Do concepts have parts? In this paper, we take the simplifying assumption that concepts can be understood as (more or less structured) bundles of features and discuss the distinction between features and concepts in terms of the distinction between parts and wholes. In doing so, we will leverage diferent ideas related to Fine's theory of parthood.
The notion of concepts has been deeply studied and developed across diferent fields, such as philosophy, logic, and cognitive science. Although the questions investigated in the diverse ifelds are somehow orthogonal (what are concepts, are they just in our mind or do they exist in the reality? How should we better represent them formally? How do humans conceptualise, categorise and combine concepts?), this has led to distinct accounts, and explanations for the role concepts have in our reasoning over, and understanding of, the world.
In the literature, the notion of concept often relies on a subsidiary one, that of feature. The
features, sometimes called attributes or characteristics, of a concept are usually assumed to play
a role in defining or describing it. In the context of psychological theories of concepts [
According to the Prototype Theory, concepts are represented in our semantic memory as
prototypes. Prototypes consist of sets, or lists, of features, each assigned a certain degree of
importance for describing the concept. This list of features encapsulates and describes the most
typical exemplars of the concept [
Most people think of birds as being feathered, two-legged creatures with wings, which fly, lay eggs in nests, and live in trees. [...] With simple, mundane knowledge, one can explain many of these features. Let’s start with flying. In order to fly, the bird needs to support its weight on wings. The feathers are important as a very lightweight body covering that also helps to create an aerodynamic form. Thus, wings and feathers enable flying. By virtue of lfying, the bird can live in nests that are in trees, because it can easily fly into and out of the trees. [1, p.143]
The distinction between features and concepts stems, arguably, from philosophy. A similar
pattern is at least endorsed by supporters of the so-called Bundle theory, proposed initially
by Hume in his Treatise [
Tho’ a particular colour, taste, and smell are qualities all united together in this apple, [...] they are not the same, but are at least distinguishable from each other" [4, pp.7-8]
This tradition has been picked up by Kant. According to Kant, a concept is a collection of
marks, or characteristics [
Not all concepts can be understood as bundles of features, nor do all agree that concepts
can be represented as some kind of feature aggregations. In philosophy, an example is Frege
discussion of the concept of relation [
The precise definition of what qualifies as a feature, and the distinction between features and concepts, is typically taken for granted. Notwithstanding this, a common thread among 1Although here Hume had in mind particular objects, the distinction applies also to simple and complex ideas, and thus translate also to the case of concepts, at least to what he calls imaginary concepts. the perspectives discussed above appears to be the idea that concepts are comprised of various features, which consequently serve as parts of the concepts.
That mental representation can be seen as constituted by parts has already been noted by
Leech [
Similarly, although not as explicitly, cognitive theories of concepts too, often, look at the features as parts of the description of the concepts:
For example, dogs have four legs, bark, have fur, eat meat, sleep, and so on. Some subset of these features might be part of the definition, rather than only one characteristic.[ 1, p.16] And, analogously, when building up concept descriptions in description logics and formal ontology, we often speak of the diferent parts composing the definition of the concept. As a matter of fact, logical operators are called concept constructors in DL, reflecting the idea that concept definition can be built from diferent concept parts.
In this paper, we take the simplifying assumption that concepts can be understood as (more
or less structured) bundles of features and discuss the distinction between features and concepts
in terms of the distinction between parts and wholes. In doing so, we will leverage diferent
ideas related to Fine’s theory of parthood [
The possibility of studying the mereology of abstract objects has been advocated by several authors, among which a place of honour is held by Fine. In Fine’s words [13, p.561], Philosophers have often supposed the notion of part only has proper application to material things or the like and that its application to abstract objects such as sets or properties is somehow improper and not sanctioned by ordinary use. But I suspect that this is something of a philosopher’s myth.
More specifically, the possibility of describing a mereology of mental representations has
already been advocated by Leech [
The idea of looking at concepts as representational devices made out of parts, possibly
contributing in diferent ways to the description of the concept as a whole, has been discussed
also in the context of Knowledge Representation, see [
We are assuming here a not too precisely specified notion of concept. The quite intuitive (and
rather vague) description would correspond to a mental representation consisting of a bundle of
2For a discussion of the representation of mereological relations in the same context see in particular [
A similar argument can be applied to formal ontologies. Concepts in the ontologies are built through definitions. Although we do not directly refer to them as composed of features, definitions can also be seen as construed by diferent parts. In this case, parts are organised and structured through the use of logical connectives.
If the relation between features and concepts is one of part-whole, what kind of part are the
features? Following Fine, parts can be classified in at least two dimensions [
If we accept that features can be parts of concepts, then it seems safe to assume that they
are parts in a basic way. Fine delineates two ways by which a parthood relationship may be
derived: through subsumption, as observed in the case of small parts, or via chaining, wherein
parthood is established through a concatenation of parthood relations [
The diference between absolute and relative parthood is not as clear. If we consider concepts as bundles of features, then necessarily concepts will be constituted by some features at any time. In this broad sense, then, features are part of the concept in an absolute way, as much as pints are parts of gallons or members are parts of sets.
Nevertheless, when considering everyday concepts, it is also quite evident that they can
change over time. Arguably, their features-parts change as well. As concepts are representations
of our reality, and reality changes over time, this doesn’t sound very surprising. A case in point
is conceptual evolution, namely the evolution of conceptual meaning over time, due to societal
or cultural changes. Consider the concept of marriage as an example. Rather obviously, the
features used to describe the concept two centuries ago were quite diferent from the ones one
would use to describe it today (at least in some parts of the world). This argument has been
used by Richards [
This phenomenon may initially appear primarily associated with the use of natural language.
Nevertheless, formal definitions of concepts within formal ontologies (or their parts) change
as well, for several reasons. For instance, one might find that the initial definition was overly
rigid and failed to encompass the intended domain of application adequately. Additionally,
revisions may be prompted by scientific advancements that necessitate a reassessment of our
understanding and, consequently, our definitions. For this reason, ontologists have developed
strategies to keep track of successive versions of their ontologies [
At least two aspects are worth noticing in the present discussion. First, conceptual evolution
opens the problem of finding appropriate identity criteria for the identification of concepts
through time. If concepts are subject to changes as are physical objects, it seems reasonable to
wonder how much a concept can change while remaining the same. How many features-parts is
it possible for a concept to lose, add or change without losing its identity? Or, in a similar fashion
but with a diferent terminology, for how long is it possible for a concept in an ontology to
maintain its identifier through ontology revisions? Diferent views exist in philosophy on similar
issues, from positions endorsing the possibility of concepts to change through maintaining
their identity to people denying this possibility (see [
A second (arguably related) aspect to note is that in many cases, although some features may change over time, others are persistent to the point of appearing as timeless parts of the concept. Let us consider the concept of game again. Arguably, although many of its features have changed over the years, it can quite stably be described as a kind of activity. Being an activity is a stable (one may say necessary) feature of game across the evolution of the concept. These kinds of features may be seen as the semantic core of concepts and could be considered to play a role in selecting the correct identity criteria for concepts.
When discussing the distinction between timeless and temporary parthood, Kit Fine introduces
two new kinds of objects/wholes, namely rigid and variable embodiments [
Rigid embodiments are introduced to exemplify objects such as sandwiches or water molecules. In such cases, Fine claims, a certain form or structure (e.g. the form of a ham sandwich, being structured as two pieces of bread with the ham between them) is embodied in the fixed, rigid, matter constituting the object (in the case of the ham sandwich, the two slices of bread and the ham). In such kinds of wholes, both the structure (or principle of rigid embodiment, in Fine’s terminology) and the fixed matter constitute timeless parts of the rigid embodiment.
Variable embodiments are instead used to describe wholes that are extended in time and have hence (possibly) temporary parts. Fine’s example is about the variable quantity of water in a river. Accordingly, there exists a principle, or function, that picks out the amount of water in the river at any time in which the river exists. There exists then a kind of whole, the variable water, that corresponds to that principle. The variable embodiment corresponds in this case to the quantity of water picked out by the principle at any given time t. The principle picking out the water is called the principle of variable embodiment, and the various objects picked out by the principle are the manifestations of the variable embodiment.
The two theories can be combined. According to Fine’s example, a particular car is, at a given time t, a rigid embodiment, constituted by several parts arranged according to a specific structure (or principle of embodiment), the one characteristic of a car. However, each of these parts can be subject to change, which then afects, and changes, the rigid embodiment as a whole as well. One can then consider the car as the variable embodiment whose manifestations are all the rigid embodiments corresponding to these changes. Moreover, each part of the particular car is constituted, in turn, by several parts (the motor is constituted by a cylinder head, pistons, etc), which are subject to change as well. The shift between rigid and variable embodiments applies to them as well.
The theory of rigid and variable embodiment has had tremendous success (although it did not
escape some criticism, e.g. [
Can the theory be applied to concepts as well? As already discussed, concepts vary in time: they can lose (or gain) some of their parts due to societal or cultural changes or even knowledge revision. Similarly to the variable water of a river, they seem to be describable in terms of variable embodiments. As some of the parts of a concept may vary, the manifestations of the concept-variable embodiment will correspond to diferent rigid embodiments at diferent times.
The question is then what would be the principle of variable embodiment in the case of
concepts. This discussion would require a dedicated paper, but one can try to give some
intuitions. Similarly to what was observed by Ferrario et al. in the context of organisations [
Even when one does not want to take a specific position over the precise structure of
concepts, it may be uncontroversial to state that concepts do have some structure, that being in
our semantic memory or our ontology. As mentioned above, if one assumes prototype theory,
the (weighted) features of concepts will be structured in the form of a list. In the case of the
knowledge view, the structure will be that of a network. In the case of concept definitions in
an ontology, some suggest employing Aristotelian definition [
We concluded the previous section by mentioning a peculiar aspect of concepts, namely the fact that some of their parts do not change over time. In other words, taking all of the possible manifestations (rigid embodiment) of concepts, some of the parts will be the same across time3.
(At least some) concepts have parts that are somehow essential to their description, or definition - features that once changed, completely change the meaning of the concept.
Cognitive theories of concepts often deny the possibility of providing clear-cut, stable definitions for everyday, common-sense concepts. In specific domain, specifically when we are not dealing with ordinary concepts, though, it is indispensable to define concepts through definitions. A case in point are scientific domains, e.g. when defining biological (or mathematical, etc) concepts. In such cases, concept definitions will have necessary parts. Definitions are, after all, often assumed to consist of set of necessary and suficient conditions for instances classification - a view that can be traced back to Aristotle. Arguably, in the case of concept definitions, changing the set of necessary and suficient conditions would lead to a completely diferent concept. One may thus imagine that, in the case of some scientific domains, concepts are more properly represented as rigid embodiments.
Nevertheless, also cognitive theories of concept which reject the possibility of providing
sharp definitions for concepts, often assume some parts of a concept description to be necessary
[
There seem to be diferent variations in the case of concepts. In some domains or contexts (e.g. mathematics, etc), one may want to treat concepts as rigid embodiments, always consisting of the same parts4. Once the parts change, the concept changes as well: it is a diferent concept. And other domains (as in the case of everyday concepts) where this would be a too-strong assumption. In such cases, concepts have both parts subject to change and timeless parts. We might call objects of this kind semi-variable embodiments: embodiments that (can) vary in almost all respects except for some essential parts.
Are concepts the only objects of this kind? There seem to be other cases.
Let us take the case of organisations. The first example involves (certain) rock bands. In many cases, all band members are replaceable except for the frontman. The departure of the frontman could lead to the dissolution of the band itself (consider, for a concrete example, the case of Nick Cave and the Bad Seeds 5: many members of the band have changed through the years, but what would happen if Nick Cave left the band?). 3One may argue that all the features describing a concept may vary as well. For instance, we argued that being an activity is an essential part of games. Nonetheless, activity is also a complex concept, whose features can change across time. What we are arguing, though, is that it is possible, stepping up in this hierarchy of concepts, to find a semantic core that is stable. 4There might be exceptions in these cases as well. Consider, for instance, how the conception of set has changed from Frege to Gödel. 5See https://en.wikipedia.org/wiki/Nick_Cave_and_the_Bad_Seeds
Another example is some dictatorships, where all major figures involved in the governments (advisers, generals, high-ranking oficials) can change, but the dictator, whose elimination would cause the cease of (at least) that specific dictatorship.
Organisations are not the only examples, as one can think of events that have essential parts as well6. Consider a birthday party. Almost every part can be variable (the guests may come and go, the location may change due to bad weather, etc), but the birthday girl is a stable, essential part of the event (if she gets sick, the birthday party is cancelled). Concerts follow a similar pattern: anyone in the audience can decide not to show up or to leave the concert, but if the singer ditches the concert, or has a problem during the concert, the event is cancelled (or ends).
As mentioned above, Fine’s discussion of variable embodiment assumes the possibility of nesting variable (and rigid) embodiment (see again the car example).
A similar nesting can (and often do) occur with concepts. Take the concept of a car. One could define it as the class of vehicles that normally have four wheels and a motor, run on roads, can seat up to eight people and are used mostly to transport people (and not cargo)7. By extending this description with more contextual features, one could see the concept of car as a (semi-)variable embodiment, that, at diferent times, can have diferent features as part (think of the evolution of cars in the last 100 years, and how the corresponding concept has changed). Nevertheless, by looking at the features of the concept car presented above, being a vehicle is also a complex concept composed of several features, and as such can be seen as a (semi-)variable embodiment, nested into the concept car.
This is a critical aspect of the representation of concepts. In the previous example, we have a concept playing the role of a feature for another concept. What is the relation of parthood in this case? And, generalising, how does the relation of parthood behave in the case of the combination of two or more concepts?
The smallest characterisation of the parthood relation is Core Mereology, which assumes the relation to be a partial ordering (a reflexive, antisymmetric and transitive relation) [ 29]. Where P is the parthood relation, this is expressed through the following axioms.
Reflexivity: ∀( ) Antisymmetry: ∀∀( ∧ =⇒ = )
Transitivity: ∀∀∀( ∧ → ) (1) (2) (3)
Applied to the case of features (and concepts) this means that: 1) each feature is part of itself; 2) if a feature 1 is part of another feature 2, and the second feature 2 is part of feature 1 as well, then the two features are the same; and that 3) if some feature 1 is part of another feature 2, and feature 2 is part of feature 3, then also feature 1 is part of feature 3. Let us focus on 6The discussion may be more tricky in the case of events, where diferent perspectives assume diferent positions on what is part of the event. 7https://en.wikipedia.org/wiki/Car axiom 38, and let us consider the example of the concept car again. We said the concept car can be described as being a vehicle with a bunch of other features. Let us take to correspond to the feature being a vehicle, and let us be the concept of car. Being a vehicle is used here as a feature of car, but it is a concept that can be described as a bunch of other features 1 , ..., . We have thus (1 < < ) and , and by transitivity . If I say that a vehicle is a machine designed for self-propulsion9, then this description become part of the concept car.
This may sound relatively unproblematic, as when we are describing the concept car as a vehicle, we want exactly to use the meaning of vehicle to describe cars. Peculiar to concepts is the possibility of combining them into more complex ones, and it is, after all, the idea behind compositionality that we can understand complex expressions based on their constituents’ meaning. Nevertheless, when considering compound concepts, the transitivity of features may sound like a too strong requirement, and in some cases blatantly false. As an example consider the concept of car bomb. Although is used mostly to transport people is one of the features that is part of the concept car, we sure wouldn’t say that it is part of the concept car bomb. The reason is that compounds often cannot be understood conjunctively as the union of the features of the two concepts, but involves more subtle semantics.
One may thus believe the case of conjunctive combination, putatively taking the union of the features of two concepts as describing the compound, to be easier to understand in terms of parthood relation. Even in the case of conjunctive combinations, however, assuming features transitivity does not work: it is easy to think of examples where transitivity fails, for instance, due to non-monotonic phenomena.
Consider a conjunctive concept such as pet fish . While, ordinarily, a pet may be described as an animal that lives in the house and is kept for a person’s company or entertainment, a ifsh does not really fit the description of an entertaining animal 10. In the combination of two concepts, some of the features may get lost, and some new features might be gained. Then we get to the point where we may want to consider the features of pet to be part of the concept pet; we may want to consider the concept pet to be part of the concept pet fish ; but then we cannot guarantee the transitivity of the features from the first to the second.
Do we have to give up on the idea of treating features as parts? Another possibility is to argue that the features are part of a concept in a diferent sense in which a concept is part of a compound 11, and this because they are the efect of diferent kinds of operations, or constructions.
When discussing his theory of parthood [
We might then try to apply similar ideas to the discussion conducted so far. In this context, we consider the features as ontological givens. We have then (at least) two kinds of operations: a “feature aggregation" operation, which aggregates features into meaningful concepts description; and (at least) a “concept combination" operation, operating on concepts (hence more complex elements, available only at a later stage).
In the following, we will distinguish between a concept/feature aggregation operation and a
concept composition operator, and discuss the CLAP principles on them [
In [
Assumptions Let us simplify by assuming that concepts consist of bundles of features, namely, they can be represented as an unordered list of features (cf. Section 1). Concept aggregation amounts to the operation of putting together the features needed to describe the concept. In other words, given the set of features = {1, ..., }, concept aggregation is the operation of defining the concept as consisting of a set of features 1, ..., . We will be using the following notation = [1, ..., ]. The result of applying the operation of concept aggregation will be called concept description.
Collapse [] = Namely, the concept formed out of a single feature (or concept) is identical to that feature (or concept). This seems to be a harmless condition when applied to concept aggregation. Leveling [[1, ..., ], , ..., ] = [1, ..., , , ..., ] Namely, the embedding of (other) concepts is irrelevant to the identity of the concept description. As discussed in Section 3, the nesting of concepts into concept descriptions does often occur (consider again the concept car being described as a vehicle). Arguably, the goal of this operation is to use the features of the nested concept in the concept description they are embedded into. The substitution of the nested concept with the features describing it wouldn’t harm the concept description. Leveling seems thus to hold to the case of concept aggregation. Absorption [1, ..., , , , , ..., ] = [1, ..., , , ..., ] Namely, the repetition of features/concepts is irrelevant to the identity of the concept description. This condition may seem too strong when considering the use of concepts in natural language. Consider someone describing wrens as small small birds, emphasising just how tiny they are. And thus one may believe that absorption may afect the identity of the whole after all. This seems however more an efect related to the use of natural language than to the representation of concepts. Consider the case where the repetition is due to the nesting of concepts among the features of the concept description, e.g. = [[1, ..., ], , ..., ]. For leveling, we would get = [1, ..., , , ..., ], and by absorption = [1, ..., , ..., ].
Permutation [1, ..., , , ..., ] = [1, ..., , , ..., ] Namely, the order of the features is irrelevant to the identity of the concept description. As we are assuming concepts to consist of unordered lists of features, permutation holds in the case of concept aggregation.
Following this discussion and starting from our assumptions, concept aggregation satisfies all
of the principles. It is thus a sum-like operation [
While in the case of concept aggregation, it is possible to make a simplifying assumption and consider concepts as relatively unstructured entities, the case of combined ones is diferent. Compounds exhibit a higher level of organization, and their meaning is intricately connected with their specific structure. Composition can occur at diferent levels of complexity. It can involve the combination of adjectives and nouns (e.g. black cats) or noun-noun combinations. Noun-noun combinations are generally assumed to have more subtle semantics12, involving complex patterns of relationship between the features of the concepts to be combined. Some of these issues are described by Wisniewski [32]. Accordingly, there exists at least three types of combination, namely: relation-linking ones, emphasising a relation between the two concepts (e.g. robin snakes are snakes that eat robins); property-mappings ones, where one of the salient properties of one of the concept is mapped onto the other (e.g. tiger spiders are spiders with stripes); and conjunctive combinations, combining the features of both concepts (e.g. poet painters have the features of both poets painters). As the contribution of the two concepts to the compounds varies in the three cases, it is not implausible to believe that the three combinations 12Although also some adjective-noun combinations are trickier than others: consider for instance fake diamonds.
Such cases escape a plain conjunctive interpretation as well.
correspond to somehow diferent kinds of operations, giving rise to diferent parthood relations.
We will simplify the discussion in what follows by analysing the case of conjunctive concepts,
and leave the discussion of the other two kinds of combination for future work.
Assumptions We assume that, as a starting point, conjunctive combinations can be described
as taking the union of the features of the constituent concepts. Leveraging on the literature
on compounds in cognitive psychology and linguistics [
We will be using the following notation to indicate the combination of concepts = (1, .., ).13 The result of applying the operation of concept combination will be called compound.
Collapse () = Namely, the compound composed of a single concept is the same as the single concept. Given the assumptions, the principle holds. The discussion of this case is analogous to what said in the context of concept aggregation.
Leveling ((, ), ) = (, , ) Namely, the embedding of compounds is irrelevant to the identity of the compound.
Given our assumption, this principle is against our intuition. One of the assumptions we have made concerns the potential loss of features during the combination referred to as inheritance failure. Let us make the slightly stronger assumption that, in the case of incompatibility of two features, both of them are lost.
Now let us assume = [1, 2], = [3, 4] and = [2, 4], and 1 is incompatible with 4. In the case of ((, ), ) we have (([1, 2], [3, 4]), ), and as 1 is incompatible with 4 we drop both of them. Now we combine the remaining concept(s) with w, namely ([2], [3], [2, 4]), and the feature 4 is thus preserved through the combination. In the case of (, , ), on the other hand, all of the occurrences of 4 would be dropped, and we would get ([2], [3], [2]). 13The assumptions made here are reminiscent of [34], which presents a formal treatment of conceptual combination in weighted DL.
Absorption (, , , ) = (, ) Namely, the repetition of concepts is irrelevant to the identity of the compound. Analogously to the case of Collapse, the principle holds given our assumptions.
Permutation (, ) = (, ) Namely, the order of the concepts is irrelevant to the identity of the compound. The principle is in contrast with the second of our assumptions, accounting for the distinction between the head and modifier concepts.
Following this discussion and starting from our assumptions, concept combination corresponds to the following pattern of principles: .
In this paper we have started discussing the mereology of concepts. By assuming the features describing a concept to be part of the concept, we have discussed what kind of parts are features, and what kind of whole are concepts.
In Section 3 we have suggested applying the theory of variable embodiment to concepts, and we have suggested the existence of semi-variable embodiments. A crucial issue, in this context, is to define the principle of (semi-)variable embodiment applying here. We have suggested the role of coherence in the classification as a possible way of constraining the principle. Further work is needed to make this idea precise, and implementable in a formal context.
The second half of the paper aims at defining an operational approach to concept construction. It diferentiate between two kinds of operations, concept aggregation and concept combination, and analyse the diferent properties of the two leveraging on Fine’s CLAP principles.
The discussion is conducted at an informal level. To develop a constructional approach to concept aggregation and combination in a formal setting is a matter of future work.
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Cognition 9 (1981) 35–58. [31] J. Fodor, E. Lepore, The red herring and the pet fish: Why concepts still can’t be prototypes,
Cognition 58 (1996) 253–270. [32] E. J. Wisniewski, When concepts combine, Psychonomic bulletin & review 4 (1997) 167–183. [33] R. Jackendof, English noun-noun compounds in conceptual semantics, The semantics of compounding (2016) 15–37. [34] G. Righetti, P. Galliani, C. Masolo, Concept Combination in Weighted DL, in: European Conference on Logics in Artificial Intelligence, Springer, 2023, pp. 385–401.