Fumiaki Toyoshima

Adrien Barton
email: adrien.barton@irit.fr

Jean-François Ethier
email: ethierj@gmail.com

Affordances and their ontological core

Keywords: Affordance, disposition, environment, image schema, intention

The notion of affordance remains elusive, notwithstanding its importance for the representation of agency, cognition, and behaviors. This paper lays down a foundation for an ontology of affordances by elaborating the idea of "core affordance" which would serve as a common ground for explaining existing diverse conceptions of affordances and their interrelationships. For this purpose, it analyzes M. T. Turvey's dispositional theory of affordances in light of a formal ontology of dispositions. Consequently, two kinds of so-called "core affordances" are proposed: specific and general ones. Inspired directly by Turvey's original account, a specific core affordance is intimately connected to a specific agent, as it is reciprocal with a counterpart effectivity (which is a disposition) of this agent within the agent-environment system. On the opposite, a general core affordance does not depend on individual agents; rather, its realization involves an action by an instance of a determinate class of agents. The utility of such core affordances is illustrated by examining how they can be leveraged to formalize other major accounts of affordances. Additionally, it is briefly outlined how core affordances can be employed to analyze three notions that are closely allied with affordances: the environment, image schemas, and intentions.

Introduction

1.1. Purpose and motivation: Looking for "core affordances"

The term "affordance" was coined by [START_REF] Gibson | The Ecological Approach to Visual Perception[END_REF] to pin down precisely the interaction between the animal and the environment in the context of ecological psychology: "The affordances of the environment are what it offers the animal, what it provides or furnishes, either for good or ill" (ibid., p. 127). To take classical examples, a step affords climbing when it is of a certain proportion relative to a person's leg length and a gap affords hiding when it is of a certain size relative to a person's body size. According to [START_REF] Kutz | The mouse and the ball: Towards a cognitively-based and ontologically-grounded logic of agency[END_REF], affordances (and image schemas, which will be introduced later) offer a first foundation upon which agents (e.g. animals) and inanimate objects (e.g. tools) can be distinguished and identified. Against this theoretical background, the notion of affordance has been investigated in a number of different agency-related domains: for example, it has been extensively utilized in robotics to foster flexible planning and behaviors of robots [START_REF] Şahin | To afford or not to afford: A new formalization of affordances toward affordance-based robot control[END_REF][START_REF] Chemero | Gibsonian affordances for roboticists[END_REF][START_REF] Min | Affordance research in developmental robotics: A survey[END_REF][START_REF] Yamanobe | A brief review of affordance in robotic manipulation research[END_REF][START_REF] Beßler | A formal model of affordances for flexible robotic task execution[END_REF][START_REF] Beßler | Foundations of the socio-physical model of activities (SOMA) for autonomous robotic agents[END_REF].

However, the nature of affordances has not been always understood properly. As a matter of fact, affordances are all too often conflated with other elements having nothing to do with them, as is illustrated by [START_REF] Norman | Affordance, conventions, and design[END_REF] remark on a certain graphic designer's confusion of an affordance with "a symbolic communication, one that works only if it follows a convention understood by the user" (ibid., p. 40). In addition, the elusiveness of affordances has yielded their multifarious theories to date (see [START_REF] Michaels | Affordances: Four points of debate[END_REF] overview of several issues regarding affordances), but the interrelationships between those theories remain underspecified. Here a formal ontology of affordances will be useful, for it will serve as a point of reference for comparing various accounts of affordances by making their implicit "ontological choices" [START_REF] Borgo | Ontological foundations of DOLCE[END_REF] transparent. An ontological analysis of affordances is thus expected to facilitate research dealing with the notion of affordance, especially in the context of information systems.

The objective of this paper is to lay down a foundation for such an ontology of affordances along with [START_REF] Toyoshima | Modeling affordances with dispositions[END_REF] and Toyoshima & Barton's (2019a[START_REF] Toyoshima | Linking image schemas with affordances: An ontological approach[END_REF] previous works. To achieve this purpose, a primary focus will be upon [START_REF] Turvey | Affordances and prospective control: An outline of the ontology[END_REF] dispositional theory of affordances. For one thing, as Heras-Escribano (2019) argues, affordances cannot be fully comprehended unless Gibson's original understanding of them within ecological psychology is well appreciated. Turvey's dispositional account of affordances is arguably one of the most straightforward and compelling interpretations of "Gibsonian affordances". In fact, a dispositional view of affordances has been recently philosophically defended by [START_REF] Vetter | Perceiving potentiality: A metaphysics for affordances[END_REF] and Heras-Escribano (2019) (albeit not necessarily exactly in the same way as by Turvey). It has been also formally explored by [START_REF] Ortmann | Affordances as qualities[END_REF], 2012) in compliance with the upper ontology the Descriptive Ontology for Linguistic and Cognitive Engineering (DOLCE; [START_REF] Masolo | Wonderweb deliverable D18 -ontology library (final)[END_REF][START_REF] Borgo | Ontological foundations of DOLCE[END_REF][START_REF] Borgo | DOLCE: A descriptive ontology for linguistic and cognitive engineering[END_REF] and, in this DOLCE-based direction of research, it has been reinterpreted and implemented in robotics [START_REF] Beßler | A formal model of affordances for flexible robotic task execution[END_REF][START_REF] Beßler | Foundations of the socio-physical model of activities (SOMA) for autonomous robotic agents[END_REF].

Quite importantly, however, it is not the contention of this paper that affordances must be always construed in [START_REF] Turvey | Affordances and prospective control: An outline of the ontology[END_REF] dispositional manner. Rather, this paper embraces a pluralistic stance on affordances. That is to say, there can be multiple, equally plausible approaches to affordances and which account of affordances is to be practically used hinges on domain-specific assumptions and needs. As will be discussed, for instance, Şahin et al. (2007) submit that the "agent perspective" and "observer perspective" on affordances are more relevant to autonomous robot control than the "environmental perspective" on them, such as Turvey's. To take another example, [START_REF] Oliver | The problem with affordance[END_REF] opines that Gibsonian affordances may be too deeply rooted in an ecological perspective on perception to be well-suited for considering people's learning and creation of contemporary technologies such as computers.

It will be nonetheless valuable to provide a "dispositional basis" for a general ontology of affordances. As was sketched out by [START_REF] Toyoshima | Modeling affordances with dispositions[END_REF] and Toyoshima & Barton (2019a) and as will be shown in more detail below, this is mainly because other major theories of affordances can be formalized as an extension of a dispositional interpretation of affordances. In this sense, "Turveian affordances" may be called "core affordances": affordances that would serve as a common ground for scrutinizing many existing kinds of affordances and hitherto unspecified interconnections among them. Briefly, this paper will spell out the hypothesis that core affordances could be advantageously conceptualized and formalized as Turveian affordances.

Scope and structure of the paper

To specify the scope and structure of this paper, it will be useful to refer to [START_REF] Galton | The formalities of affordance[END_REF] three distinct goals that a complete theory of affordances must achieve:

• The upper-level goal is to answer what may be called "ecological" or "environmental" questions concerning the role of affordances in the life of an individual, how they can be used to explain features of human and animal behaviour, and how they can be exploited for the better design of environments.

• The middle-level goal is concerned with characterising exactly what affordances are: this may be called the "ontological" question. How is an affordance defined, and what is the logical relationship between statements about affordances and other statements about the world? • The lower-level goal is the answer the "aetiological" question of where affordances come from, exactly how the physical layout of surfaces determines the affordances it has for any given class of creatures. As is clear from Section 1.1, this paper aims primarily at the middle-level goal to answer an ontological question of how to characterize affordances. As a starting point to pursue it, Section 2 provides preliminaries by presenting a basic ontological framework and a theory of dispositions that both underlie the inquiry of this paper. Section 3 develops a theory of core affordances based on [START_REF] Turvey | Affordances and prospective control: An outline of the ontology[END_REF] dispositional account of affordances. Section 4 explicates how this account of core affordances can be extended to other well-known interpretations of affordances. Now it is reasonable to think that an adequate answer to the ontological question about the characterization of affordances will contribute to an in-depth inquiry into the ecological question of how to consider the role of affordances in connection with agents and the environment. With this train of thoughts, Section 5 turns to the upper-level goal and discusses, from the viewpoint of core affordances, three entities that are intimately connected to this goal: the environment, image schemas, and intentions. Finally, the lower-level goal is basically off the table in this paper, but Section 6 concludes the paper with some brief remarks on future work, including a possible direction of research in which the lower-level goal is to be investigated.

Preliminaries

This section offers preliminaries to the development of a theory of core affordances. Section 2.1 specifies a basic ontological framework for the overall study of this paper. Section 2.3 explains an ontology of dispositions by presenting their general idea, based on [START_REF] Röhl | Representing dispositions[END_REF] and Barton et al.'s (2018) works (Section 2.3.1) and Barton et al.'s (2017) theory of parthood between dispositions (Section 2.3.2). Note that the reader who does not want to enter into the details of an ontology of dispositions may skim through Section 2.3 (especially Section 2.3.2) and return to it later when necessary.

Basic ontological framework

First of all, the formal investigation of this paper will be conducted in first-order logic (with identity). Indeed, another language such as the Web Ontology Language (OWL; [START_REF] Horrocks | OWL: A description-logic-based ontology language for the semantic web[END_REF] is often used in practical ontological modeling, but it may lack the expressiveness to axiomatize the intended meaning of affordances. One may think that first-order modal logic [START_REF] Braüner | First-order modal logic[END_REF] would be more suitable for representing affordances (e.g. the climbability of steps) because they are closely related to agents' possible actions. Consider, for instance, [START_REF] Galton | The formalities of affordance[END_REF] usage of possibility operators ("◊") in his formalization of affordances. Nonetheless, first-order logic will be employed in this paper (but see [START_REF] Borgo | A first-order formalization of event, object, process, and role in YAMATO[END_REF] discussion on other kinds of formalizations of ontologies than first-order modal logic and OWL). For one thing, first-order logic would have the practical virtue of higher computational tractability, even if some first-order modal logics are translatable into first-order (nonmodal) logics [START_REF] Braüner | First-order modal logic[END_REF].

This paper leverages categories and relations that are relatively widely accepted in upper ontologies. In particular, it avails of the general structure of upper ontologies that have the category of disposition (which is vital for theorizing on core affordances) such as Basic Formal Ontology [START_REF] Arp | Building Ontologies with Basic Formal Ontology[END_REF][START_REF] Otte | BFO: Basic Formal Ontology[END_REF] and the Unified Foundational Ontology [START_REF] Guizzardi | Ontological foundations for structural conceptual models[END_REF][START_REF] Toyoshima | Towards ontological foundations for conceptual modeling: The unified foundational ontology (UFO) story[END_REF][START_REF] Guizzardi | UFO: Unified Foundational Ontology[END_REF]. In the first place, a distinction is drawn between instances (synonym: "particular", "individual") and classes (synonym: "universal", "kind"). In the text, terms for instances and classes will be italicized and boldified, respectively: e.g. this particular person Mary and the human class Human. In formalization, free variables will be assumed to be universally quantified and classes will be represented by unary predicates without being reified: e.g. HUMAN(x)→ ANIMAL(x) ("Every human is an animal").

Instances fall into two categories: continuants (aka "endurants") and occurrents (aka "perdurants"). Roughly speaking, continuants persist in time: they exist at one time and also exist at another different time (refer to [START_REF] Toyoshima | Foundations for ontology of persistence: Beyond talk of temporal parts[END_REF] for details on an ontology of persistence). Continuants can be further divided into independent continuants and dependent continuants. As for the relationship between them, we will assume here that a dependent continuant inheres (INH) in exactly one independent continuant, where inherence is a form of existential dependance (refer to [START_REF] Borgo | Ontological foundations of DOLCE[END_REF] for details on this inherence relation; and as we will see, there can be relations of existential dependence different from inherence). For instance, Mary's mass depends on and also inheres in Mary. One paradigmatic example of independent continuants (IND) is objects such as steps. Among objects are agents (AGE) such as Mary.

In contrast, occurrents extend through time and they have temporal parts. This paper will focus on a subtype of occurrents called "process" (PRO): occurrents that independent continuants (especially objects) participate in. Among processes are actions (ACT) such as Mary's climbing a step (see, for details, [START_REF] Trypuz | Formal ontology of action: A unifying approach[END_REF] formal ontology of actions). Note that this differs from the view of processes as so-called "dynamic occurrents" or "continuant-like occurrents" (refer to [START_REF] Toyoshima | How do processes work?[END_REF] for more detailed and critical thoughts on these alleged "dynamic processes"). Formal constraints on those basic categories and relations are outside the scope of this paper.

As for the parthood (P) relation, [START_REF] Casati | Parts and Places. The Structures of Spatial Representation[END_REF] Closed Extensional Mereology (CEM) will be assumed for all practical purposes (see, for instance, [START_REF] Borgo | A first-order formalization of event, object, process, and role in YAMATO[END_REF] usage of CEM in formalizing an upper ontology). It is a binary relation between processes or independent continuants, above all objects (refer to [START_REF] Masolo | Collectives, composites and pluralities[END_REF] for details on parthood involved in objects). Other mereological relations and operators will be defined as usual in CEM, such as O (O(x,y) for "x and y overlap"), PP (PP(x,y) for "x is a proper part of y"), and SUM (SUM(z,x,y) for "z is the sum of x and y").

Overview of classes and relations introduced in the paper

We will use in the core of this paper (Section 3) the taxonomy as pictured on Figure 1. 1 will be accepted, that is, if A is a subclass of B, we accept A(x) → B(x). We will also introduce the relational terms with the domains and ranges in Table 1. Here again, all the associated axioms will be accepted: if B is a binary relation with domain D and range R, we accept the axiom B(x,y) → D(x) ∧ R(y).

Ontology of dispositions

General idea about dispositions

This paper proposes an ontology of dispositions theoretically underpinned by [START_REF] Mckitrick | Dispositional Pluralism[END_REF] causal approach to dispositions (which she calls "disposition pluralism"). This approach is pragmatically motivated, as it is useful for representing multifarious entities dispositionally (see Toyoshima et al.'s (2021) detailed examination of her theory). She embraces a broad conception of dispositions according to which expressions of the form "the disposition to bring about R if T holds" serve, at least partially, as a reliable guide for identifying dispositions, where R is a realization (e.g. the glass-breaking) and T is a triggering condition (e.g. the glass being forcefully pressed) (but see also [START_REF] Vetter | Potentiality: From Dispositions to Modality[END_REF] possibilitybased, rather than causal, account of dispositions solely in terms of their realizations, which will not be discussed in this article).

More specifically, a characterization of dispositions in the line of [START_REF] Röhl | Representing dispositions[END_REF] and [START_REF] Barton | The identity of dispositions[END_REF] will be utilized. To explain it, two major subtypes of dependent continuants will be introduced: qualities (sometimes called "categorical" properties) and dispositions (refer to [START_REF] Choi | Dispositions[END_REF] for details on this categorical/dispositional distinction). A disposition is a causal property that can be linked to a realization, namely a specific process in which an independent continuant that is the bearer of the disposition participates. To be realized in a process, a disposition needs to be triggered by some other process, assuming a processual view of triggers of dispositions [START_REF] Mckitrick | Dispositional Pluralism[END_REF]. REAL(x,y) and TRIG(x,y) state that y is a realization and trigger of x, respectively. Classical examples include fragility (the disposition to break when pressed with a force), solubility (the disposition to dissolve when put in a solvent), and flammability (the disposition to get combusted when ignited). Dispositions may exist even if they are not realized or even triggered: for example, a glass is fragile even if it never breaks or even if it never undergoes any shock.

In addition, the notion of "categorical basis" of dispositions will be introduced [START_REF] Barton | The identity of dispositions[END_REF]. Synonyms include "causal basis" [START_REF] Prior | Three theses about dispositions[END_REF], "base" [START_REF] Röhl | Representing dispositions[END_REF], and "categorical base" [START_REF] Choi | Dispositions[END_REF]. A categorical basis of a disposition is a quality of the disposition bearer such that the disposition exists in virtue of this quality. Apart from a very special kind of dispositions (which McKitrick (2018) calls "ungrounded dispositions"), dispositions have some categorical basis [START_REF] Toyoshima | Towards a unified dispositional framework for realizable entities[END_REF]. CBA(x,y) means that y is a categorical basis of x.

For instance, a categorical basis of the fragility of this glass is a molecular structure (a quality) of the glass such that the structure makes the glass fragile. In this paper, a main focus will be upon "sure-fire dispositions" [START_REF] Röhl | Representing dispositions[END_REF] whose realizations necessarily occur once the disposition has been triggered; and also upon "single-track dispositions" [START_REF] Röhl | Representing dispositions[END_REF] which have one (sufficiently specific) kind of realizations and one (sufficiently specific) kind of triggering processes, such as the fragility of a glass, which has as realizations instances of the classes Breaking of a glass and as triggers instances of Pressing a glass. The term "disposition" will henceforth refer to a sure-fire and single-track disposition unless otherwise stated (refer to [START_REF] Röhl | Representing dispositions[END_REF] for a formalization of such dispositions and [START_REF] Barton | The identity of dispositions[END_REF] for their identity conditions).

A theory of parthood between dispositions

Finally, Barton et al.'s (2017) theory of part-whole relations (DP) between dispositions will be presented. We will focus on two subtypes of disposition-parthood: mod-parthood and add-parthood. Note that proper mod-parthood (MOD-PP) and proper add-parthood (ADD-PP) can be straightforwardly defined, just as proper parthood (PP) is defined in terms of parthood (P) in CEM.

First, mod-parthood represents several possible pathways, or modes, of realizations of dispositions. For example, the ferromagnetic disposition of this magnet has two mod-parts, i.e. its disposition to attract another magnet when facing an unlike pole and its disposition to repulse another magnet when facing a like pole. Note that a disposition that has several proper mod-parts is a "multi-track disposition" [START_REF] Barton | A modelling pattern for multi-track dispositions for life-science ontologies[END_REF][START_REF] Barton | The identity of dispositions[END_REF] which have different kinds of realizations according to different kinds of triggers. Second, add-parthood represents the additive character of dispositions. For example, the solubility disposition of this whole tablet has two add-parts, i.e. the solubility disposition of the left half of the tablet and the solubility disposition of the right half of the tablet.

Formal constraints on those two subtypes of disposition-parthood will be provided. In the first place, disposition-parthood in general (whether mod-or add-) can be characterized in terms of the bearer of the disposition.

( Next, mod-parthood satisfies the following axioms about the realization and trigger of the "mod-(disposition-)complex". That is, a mod-complex is realized in a process if and only if at least one of its proper mod-parts is realized in this process; and a mod-complex is triggered by a process if and only if at least one of its proper mod-parts is triggered by this process.

(MOD-PP-REAL)

[∃d1 MOD-PP(d1,d2) ∧ REAL(d2,x)] ↔ ∃d3[MOD-PP(d3,d2) ∧ REAL(d3,x)]
d2 has a proper mod-part and d2 is realized in x if and only if there is some d3 such that d3 is a proper mod-part of d2 and d3 is realized in x.

(MOD-PP-TRIG) [∃d1 MOD-PP(d1,d2) ∧ TRIG(d2,x)] ↔ ∃d3[MOD-PP(d3,d2) ∧ TRIG(d3,x)]
d2 has a proper mod-part and d2 is triggered by x if and only if there is some d3 such that d3 is a proper mod-part of d2 and d3 is triggered by x. In the magnet example, the ferromagnetic disposition of this magnet is realized in a process of the magnet attracting or repulsing another magnet if and only if its attraction disposition or its repulsion disposition is realized in this process. Similarly, the ferromagnetic disposition of this magnet is triggered by a process of another magnet facing an unlike or like pole if and only if its attraction disposition or its repulsion disposition is triggered by this process.

Then, add-parthood satisfies the following axioms about the realization and trigger of the "add-(disposition-)complex". That is, if an add-complex is realized in a process, then any of its add-parts is realized in a part of this process; and if an add-complex is triggered by a process, then any of its add-parts is triggered by a part of this process.

(ADD-PP-REAL) REAL(d1,x) ∧ ADD-PP(d2,d1) → ∃y[REAL(d2,y) ∧ P(y,x)] If d1 is realized in x and d2 is a proper add-part of d1, then there is some y such that d2 is realized in y and y is part of x. (ADD-PP-TRIG) TRIG(d1,x) ∧ ADD-PP(d2,d1) → ∃y[TRIG(d2,y) ∧ P(y,x)]

If d1 is triggered by x and d2 is a proper add-part of d1, then there is some y such that d2 is triggered by y and y is part of x. In the tablet example, if the solubility disposition of this whole tablet is realized in a process of dissolving, then all of its add-parts (i.e. the solubility dispositions of the right and left half tablets) are realized in a part of this process. Similarly, if the solubility is triggered by a process of the tablet being putting in water, then all of its add-parts are triggered by a part of this process.

A theory of core affordances

This section develops a theory of core affordances by expounding on [START_REF] Toyoshima | Modeling affordances with dispositions[END_REF] and Toyoshima & Barton's (2019a) preceding works. Section 3.1 presents the basic idea of [START_REF] Turvey | Affordances and prospective control: An outline of the ontology[END_REF] dispositional account of affordances. To examine it, Section 3.2 elaborates the notion of "reciprocal disposition". Then Section 3.3 conceptualizes and formalizes core affordances as a special kind of dispositions. Section 3.4 provides a model of our theory of affordances. Section 3.5 is devoted to the discussion on core affordances, in particular on their (in)definability (Section 3.5.2) and on the processual view of their realizations (Section 3.5.3).

The climbability of a step will be used as a driving example for explaining affordances. For one thing, it is a paradigmatic example of affordances taken from the experimental evidence gathered by ecological psychology [START_REF] Warren | Perceiving affordances: Visual guidance of stair climbing[END_REF] and, as Heras-Escribano (2019) says, it is quite useful for illustrating the nature of Gibsonian affordances, which a theory of core affordances focuses primarily on.

Turvey's dispositional account of dispositions

3.1.1. Basic idea: Affordances as reciprocal dispositions with effectivities [START_REF] Turvey | Affordances and prospective control: An outline of the ontology[END_REF] dispositional view of affordances is motivated by an understanding of animal activity in terms of prospective control: "control concerned with future events, usually interpretable as goals to be realized" (ibid., p. 174). To walk across a cluttered room, for instance, an agent needs to know what (bodily movement) is possible. An ecological approach to prospective control thus requires that affordances should be closely linked with agents' behaviors enabled by the environment with respect to which prospective control is conducted.

The key ideas of Turvey's dispositional approach to affordances are well articulated by his following statements:

An affordance is a particular kind of disposition, one whose complement is a dispositional property of an organism. (Turvey, 1992, p. 179) Given that a dispositional property is not defined (i.e., it is a nonexistent property) when there is no complement, then an affordance is not defined (i.e., is nonexistent) without a complementing animal property and, in like fashion, an effectivity is not defined (i.e., is nonexistent) without a complementing environment property. (ibid., pp. 179-180) Therefore, an affordance is a dispositional property found in the environment such that the affordance is a "complement" of the organism's dispositional property which Turvey calls "effectivity". For instance, the climbability affordance of this step is its disposition to move Mary upward and its complement is Mary's disposition (effectivity) to move upward. To develop an account of core affordances, an affordance and its "complementary" effectivity need to be formalized. For this purpose, Section 3.2 introduces the general notion of "reciprocal disposition" so that affordances and effectivities (which will be named "specific core affordances" and "specific core effectivities") can be analyzed as reciprocal dispositions. (Note that the subject matter for now is the climbability of a step, rather than its "descendability" or both. However, it would be possible to consider the affordance of a step as its disposition to move Mary upward or downward. As will be shown in Section 3.5.1, this construal of the affordance of a step can be modeled as an extension of the climbability affordance of the step.)

Dispositions for affordances: Are they ontic or predicatory properties?

Before the introduction of reciprocal dispositions in Section 3.2, it is necessary to understand accurately what [START_REF] Turvey | Affordances and prospective control: An outline of the ontology[END_REF] means by the term "disposition" or "dispositional property". He defends what he calls "property realism" and conceives dispositions as "real properties" (ibid., Section 3). To clarify this point, it will be helpful to consider [START_REF] Bird | Overpowering: How the powers ontology has overreached itself[END_REF] argument that the term "disposition" is open to both "ontic" and "predicatory" interpretations. Interpreted ontically, it refers to a property with dispositional essence that plays a distinctive ontological role. Interpreted predicatorily, by contrast, it is an ontologically uncommitted, mere façon de parler. To take one example, the statement "This fragile glass is disposed to break if it is pressed with a certain force" does not ipso facto entail dispositions in the ontic sense of the term. From Bird's perspective, Turvey's view of dispositions as real properties would mean that dispositions should be construed ontically, rather than predicatorily.

However, it may suffice to have a predicatory understanding of dispositions in order to adopt a dispositional approach to affordances. [START_REF] Bird | Overpowering: How the powers ontology has overreached itself[END_REF] maintains that an ontology of dispositions (in the ontic sense of the term) is not necessarily vindicated by an illuminating dispositional account of something, such as causation [START_REF] Bird | A dispositional account of causation, with some remarks on the ontology of dispositions[END_REF]. Furthermore, Heras-Escribano's (2019) dispositional theory of affordances is based on [START_REF] Ryle | The Concept of Mind[END_REF] "non-factualist" interpretation of dispositions and consequently, affordances are "not entities per se" but "a special way of explaining the complementarity relationship between the organism and environment" (Heras-Escribano, 2019, p. 88). This would be equivalent to a predicatory construal of dispositions.

It would be therefore a viable modeling option to combine Turvey's dispositional view of affordances with a predicatory interpretation of dispositions. In this case, a different general ontological framework from the one presented in Section 2.1 may be presupposed, especially a framework that does not adopt the categorical quality/disposition distinction. Notably, this approach could mesh with upper ontologies (e.g. DOLCE) that do not have the category of disposition. Consider, for instance, [START_REF] Ortmann | Affordances as qualities[END_REF], 2012) formalization of Turveian affordances as a subtype of the DOLCE category of quality.

Nevertheless, there may be some considerations in favor of a dispositional theory of affordances relying on an ontology of dispositions. For instance, an ontology of dispositions can meet some modeling purposes that a predicatory understanding of dispositions would fail to, such as the building of a realist ontology of the environment [START_REF] Smith | Toward a realistic science of environments[END_REF] (see Section 5 for discussion on affordances and the environment). An ontology of dispositions has been undoubtedly subject to criticism. To take one example, [START_REF] Guarino | BFO and DOLCE: So far, so close…[END_REF] would seem to think that laws of nature could fulfill the ontological role that dispositions can. However, dispositions are as relevant to scientific ontologies in general, as causation, laws of nature, and counterfactuals are (Toyoshima, 2020a). As for Guarino's worry, [START_REF] Turvey | Affordances and prospective control: An outline of the ontology[END_REF] emphasizes the vital role of laws of nature in his theorizing on affordances (ibid., Section 4). Therefore, dispositions and laws of nature may be complementary with each other in Turvey's dispositional approach to affordances (for more thoughts, see [START_REF] Kistler | Powers, dispositions and laws of nature[END_REF] and [START_REF] Ioannidis | No laws and (thin) powers in, no (governing) laws out[END_REF] philosophical arguments for dispositions and primitive lawhood in a governing account of laws of nature).

Interlude: Reciprocal disposition

Basic idea

The idea of reciprocal disposition can be attributed to [START_REF] Martin | The Mind in Nature[END_REF]. In formal ontology, it has been examined under labels such as "complementary disposition" [START_REF] Goldfain | Dispositions and the infectious disease ontology[END_REF] and "reciprocal dependence" among dispositions (Arp et al., 2015, p. 104). Classical examples include a key (say key1) and a lock (say lock2) such that key1 opens lock2. Intuitively, key1 has the disposition dkey1 to open lock2, and lock2 has the disposition dlock2 to be opened by key1. Those two dispositions, dkey1 and dlock2, have something in common. That is, they have the same class of realization processes (say lock2-opening-by-key1) and they have the same class of triggering processes (say key1-pivoting-in-lock2) (but see Section 3.5.3 for discussion on this processual view of realizations of dispositions in general). This key/lock scenario will be used as a guiding example for characterizing reciprocal dispositions formally.

A brief philosophical remark on reciprocal dispositions will be made here. As a matter of fact, [START_REF] Martin | The Mind in Nature[END_REF] proposes his model of reciprocal dispositions as an alternative to the traditional "stimulusresponse" model of dispositions as presented in Section 2.3.1 (see [START_REF] Anjum | Mutual manifestation and Martin's two triangles[END_REF] detailed discussion on Martin's model). This new model may take a trigger of a disposition to be its reciprocal disposition rather than some process: for instance, the solubility disposition of this salt and the dissolving disposition of this water are triggered by each other and mutually realized in a process of the salt dissolving in the water. In this paper, by contrast, the notion of reciprocal disposition is developed in the context of a trigger-realization account of dispositions. As will be explained in detail below, it turns out to be a specific interpretation of "extrinsic dispositions" [START_REF] Mckitrick | A case for extrinsic dispositions[END_REF][START_REF] Mckitrick | Dispositional Pluralism[END_REF].

To formalize reciprocal dispositions, the "reciprocity relation" (RECI) between dispositions is introduced (e.g. RECI(dkey1, dlock2)) and it is characterized by the following axioms:

A1 RECI(d,d′) → RECI(d′,d) The reciprocity relation is symmetrical. A2 RECI(d,d′) ∧ RECI(d,d′′) → d′=d′′
The reciprocity relation is functional.

A3 RECI(d,d′) → [REAL(d,x) ↔ REAL(d′,x)]
Two reciprocal dispositions have the same realizations.

A4 RECI(d,d′) → [TRIG(d,x) ↔ TRIG(d′,x)]
Two reciprocal dispositions have the same triggers.

A5 [RECI(d,d′) ∧ INH(d,x) ∧ NH(d′,y)] → ¬O(x,y)
The bearers of two reciprocal dispositions have no common part. (If d is reciprocal with d′, d inheres in x, and d′ inheres in y, then x and y do not overlap.) A5 trivially implies theorem T1:

T1 ¬RECI(d,d)
The reciprocity relation is non-reflexive. A5 excludes the possibility that the disposition of the door closed by lock2 to be opened by key1 (in contrast to the disposition of lock2 itself to be opened by key1) would be reciprocal of dlock2 (the disposition of lock2 to be opened by key1), which would be at odds with the intended meaning of reciprocal dispositions.

Turning to the categorical bases of reciprocal dispositions, let qkey1 be the categorical basis of dkey1: the quality of key1 that make it fit with lock2 (for the sake of simplicity, we assume there is only one such quality). Let qlock2 be the categorical basis of dlock2: the quality of lock2 that make it fit with key1. One defining characteristic of dkey1 and dlock2 is that dkey1 depends existentially not only on its categorical basis qkey1 that inheres in its bearer key1, but also on qlock2 which is external to key1. Similarly, dlock2 depends existentially not only on its categorical basis qlock2 that inheres in its bearer lock2, but also on qkey1, which is external to lock2.

To account for this, we can differentiate two kinds of categorical bases, namely "internal" and "external" bases. Internal bases are categorical bases of a disposition that inhere in the bearer of this disposition. External bases are categorical bases that inhere in something that does not overlap the bearer of this disposition (see definitions D1 and D2 below). Thus, since qlock2 does not inhere in key1, it is an external basis of dkey1; and qkey1 is an external basis of dlock2 (see also Barton et al.'s (2014) discussion about external bases of dispositions).

Individual-and kind-directed dispositions

However, dispositions such as dkey1 and dlock2 may be non-orthodox from the standpoint of an ontology of dispositions. To explain this, we will introduce the distinction between intrinsic and extrinsic properties. A property (particular) is intrinsic if that property exists and inheres in its bearer purely in virtue of the way the bearer is; and a property is extrinsic otherwise: it exists and inheres in the bearer (at least partially) in virtue of the way the world that is external to the bearer is (refer to [START_REF] Marshall | Intrinsic vs. extrinsic properties[END_REF] for details). Dispositions are traditionally taken to be intrinsic properties, as is illustrated by BFO characterizing them as "internally grounded realizable entities", and UFO classifying them as a subtype of "intrinsic moments", respectively (see Toyoshima et al.'s (2021) discussion on dispositions in BFO). For example, the fragility of a certain glass is intrinsic because the glass is fragile under any external circumstances, even when packed in a bubble wrap.

On the contrary, dkey1 and dlock2 are extrinsic, as whether dkey1 inheres in key1 depends also on its external basis qlock2, and similarly, whether dlock2 inheres in lock2 depends also on its external basis qkey1. This is in accordance with [START_REF] Shoemaker | Causality and properties[END_REF] examination of the key/door example based on [START_REF] Geach | God and the Soul[END_REF] notion of "mere Cambridge change": roughly, a property change that does not involve any intrinsic physical change. As a result, dkey1 and dlock2 are "extrinsic dispositions" [START_REF] Mckitrick | A case for extrinsic dispositions[END_REF]; but see [START_REF] Contessa | Do extrinsic dispositions need extrinsic causal bases?[END_REF] argument that extrinsic dispositions do not need extrinsic bases). According to [START_REF] Mckitrick | Dispositional Pluralism[END_REF] disposition pluralism, there are many different kinds of dispositions such as "extrinsic dispositions" and "intrinsic dispositions" (see Toyoshima et al.'s (2021) formal analysis of extrinsic dispositions). As Vetter (2018) claims, [START_REF] Turvey | Affordances and prospective control: An outline of the ontology[END_REF] dispositional account of affordances (which will be scrutinized in Section 3.3) entails an ontological commitment to extrinsic dispositions.

In this way of thinking, there is a modeling choice to introduce dispositions that are intimately connected to but different from dkey1 and dlock2. Let Key1 be the class of keys that can open lock2 (in particular, key1 is an instance of Key1) and Lock2 the class of locks that can be opened by key1 (in particular, lock2 is an instance of Lock2). Let Q1 be the class of qualities that characterize the structure of instances of Key1 that enable them to open lock2, and Q2 be the class of qualities that characterize the structures of instances of Lock2 that enable them to be opened by key1, such that qkey1 is an instance of Q1 and qlock2 is an instance of Q2. Now consider the following two dispositions:

• the disposition dkey1′ of key1 to open a lock instance of Lock2

• the disposition dlock2′ of lock2 to be opened by a key instance of Key1 Notably, dkey1′ and dlock2′ are different from dkey1 and dlock2 in several ways. For one thing, unlike dkey1, dkey1′ does not depend existentially on lock2 or on any individual lock similar to lock2. Likewise, unlike dlock2, dlock2′ does not depend existentially on key1 or on any individual key similar to key1. That is to say, neither dkey1′ nor dlock2′ has any external bases, and dkey1′ and dlock2′ are intrinsic dispositions, whereas dkey1 and dlock2 are extrinsic dispositions. For another, dkey1 and dlock2 are reciprocal with each other, while dkey1′ and dlock2′ are not reciprocal from each other. Indeed, dkey1′ and dlock2′ do not have the same class of realizations or the same class of triggers: for example, there can be a process in which dkey1′ is realized but dlock2′ is not, such as a process of key1 opening another lock instance of Lock2 than lock2. Intuitively, dkey1 is a disposition that is directed at the individual lock2, whereas dlock2′ is a disposition that is directed at the kind Lock2. Similarly, dlock2 is directed at the individual key1, whereas dlock2′ is directed at the kind Key1. In other words, dkey1 and dlock2 are "individual-directed dispositions" and dkey1′ and dlock2′ are "kinddirected dispositions" (where the latter is a renaming of Toyoshima & Barton's (2019a) "family-directed dispositions").

The "having as internal basis" relation (IBA) and "having as external basis" relation (EBA) are formally defined by definitions D1 and D2. A6 states that every disposition has at least one internal basis. A7 specifies the relationship between the internal bases of a disposition and some external bases of its reciprocal disposition. Individual-directed dispositions (IDD) and kind-directed dispositions (KDD) are characterized in terms of external bases with the axioms A8 and A9, respectively (and as a reminder, RECI was defined earlier as holding on IDD).

D1 IBA(d,q):=def CBA(d,q) ∧ ∃x [INH(d,x) ∧ INH(q,y)]
An internal basis of a disposition is a categorical basis of this disposition that inheres in the disposition bearer. (d has q as external basis, and there is some x in which both d and q inhere.)

D2 EBA(d,q):=def CBA(d,q) ∧ ∃x,y [INH(d,x) ∧ INH(q,y) ∧ ¬O(x,y)]
An external basis of a disposition is a categorical basis of this disposition that inheres in something that does not overlap the disposition bearer. (d has q as external basis, and there is some x and y that do not overlap such that x inheres in x and q inheres in y.)

T2 [EBA(d,q) ∧ INH(d,x)] → ¬INH(q,x)
An external basis of a disposition does not inhere in the bearer of the disposition. (If d has q as external basis and d inheres in x, then q does not inhere in x.) Proof: Suppose that EBA(d,q) and INH(d,x). Assume for reductio that INH(q,x). By D2, ¬O(x,x). This contradicts the axiom that anything overlaps with itself. QED.

A6 DIS(d) → ∃q IBA(d,q)

Every disposition has at least one internal basis. (If d is a disposition, then there is some q such that d has q as internal basis.) A7 RECI(d,d′) ∧ IBA(d,q) → EBA(d′,q)

Any internal basis of a disposition that has a reciprocal disposition is an external basis of this reciprocal disposition (If d is reciprocal with d′ and d has internal basis q, then d′ has external basis q) Note that we do not suppose the contraposed: the external basis of a disposition d that has a reciprocal disposition d′ might not be an internal basis of d′. Indeed, there are several ways for a disposition to have an external basis, and having a reciprocal disposition is only one of those ways (refer to [START_REF] Barton | Newtonian forces and causation: A dispositional account[END_REF] for alternative ways).

We can prove that any reciprocal disposition has an external basis: T3 RECI(d,d′) → ∃q EBA(d,q) Proof: Suppose that RECI(d,d′). By A6, there is an internal basis q of d′. By A7, q is an external basis of d. QED. We actually accept the axiom A8 that any individual-directed has an external basis (which is stronger than T3, as we do not exclude there might be some individual-directed dispositions that are not reciprocal dispositions):

A8 IDD(d) → ∃q EBA(d,q) An individual-directed disposition is a disposition and has an external basis. (If d is an individualdirected disposition, then d is a disposition and there is some q such that d has q as external basis.) A9 KDD(d) → ¬∃q EBA(d,q)

A kind-directed disposition is a disposition and has no external basis. (If d is a kind-directed disposition, then d is a disposition and there is no q such that d has q as external basis.) T3 and A9 yield together the theorem T4 that is relevant to the nature of kind-directed dispositions. T4 KDD(d)→ ¬∃d′ RECI (d,d′) No kind-directed disposition is reciprocal with anything. (If d is a kind-directed disposition, then there is no d' such that d is reciprocal with d'.) Proof: Suppose that KDD(d). Then by A9 there is no x such that EBA(d,x). Suppose for reductio that there is some d′ such that RECI(d, d′). Then by T3, there is some q such that EBA(d,q). Contradiction. Therefore, ¬∃d′ RECI(d,d′). QED.

Formalizing core affordances

The affordance/effectivity complex

As was explained in Sections 3.1 and 3.2, the crux of [START_REF] Turvey | Affordances and prospective control: An outline of the ontology[END_REF] dispositional approach to affordances can be formulated, in the terminology of this paper, as the statement that affordances and effectivities are reciprocal dispositions from each other, and are therefore individual-directed dispositions (which are extrinsic). Additionally, it was shown that there are also dispositions that are intimately related with individual-directed ones: kind-directed dispositions (which are intrinsic). In what follows, Turvey's account of affordances will be more fully presented and be formalized by means of this ontology of dispositions. Its central idea is that an affordance is to be understood not only in terms of an effectivity (the reciprocal disposition with the affordance) but also in terms of a third property that is borne by the "system" composed of the affordance bearer and the effectivity bearer. This "third property" of a system will be explored below (for more theoretical details on extrinsic dispositions, see [START_REF] Vetter | Potentiality: From Dispositions to Modality[END_REF] systemic account of extrinsic dispositions, which she calls "extrinsic potentialities", and its formal examination by [START_REF] Toyoshima | Towards a unified dispositional framework for realizable entities[END_REF]).

Imagine a scenario in which a person (say Mary) climbed this step (say step0). This concrete example will be called "CLIMB". Let a0 be the climbability affordance of step0 and let e0 be Mary's corresponding effectivity such that RECI(a0,e0)holds. That is to say, a0 and e0 have the same class of realizations (say Mary-climbing-up-step0) (A3), they also have the same class of triggers (say Mary-contracting-her-legson-step0) (A4), and they are different because their bearers do not overlap (A5). Consider now a system (say the Mary/step0 system) composed of Mary and step0. A system is, roughly, a complex object (or an aggregate of objects) with specific kinds of parts and relationships between them. However, an accurate ontological characterization of systems is beyond the scope of this paper (refer to [START_REF] Röhl | Mechanisms in biomedical ontology[END_REF] for some considerations) and the Mary/step0 system will be assumed to be analyzable simply as the sum of Mary and step0, namely an entity written Mary+step0 such that SUM(Mary+step0, Mary, step0)holds. According to Turvey's (1992, p. 180) semi-formal explication, Mary+step0 bears neither a0 nor e0 and there is a third property instance (say ae0) such that Mary+step0 bears ae0 and neither Mary nor step0 bears ae0. The bearer (i.e. Mary+step0) of ae0 is the sum of the bearers (i.e. Mary and step0) of a0 and e0. Accordingly, ae0 would be formalized as being in some sense the "sum" of a0 and e0.

To formalize this third property, a ternary relation (CES)between dispositions will be introduced such that CES(d1,d2,d3) is read as "d1 is a causally equivalent sum of and d2 and d3" and e.g. CES(ae0,a0,e0)holds. This relation can be characterized by the axioms A10 to A14, and the theorems T5 and T6 trivially follow from the preceding axioms.

A10 CES(d1,d2,d3) → DIS(d1) ∧ DIS(d2) ∧ DIS(d3) ∧ RECI(d2,d3) If d1 is a causally equivalent sum of d2 and d3, then d1, d2, and d3 are dispositions and d2 is reciprocal with d3. A11 CES(d1,d2,d3) ↔ CES(d1,d3,d2)

The causally equivalent sum of d2 and d3 is the same as that of d3 and d2.

A12 CES(d1,d2,d3) → [REAL(d1,x) ↔ REAL(d2,x)]
A causally equivalent sum has the same realizations as the first of its two reciprocal dispositions.

A13 CES(d1,d2,d3) → [TRIG(d1,x) ↔ TRIG(d2,x)]
A causally equivalent sum has the same triggers as the first of its two reciprocal dispositions.

T5 CES(d1,d2,d3) → [REAL(d1,x) ↔ REAL(d2,x) ↔ REAL(d3,x)]
A causally equivalent sum has the same realizations as its two reciprocal dispositions. Proof. Omitted. It trivially follows from A3, A10 and A12. T6 CES(d1,d2,d3) → [TRIG(d1,x) ↔ TRIG(d2,x) ↔ TRIG(d3,

x)]]

A causally equivalent sum has the same triggers as its two reciprocal dispositions.

Proof. Omitted. It trivially follows from A4, A10 and A13.

A14 [CES(d1,d2,d3) ∧ INH(d1,x) ∧ INH(d2,y) ∧ INH(d3,z)] → SUM(x,y,z)
The bearer of a causally equivalent sum is the sum of the bearers of the two reciprocal dispositions.

(If d1 is a causally equivalent sum of d2 and d3, d1 inheres in x, d2 inheres in y, and d3 inheres in z, then x is a sum of y and z.) Moreover, this CES relation can be further elucidated in terms of add-parthood between dispositions. For instance, ADD-PP(a0, ae0)and ADD-PP(e0, ae0)) would both hold, as hinted by the fact that a0 and ae0, as well as e0 and ae0, satisfy the three axioms characterizing add-parthood, respectively:

(DP-BEA) The bearers of a0 and e0 (i.e. Mary and step0) are (proper) parts of the bearer (i.e. Mary+step0) of ae0.

(ADD-PP-REAL) If ae0 is realized in a process of Mary climbing up step0, then both a0 and e0 are realized in a part of this process (namely, this very process).

(ADD-PP-TRIG) If ae0 is triggered by a process of Mary positioning her feet on step0, then both a0 and e0 are triggered by a part of this process (namely, this very process). In general, the following axiom holds:

A15 CES(d1,d2,d3) → ADP(d2,d1) ∧ ADP(d3,d1) A causally equivalent sum has as add-parts its two reciprocal dispositions. (If d1 is a causally equivalent sum of d2 and d3, then d1 is an add-part of d1 and d3 is an add-part of d1.) All these considerations can lead to the conceptualization of what [START_REF] Turvey | Affordances and prospective control: An outline of the ontology[END_REF] calls "third property" as an "affordance/effectivity complex": the "causally equivalent add-whole" that has as add-parts an affordance and its reciprocal effectivity. An affordance (e.g. a0), its corresponding effectivity (e.g. e0), and their affordance/effectivity complex (e.g. ae0) have the same realizations and triggers. In this sense, all those three dispositions may be informally said to play the same causal role. Nevertheless, they have different bearers, as the bearers of add-parts are a part of the bearer of the add-complex. In other words, two reciprocal dispositions can be seen as two "causal aspects" of a more encompassing disposition, when viewed from the perspective of a part of the bearer of this disposition. Consider, for example, the disposition d3 of the "key1/lock2 system" which is the causally equivalent sum of dkey1 and dlock2. Then dkey1 would be a causal aspect of d3, when viewed from the perspective of key1, and dlock2 would be a causal aspect of d3, when viewed from the perspective of lock2.

Additionally, an affordance, its reciprocal effectivity, and their affordance/effectivity complex have different categorical bases. We can assume that the categorical basis of ae0 is closely related to the categorical bases of a0 and e0. (However, the general axiomatization of this claim requires a theory of parthood between qualities, which is outside the scope of this paper; for an account of parthood that encompasses parthood between qualities, refer to [START_REF] Borgo | Ontological foundations of DOLCE[END_REF]) An affordance, its reciprocal effectivity, and their affordance/effectivity complex are different because they do not have the same internal and external bases. This accords well with Turvey's view of a "third property" for affordances (and effectivities).

Two kinds of core affordances

There are two major ways of formalizing core affordances depending on whether they are taken as individual-directed or kind-directed dispositions. On the one hand, a straightforward interpretation of [START_REF] Turvey | Affordances and prospective control: An outline of the ontology[END_REF] dispositional account of affordances yields the triad of an affordance (an individualdirected disposition), an effectivity (the reciprocal disposition with the affordance), and an affordance/effectivity complex (the causally equivalent sum of the affordance and the effectivity). As said, affordances as construed this way are "extrinsic dispositions", as they are dispositions that have an external basis. This interpretation may well capture the "specificity" of affordances, which Heras-Escribano (2019) regards as one of the distinguishing features of Gibsonian affordances: there is a strong correlation between "ecological information" of the environment and the available affordances for the agent. In this sense, affordances conceived as individual-directed dispositions may coincide better with Gibsonian affordances than when conceived as kind-directed dispositions.

In this line of thinking an affordance in the case of an individual-directed disposition will be called "specific core affordance" (SCA), and its corresponding term "specific core effectivity" (SCE) is introduced. They are inherently related to an affordance/effectivity complex (AEC). The axioms A16 to A21 characterize specific core affordances and effectivities, affordance/effectivity complexes, and their interrelations. The theorems T7 and T8 say that specific core effectivities and affordance/effectivity complexes are realized in actions (ACT) as well.

A16 SCA(x) → ∃y[IND(y) ∧ ¬AGE(y) ∧ INH(x,y)]
A specific core affordance inheres in a non-agentive object.

A17 SCE(x) → ∃y[AGE(y) ∧ INH(x,y)]

A specific core effectivity inheres in an agent. A18 [SCA(x) ∧ REAL(x,y)] → ACT(y)

A specific core affordance is realized only in actions. A19 SCA(x) → ∃y,z[SCE(y) ∧ AEC(z) ∧ CES(z,x,y)]

For any specific core affordance, there is some specific core effectivity and some affordance/effectivity complex that is the causally equivalent sum of the affordance and the effectivity.

A20 SCE(x) → ∃y,z[SCA(y) ∧ AEC(z) ∧ CES(z,x,y)]
For any specific core effectivity, there is some specific core affordance and some affordance/effectivity complex that is the causally equivalent sum of the effectivity and the affordance.

A21 AEC(x) → ∃y,z[SCA(y) ∧ SCE(z) ∧ CES(x,y,z)]
For any affordance/effectivity complex, there is some specific core affordance and some specific core effectivity that are causally equivalent add-parts of the affordance/effectivity complex. T7 [SCE(x) ∧ REAL(x,y)] → ACT(y) Specific core effectivities are only realized in actions.

Proof. Suppose that SCE(x) ∧ REAL(x,y). Then by SCE(x) and A20, there exist z and w such that SCA(z) ∧ AEC(w) ∧ CES(w,x,z). Then by REAL(x,y), CES(w,x,z), and T5, we have REAL(z,y).

From SCA(z), REAL(z,y), and A18, ACT(y) can be deduced. QED. T8 [AEC(x) ∧ REAL(x,y)] → ACT(y)

Affordance/effectivity complexes are only realized in actions.

Proof. Suppose that AEC(x) ∧ REAL(x,y). Then by AEC(x) and A21, there exist z and w such that SCA(z) ∧ SCE(w) ∧ CES(x,z,w). Then by REAL(x,y), CES(x,z,w), and T5, REAL(z,y). From SCA(z), REAL(z,y), and A18, ACT(y) can be deduced. QED. On the other hand, kind-directed dispositions would characterize what will be called "general core affordances" (GCA) and "general core effectivities" (GCE). For instance, step0 has a general core affordance a0′ to let instances of a class Person0 climb step0, where all instances of Person0 can climb step0. Relatedly, Mary has a general core effectivity e0′ to climb instances of a class Step0, where all instances of Step0 can be climbed by Mary. Although they may be somewhat distant from Turveian affordances, general core affordances can be useful for formalizing [START_REF] Reed | Encountering the World: Toward an Ecological Psychology[END_REF] and [START_REF] Chemero | An outline of a theory of affordances[END_REF] accounts of affordances (see Section 4) and image schemas (see Section 5). General core affordances and effectivities are characterized by the axioms A22 to A25.

A22 GCA(x) → ∃y[IND(y) ∧ ¬AGE(y) ∧ INH(x,y)]
A general core affordance inheres in a non-agentive independent continuant.

A23 GCE(x) → ∃y[AGE(y) ∧ INH(x,y)]

A general core effectivity inheres in an agent. A24 [GCA(x) ∧ REAL(x,y)] → ACT(y)

A general core affordance is realized only in actions.

A25 [GCE(x) ∧ REAL(x,y)] → ACT(y)

A general core effectivity is realized only in actions. To specify the interlink between general core affordances and effectivities on the one hand and specific core counterparts on the other, it will be useful to employ Toyoshima et al.'s (2021) "existential reliance" relation (REON for "relies on") between an intrinsic disposition and an extrinsic disposition. For instance, dkey1 relies existentially on the intrinsic disposition dkey1′ of key1 to open any instance (including lock2) of the type Lock2. Hence: REON(dkey1, dkey1′). We can accept the following axiom (inspired by Toyoshima et al.'s (2021) axiom a5): if d relies on d′, then every internal basis of d is an internal basis of d′:

A26 REON(d,d′) ∧ IBA(d′,q) → IBA(d,q) A disposition has as internal basis any internal basis of a disposition that relies on it. From this, we can show the theorem:

T9 REON(d,d′) ∧ INH(d′,o) → INH(d,o)
A disposition inheres in the bearer of a disposition that relies on it.

Proof: Suppose that d relies on d′ and d′ inheres in o. By A6, d′ has some internal basis q, which therefore inheres in o. By A26, q is an internal basis of d. Thus, the bearer of d is the bearer of q, namely o. QED.

The contraposed is then a trivial consequence, using the unicity of the bearer of a dependent continuant:

T10 REON(d,d′) ∧ INH(d,o) → INH(d',o)
A disposition inheres in the bearer of a disposition on which it relies. We can also accept the following axioms (corresponding to Toyoshima et al. ( 2021)'s axioms a3 and a4): if d relies on d′, then every realization of d is a realization of d′, and every trigger of d is a trigger of d′:

A27 REON(d,d′) ∧ REAL(d,p) → REAL(d′,p) A realization of a disposition is a trigger of the disposition on which it relies. A28 REON(d,d′) ∧ TRIG(d,p) → TRIG(d′,p)

A trigger of a disposition is a trigger of the disposition on which it relies. We can ground the relation between specific core affordances/effectivities and generic ones on the "relies on" relation, with the following axioms:

A29 SCA(x) → ∃y (GCA(y) ∧ REON(x,y))

A specific core affordance relies on some general core affordance. A30 SCE(x) → ∃y (GCE(y) ∧ REON(x,y))

A specific core effectivity relies on some general core effectivity. From those axioms, T10, and the axiom that every dependent continuant inheres in some independent continuant, we can deduce trivially the following more specific assertions:

T11 SCA(x) → ∃y,o (GCA(y) ∧ REON(x,y) ∧ INH(x,o) ∧ INH(y,o))
A specific core affordance relies on some general core affordance that inheres in the same bearer.

T12 SCE(x) → ∃y,o (GCE(y) ∧ REON(x,y) ∧ INH(x,o) ∧ INH(y,o))

A specific core effectivity relies on some general core effectivity that inheres in the same bearer. That is, reusing Toyoshima et al.'s (2021) vocabulary, every specific core affordance has an intrinsic dependee which is a general core affordance, that we will call its "general core affordance dependee", inhering in the same bearer; and every specific core effectivity has an intrinsic dependee which is a general core effectivity (its "general core effectivity dependee"), inhering in the same bearer. To illustrate, REON(a0,a0′) and REON(e0,e0′) hold: a0′ is an intrinsic dependee of a0, and e0′ is an intrinsic dependee of e0.

From axioms A27-A30, we can derive easily the following four theorems: T13 SCA(x) ∧ GCA(y) ∧ REON(x,y) ∧ REAL(x,p) → REAL(y,p) When a specific core affordance is realized, then its general core affordance dependee is realized in the same process. T14 SCA(x) ∧ GCA(y) ∧ REON(x,y) ∧ TRIG(x,p) → TRIG(y,p)

When a specific core affordance is triggered, then its general core affordance dependee is triggered by the same process. T15 SCE(x) ∧ GCE(y) ∧ REON(x,y) ∧ REAL(x,p) → REAL(y,p) When a specific core effectivity is realized, then its general core effectivity dependee is realized in the same process. T16 SCE(x) ∧ GCE(y) ∧ REON(x,y) ∧ TRIG(x,p) → TRIG(y,p)

When a specific core effectivity is triggered, then its general core effectivity dependee is triggered by the same process. We might also add that every categorical basis of the general core affordance (alternatively: effectivity) dependee of some specific core affordance (alternatively: effectivity) is a categorical basis of the latter.

It is worth noting that some above-given axioms about specific core affordances/effectivities turn out to be theorems that can be derived from axioms about general core counterparts (such as A29 and A30) and axioms regarding this existential reliance relation (REON). To take one example, the axiom A18 follows from A24, A27, and A29. Here, we formalized first a theory of specific core affordances based on [START_REF] Turvey | Affordances and prospective control: An outline of the ontology[END_REF] extrinsically dispositional view of affordances. As a result, specific core affordances and effectivities were formalized primarily in terms of axioms (as illustrated by A18) rather than theorems derivable from axioms on general core affordances and effectivities.

Model

We have presented a logical theory including definitions D1 and D2 and axioms A1 to A30. The theory was implemented in Alloy4 [START_REF] Jackson | Software Abstractions: Logic, Language, and Analysis[END_REF] and the model represented in Figure 2 was found, proving its consistency (facts involving ADP, CBA and O are omitted): As was announced, this subsection is devoted to the discussion about an alternative formulation of an affordance of a step (Section 3.5.1), the (in)definability of core affordances (Section 3.5.2), and the formalization of core affordances with dispositions and processes (Section 3.5.3).

An alternative formulation of an affordance of a step

Regarding the CLIMB example, it is interesting to note that one may want to consider the affordance of step0 not in terms of climbability alone but in terms of climbability and "descendability". When specific core affordances are concerned, one may want to characterize the affordance of step0 by dint of its disposition a0 to move Mary upward and its disposition (say a0*) to move Mary downward. This more general, specific core affordance (say a0+a0*) of step0 would be a mod-complex of those two dispositions, as is hinted by the fact that it satisfies the three axioms relating to mod-parthood:

(DP-BEA) The bearer (i.e. step0) of a0+a0* is an improper part of the bearers (i.e. step0) of a0 and a0*.

(MOD-PP-REAL) a0+a0* is realized in a process of Mary moving using step0 if and only if at least one of its proper mod-parts (i.e. a0 or a0*) is realized in this process.

(MOD-PP-TRIG) a0+a0* is triggered by a process of Mary positioning her feet on step0 if and only if at least one of its proper mod-parts (i.e. a0 or a0*) is realized in this process.

The (in)definability of core affordances

Definitions are crucial for ontology construction [START_REF] Smith | New desiderata for biomedical ontologies[END_REF]; see also [START_REF] Seppälä | The functions of definitions in ontologies[END_REF] detailed discussion). Definitions are composed of at least two types of features. First, the main part of a definition is known as the genus and it refers to the type into which the defined entity falls. Second, further one or more parts of the definition are the differentia(e) and they distinguish the defined type from other types of the same genus. For instance, the definition of a human being as a rational animal consists of the genus part of being an animal and the differentia part of being rational. Definitions can be also divided into two kinds in respect of their form: textual definitions, which are human-readable by being written in natural language, and logical definitions, which are machine-readable by being written in formal language such as OWL. To ensure the consistency of ontology development and use, ontologies are advised to include systematically both textual and logical definitions that convey the same content of the term.

One may then expect an ontological definition of affordances from an account of core affordances. However, it is not straightforward to define core affordances explicitly. Consider, for instance, specific core affordances. Given their logical axioms presented above, a first attempt to provide their textual definition may be the following (while leaving aside the notion of affordance/effectivity complex for simplicity): specific core affordance =def. A disposition that inheres in an environment within an agentenvironment system and that is realized by this agent interacting with that environment. specific core effectivity =def. A disposition that inheres in an agent within an agent-environment system and that is realized by this agent interacting with that environment. Nonetheless, to be non-circular, such definitions would require providing explanations of "agent" and "environment" that do not make a substantive appeal to the notions of "affordances" and "effectivities", but this is a non-trivial goal that exceeds the purpose of this paper (see also Section 5.1 for the relationship between affordances and the environment).

Moreover, core affordances (whether specific or general) may be in nature indefinable and they may be at best elucidated with informal explanation, formal axioms, and/or illustrative examples, as is often the case with upper-level entities (consider e.g. continuants and occurrents, cf. Arp et al., 2015, p. 89). To see this point, consider [START_REF] Sanders | An ontology of affordances[END_REF] opinion about affordances which is motivated by his criticism of [START_REF] Turvey | Affordances and prospective control: An outline of the ontology[END_REF] reductive dispositional analysis of Gibsonian affordances. Its main claim is: "affordances themselves seem like excellent candidates not only for primitive terms in the explanation of perception, but as primitive to the deeper tasks of ontology itself" (Sanders, 1997, p. 111). Consequently, for Sanders, affordances may be more fundamental than objects and processes, for example.

On the one hand, Sanders's view of affordances as fundamental primitives would be too radical to mesh with the general ontological background of this paper. On the other hand, the present theory of core affordances could subscribe to a weaker version of his thesis: that is, affordances should be treated ontologically rather than epistemically or semantically (see Section 3.1.2), and more specifically, as a basic component of an ontology of cognition, agency, and actions à la [START_REF] Kutz | The mouse and the ball: Towards a cognitively-based and ontologically-grounded logic of agency[END_REF]. In this respect, it may be reasonable to take core affordances as something that would defy definition but whose nature can still be elucidated through additional clarifications.

Formalizing core affordances with dispositions and processes

The current theory of core affordances stipulates, partly for the sake of simplicity, that core affordances are sure-fire dispositions (which will be necessarily realized when triggered) whose realizations are processes (see Section 2.3.1). Doubts can be nevertheless raised over the plausibility of this claim. Using the CLIMB example, suppose that Mary positioned her feet on step0. In particular, she contracted her muscles adequately in an effort to climb up step0. However, a gust of wind blew so strongly that Mary failed to climb up step0. Following [START_REF] Williams | The Powers Metaphysic[END_REF][START_REF] Williams | What are manifestations?[END_REF], a dilemma can arise as to whether the specific core affordance a0 (which is a sure-fire disposition) of step0 was realized or not. On the first horn of the dilemma, if a0 was realized, then a whole process of Mary climbing up step0 must have occurred. On the second horn, if a0 was not realized, then it contradicts the fact that a0 was indeed triggered by a process of Mary contracting her legs on step0.

Being inspired by [START_REF] Williams | The Powers Metaphysic[END_REF][START_REF] Williams | What are manifestations?[END_REF], one way to resolve this dilemma is to introduce powers being manifested in states of affairs [START_REF] Armstrong | A World of States of Affairs[END_REF] by analogy with dispositions being realized in processes. Construed within the present general ontological framework, powers may form a subtype of Dependent continuant (just as dispositions are) and a state of affairs would be, very roughly, an independent continuant bearing some dependent continuant(s) (including powers) such that this state of affairs "holds" or "obtains" at instants and for longer, such as the state of affairs of Mary's being six feet tall and the state of affairs of Alice loving Bob.

In Williams's view, a processual realization of a disposition can be roughly seen as a causal sequence of states of affairs which are manifestations of powers. In the current "failed affordance" version of the CLIMB example, the realization of a0 (namely, a process of Mary climbing up step0) would be analyzed as a causal series of many states of affairs (e.g. of Mary being in a certain physical posture) the last of which is the state of affairs (say soa1) of Mary being on step0, but soa1 did not hold because the strong wind led to a different causal chain of states of affairs. To borrow Williams's expression, dispositions are "process initiators": "To have a disposition is for it to be the case that the powers of the constituent entities are poised to bring about one of a range of process types" (Williams, 2019, p. 231).

This line of reasoning may motivate the avoidance of the usage of sure-fire dispositions in theorizing upon affordances. [START_REF] Turvey | Affordances and prospective control: An outline of the ontology[END_REF] dispositional approach to affordances might then be alternatively articulated based on powers and states of affairs, perhaps with the help of a previous formal ontology of states (of affairs) in first-order logic [START_REF] Benevides | States, events, and truth-makers[END_REF][START_REF] Masolo | Observations and their explanations[END_REF][START_REF] Masolo | The interplay between models and observations[END_REF]. However, the present account of core affordances presupposes the idea of dispositions being realized in processes while leaving examination of their "sure-fireness" for future work. For one thing, the nature of states (of affairs) may be controversial because it remains unclear whether states are classified as a subtype of continuants or occurrents (see, for thoughts, [START_REF] Galton | Processes as patterns of occurrence[END_REF] ontological analysis of states). This can be problematic, because a theory of core affordances aims to serve as the most general ontology of affordances. Furthermore, states of affairs might yield a departure from the currently accepted, traditional continuant/occurrent distinction (refer to [START_REF] Kassel | Processes endure, whereas events occur[END_REF][START_REF] Kassel | Physical processes, their life and their history[END_REF] for this line of inquiry). For another, modelers of affordances would be typically interested in processes (e.g. Mary climbing up step0) that are successfully (i.e. with no interruption) brought about by affordances (e.g. a0), as illustrated by [START_REF] Michaels | Affordances: Four points of debate[END_REF] view of "affordances as actions" and Moralez's (2016) view of "affordances as perceptual events". Thus, it may be practically virtuous to formalize affordances in terms of dispositions and processes.

Extension

This section examines the articulation between our theory of core affordances and other major interpretations of affordances by drawing on [START_REF] Toyoshima | Modeling affordances with dispositions[END_REF] and Toyoshima & Barton's (2019a) preliminary studies. For this purpose, Şahin et al.'s (2007) overview of accounts of affordances will be leveraged. It will be scrutinized how the dispositional account of core affordances can be expanded towards three different perspectives from which to characterize the affordance notion: the environmental perspective (Section 4.1), the agent perspective (Section 4.2), and the observer perspective (Section 4.3).

Environmental perspective

Firstly, the environmental perspective "attaches affordances over the environment as extended properties that [are] perceivable by the agents" (Şahin et al., 2007, p. 457). For instance, Şahin et al. (2007) classify Turvey's (1992) dispositional account of affordances (around which a theory of core affordances centers) as an environmental approach to them. However, it is contentious whether and to what extent perceptions are pivotal to Turveian affordances. The environmental perspective on affordances may well be thus interpreted more widely vis-à-vis those properties of the environment which have to do with agents.

Reed's "affordances as resources"

There may be nonetheless other environmental understandings of affordances than [START_REF] Turvey | Affordances and prospective control: An outline of the ontology[END_REF] that Şahin et al. (2007) do not bring up for discussion. Among them is [START_REF] Reed | Encountering the World: Toward an Ecological Psychology[END_REF] conception of affordances (which will be called "Reedian affordances") as resources in the environment that are linked with natural selection:

The fundamental hypothesis of ecological psychology (…) is that affordances and only the relative availability (or nonavailability) of affordances create selection pressure on the behavior of individual organisms; hence, behavior is regulated with respect to the affordances of the environment for a given animal. (Reed, 1996, p. 18) The resources encountered by an animal are the affordances of the environment. (…) Resources -or, to be more precise, special combinations of resources commonly found in certain habitats -become affordances when natural selection works to evolve such specific regulatory activity. (ibid.) Moreover, Reed emphasizes that "affordances exist independently of organisms" (ibid., pp. 26-27), in contradistinction to Şahin et al.'s own characterization of the environmental perspective on affordances that relates to their perceivability by agents (including organisms).

Consider how Reed's resource view of affordances can be conceptualized with recourse to a theory of core affordances. Since Reedian affordances do not depend existentially on individual organisms, their conceptualization would require the usage of general core affordances (which are intrinsic dispositions). The notion of resource has been utilized in formal ontology, e.g. in the Resource-Event-Agent (REA) ontology in which business processes are modeled upon the exchange of resources between agents [START_REF] Hruby | Model-Driven Design Using Business Patterns[END_REF][START_REF] Laurier | REA 2 : A unified formalisation of the Resource-Event-Agent ontology[END_REF] as well as in the manufacturing domain [START_REF] Sanfilippo | Resources in manufacturing[END_REF]. The general nature of resources remains nebulous, however. Although resources (e.g. minerals) are usually taken to be something to be used to meet a particular purpose (e.g. for good health), resources in Reed's sense of the term could be identified with dispositions, or more specifically general core affordances. Since resources in the environment "become" affordances when they pertain to natural selection, one possible textual definition of Reedian affordances may be given as follows:

Reedian affordance =def. A general core affordance that creates selection pressure on the behavior of organisms. Note that all resources in Reed's sense of the term may be general core affordances, yet they "become" Reedian affordances only when they create such selection pressure. It is also interesting to remark that Reed's resource-based account of affordances may lack practical utility owing to its appeal to evolutionary theory, especially when it comes to the modeling of affordances of artifacts [START_REF] Cosentino | Artifacts and affordances[END_REF]. To take the CLIMB example, the climbability affordance a0 of step0 would not be Reedian because step0 is not a product of natural selection.

Agent perspective

Stoffregen's "affordances as emergent properties"

In contradistinction with the environmental perspective, the agent perspective takes affordances to be something that resides "within the agent interacting in the environment through his [sic] own behaviors" (Şahin et al., 2007, p. 457). To take one of Şahin et al.'s (2007) examples, [START_REF] Stoffregen | Affordances as properties of the animal-environment system[END_REF] argues that affordances are emergent properties of the animal-environment system. In more detail, an affordance is a property that is "defined" by the relation between a property of the animal and a property of the environment such that this affordance is borne by the animal-environment system, but neither by the animal nor by the environment.

Stoffregen's account of affordances may admit of multiple interpretations, mainly owing to the meaning of the "defining" of a property by some relation. Seen from the standpoint of core affordances, however, a "Stoffregenian affordance" could be formalized as an affordance/effectivity complex (such as ae0 in the CLIMB example; see Section 3.3.1). For it is "defined" in some sense by the relationship between a property (i.e. a specific core affordance such as a0) of (an object in) the environment and a property (i.e. a specific core effectivity such as e0) of an agent, and it inheres in the agent-environment system, but not in the environment or in the agent.

Chemero's "affordances as relations"

As for Şahin et al.'s other example of the agent perspective, [START_REF] Chemero | An outline of a theory of affordances[END_REF] thinks of affordances as relations between the abilities of animals and features of the environment. This relational account of affordances has been widely employed, as in [START_REF] Fiebich | Perceiving affordances and social cognition[END_REF] analysis of the role of social cognition for the perception of ecological affordances. According to his semi-formal characterization of affordances:

Affordances (…) are relations between the abilities of organisms and features of the environment. Affordances, that is, have this structure:

Affords-φ (feature, ability). (Chemero, 2003, p. 189) To borrow Chemero's illustration, a climbability affordance involved in the CLIMB scenario is a relation between the height of step0 and Mary's ability to raise her legs. Furthermore, Chemero argues for the ontological autonomy (which is not his term, though) of affordances in the following sense:

There will be affordances in which that feature [of the environment] takes part as long as some animal exists with the appropriate ability [of the animal]. This is the case even if that animal is nowhere in the vicinity of the situation that affords something to it. Affordances do not disappear when there is no local animal to perceive and take advantage of them. (Chemero, 2003, p. 193) Consider how Chemero's view of affordances as binary relations (CAF) can be formalized by means of a theory of core affordances. To characterize the first relatum of a "Chemeran (relational) affordance", namely what he calls "feature of the environment", general core affordances will be helpful because they are properties of (objects in) the environment and, unlike specific counterparts, exist independently of individual agents, in line with his emphasis on the ontological autonomy of affordances.

It would then be natural to analyze the ability of agents (which is the second relatum of a Chemeran affordance) in terms of a general core effectivity, which is an agentive counterpart of a general core affordance. This would imply that abilities are a subtype of dispositions. Nevertheless, Chemero objects to a dispositional understanding of abilities and proposes the view of abilities as what he calls "functional properties":

A better way to understand abilities [than as dispositions] is as functional properties of animals. Functional properties of animals depend on the individual animal's developmental history or the evolutionary history of the species. They are properties of the animal that came to play the role they do in the behavioral economy of the animal because, at some point in the past, they were helpful in helping the animal (or its ancestor) survive, reproduce, or flourish. (Chemero, 2003, pp. 189-190) Pace Chemero, however, abilities of agents can be construed dispositionally while retaining his understanding of them as "functional properties". Given an ontology of dispositions, abilities (synonym: "capacity", "capability") can be represented as a subtype of dispositions in formal ontology [START_REF] Miranda | An ontological analysis of capability modeling in defense enterprise architecture frameworks[END_REF][START_REF] Merrell | Mental capabilities[END_REF][START_REF] Merrell | Capabilities[END_REF], in spite of some philosophical challenges involved [START_REF] Vetter | Dispositional accounts of abilities[END_REF][START_REF] Vetter | Are abilities dispositions?[END_REF]. Moreover, functional properties in his sense of the term could also be dispositionally characterized, along with Spear et al.'s (2016) general account of functions as dispositions of the bearers with a specific kind of historical development (which has been criticized by [START_REF] Röhl | Why functions are not special dispositions: an improved classification of realizables for top-level ontologies[END_REF] and Jansen (2018), though).

In short, a Chemeran affordance can be analyzed as a relation between a general core affordance and a general core effectivity based on a theory of core affordances, hence the following axiom:

CAF(x,y) → GCA(x) ∧ GCE(y) A Chemeran affordance is a relation between a general core affordance and a general core effectivity. Further axioms can also be added to specify the relationality of Chemeran affordances with connection with general core affordances:

GCA(x) → ∃y CAF(x,y) If x is a general core affordance, then there is some y such that x bears the Chemeral affordance relation towards y.

GCE(y) → ∃x CAF(x,y) If y is a general core effectivity, then there is some x such that x bears the Chemeral affordance relation towards y.

Observer perspective

Finally, the observer perspective "is used when the interaction of an agent with the environment is observed by a third party" and "one must also have the capability of taking the observer perspective when perceiving affordances, at least for the agents of the same species as the observer" (Şahin et al., 2007, p. 457). Şahin et al. (2007) argue for the relevance of the observer perspective to affordance-based robot control. Although not being referred to in their argument, examples to illustrate the observer perspective may include [START_REF] Ortmann | Affordances as qualities[END_REF], 2012) extension of their ontology of observation to [START_REF] Turvey | Affordances and prospective control: An outline of the ontology[END_REF] dispositional account of affordances and its application to what [START_REF] Ortmann | An egocentric semantic reference system for affordances[END_REF] call an "egocentric semantic reference system for affordances"; as well as Beßler et al.'s (2020) definition of affordances as "descriptions of dispositions" (rather than as dispositions themselves) in ameliorating autonomous robot control [START_REF] Beßler | Foundations of the socio-physical model of activities (SOMA) for autonomous robotic agents[END_REF].

While a theory of core affordances aims at a foundation for an ontology of affordances, the observer perspective may rather pertain to epistemology of affordances, since it is about how the agentenvironment interaction is observed by a third agent outside it. It might be indeed worthwhile to consider the observer perspective in terms of core affordances, mainly because of its importance for the practical application of affordances, as in robotics. There are nonetheless so multiple possible ways of modeling the observer perspective depending on its more specific interpretations that their complete articulation is outside the purview of this paper. To take one example, the notion of observer can be analyzed in terms of roles (e.g. students), as is indicated by Şahin et al.'s own expression "observer role". However, the understanding of roles varies greatly from one upper ontology to another [START_REF] Toyoshima | Roles and their three facets: A foundational perspective[END_REF] and clarification of the observer perspective on affordances will require choosing a particular account of roles based on some additional foundational and domain-specific assumptions.

Application

This section discusses, from the viewpoint of core affordances, three entities that are closely intertwined with affordances: the environment (Section 5.1), image schemas (Section 5.2), and intentions (Section 5.3). Since these notions each merit careful investigation, the goal of this section is not to provide a full ontological analysis of them, but rather to demonstrate that the theory of core affordances developed above has the potential to serve as a solid basis for their ontologies. Note that an affordance-based approach (to appear below) to the environment, image schemas, and intentions is preliminarily considered by [START_REF] Toyoshima | Modeling affordances with dispositions[END_REF], Toyoshima &[START_REF] Toyoshima | Linking image schemas with affordances: An ontological approach[END_REF][START_REF] Toyoshima | Foundations for an ontology of belief, desire and intention[END_REF], respectively.

Environment

The notion of environment plays such a vital role in contemporary society that its ontological characterization will be beneficial to many domains [START_REF] Smith | Toward a realistic science of environments[END_REF][START_REF] Bennett | Foundations for an ontology of environment and habitat[END_REF]. For instance, the Environmental Ontology (EO; [START_REF] Buttigieg | The environment ontology in 2016: Bridging domains with increased scope, semantic density, and interoperation[END_REF] has been developed to satisfy the needs for environmental semantics in alignment with BFO and the Open Biomedical Ontologies (OBO) Foundry [START_REF] Smith | The OBO Foundry: Coordinated evolution of ontologies to support biomedical data integration[END_REF]: a collaborative project to coordinate ontologies to support biomedical data integration. Although the EO includes the term "environmental system" (ENVO_01000254), its current definition "A system which has the disposition to environ one or more material entities" leaves room for improvement partly owing to the "environing relation" therein under development. Other environment-related terms are also of growing importance in formal ontology and other domains. Examples of such terms include "niche" [START_REF] Smith | The Niche[END_REF][START_REF] Keet | Representations of the ecological niche[END_REF][START_REF] Corris | Defining the environment in organism-environment systems[END_REF], "habitat" [START_REF] Bennett | Foundations for an ontology of environment and habitat[END_REF], "Umwelt" [START_REF] Ortmann | Modelling Umwelten[END_REF], "place" [START_REF] Jordan | An affordance-based model of place in GIS[END_REF][START_REF] Scheider | Place reference systems: A constructive activity model of reference to places[END_REF], and "locality" [START_REF] Garbacz | Identity criteria for localities[END_REF][START_REF] Garbacz | Identity of historical localities in information systems[END_REF].

A theory of core affordances may help to understand better the nature of the environment. In fact, an affordance-based representation of places has been advanced in the field of Geographic Information Systems (GIS) [START_REF] Jordan | An affordance-based model of place in GIS[END_REF][START_REF] Scheider | Place reference systems: A constructive activity model of reference to places[END_REF] and an intimate connection between affordances and the Umwelt has been pointed out [START_REF] Ortmann | Modelling Umwelten[END_REF]. On a more practical side, it is argued that affordances can serve as a guiding heuristic for environmental policy [START_REF] Kaaronen | Affording sustainability: Adopting a theory of affordances as a guiding heuristic for environmental policy[END_REF].

In the current framework, the environment (ENV) can be approached in terms of specific core affordances, since they are inherently linked with the environment in the sense of emerging from the system constituted of the object and the agent. More specifically, it would be an entity that has as part a bearer of some specific core affordance, hence the following axiom:

ENV(x) → ∃y,z(SCA(y) ∧ INH(y,z) ∧ P(z,x)) The environment is an entity that has as part a bearer of a specific core affordance. In the CLIMB example, the environment would be something that has as part a bearer (namely, step0) of the climbability specific core affordance (namely, a0). One apparent difficulty with this view of the environment is that it is explanatorily uninformative, for the very idea of specific core affordance is understood only in the context of the animal-environment system. In this sense, the notion of environment (as with core affordances) may be at best "elucidated" with some additional explanations, axioms, and examples (see Section 3.5.2) and this issue is to be grappled with in the future.

Image schema

As was alluded to in Section 1.1, the notion of image schema (as well as affordances) provides a first foundation upon which agents and inanimate objects can be distinguished and identified [START_REF] Kutz | The mouse and the ball: Towards a cognitively-based and ontologically-grounded logic of agency[END_REF]. Usually attributed to [START_REF] Lakoff | Women, Fire, and Dangerous Things: What Categories Reveal about the Mind[END_REF] and [START_REF] Johnson | The Body in the Mind: The Bodily Basis of Meaning, Imagination, and Reason[END_REF], image schemas are nowadays used in a number of different domains, ranging from cognitive linguistics and developmental psychology to artificial intelligence and robotics [START_REF] Hedblom | Dynamic action selection using image schema-based reasoning for robots[END_REF]. They may be explained somewhat differently in different contexts, for example:

• "An image schema is a recurring, dynamic pattern of our perceptual interactions and motor programs that gives coherence and structure to our experience." (Johnson, 1987, p. xiv) • "an image schema is a condensed redescription of perceptual experience for the purpose of mapping spatial structure onto conceptual structure." (Oakley, 2010, p. 215) • "Image schemas are generally viewed as redescriptions of perceptual events, or even more broadly as generalizations over perceived similarities." (Mandler & Cánovas, 2014, p. 526) Nevertheless, image schemas can be generally characterized as mental patterns or "conceptual building blocks" that are extracted from the sensory and motile experiences. They are presumed to be learnt during early infancy [START_REF] Mandler | How to build a baby: II. Conceptual primitives[END_REF] and their complete understanding requires considering carefully prelinguistic conceptual development [START_REF] Mandler | On defining image schemas[END_REF]. They are also multimodal (just as embodied experiences are), as they are extracted from all sensorimotor inputs [START_REF] Hampe | Image schemas in cognitive linguistics: Introduction[END_REF]. Quite often, however, image schemas are practically associated with generic spatiotemporal relationships that are learnt from the repetitive interactions with the environment (and the objects therein), as is illustrated by Kuhn's (2007) simple explanation of image schemas as "patterns abstracting from spatio-temporal experiences" (ibid., p. 155).

One of the most intensively studied image schemas is CONTAINMENT: broadly, the recognition that objects can be inside other objects or (container-shaped) places [START_REF] Bennett | Corpus guided sense cluster analysis: A methodology for ontology development (with examples from the spatial domain)[END_REF][START_REF] Davis | Commonsense reasoning about containers using radically incomplete information[END_REF][START_REF] Hedblom | In, out and through: Formalising some dynamic aspects of the image schema Containment[END_REF]Pomarlan & Bateman, 2020). (Note that image schemas are conventionally written in upper case letters in the relevant literature.) CONTAINMENT is most basically defined as the relationship between an inside, an outside, and the border between them [START_REF] Johnson | The Body in the Mind: The Bodily Basis of Meaning, Imagination, and Reason[END_REF]. Seen dynamically, however, CONTAINMENT could be characterized in terms of more "fine-grained" image schemas such as INTO and OUT OF [START_REF] Mandler | On defining image schemas[END_REF]. Other paradigmatic examples of image schemas include SUPPORT (which denotes a relationship between two objects in which one provides support to the other), PATH (which represents movement of objects from one point to another), and LINK (an enforced connection between objects where one linked object reacts to the stimuli of the other).

However, image schemas are still ill-defined and there is an increasing demand for a formal approach that would provide their unifying conception devoid of terminological confusion (Hedblom et al., 2015a[START_REF] Hedblom | Choosing the right path: Image schema theory as a foundation for concept invention[END_REF]. Here it may be well worth examining possible (formal) connections between image schemas and affordances, as these two notions are the twin pillars on which general theorizing on agency is built [START_REF] Kutz | The mouse and the ball: Towards a cognitively-based and ontologically-grounded logic of agency[END_REF]). To take one example, Kuhn (2007) develops an image-schematic and algebraic account of affordances and it has been widely applied e.g. in Cunha et al.'s (2018) visual representation of concepts. More specifically, the main focus will be upon the kind of image schemas that pertain to affordances, which will be called "image schemas of the affordance-based type" or more simply "Type A image schemas" (where "A" stands for affordances).

To consider Type A image schemas, [START_REF] Galton | The formalities of affordance[END_REF] following concise statement will serve as a useful start point:

Examples of image schemas include CONTAINER and PATH: the link with affordances is obvious, since to be a container is precisely to afford containment, while to be a path is to afford passage. Thus at least in many cases image schemas may be characterized in terms of the affordances of actual exemplars of those schemas. (Galton, 2010, p. 1) Seen from the viewpoint of core affordances, Type A image schemas would be better analyzed in terms of general core affordances than specific core affordances; for, image schemas are "general" in the sense of being extracted from individual sensorimotor experience, as Kuhn (2007) argues for the following thesis as an "ontological property" of image schemas: "Image schemas generalize over concepts (e.g., the CONTAINMENT schema abstracts container behavior from concepts like cups, boxes, or rooms)" (ibid., p. 156).

To illustrate this point with CONTAINMENT (as it is one of the most typical image schemas), take for example the "containment affordance", taking a cue from [START_REF] Galton | The formalities of affordance[END_REF] foregoing remark. Consider now the CONTAINMENT image schema of the Type A, especially John's CONTAINMENT image schema of the Type A (which will be termed "CONJohn"). Assuming that a gap is a kind of container, several affordances and effectivities can be considered to explicate CONJohn: first, the specific core affordance a1 of a gap (say gap1) to contain a person John and its reciprocal, specific core effectivity e1 of a person John to be contained in gap1; second, the general core affordance a1′ of gap1 to contain any instance (e.g. John) of a general class Agent1 (namely, the class of agents with the appropriate dimensions to be contained in gap1) and the general core effectivity e1′ of John to be contained in any instance (e.g. gap1) of a general class Gap1 (namely, the class of gaps with the appropriate dimensions to contain John). Since CONJohn is a Type A image schema, it is general and should be construed vis-à-vis a1′ and e1′, rather than vis-à-vis a1 and e1.

However, there is a second dimension according to which general core affordances and effectivities that relate to Type A image schemas can be further generalized. Indeed, a1′ is an instance of the class A1′ whose instances are general core affordances of gaps (not necessarily gap1) to contain any instance of agent that can fit in. Similarly, e1′ is an instance of the class E1′ whose instances are general core effectivities of agents (not necessarily John) to be contained in gaps of appropriate dimensions. Then, CONJohn would be formed not only from John's experience (perceptual or cognitive) with the gaps where he could fit, but also from his recognition of the class Gap and of the agents that could fit in each particular gap. To generalize, the image schema CONTAINMENT of the Type A would be construed in terms of such classes of general core affordances and effectivities as A1′ and E1′.

Even this view would be nonetheless too restrictive to accommodate the general notion of image schema. Indeed, the image schema CONTAINMENT may well be characterizable with the containment of material objects (not only agents) of appropriate dimensions, for it is reasonable to think that CONJohn is formed from John's recognition of the class of gaps and the class of all kinds of material objects that could fit in each of those gaps. Since many material objects are not agents, but affordances and effectivities always involve some agent, such containment features exceed the framework of affordances or effectivities. That is, Type A image schema might have to be construed in terms of classes of dispositions that are neither affordances nor effectivities.

In addition, Type A image schemas might be plausibly taken to be about (synonym: "refer to") some entities. This is because image schemas can be generally seen as "redescriptions" of perceptual experience, as is illustrated by Oakley's (2010, p. 215) and Mandler & Cánovas's (2014, p. 526) explanations cited above, and the notion of description has a referential dimension. Building on what has been just said, some Type A images schemas might be about affordances and effectivities. In particular, CONJohn would typically be about A1′ and E1′. Following the remarks above, though, some Type A images schemas might be about (classes of) dispositions that are neither affordances nor effectivities. Additionally, some Type A image schemas, such as THING, might be about non-dispositional entities [START_REF] Toyoshima | Linking image schemas with affordances: An ontological approach[END_REF]. Finally, it must be emphasized that the aboutness of Type A image schemas does not need to be generalized into the aboutness of the traditional notion of image schema. Indeed, image schemas can be minimally characterized as abstract patterns emanating from repeated experiences with the external world and aboutness may not be a key feature of all image schemas.

Nonetheless, analysis of Type A image schemas should be helpful in developing a successful unifying theory of image schemas. Even though it is presently restricted to Type A image schemas, the core affordance-based approach may suggest a novel perspective on the nature of image schemas. For instance, one of [START_REF] Tseng | Exploring image schemas as a critical concept: Toward a critical-cognitive linguistic account of imageschematic interactions[END_REF] features of image schemas is that they are static and dynamic, although that may sound contradictory: "Image schemas can be experienced as states of being or as a process. For example, the PATH schema can be experienced in a dynamic way -the process of moving from one place to another. Or it can be realized 'as a static thing', the road, track or passage that has been traversed" (ibid., p. 143) (see also Hedblom et al.'s (2015b) ontological structuring of PATH). Similarly, Kuhn (2007) contends: "[image schemas] are internally structured (e.g., the CONTAINMENT schema involves behavior associated with an inside, an outside, a contained entity, and possibly a boundary)" (ibid., p. 156) (see also Hedblom et al.'s (2018) formalization of dynamism involved in CONTAINMENT).

As far as Type A image schemas are concerned, their paradoxical nature may be (at least partially) explained by the claim that they are about general core affordances and effectivities, or more generally about dispositions. As a matter of fact, dispositions have both static and dynamic dimensions in the sense that: "dispositions connect the static structure of the world, i.e. the natural kinds of continuants, with the dynamical structure, i.e. the types of possible and actual causal processes" (Röhl & Jansen, 2011, p. 3, italicization added). For instance, the PATH image schema of the Type A can be about the class of general core affordances (i.e. dispositions) inhering in (static) material pathways that can be realized by an agent (dynamically) moving along such pathways.

There are still some challenges to be met in order for this core affordance-based analysis of Type A image schemas to be more meticulous. On the formal side, because the present approach to image schemas says that many Type A image schemas would be about classes of secondary core affordances and effectivities, a first-order representation of image schemas would require treating classes as firstorder entities, which goes beyond the scope of this paper. This line of inquiry is to be pursued together with careful consideration of existing formalizations of image schema [START_REF] Hedblom | Between contact and support: Introducing a logic for image schemas and directed movement[END_REF][START_REF] Hedblom | In, out and through: Formalising some dynamic aspects of the image schema Containment[END_REF]Pomarlan & Bateman, 2020) and of the vexed problem of how to represent type-level entities in ontologies [START_REF] Barlatier | A type-theoretical approach for ontologies: The case of roles[END_REF]Schutz et al., 2014;[START_REF] Brodaric | Particular types and particular dependence[END_REF].

On the conceptual side, the elusive notion of aboutness needs to be properly conceptualized and formalized, notwithstanding some previous works in philosophy [START_REF] Yablo | Aboutness[END_REF][START_REF] Hawke | Theories of aboutness[END_REF] and in formal ontology [START_REF] Ceusters | Aboutness: Towards foundations for the Information Artifact Ontology[END_REF][START_REF] Biccheri | Needs and intentionality -An ontological analysis and an application to public services[END_REF]; see also [START_REF] Sanfilippo | Ontologies for information entities: State of the art and open challenges[END_REF] brief discussion on aboutness of information entities). Here it is interesting to note growing awareness of the relevance of [START_REF] Peirce | The Essential Peirce[END_REF] semiotics to formal ontology [START_REF] Sowa | Signs and reality[END_REF], especially in respect of closer examination of aboutness [START_REF] Bateman | Ontology, language, meaning: Semiotic steps beyond the information artifact[END_REF]. As a matter of fact, Ortmann & Kuhn (2012) outline a semiotic account of affordances while highlighting the commonalities between Peirce's theory of semiotics and [START_REF] Gibson | The Ecological Approach to Visual Perception[END_REF] theory of affordances. Semiotic considerations may be thus expected to strengthen an aboutness-based linkage between core affordances and Type A image schemas.

Intention

One of the traditional approaches to cognition and actions is the BDI (Belief-Desire-Intention) model of agency [START_REF] Wooldridge | Reasoning about Rational Agents[END_REF]. Inspired by [START_REF] Bratman | Intention, Plans, and Practical Reason[END_REF] philosophy, the BDI model recognizes the primacy of the BDI entities in practical reasoning and rational actions. It has been utilized in formal ontology of mind [START_REF] Ferrario | Towards a computational ontology of mind[END_REF]) and actions [START_REF] Trypuz | Formal ontology of action: A unifying approach[END_REF]. Relatedly, the notion of goal and related entities (e.g. trying, success, and failure) have been formally investigated, as it plays a vital role in the BDI model [START_REF] Hobbs | Goals in a formal theory of commonsense psychology[END_REF]). An important motivation behind the BDI model is that belief, desire, and intention (which will be called "BDI entities") are valuable for examining cognition and actions, at least in commonsense psychology [START_REF] Kashima | The category of the mind: Folk psychology of belief, desire, and intention[END_REF]. It will be therefore promising to couple a theory of (core) affordances with an ontology of the BDI entities. For instance, [START_REF] Joo | Perception and BDI reasoning based agent model for human behavior simulation in complex system[END_REF] develops a human agent model for behavior simulation in a human-environment complex system by combining the BDI model with [START_REF] Turvey | Affordances and prospective control: An outline of the ontology[END_REF] theory of affordances (and effectivities). This line of inquiry nonetheless remains largely unexplored.

Here it will be a useful starting point to refer to Heras-Escribano's (2019) sketch of an ecological approach to agency. He argues that [START_REF] Reed | An outline of a theory of action systems[END_REF][START_REF] Reed | Encountering the World: Toward an Ecological Psychology[END_REF] theory of action systems can provide a basis for such a framework. The central idea of this theory is that actions stem from an "action system" that is constituted by the agent and the environment, and intentions are "shaped" by this action system in the sense that a particular action is understood as motivated or intentional with respect to some affordance emerging from the action system. This Reedian understanding of agency would indicate that, so as to integrate core affordances and the BDI entities, it will be necessary to consider how specific core affordances and effectivities (emerging from the agent-environment system) can be linked with intentions and (intentional) actions. To take some first steps along this line, it will be helpful to avail of Toyoshima et al.'s (2020) ontological analysis of intentions as mental dispositions that can be realized in actions: for example, Mary's intention to climb up step0 is a disposition to be realized in a process of Mary climbing up step0 (see also their discussion on triggers of dispositional intentions). The term "mental disposition" roughly refers to a disposition whose bearer is an agent (e.g. Mary) and whose categorical basis is some mental structure of the bearer (e.g. that part of Mary's cerebral structure which is relevant to her mental functioning).

There are many different ways of considering the relationship between intentions and specific core affordances or effectivities, depending on which specific theory of intentions and (intentional) actions is endorsed. (For the sake of readability, the terms "affordance" and "effectivity" refer to a specific core affordance and effectivity, respectively, until the end of this section.) Discussed below is the representation of the connection between intentions (INT) and effectivities (SCE), for it will be a central topic in this direction of research how to specify the link between those two kinds of dispositions while they both inhere in agents. One of the simplest observations is arguably that the realization of an agent's intention is also the realization of some effectivity inhering in the agent. Using the CLIMB example, the realization of Mary's intention to climb up step0 is the process of Mary's climbing and this process is also the realization of Mary's effectivity e0 (and so is the realization of its reciprocal affordance a0 of step0 to move Mary upward). This claim can be generally axiomatized as follows.

INT(x) ∧ INH(x,y) ∧ REAL(x,z) → ∃w(SCE(w) ∧ INH(w,y) ∧ REAL(w,z)) The realization of an intention is the realization of some specific core effectivity inhering in the bearer of the intention. (If x is an intention, x inheres in y, and x is realized in z, then there is some w such that w is a specific core effectivity, w inheres in y, and w is realized in z.) The next steps along this line of exploration should include meticulous analysis of many relevant notions, such as actions [START_REF] Trypuz | Formal ontology of action: A unifying approach[END_REF] and the mental objects of intentions [START_REF] Biccheri | Needs and intentionality -An ontological analysis and an application to public services[END_REF].

Conclusion and future work

Going back to [START_REF] Galton | The formalities of affordance[END_REF] three distinct goals that a full-fledged account of affordances must attain (see Section 1.2), this paper focused mainly on the middle-level goal of how to characterize affordances accurately. To answer this ontological question, a theory of core affordances was developed based on [START_REF] Turvey | Affordances and prospective control: An outline of the ontology[END_REF] dispositional approach to affordances. As a result, two kinds of core affordances were proposed. On the one hand, specific core affordances are individual-directed and extrinsic dispositions that are reciprocal with counterpart effectivities and that emerge from the agent-environment system. On the other hand, general core affordances are kind-directed and intrinsic dispositions. The article then examined whether and how this theory of core affordances can be extended to Şahin et al.'s (2007) three perspectives on affordances (viz. environmental, agent, and observer). Finally, so as to undertake the upper-level goal to consider the role of affordances in the realm of agents and the environment, it was outlined how the theory of core affordances can be applied to the analysis of three entities that are closely related to this goal: the environment, image schemas, and intentions.

There are many future directions of research in which the present theory of core affordances will be able to proceed. On the theoretical side (having to do with Galton's (2010) three goals), the middle-level goal is to be further explored. For instance, the temporal dimension of affordances [START_REF] Galton | The formalities of affordance[END_REF] will need to be taken into account, as core affordances are something to be realized in actions and actions are usually reckoned to be temporally extended. To take another example, [START_REF] Steedman | Formalizing affordance[END_REF] formalizes affordances based on linear dynamic event calculus, i.e. "a formalism for reasoning about causal relations over events" (ibid., p. 834), where his term "event" would be synonymous with the term "process" used in this paper. It will warrant scrutiny how this first-order theory of core affordances can harmonize with his linear logical formulation of affordances. Pointers to this line of inquiry may include [START_REF] Borgo | A first-order formalization of event, object, process, and role in YAMATO[END_REF] illustrative usage of linear logic [START_REF] Girard | Linear logic[END_REF] in ontologies and the preceding discussion on the relationship between dispositions and causation (Toyoshima, 2020a;[START_REF] Toyoshima | Towards a unified dispositional framework for realizable entities[END_REF].

In addition, the upper-level goal will require future investigation as well (see, for instance, Toyoshima & [START_REF] Toyoshima | Linking image schemas with affordances: An ontological approach[END_REF] detailed discussion on the core affordance-based approach to Type A image schemas as sketched in Section 5.2).

Lastly, the lower-level goal (addressing the aetiological question of where affordances come from) will need to be within the scope of this theory of core affordances. A key to addressing this question would be a closer examination of the notion of categorical basis of dispositions, because a core affordance (which is a disposition) is "causally based" on some qualities of the affordance bearer.

On the practical side, an affordance-based study of learning and creativity is well worth pursuing, as there is a growing demand for such an approach e.g. in communication technology [START_REF] Nagy | Imagined affordance: Reconstructing a keyword for communication theory[END_REF][START_REF] Tenenboim-Weinblatt | Temporal affordances in the news[END_REF]. In this direction, the present theory of core affordances will require further refinement in order to accommodate people's (especially children's) exploratory behaviors to discover "non-canonical affordances" [START_REF] Glăveanu | What can be done with an egg? Creativity, material objects, and the theory of affordances[END_REF][START_REF] Withagen | An ecological approach to creativity in making[END_REF]. This work may contribute to the development of mechanical systems (e.g. softwares) guiding humans into cognitively enhanced and imaginative acts [START_REF] Asprino | An ontology design pattern for supporting behaviour arbitration in cognitive agents[END_REF]). To take another example, given the importance of affordances to agency and actions, this theory of core affordances can be expected to facilitate an ontology-based approach to the integration of diverse theories in behavioral science [START_REF] Hastings | Theory and ontology in behavioural science[END_REF] and also to behavior change interventions, as they are key elements of health policy and practice [START_REF] Michie | The Human Behaviour-Change Project: Harnessing the power of artificial intelligence and machine learning for evidence synthesis and interpretation[END_REF]. Finally, one can mention Barton et al.'s (2020) formalization of Baldoni et al.'s (2006) view of "roles as affordances" by means of specific core affordances (and effectivities), the possible application of core affordances to the analysis of functions and artifacts [START_REF] Cosentino | Artifacts and affordances[END_REF], and how this theory of affordance can be leveraged to account for the nature of nudges, namely arrangements of the environment that are designed to influence people's behavior in order to contribute to people's welfare [START_REF] Barton | Foundations for an ontology of nudges[END_REF].

Figure 1 .

 1 Figure 1. Taxonomy of entities and associated unary predicates.

  DP-BEA) [DP(d,d') ∧ INH(d,x) ∧ INH(d',y)] → P(x,y) If d is a disposition-part of d', d inheres in x, and d' inheres in y, then x is part of y.

Figure 2 .

 2 Figure 2. A model of the theory.

Table 1 .

 1 Binary relations

	Binary relation	Predicate	Domain	Range
	has realization	REAL	DIS	PRO
	has trigger	TRIG	DIS	PRO
	reciprocal of	RECI	IDD	IDD
	inheres in	INH	QUA or DIS	IND
	proper part of	PP	IND	IND
	overlaps	O	IND	IND
	has categorical basis	CBA	DIS	QUA
	has internal basis	IBA	DIS	QUA
	has external basis	EBA	DIS	QUA
	add-part of	ADP	DIS	DIS
	All the axioms reflecting the taxonomic structure in Figure

Corresponding author. E-mail: toyo.fumming@gmail.com.

Acknowledgements

We thank Maria M. Hedblom for her valuable comments on our discussion as to image schemas. FT acknowledges financial support by the SPOR Canadian Data Platform (CIHR).