Towards a teleo-semiotic theory of individuation

1.1 The ontological square

In order to gain some initial clarity and organization to facilitate our progress, we will invoke and assume axiomatically the Aristotelian so-called “ontological square” as a guide to the most basic categories of what is. The ontological square, proposed and discussed recently, by E. J. Lowe (2006), Luc Schneider (2009), and Katherine Munn and Barry Smith (2008), among others, consists of four basic divisions or categories. Fixing our terminology:

1. objects (e. g., Nelson’s column) individual beings – Latin, ens;

2. modes (e. g., the greyness of Nelson’s column), the particular properties that characterize particular objects;

3. kinds (e. g., statues), the sorts of objects;

4. attributes (e. g., grey), the sorts of modes

Certain formal ties bind the categories together: kinds and attributes are instantiated by objects and modes respectively; whereas modes characterize objects (Nelson’s column is grey), and attributes characterize kinds (statues are colored).

Following E. J. Lowe’s (2006) nomenclature, we will reserve the word “individual” to refer to the first two of these four categories, i. e., for objects and modes, and the word “sortal” to refer to the last two, i. e., for kinds and attributes. Our task is to unearth semiotic individuation conditions for things of each of the four categories here enumerated. Part of the semiotic account will have to be a demonstration of the grounds by which it is warranted to propose that there are these four, and just these four, ontological categories, which will, of course, amount to an account of the individuation conditions of these four categories themselves.

1.2 “The ‘one’ is not”

One further controversial presupposition should be made explicit at the outset: Alain Badiou’s dictum that “the ‘one’ is not” (Badiou 2001: 25). This is here taken to mean that being as such, indeterminate being, to use Hegel’s vocabulary, is not differentiated into distinct or individuated entities, but is rather undifferentiated – a “pure multiplicity,” as Badiou calls it. Only a subsequent operation, an operation the basic procedures of which we shall here attempt to sketch out, renders being determinate, that is, distinguishes or individuates one being from another. This presupposition is taken as it is presented by Badiou: as a necessary corollary of the removal of the creative power of God from the conceptuality of the philosophical enterprise. Invoking Plato’s metaphor in the Phaedrus (Plato 2003: 265e), metaphysical realists such as Theodore Sider (e. g., 2011) frequently talk of basic objects, or, in any case, “the fundamental furniture of the universe,” as individuated by nature’s “joints.” Philosophically, this will not do. In the first place, this is, of course, merely a metaphor and cannot pass as an adequate foundational explanation without substantial cashing out. And how such cashing out could be effected is, at least to this author, profoundly unclear. In the second place, it begs the question: what individuates the joints? Why are certain bits of nature – its joints – individuated in the first place, subsequently to individuate objects? Simply to call them fundamental is to re-state rather than answer the question. Thirdly, even if it does make sense to talk of nature’s joints, why should these joints be taken to have any bearing on what there is, or any bearing on our truth-claims about what there is? Why such a grounding relation? This must be explained rather than presupposed. And, after all, what does this metaphor of “joints” reveal about our understanding of individuation and objecthood? That it is based upon the possibility of relative motion? But why should we accept that sort of account of objecthood without justification? Badiou’s thought is that, without recourse to a concept of God, there can be no philosophical justification for the claim that nature has any joints. If there is no God, there is no designer: there are no joints that can claim ultimate reality.

2.1 Projection

We start with the formal structure of teleology. For my purposes here, I take this to be, in general, a basic, unqualified, unidirectional, asymmetric relationship of projection (in the sense of its etymological origin as “thrown forth”). It can be conceived as a simple negation, and thus as a pure differential opposition. And can thus be schematized as per Figure 1, where, using some of the vocabulary of category theory, A is the source and B is the target. We will refer to this basic relation of projection as the teleological functor. In laymen’s terms, it would be the most general structure of desire, where this should be taken to include the whole variety of teleo-psychological phenomena, some of which we mentioned earlier.

Figure 1:

Teleological functor.

On the frame of this original relationship, two further relationships are founded, based on the possibility of composition: the useful or means, and the useless, or non-means. These can be schematized around the teleological functor as per Figure 2, where C is a useful means that commutes the teleological functor such that

where this reads: h after g is a realization of tf; and D is a useless non-means that fails to commute it.

Figure 2:

Teleological syntagm.

Call this four-way distribution of determinacy a teleological syntagm. In general, a teleological syntagm comprises nodes and arrows (the latter called “morphisms” by category theorists) arranged around an axial teleological functor. The nodes are determinate, but only insofar as they stand in relationships to other nodes. Their determinacy is just the sum of these morphic relationships.

2.2 Polysemy

It can then be demonstrated that any given node is polysemic, since it can support further multiple relationships. These are the result of the composibility and overlapping of distinct teleological syntagms.

2.2.1 Vertical polysemy: Composibility

Within any given node of a teleological syntagm, there may be further functions, as per Figure 3, where a function, i, is enmeshed within C. Category theory uses the word composibility to describe this structure. Further, the target, B, can itself be composed in some further syntagm insofar as it is itself a means to some further target, as per Figure 4.

Figure 3:

Composibility 1.

Figure 4:

Composibility 2.

In principle, such composibility or “enmeshment” is infinitely extendable in both directions. This is what we can call the vertical dimension of polysemy: in general, a node occupying a position within a syntagm is determined by other syntagms in which its syntagm is composed and/or by other syntagms which are composed in its syntagm.

2.2.2 Horizontal polysemy: Overlapping

On the other hand, polysemy has a horizontal dimension. Horizontal polysemy is due to the overlapping of teleological syntagms. A given object or node can be the target of more than one teleological functor (Figure 5).

Figure 5:

Overlapping 1.

And a given object or node can be the means of more than one syntagm (Figure 6).

Figure 6:

Overlapping 2.

In both cases, the determination of the means and target is a function of the sum of their relationships. Again, in principle, the overlapping of nodes of syntagms is unrestricted. A given node may function within any amount of overlapped syntagms.

3.1 The individuation of individuals

The first two theses to be proposed here are that:

1. Objects are to be defined as the functional nodal invariances of concrete, polysemic, teleological syntagms. Sources, targets, means and non-means in concrete syntagms are all objects. The criterion of objectual identity that would accompany this account of individuation would therefore be the following: objects A and B are identical if, and only if, A and B bear the same teleological syntagmatic relationships. ^[7]

2. Modes are to be defined as the functional morphisms conformed between the nodes of concrete, polysemic teleological syntagms. The corresponding criterion of modal identity would be: modes f and g are identical if, and only if, f and g form the same relationship between the same nodes in the same teleological syntagms.

On our proposal, then, what accounts for the individuation of objects – such as the Atlantic Ocean, for example – and their modes, is originally their functional roles in teleological syntagms. ^[8] The well-known semiotic tool of commutation testing is needed to illuminate the operation.

The commutation test has generally been conceived as a tool for analyzing the units, classes and structures of systems – in the semiotic case, of significative systems. Systematic substitution, transposition, addition and deletion of the matter of the system – phonemes, for example – enable one to identify and classify the units significant to the functioning of the system. Given a primitive or originary teleological functor, and a field of indeterminacy, sources, targets, means and non-means are individuated as commutable, that is, as significant elements, unities, individuals, and thus constitute a functional syntagm. They are distinguished or determined – unified – from the indeterminacy of “pure multiplicity” by the commutable relevance they bear to the teleological functor projected. By this originary operation, they are individuated.

To take our previous example of the Atlantic Ocean: from the indeterminacy of the regions of pure multiplicity in general, it is individuated as a watery expanse to be explored, as a means of obtaining food, as a target for sailors, as something to be mapped for geographers, as an obstacle to getting to the New World, as a desperate watery desert, etc. Its modes are correspondingly individuated as the relations that determine it as a means, target, etc. So it is abundant in food, a challenge to rowers and sailors, lying in between Eurasia and Africa to the east and the Americas to the west, too large, inhospitable, wet and cold, etc. The set of properties of the ocean are the set of these morphic relations within the relevant concrete teleological syntagms. The objecthood and modes of the Atlantic Ocean are subsequently fixed and secured by language and convention. ^[9]

For those who like formal languages, the characterization of an object by a mode can be expressed approximately, using the notation of first order logic, as a straightforward disjunction – a disjunction of two conjunctions:

For all x belonging to the syntagm S, to say that x has property or mode G is to say that there exists a y such that, if x has G, then y, and, if x does not have G, then not y, or, if x has G, then not y, and, if x does not have G, then y. ^[10]

I say “approximately” since this disjunction would not be sufficient were x and y not related as terms of a teleological syntagm, but first-order logic has, of course, no proper resources to express this. I have tried to articulate the restriction by importing the set-theoretic notion of belonging (in this case to a syntagm).

On this account there can be no “bare particulars,” contra some commentators’ construal of Locke (cf. Locke 1979: Bk. II, Ch. xxiii, § 2), that is, objects without modes, just as there can be no “free-floating” or unattached modes.

3.2 The individuation of sortals

In order to incorporate sortal concepts into our model, whether they be kinds, which find their instantiation in objects, or attributes, which find their instantiation in modes, we invoke once more the much-used semiotic tool of commutation.

Applied to a teleological syntagm, such that the nodes and morphisms constituting the scheme are variously but systematically substituted, re-ordered, deleted and added, commutation classes can be constructed of both objects and modes. Similarity of commutation is ultimately a guide to similarity of function within a given scheme. Sortals are sets of commutation functions:

1. The commutation classes of objects are called kinds. Take teleological functor AB and means C. If D can be substituted for C without affecting the functionality of the scheme, then C and D – with respect to teleological functor AB – belong to the same commutation class. They are, in this respect, of the same kind.

2. The commutation classes of modes are called attributes. Take teleological functor AB and means C, where f is the mode that makes C a means to B. If D can be substituted for C because it has mode g, then modes f and g belong to the same commutation class with respect to teleological functor AB. They are, in this respect, of the same attribute.

The construction of two commutation classes out of the two elements of teleological syntagms – morphisms and nodes – grounds the appropriateness of the four category ontology of the ontological square. In this case, we have provided an account of the individuation of just these four categories.

4.1 Persistence and change

It may be objected to this account that if identity is tied so closely to teleo-semiotic function, then any change in the syntagm will result in new individuals altogether: any change in the morphic relationships will produce identity changes in the nodes constituting the syntagm. Thus any change in qualitative identity would result in a change in numerical identity. This would be a particularly counterintuitive result. It would mean, for example, that my desk could not get a scratch on it. For, given that “scratched” is a mode, being scratched would bring with it a new desk entirely. (Note that this is true of objects but not of modes: modes cannot change their properties without being replaced by different modes, for their properties are their numerical identity. The color of my desk cannot change color without becoming a different color.)

In order to avoid this consequence, one needs an account of how an entity can change. Traditional – Aristotelian – accounts refer to a distinction between essential and accidental properties, ^[11] the basic idea being that an object can survive changes in its accidental properties, but not in its essential ones. But then one requires a non-question-begging and grounded account of this distinction. A currently popular alternative invokes perdurance antipodal to endurance, and argues that objects have distinct temporal parts. ^[12] But this four-dimensionalism, as it is known, is so unintuitive that one commentator thereof remarked: “I simply do not understand what [distinct temporal parts] are supposed to be, and I do not think this is my fault. I think that no one understands what they are supposed to be, though of course plenty of philosophers think they do” (Van Inwagen 1981). Roderick Chisholm (1969) offers a further account, invoking a distinction between “loose and popular” and “strict and philosophical” senses of identity. But this ends him with the unpalatable consequence that, in the “strict and philosophical” sense, an entity cannot change any of its modes without thereby becoming something numerically different. In any case, the model we have developed so far lends itself to another approach.

Objects are determined by the modes of indefinitely many teleological syntagms. Each of these bears on the object’s identity in a different way. Take a hat. It may be at least determined by syntagms oriented around a need to keep warm, those oriented around a desire to look sharp, and those oriented around a preference for a particular color. Change occurs when one or more syntagms become dysfunctional while others continue to determine the entity. So, if the hat gets a hole in it, it may no longer support the desire to look sharp, but its capacity to provide warmth and its colors remain much the same.

Now kinds have been construed as sets of syntagmatic commutation functions. On this account, with the collapse of a syntagm, an entity is changed in respect of its kind. If the hat’s brim falls off, for example, it ceases to belong to the kind brimmed hats. It nevertheless continues to belong to the kind hat. Such changes may accrue to a point at which we are no longer inclined to consider it a hat – perhaps it gets hopelessly torn or crushed. Nevertheless, it continues to exist subsumed under other kinds – spare felt, for example, or clutter.

An entity changes, then, when the set of syntagmatic relationships that determine it changes. Since each of these relationships bestows upon it membership of some kind or other, a change of an entity is always a change of its kind memberships. ^[13] Correlatively, when we say an object persists through change, we refer to the maintenance of some of its syntagmatic functionalities through collapses of other of its syntagmatic functionalities. A hat persists through the removal of its brim; a piece of felt persists through its manufacture into the shape of a hat, and the later destruction of that shape. ^[14]

As an account of the persistence or identity over time of everyday physical objects, this account differs from the standard contemporary approach that invokes, in one sense or other, spatiotemporal continuity (cf. note 2). Despite its popularity, however, the latter approach is unfit for purpose, for its explanans and its explanandum are at bottom the same: continuity through time is surely at least partly what is meant by identity over time. The semiotic approach offered here avoids the circularity.

4.2 Vagueness

No rigorous distinction between essential and accidental properties is drawn here. But what is bought for this price is the ability to incorporate into the account ontological vagueness, and to do so in a way alternative to epistemicism, super-/sub-valuationism, contextualism, the elaboration of many valued logics, or its reduction to linguistic vagueness. Mark Heller’s so-called “step argument” demonstrates the difficulties in the naive metaphysical realist approach to vagueness (Heller 1996). On the other hand, Timothy Williamson’s epistemicism is committed to the view that any apparent vagueness in the truth conditions of a given assertion is the result of ignorance and could properly be resolved only by epistemological improvement (even if such improvement is not possible in principle; Williamson 1994). The problem here is that a cursory analysis of natural languages reveals that the truth conditions of at any rate a great many assertions are established by socio-cultural conventions that are inherently imprecise. Take the conventions which have named different regions of the world’s oceans “Atlantic” and “Pacific.” No doubt there are regions of water concerning which it seems vague as to whether they are part of the Atlantic or part of the Pacific. However, a closer analysis or measurement of these regions cannot resolve the issue, because that relative to which an answer could be found is itself vague. The world cannot resolve for us vagueness in this case in principle. It makes very little sense, therefore, to characterize vagueness as purely an issue of epistemology. But this should not force us to the anti-realist dispositions of those who conceive vagueness as fundamentally a function of language.

On the contrary, our account is a sort of semiotic extension of Delia Graff’s “interest-relative” solution to sorites-type paradoxes: “the semantics of vague expressions renders the truth conditions of utterances containing them sensitive to our interests, with the result that vagueness in language has a traceable source in the vagueness of our interests” (Graff 2000). Doing without a truth-conditional semantics, we can locate the source of the vagueness of expressions within the effect that the vagueness of our interests has on the ontological schemes of the relevant syntagms. Suppose I order from a bar two Bloody Marys, one for me, and one for a friend. The bartender puts very little vodka and very little flavoring in the tomato juice, such that, when she takes a sip, my friend exclaims that it is not a Bloody Mary at all. I, on the other hand, liking my cocktails weak, declare that it is. The appropriateness of our respective claims is clearly relative to our interests, preferences, and desires. It is a Bloody Mary for me – for my taste – but not for my friend. As the bartender added vodka, etc., to the tomato juice, there came a point at which I was happy to declare the drink no longer a tomato juice but rather a Bloody Mary, whereas, for my friend, that point did not arrive – although presumably it would have done eventually had the bartender continued. The identity of being a Bloody Mary is thus vague, and determinate only relative to teleological structures, to a given set of interests, preferences, etc. ^[15]

4.3 Coincidence and composition

We have here also a response to the standard problematic of coincidence and composition: why is it that a statue can lose one of its gold atoms and still be the same statue, whereas the set of gold atoms that constitutes the statue at time t cannot lose any atoms without becoming thereby a different set? The statue can change in this particular way, but the set of atoms constituting it cannot. Conversely, if we melt the statue down, we lose the statue, but we retain the set of gold atoms. Here we have to do with two different objects, distinct both numerically and qualitatively, that have different persistence conditions because they support very different syntagmatic functions. They do happen to occupy the same space, of course, but spatiality is, on the present account, indeterminative of identity.

4.4 Wholes and parts: A mereology

Peter Van Inwagen (1987) asked a question that generated a lot of literature: how it is that when some objects are composed together, a new object emerges, and when other objects are composed together, or when the first objects are composed together in the “wrong way,” nothing more than an assemblage of objects results. So, for example, when a bunch of metal components are fitted together in an appropriate way, you get a bicycle, but when a bunch of scrap metal is thrown into a yard, there is nothing but bits of scrap metal. This raises the traditional problem of wholes and parts that so bothered Aristotle and Plato.

The account of individuation offered here may provide one solution to this issue: what would determine whether or not an assemblage constitutes a single object rather than merely an assemblage of objects is whether it supports some functional role within a teleological syntagm. So, for example, put together, in the right way, an assemblage of metal bits and a bicycle emerges, because a bicycle is a useful means of transport. Throw scrap metal into a yard and nothing especially functional results. Some new object could result, of course: an artwork, for example; but it could only be something of some teleo-semiotic value.

Similar considerations perhaps enable our model to account for the whole-part distinction in general. If a rock breaks into pieces, one gets fragments of rock, or, better, a bunch of new smaller rocks. One doesn’t get several parts of one rock. This must be the case given what we know about geology and erosion, for otherwise all rocks would be termed parts of rocks. Take apart a bicycle, on the other hand, and one gets a bunch of parts or components of a bicycle – at least if one does so carefully, respecting the boundaries of the partially autonomous structures that constitute it. Reference to the concept of teleological function could, again, explain this: whether or not an object can be taken to be a part of a larger whole would depend upon whether it contributes in some functionally individuated way to the functionality of the whole. Thus, a bicycle wheel can be taken to be part of a bicycle, whereas a rock fragment cannot be taken to be part of larger rock. The fact that the latter might not be true in the case of a stone sculpture or statue would only serve to confirm the thesis.

Just as any adequate mereology must, this account neatly preserves transitivity in relation to parts: any part of any part of a thing is also thereby a part of that thing.

A possible objection to this account of parts might argue that such functional roles cannot be necessary for parthood since some proper parts of some things do not have such roles. For example, the part of my birthday cake that I am going to offer to you has, prima facie, no functional role distinct from any other equally sized segment of the cake. Be that as it may, a functional role within some teleo-semiotic syntagm is introduced just insofar as I decide to offer you a certain segment of the cake, or insofar as you choose such a segment. The part of the cake that I offer you thus comes into existence as a distinct part of the cake just insofar as it becomes teleo-semiotically functional. Before my decision or your choice, it has no part-hood.

A further standard issue for mereological theories concerns explanation of talk of parts that do not appear to be spatially or temporally distinct. Is the vermouth part of my martini? Given that teleo-semiotic function is not dependent on spatial or temporal distinctness, such an example poses no special problem for my account: given that the vermouth adds a distinct flavor to the gin, it contributes in a functionally individuated way to the whole. It should therefore be considered a part of the cocktail.

It should be remarked that the account of vagueness offered above is intended to apply equally to our mereology: it is sometimes apparently vague as to whether something is a part of something – whether, to use a well-known example, a particular peripheral water molecule belongs to cloud. In this sort of case, again, vagueness would be the result of the relevant syntagms being conformed relative to interests.

4.5 The problem of universals

When confronted by the age-old problem of universals, the model presented here forges a middle way between the two traditional extremes of realism and anti-realism. When we predicate attributes to individuals – “The statue and the squirrel are grey” – are we invoking another, universal entity – greyness – besides the individuals? Or is there no additional referent here but merely a conventional way of talking?

The realist, traditionally, will postulate the existence of universals above and beyond the existence of particulars and their particular instantiations of attributes. ^[16] But such a postulation strikes many metaphysicians as queer: where are these universals? In a different realm or world altogether? Besides their instantiation in individuals, what are they like? What explains their origin and individuation? Don’t the pressures of conceptual economy press us hard to avoid admitting such entities if at all possible? And do they provide sufficient explanation in any case? The famous “third man argument” of Plato’s Parmenides asks what would explain why the universal “grey” is grey except a higher form, and so on (Plato 1996).

Problems such as these might motivate the turn to an anti-realism about universals. Here, the predicate is conceived merely as a referent-less linguistic or conceptual convention. The illusion of universal attributes arises because it is appropriate to use the same string of words or letters to describe different individuals. But then the problem is just pushed further back: what is it about the individuals – the statue and the squirrel, for example – that makes it appropriate to use the same particular conceptual or linguistic convention in regard to them? Without pointing to some mode common to both, this seems inexplicable, yet doing so is just to restate the issue once more, if it is not to collapse into realism. Attempting to escape the problem by referring the conventions to resemblance or similarity is again to restate rather than to solve it: what constitutes the resemblance?

The teleo-semiotic account of these properties provides an alternative to this polar opposition. It has no need to posit absolute or wholly objective universals. It cannot, therefore, be accused of queerness – at least in this sense. On the other hand, it does provide an extra-conceptual/linguistic account of the grounds of commonality between individuals. Thus it does provide a horizon for explanation, and therefore a potential solution to the problem, as opposed to a disguised restatement of it.

It is because they share a commutable teleo-semiotic function – that is, a mode that belongs to the same teleo-semiotic commutation class – that the statue and the squirrel can both appropriately be called “grey.” On this analysis, universals are to be construed as commutation classes of teleo-semiotic functions. They have no individuated reality apart from such syntagmatic functions; on the other hand, they give meaning and reference to the linguistic uses of predicates, and are thus not reducible to linguistic functions. Such a theory also offers an easy account of the notorious relation of “participation,” the relation between particulars and universals: Given the organization into commutation classes of the nodes and morphisms in terms of their respective teleo-semiotic functions, individuals are necessarily associated with sortals.

4.6 Necessity and sufficiency

To what extent are these conditions necessary or sufficient? I submit that they are sufficient: any bit of the world that takes on semiotic value as source, means, non-means or target precisely thereby is individuated. Further, the property or properties that in this case determine it as source, means, non-means or target are also precisely thereby individuated. Given that it is unrestricted as to what the teleological functor relates together, then, in virtue of these conditions, any segment or property of reality can be individuated. On the other hand, very strange objects or properties might be particularly transient. The socio-cultural question of how an object or a property becomes conventional, assigned a linguistic value, and thus ensured of its individuation for a linguistic community is one I will not attempt to broach here.

But I do not want to suggest that these conditions are necessary. I have explicitly exempted persons and organisms and correlative, particularly personal and organic properties from the analysis. But what about the constituents of the natural world – not oceans, but water molecules, hydrogen atoms, electrons, etc.? Can we claim that these are individuated only insofar as they receive a semiotic value? The model advocated here sanctions so-called instrumentalism about science. Humans evidence a desire for knowledge and explanation. The fundamental concepts of the respective natural sciences accord semiotic value to particles, compounds, laws and the like as means to such explanation and knowledge. The scientific endeavor is teleological: an attempt to furnish us with concepts sufficient for the explanation and prediction of the empirical world. Scientific experiments confirm our progress towards this goal and indicate where our concepts are inadequate, inutile or lacking altogether. The aim is then to improve our conceptual resources as means to the target. To make sense of this endeavor, we do not need to postulate anything individuated apart from it.