, un-
derextension, where we include less instances than belong to the (intersection
of the) defining concepts. One way of making ‘less’ precise is to collect only the
(globally) most typical members of the intersection (see below).
Finally, consider a houseboat. A houseboat is both a house and a boat, how-
ever, it is a rather special house, being able to swim and lacking a solid fun-
dament, and it is a rather special boat, being able to be inhabited. This is an
example of, on the one hand, overconstraining the conjunction (as in underexten-
sion), as well as prioritising certain aspects of a concept over others. In this case,
we obtain an example of conceptual blending [Fauconnier and Turner, 2003].3
Such blends can be viewed as examples for the third of our three classification
problems, dominance, where we include instances that are primarily a C (the
boat is the head of the concept) and secondly a D (the house is the modifier,
adding unusual aspects to a house).
Such complex aspects of how humans form and combine concepts, which are
notoriously difficult to capture in Cognitive Science, are largely absent in logic-
based approaches for the representation of concepts. We base the approach of this
paper on the work of Hampton [Hampton, 1987, 1988a,b, 2017] and we develop
an abstract setting to represent the three particular concept formation aspects
just discussed: overextension, when a concept classifies more than what is con-
strained by its parts, underextension, when it classifies less, and dominance,
when some parts of a concept dominate the classification question. The moti-
vation for this work is to bridge the traditional techniques for doing knowledge
representation with cognitively grounded views of human classification.
of Wakanda, ruled by the Black Panther (see https://www.nationalgeographic.
com/animals/2019/02/rare-black-panther-spotted-in-africa-century/).
3
In conceptual blending, the combination of two concepts is both a merge of common
properties, together with emergent properties [Hedblom et al., 2016; Eppe et al.,
2018], which is not fully covered by our present approach in this paper (consider the
case of a ‘boathouse’ which needs large enough doors so that boats can fit through
them.).
3
This paper contributes to the project of extending Description Logics (DLs)
to model cognitively relevant features of classification [Porello et al., 2019]. We
start from familiar DL formalisms (in particular from ALC), which is an impor-
tant logical language to model concepts and concept combinations in knowledge
representation. We introduce a family of operators which apply to sets of concept
descriptions and return a composed concept whose instances are those that sat-
isfy ‘enough’ of the listed concept descriptions. To provide a meaning of ‘enough’,
the operator takes a list of weighted concepts as argument, as well as a thresh-
old. The combined concept applies to an instance if the sum of the weights of
the concepts it satisfies meets the threshold. Using a threshold, the presentation
focuses on crisp categorisations. Although the framework of weighted concepts
easily adapts to a many-valued setting, we do not admit degrees of classification
here.
The approach to weighted logics that we follow here takes inspiration from
the use of sets of weighted proposition for representing utility functions in Uck-
elman et al. [2009]. Extensions of that approach to DLs have been developed
in Ragone et al. [2009]. Two related articles are [Masolo and Porello, 2018] and
[Masolo and Porello, 2016], where cognitive features of categorisation have been
modelled by means of sets of weighted predicative formulas. The main difference
in the present approach is that we study weighted combinations of concepts by
explicitly introducing syntactic operators on concepts extending the basic con-
cept languages, and employing their logical expressivity for cognitive modelling.
We introduce the weighted logic in Section 2, introduce in more detail the
three classification problems in Section 3, discuss the formalisation of the prob-
lems in Section 4, and conclude in Section 5.
2 Weighted concept combination: The ∇
∇-operator logic
We introduce a class of m-ary operators, denoted by the symbol ∇∇ (spoken
‘tooth’), for combining concepts. Each operator works as follows: i) it takes a
list of concept descriptions, ii) it associates a vector of weights to them, and
iii) it returns a complex concept that applies to those instances that satisfy a
certain combination of concepts, i.e., those instances for which, by summing up
the weights of the satisfied concepts, a certain threshold is met.
An initial technical study of ALC ∇∇ was undertaken in Porello et al. [2019],
including a number of valid principles, expressivity issues, as well as counting
and maximisation principles. We here briefly introduce the basic syntax and
semantics, before we employ ALC ∇∇ to study the above introduced classification
problems formally.
Basic Syntax and Semantics of the ∇ ∇-Logic. The new logic is denoted
by ALC ∇∇R , where weights and thresholds range over real numbers r ∈ R. In
the following we will refer to the languages for brevity just as ALC ∇∇ . To de-
fine the extended language of ALC ∇∇ , we add combination operators as follows,
which behave syntactically just like m-ary modalities. We assume a vector of
4
m weights w ∈ Rm and a threshold value t ∈ R. Each pair w, t specifies an
operator: if C1 , . . . , Cm are concepts of ALC, then ∇∇tw (C1 , . . . , Cm ) is a concept
of ALC ∇∇ . Note that in this basic definition, the possible nesting of the operator
is excluded.4
For Ci0 ∈ ALC, the set of ALC ∇∇ concepts is then described by the grammar:
C ::= A | ¬C | C u C | C t C | ∀R.C | ∃R.C | ∇∇tw (C10 , . . . , Cm
0
)
The semantics of the operator is obtained by extending the definition of the
semantics of ALC as follows. Let I = (∆I , ·I ) be an interpretation of ALC.
We define the value of a d ∈ ∆I under a ∇∇-concept C = ∇∇tw (C1 , . . . , Cm ) by
setting:
X
vCI (d) = {wi | d ∈ CiI } (1)
i∈{1,...,m}
The interpretation (i.e., the extension) of a ∇∇-concept in I = (∆I , ·I ) is then:
∇tw (C1 , . . . , Cm ))I = {d ∈ ∆I | vCI (d) ≥ t}
(∇ (2)
To better visualise the weights an operator associates to the concepts, we some-
times use the notation ∇∇t ((C1 , w1 ), . . . , (Cm , wm )) instead of ∇∇tw (C1 , . . . , Cm ).
In the following examples, we will consider the value of an object name a (aka
individual constant) wrt. a ∇∇-concept for interpretations that satisfy a certain
knowledge base K (i.e., a set of formulas).
Definition 1 (Weights relative to a knowledge base). Let a be an object
name of ALC and K an ALC knowledge base. We set
X
vCK (a) := {wi | K |= Ci (a)}
i∈{1,...,m}
I.e., vCK (a) gives the accumulated weight of those Ci that are entailed by K to
satisfy a.
Note that for positive weights, a given name a and a fixed interpretation I such
that I |= K, we always have that vCK (a) ≤ vCI (aI ).
Example 1. Consider the set of concepts C = {Red, Round, Coloured} and the
concept C defined by means of the ∇∇ operator
∇t ((Red t Round, w1 ), (∃above.Coloured, w2 ))
C=∇
4
In a more fine-grained definition ALC i∇∇K , i ≥ 0, is the logic with i levels of allowed
nesting and where weights and thresholds range over K; we will comment on this
further below.
5
The definition of C means that the relevant information to establish the cate-
gorisation under C of an object is whether (i) it is red or round, and (ii) it is
above a coloured thing.
Consider the following knowledge base K = {Red(a), ∃above.Blue(a), Blue v
Coloured}, i.e., an agent knows that the object a is red and it is above a blue
thing and that blue things are coloured, Blue v Coloured.
The value of a returned by vCK is computed as follows. Firstly, if a satisfies Red,
then a satisfies Red t Round, so the weight w1 can be obtained. Moreover, since
Blue v Coloured ∈ K and a satisfies ∃above.Blue, then a satisfies ∃above.Coloured,
so also the weight w2 can be obtained. Thus, vCK (a) is w1 + w2 . If w1 + w2 ≥ t,
then a is classified under C.
Having defined the basic ∇∇ operator, a number of derived operations are
easily obtained. First, it is possible to define the set of instances that at most
reach a given threshold:
∇≤t ((C1 , w1 ), . . . (Cm , wm )) ≡ ∇∇−t ((C1 , −w1 ), . . . , (Cm , −wm ))
∇ (3)
Given this, we can define the concept of instances that exactly score a certain
threshold value t:
∇=t ((C1 , w1 ), . . . (Cm , wm )) ≡
∇
∇t ((C1 , w1 ), . . . , (Cm , wm )) u ∇∇≤t ((C1 , w1 ), . . . (Cm , wm ))
∇ (4)
Finally, it is interesting to consider the set of entities that maximally satisfy
a combination of concepts C1 , . . . Cm of ALC. That is, we may define an operator
with the following semantics:
max
I I I 0 0
∇
∇ ((C1 , w1 ), . . . (Cm , wm )) = {d ∈ ∆ | vC (d) ≥ vC (d ) for all d ∈ ∆} (5)
Defining ∇ ∇max in terms of ∇∇t would require to use a universal role, which
significantly increases the expressive power of ALC [Hemaspaandra, 1996], as
shown in Porello et al. [2019].
3 Three Problems in Classification
3.1 Overextension
What happens to ordinary concepts when they are subjected to the logical op-
eration of conjunction? In the 80s Hampton started a series of experiments to
evaluate how people interpret conjunctive phrases such as sports witch are also
games or games which are also sports ([Hampton, 1988a]). The classical semantic
interpretation of these cases would interpret their meaning in terms of conjunc-
tion: the extension of the two phrases would correspond to the intersection of
the set of games and the set of sports. However, experimental results showed
something different. In his experiment, Hampton first provided a list of activi-
ties and asked people to evaluate which items in the list were sports and which
6
were games (with the possibility of overlapping). After a couple of weeks, people
were asked again to evaluate which items in the list could have been considered
sports which are also games and, conversely, games which are also sports, and
the results were interpreted to verify if people followed a rule corresponding to
an intersection or conjunctive interpretation. The answer is no: a number of
sports (games) that were not considered games (sports), according to the first
classification, were considered as members of the conjunction in the second part
of the experiments. This phenomenon is called overextension in conjunction, but
the problem of non-extensional combination applies equally to disjunctions (and
negation). For instance, most people do not categorise a mushroom as fruit, nor
as vegetable, but then categorise it as an instance of fruit or vegetable [Hampton,
2017].
Hampton’s proposal is to model the combination of such concepts in terms
of intensional composition which takes into account the features that define the
combined concepts. When a complex concept arises as a conjunction, the features
that define it
will be derived in a compositional fashion as a disjunction of the features
of the two conjuncts [Hampton, 2017, p. 98].
Conjunctive phrases such as sports which are also games should then return a
composite prototype resulting from aggregating all the features commonly associ-
ated with either of the two conjuncts (p. 98). The new set of features would then
be modified according with Hampton’s model for attribute inheritance [Hamp-
ton, 1987]. This model of the construction of a conjunctive concept would also
predict overextension in conjunctions. Roughly, Hamptons argument runs as fol-
lows: let us imagine that, in order to be categorised under a concept, an object
must show at least two out of three features of that concept. So, let us suppose
that to be categorised in a conjunction, it should show four out of six features of
the component concepts. If an object presents three features of the first object
but just one feature of the second object it would be categorised as an instance
of the conjunction, even if it is not an instance of both components. This would
reflect overextension in conjunctions [Hampton, 2017].
Interestingly, Hampton’s experiments also showed that this kind of relative
clause form is non-commutative: the concept games which are also sports is not
identical to the concept sports which are also games. The tooth operator can
capture some of these insights and distinctions, as we will show in more detail
in Section 4.
3.2 Underextension
Overextension’s specular, or dual, phenomenon is that of underextension, that
is, when considered in the context of conjunction, the phenomenon in which
objects categorised as instances of two categories are not classified as belonging to
the (underextending) conjunction of the categories. For instance, an item could
be categorised as sport and as game, but could be not considered an instance
7
of sports which are also games [Hampton, 1988a]. Actually, when related to
underextension, Hampton’s experiments were mostly directed to examine the
disjunction of common sense concepts, and to verify the consistency of people’s
judgements with respect to the set union rule [Hampton, 1988b]. So, in that
context, underextension would happen when an item categorised under a concept
is not categorised as an item of the disjunction of that concept and another
concept. For instance, an item could be categorised as a game, but then could
be not considered as an instance of game or sport (maybe because it was a
very atypical game). Here, we are dealing with another intuition. The idea is
that underextension in conjunctions would mean to identify particularly typical
members of the intersection of two concepts. As a particular case, we can then
define a prototype concept of the regular intersection of the concepts.
3.3 Dominance
Hampton’s experiments showed another interesting phenomenon, namely the
dominance effect. Roughly speaking, in the simplest case of the composition of
just two concepts, one of the two concepts ‘weighs more’ than the other in the
definition of the conjunction. More precisely, the dominance effect was analysed
with respect to typicality. Hampton’s aim was in fact to predict the typicality
of an item in a conjunction of two classes on the basis of the typicality of the
item in each of the conjuncts. In his analyses, he discovered that the weights of
the two constituent concepts in the determination of an item’s typicality were
usually not equal. On the contrary, dominance of a concept with respect to a
second was frequently observed. For instance, the weight of the concept sports
was almost twice the weight of games in determining the typicality of items
in both of the concepts sports which are also games and games which are also
sports [Hampton, 1987]. Following these insights, one can distinguish between a
concept Head (sports in the example), and a concept Modifier (e.g. games).
We can naturally model such phenomenon assigning a higher weight to the
‘dominant concept’ (the Head), or specific features thereof, or by considering
only the most prototypical instances of the dominant concept, as discussed in
more detail below.
4 ∇
∇-Modelling of Over/Underextension and Dominance
We start by proposing how to represent prototype-based classifications by means
of the ∇∇ operator. We then proceed to, firstly, show how we can capture simple
intersections of concepts defined in terms of prototypes. Secondly, we present
a simple modelling of all three phenomena that we have previously discussed
(overextension, underextension, and dominance effect).
The general picture is informally illustrated in Fig. 1, the formal modelling
of which we undertake in the this section.
The main motivation is the simple elegance of the approach, which allows to
compactly define specific instances of the three problems. Although not strictly
8
Fig. 1. The interaction of two graded concepts defined via prototype theory: the cir-
cular areas inside C, D correspond to the cores of the concepts (the most prototypical
instances); the classic intersection is inside the dark overlapping part.
more expressive, replacing extremely large disjunctions with short tooth-based
expressions may be arguably seen as a more cognitively adequate modelling.
Another, related point, is that the relative weights of the participating features
are being made explicit.
The ∇ ∇ operators can represent the cognitive approach to concepts based on
prototypes, cf. Porello et al. [2019]. Following Smith et al. [1988], a “prototype is
a prestored representation of the usual properties associated with the concept’s
instances”[Smith et al., 1988, p.487]. In our setting, we can introduce a concept
C by using a ∇ ∇ operator that directly considers the usual properties associated
to its instances:
C = ∇∇t ((P1 , w1 ), . . . , (Pm , wm )) (6)
The classification under C applies to the objects that have “enough” features in
common with the prototype, so to exceed the threshold t. This is a very simple
way to model the idea of being close enough with respect to the prototype. We
can also individuate the prototypical instances of C as the objects (if they exist)
that satisfy all the Pj in C. A different and more liberal approach is to collect
all those instances that reach the maximal possible value confirming a number
of the Pj : we call this set the core of C and compute it as ∇∇max (C).
In the following paragraphs, we assume that each concept is defined by means
of a prototype, cf. Eq. (6). We focus on two concepts, which are thus defined by:
C=∇ ∇s ((P1 , w1 ), . . . , (Pm , wm )) and D = ∇∇t ((Q1 , v1 ), . . . , (Qn , vn )).
4.1 Simple intersection
The logical conjunction of two concepts, which provides the intersection of the
extension of the concepts, can be simply defined by means of the u operator of
ALC:
9
∇s ((P1 , w1 ), . . . , (Pm , wm )) u ∇∇t ((Q1 , v1 ), . . . , (Qm , vm ))
∇ (7)
Example 2. Suppose now we define the two concepts Sport and Game as follows:
∇s ((DoneForFun, w1 ), (IsCompetitive, w2 ),
Sport = ∇
(DoneToGetFit, w3 ), (NeedsAthleticAbilities, w4 )) (8)
∇t ((DoneForFun, w1 ), (IsCompetitive, w2 ),
Game = ∇
(NotSeriousGoal, w5 ), (InvolvesConcentration, w6 )) (9)
Let us further assume the following knowledge base:
K1 = {DoneForFun(f), IsCompetitive(f), NeedsAthleticAbilities(f), NotSeriousGoal(f)}
It allows one to categorise ‘football’ (f) as an instance of both Sport and Game
(i.e., w1 + w2 + w4 ≥ s and w1 + w2 + w5 ≥ t). In this case, football would get
both the weight for being a Sport and the weight for being a Game, therefore
belonging to the simple intersection of the two concepts as defined above.
On the other hand, consider a different knowledge base, about beerpong (bp):
K2 = {DoneForFun(bp), IsCompetitive(bp), NotSeriousGoal(bp)}
In this case, we could imagine that w1 + w2 + w5 ≥ t but w1 + w2 ≤ s: beerpong
would then not enter in the simple intersection of Sport and Game.
4.2 Modelling Overextensions
A combination of concepts that returns a set larger than the intersection of the
two concepts can be approached in a number of ways. We discuss two strategies.
Firstly, we can lower the threshold for the classification under C or under D.
That is, in the combination of C and D the importance of the list of features
defining C and D is relaxed. So, more instances shall fall under either concepts,
returning a larger intersection. For example, when combining boat and house,
into a house boat we are not requiring that the instance of the combined concept
satisfy all the attributes of the composing concepts, only a sufficient number.
Given C = ∇ ∇s ((P1 , w1 ), . . . , (Pm , wm )) and D = ∇∇t ((Q1 , v1 ), . . . ,
(Qn , vn )), a family of overextending compositions of C and D can be introduced:
C uO D = ∇
∇x ((P1 , w1 ), . . . , (Pm , wm )) u ∇∇y ((Q1 , v1 ), . . . , (Qn , vn )) (10)
where s ≥ x or t ≥ y or both. In this case, as for the standard intersection, we
assume that the (marginal) contribution of the two concepts C and D is equal
(they both have to be satisfied). This reflects that the overextending composition
is in fact a composition of the two concepts.
The effect of lowering the threshold for classifying an entity under the com-
posing concepts has the consequence of raising the importance of a (number of)
10
features for the composed concept (whether it is their salience or their diagnos-
ticity). Globally lowering the threshold amounts to assume no information about
which specific feature is important in the combination, whereas directly raising
the weight of a number of features entails knowing that those are the important
bits, when combining the concepts. Thus, a second strategy for combining con-
cepts that overextends the intersection is by raising the weights of some of the
features in C and D.
Example 3. Suppose we define again the two concepts Sport and Game as in
Equation 8 and Equation 9. Let us also assume the following knowledge base:
K3 = {DoneForFun(w), IsCompetitive(w), NeedsAthleticAbilities(w)}
which allows one to categorise ‘wrestling’ (individual w) as an instance of Sport
(i.e., w1 + w2 + w4 ≥ s) but does not allow to categorise it as a Game (i.e.,
w1 + w2 < t). The simple intersection would not allow to represent ‘wrestling’ as
an instance of the intersection of Sport and Game. To overextend the intersection
of the two concepts in our setting, one should represent the intersection as follow:
Sport uO Game = ∇ ∇x (DoneForFun, w1 ), (IsCompetitive, w2 ), (DoneToGetFit, w3 ),
∇y (DoneForFun, w1 ), (IsCompetitive, w2 ),
(NeedsAthleticAbilities, w4 ) u ∇
(NonSeriousGoals, w3 ), (InvolvesConcentration, w4 ) (11)
where the threshold y is set in a way that allows to classify ‘wrestling’ as a
Game.
4.3 Modelling Underextensions
In a similar manner, we can treat the case of underextensions, that is, by raising
the threshold required for classifying the composing concepts. One interpretation
of this case is that when combining two concepts, only instances that are more
typical of the two concepts can enter the combination of the two concepts. By
means of the ∇ ∇max , we can define underextension operators as follows. Given
C=∇ ∇ ((P1 , w1 ), . . . , (Pm , wm )) and D = ∇∇t ((Q1 , v1 ), . . . , (Qn , vn )), we set:
s
C uU D = ∇
∇max ((P1 , w1 ), . . . , (Pm , wm )) u ∇∇max ((Q1 , v1 ), . . . , (Qn , vn ))
(12)
Example 4. Suppose we define the two concepts Sport and Game as before in
Equation 8 and Equation 9. Let us also assume that both ‘basketball’ (individual
bb) and ‘ping pong’ (individual pp) are in the intersection of Sport and Game,
according to the following knowledge base
KB = {DoneForFun(bb), IsCompetitive(bb), NeedsAthleticAbilities(bb),
DoneToGetFit(bb), NotSeriousGoal(bb), InvolvesConcentration(bb),
DoneForFun(pp), IsCompetitive(pp), NeedsAthleticAbilities(pp),
NotSeriousGoal(pp), InvolvesConcentration(pp)}
11
Suppose that we define underextending composition of Sport and Game as in
Equation (12). Then, basketball would be selected by our operator, as it max-
imises both Sport and Game, while ping pong will not, as it does not maximise
Sport (by failing to satisfy DoneToGetFit).
Another way of introducing underextension is to constrain the intersection
of C and D by imposing that the instances must maximise the combination of the
features of C and D: CuD u∇ ∇max (P1 , w1 ), . . . , (Pm , wm ), (Q1 , w1 ), . . . , (Qn , vn ) .
This definition is non-equivalent to (12) only in the case of mutually incom-
patible attributes defining C and D. We leave this aspect for future work.
4.4 Modelling Dominance Effects
In the case of dominance, as we have seen, the head of the combination out-
weighs the other composing concepts. Given C = ∇∇s ((P1 , w1 ), . . . , (Pm , wm ))
and D = ∇ ∇t ((Q1 , v1 ), . . . , (Qn , vn )), we present three strategies to model dom-
inance operators. A first strategy is to define dominance operators as follows:
0 0
C uD ∇s (P1 , w1 ), . . . , (Pm , wm ) u ∇∇t (Q1 , v1 ), . . . , (Qn , vn )
1 D=∇ (13)
where s0 ≥ s and t0 ≤ t.5 In this case, globally raising the threshold of the domi-
nant concept amounts to include the typical instances of C into the composition,
whereas globally lowering the threshold of the non-dominant concept includes
possibly non-D instances. This means that a sport that is also a game must be
primarily a sport.
A second strategy is to unpack (or flatten) the definitions of C and D and
raise the weights of the most salient features ∅ 6= S(C) ⊆ {1, . . . , m} of the
dominant concept C for the composition, whilst including some required elements
R(D) ⊆ {1, . . . , n} of D to compute a new minimal threshold. We may set:
C uD ∇π(C,D) (P1 , w1π ), . . . , (Pm , wm
π
), (Q1 , v1π ), . . . , (Qn , vnπ )
2 D=∇ (14)
where π is a function that re-assigns weights such that π(wi ) = wiπ > wi for all
i ∈ S(C) (and π(x) = x otherwise) and π(C, D) = Σi∈S π(wi ) + Σj∈R π(vj ) is a
new threshold that accumulates the inflated weights for the salient features of C
and the required weights from D.
A third strategy to define a dominant concept would be to collect those
instances that maximise the sum of weights of the dominant concept (using e.g.
the ∇∇max operator).
Notice that these types of compositions are generally non-commutative, as
the first combined concept (the head) is more important than the second.
Example 5. Suppose we define again the two concepts Sport and Game as in
Equation 8 and Equation 9, and consider again the knowledge base in Example
5
Here the choice of making the first concept dominant is merely formal, we are not
making any assumption on the linguistic phrasing of this complex concept.
12
3. If we consider the concept Sport as the dominant one, we could apply Equa-
tion 13 to include wrestling in the intersection of the concepts Sport and Game,
simply lowering the threshold for being included in Game. This is again a case
of overextension, but the idea behind this choice here is that having the most
important features of the concept Sport ‘count more’ to enter in the conjunctive
concept.
We have a similar result if we apply Equation 14:
Sport uD ∇π(Sport,Game) (DoneForFun, w1π ), (IsCompetitive, w2π ),
2 Game = ∇
(DoneToGetFit, w5π ), (NeedsAthleticAbilities, w6π )(DoneForFun, v1π ),
(IsCompetitive, v2π ), (NotSeriousGoal, v3π ), (InvolvesConcentration, v4π )
(15)
where we set π(Sport, Game) = π(w1 )+π(w2 )+π(w6 )+π(v1 )+π(v2 ). It is worth
noticing here that just the weights associated to the concept Sport are raised by
the function π, as required in Eq. 14: since Sport is the dominant concept, its
salient features are considered the most important in this combination.
5 Conclusions and Future Work
We studied three cognitively and psychologically interesting and important phe-
nomena that address classification problems with a focus on conjunctive oper-
ations on concepts, namely overextension, underextension, and dominance. To
address these phenomena from a formal modelling point of view, we employed
the logic ALC ∇∇ introduced in [Porello et al., 2019], where a class of concept-
forming operators was introduced which can define complex concepts by means
of accumulating the weights of defining attributes and comparing this value to
a given threshold. We here introduced basic modelling approaches for the three
classification problems, discussed several relevant examples, and outlined the
remaining challenges for the formal approach.
Future work will be dedicated to further investigate the logical properties of
the operators and their natural extensions, and to employ them in a systematic
fashion to the modelling of salient cognitive features of concepts, concept com-
binations in general, and conceptual blending in particular [Eppe et al., 2018;
Confalonieri and Kutz, 2019]. Moreover, the landscape of over/under-extension
and dominance should be analysed in a more finegrained manner, considering
interdependencies, special cases, and the introduction of further (conjunctive)
classification problems. Eventually, we will aim to justify the formal fine-theory
of conjuctive (and disjunctive etc.) classification in a detailed empirical study.
Another line of research will deepen the comparison with the formal studies
on typicality (e.g. Lieto and Pozzato [2018]) and the relation and combination
with similarity frameworks based on a notion of distance (e.g. Confalonieri et al.
[2017]; Sheremet et al. [2007]; Hois and Kutz [2008]).
Further, we hope to pursue further a deeper exploration of the correspon-
dence theory between tooth logic, statistical learning and classification models,
and concept learning in DL, as begun in Galliani et al. [2019]. Such a corre-
spondence will not only contribute to these fields individually, but will allow to
13
more systematically learn the relevant thresholds for complex tooth concepts,
and it will support hybrid frameworks where e.g. statistically learned concepts
can be naturally represented in DL knowledge bases, thereby contributing also
to neural-symbolic integration and to the AI explainability challenge [Garcez
et al., 2019].
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