A new perspective on criteria of identity

Abstract

Are identity criteria grounding principles? The recent debate over this issue seems to indicate a definitively negative answer. Recognising various objections to identity criteria as grounds for identity facts, one may wonder whether the former are capable of playing any salient role in ontology. I argue in this paper that they are, provided that one interprets them not as ontological explanations of identity facts but as ontological specifications thereof. I attempt to elaborate this view in terms of the well-establish distinction between determinables and their determinates, where identity criteria are to play the role of the determinates and the relation of identity is to be their determinable.

1 Introduction

Identity fascinates and puzzles philosophers, particularly those fond of doing metaphysics with the help of formal logic. Following Quine (1969), some argued or stipulated that only entities with clear identity criteria are ontologically respectable or at least more respectable than those without. Others pointed out that the search for identity criteria is, more often than not, an exercise in futility—a few years ago Carrara and Giaretta (2004) provided an extended discussion of various, more or less specific, issues one may encounter in this pursuit.

A criterion of identity is usually understood as a principle that is supposed to account for the identity of a thing. Following (Williamson, 1990, pp. 144–153) two kinds of such principles are distinguished. A one-level criterion of identity is expressed as a sentence of the form:

$$\begin{aligned} \forall x, y [{\kappa }(x) \wedge {\kappa }(y) \rightarrow (\rho (x,y) \rightarrow x=y)] \end{aligned}$$

(1)

where ‘${\kappa }$’ denotes an (ontological) kind or sort of objects and ‘$\rho$’ stands for a certain relation between x and y and $\rho$ is usually taken to have either the form of R(x, y) or the form of R(f(x), f(y)) as those forms correspond, respectively to, one-level and two-level criteria of identity. For example, the axiom of extensionality may be seen as a one-level criterion of identity for sets:

$$\begin{aligned} \forall x, y [\texttt{Set}(x) \wedge \texttt{Set}(y) \rightarrow (\texttt{SameMembers}(x, y) \rightarrow x=y)] \end{aligned}$$

(2)

where ‘$\texttt{SameMembers}$’ is defined as below:

$$\begin{aligned} \forall x, y [\texttt{SameMembers}(x,y) \equiv \forall z (z \in x \equiv z \in y)] \end{aligned}$$

(3)

A two-level criterion of identity is expressed as a sentence of the form:

$$\begin{aligned} \forall x, y [{\kappa }(x) \wedge {\kappa }(y) \rightarrow (\rho (x,y) \rightarrow \delta (x)=\delta (y))] \end{aligned}$$

(4)

where ‘${\kappa }$’ and $\rho$ are understood as in the previous case and ‘$\delta$’ denotes a function from objects within one ontological level to objects from another level. The paradigmatic example of such a criterion is the following: the directions of lines are identical if the lines are parallel:

$$\begin{aligned} \forall x, y [\texttt{Line}(x) \wedge \texttt{Line}(y) \rightarrow (\texttt{Parallel}(x, y) \rightarrow \texttt{dir}(x)=\texttt{dir}(y))] \end{aligned}$$

(5)

Usually a criterion of identity, say relation $\rho$, is supposed to satisfy some additional constraints (Carrara & Gaio, 2012):

• IC1 formal properties of identity: $\rho$ is an equivalence relation;

• IC2 informativity: criterion $\rho$ (for objects of kind $\kappa$) should contribute to the nature of these objects;

• IC3 non-circularity: criterion $\rho$ (for objects of kind $\kappa$) cannot make use of the concept of $\kappa$;

• IC4 non-tautologicity: $\rho$ is a proper subset of $K \times K$ (where K is the set of all $\kappa$-objects);

• IC5 non-vacuousness: $\rho$ cannot have parts that are vacuously satisfiable, i.e., $\rho$ hold between elements of a certain kind such that all of them are alike with respect to the properties associated to that kind;

• IC6 partial-exclusivity: criterion $\rho$ (for objects of kind $\kappa$) cannot be so general that it can be applied to other kinds of objects;

• IC7 minimality: criterion $\rho$ (for objects of kind $\kappa$) specifies the smallest number of determinables, i.e., aspects, such that determinates that fall under them turn out to be necessary and sufficient to ensure the identity of two objects of kind $\kappa$

• IC8 $\kappa$-maximality: $\rho$ is the $\subseteq$-greatest relation that makes condition 1 above true;

• IC9 uniqueness: $\rho$ is the unique identity criterion for $\kappa$.¹

Incidentally, let me note that the phrase ‘criterion of identity’ may refer either to whatever $\rho$ refers to or to any sentence of the form given by 1—in this paper I am happy to follow this tradition and indulge myself in this ambiguity except when the clarity in this respect will be essential.²

Also it is often recognised now that such principles can play three different roles in philosophy. If we assume for a moment that a criterion of identity may, either directly or indirectly, answer questions of the form: “How can we know whether a is identical to b?”, then this question may be construed as:

• 1. an ontological question when we ask what it is for one object to be identical to another;

• 2. an epistemic question when we ask how can we know that one object is identical to another;

• 3. a semantic question when we ask whether one name refers to the same object as another. (Carrara & De Florio, 2020, p. 3152)

Although all three interpretations may raise various concerns, recently the ontological interpretation was the one that caught particular attention; or, I should say, an ontological interpretation of criteria of identity was put under scrutiny. The specific interpretation I have in mind construes identity criteria as ontological explanations or grounds for identity claims or facts. For instance, suppose that there exist sets known by the names of ‘ABC’ and ‘BAC’ (I neither assume that $ABC = BAC$ nor that $ABC \ne BAC$). According to this interpretation, if it is the case that $\texttt{SameMembers}(ABC, BAC)$, this fact explains or grounds the fact that $ABC=BAC$ or, simply put, $ABC=BAC$ because $\texttt{SameMembers}(ABC, BAC)$.

Section 2 below summarises, in chronological order, three major recent objections to this interpretation. Section 3 points to a different, although related, issue. Namely, discussing whether identity criteria ground identity facts or not, we may come across the problem of what kind of things identity criteria and identity facts really are. Since the aforementioned objection implicitly casts doubt on the ontological credentials of identity facts, I propose in this section to reconceptualise the problem and replace the ‘fact-focused discourse’ with the ‘relation-focused discourse’. This reconceptualisation is detailed in Sect. 4, where I outline an alternative ontological interpretation of identity criteria, which is immune to these objections, or at least I claim so. Section 6 shows how my interpretation may challenge some well-entrenched beliefs about identity and how this challenge may be met.

2 Identity criteria fail to ground

Fine argues in Fine (2016) that if a criterion of identity is to account for identity statements, i.e., if the former is to ground the latter, we need to construe it in generic, not general, terms. The rationale behind this claim stems from Fine’s attempt to make ontological sense of the fact that identity criteria are universally quantified formulas. He argues that the usual, perhaps implicit, interpretation of these criteria takes them as general principles that we can apply to particular cases by instantiating bounded variables in such formulas by constants. In his view this interpretation may in some cases undermine the very idea of identity criteria when we consider particular instances thereof, which may occur in specific ontological contexts. His arguments focus on the differences between generic formulas like 2 and their particular instances. These differences boil down to two types:

• There are instances of the generic identity criteria that do not provide a proper basis for the latter.

• There are some special instances of the generic identity criteria that do not hold, i.e., are not grounds for identity facts.

Let me start with the first type. Consider the following restricted version of Principle 2:

$$\begin{aligned} \forall x [\texttt{Set}(x) \rightarrow (\texttt{SameMembers}(x,x) \rightarrow x=x)] \end{aligned}$$

(6)

Fine imagines an intensional conception of sets where the members of a given set can vary from world to world. Then $\texttt{SameMembers}(x,x)$ will not ground set identity because sets’ having the same members in the actual world will not in general (i.e., across possible worlds) make them the same. Still this conception is consistent with a general ontological view to the effect that an entity being some way grounds its self-identity. Suppose that ‘a’ is an entity that satisfies 7:

$$\begin{aligned} \texttt{Set}(a) \rightarrow (\texttt{SameMembers}(a,a) \rightarrow a=a)] \end{aligned}$$

(7)

Then $\texttt{SameMembers}(a,a)$ implies that a is in some way and thus $a=a$. Still $a=a$ because a is in some way and not because $\texttt{SameMembers}(a,a)$. Fine concludes that, given the aforementioned intentional view on sets, Principle 6 does not provide a proper basis for the generic reading of Principle 2, despite the fact that the former is satisfied in this scenario.

Also when you consider the dual version of Formula 6:

$$\begin{aligned} \forall x, y [\texttt{Set}(x) \wedge \texttt{Set}(y) \wedge x \not = y \rightarrow (\texttt{SameMembers}(x,y) \rightarrow x=y)] \end{aligned}$$

(8)

you will not, Fine argues, arrive at a basis for the generic reading of 2. He thinks that one accepts 8 because one accepts 2 and not the other way round. In other words, Principle 2 is a basis for Principle 8 despite the fact that the latter is less general than the former.

To show that some instances of the generic identity criteria do not hold, i.e., are not grounds for identity facts, in some, rather specific situations, Fine considers a non-well-founded set theory in which there is set ss, whose sole member is itself, i.e., $ss=\{ss\}$. He seems to assume that in this particular case one may come up with the following criterion of identity (for ss):

$$\begin{aligned} \forall x [x \in x \wedge \forall z (z \in x \rightarrow x=z) \rightarrow x =ss] \end{aligned}$$

(9)

This ‘plausible’, as Fine calls it, criterion fails in the case where $x=ss$:

$$\begin{aligned} ss \in ss \wedge \forall z (z \in ss \rightarrow z=ss) \rightarrow ss=ss. \end{aligned}$$

(10)

because $ss \in ss$ is, according to him, partially grounded in $ss=ss$ and not vice versa.

In order to address these issues, Fine suggests that we need (i) to cast criteria of identity in terms of the so-called arbitrary objects and (ii) to define the notion of generic ground, which applies to these objects. This, however, increases the ontological commitment of any ontological position that requires criteria of identity as you need to acknowledge arbitrary objects in your ontological inventory.

Carrara and De Florio (2020) put forward two other issues implied by the view that identity criteria ground identity facts. One issue has to do with the discrimination of facts involved in identity criteria, and the other issue concerns the logical structure of the identity facts themselves.

Consider again a criterion of identity of the form given by 1 or rather by 11:

$$\begin{aligned} \forall x, y [{\kappa }(x) \wedge {\kappa }(y) \rightarrow (\rho (x,y) \equiv x=y)] \end{aligned}$$

(11)

What is the relation between the fact referred to by ‘$\rho$’ and the fact referred to by ‘$x=y$’? Or, to be more specific, what is the relation between the fact that ABC has the same members as BAC and the fact that $ABC = BAC$? Carrara and De Florio argue that these facts are identical and, for this reason, the former cannot ground the latter (provided that we understand the relation of ground as irreflexive.)³ Let us reflect on what it means that it is a fact that $\texttt{SameMembers}(ABC, BAC)$. Very likely, this is a (mathematical) scenario in which at least two sets are in a certain set-theoretical relation. Nonetheless in this specific scenario, ABC is BAC. Then, Carrara and De Florio claim, it seems that there exists just one fact, let us dub it an identity fact, which is not grounded in any other fact, so there is nothing capable of grounding the identity of these sets. We refer to this fact in two different ways, but the fact is singular. Given the ireflexivity of grounding, one must not take a position that a fact may be grounded in itself. Giving up on the irreflexivity one probably needs to explain whether all facts are grounded in themselves and if not, what it is that makes identity facts with identity criteria being grounded in themselves.

The second worry expressed in Carrara and De Florio (2020) is that if identity criteria were to ground identity facts, we would not be able to interpret the latter as indiscernibility facts. They start with Leibniz’s law in the form of:

$$\begin{aligned} x=y \equiv \forall A (A(x) \equiv A(y)). \end{aligned}$$

(12)

The left-hand side of 12 is to refer to identity facts, while the right-hand side points to indiscernibility facts. Carrara and De Florio state that it is plausible to take the latter facts as explanations of the identity facts. But then, if a criterion of identity explains identity, we must not infer that it explains the respective indiscernibility facts. For example, it is plausible that $\forall A (A(x) \equiv A(y))$ grounds that $x=y$. Even if we suppose that $\texttt{SameMembers}$ grounds $x=y$, we cannot infer that $\texttt{SameMembers}$ grounds $\forall A (A(x) \equiv A(y))$. We could infer that only provided that we assume that identity facts ground indiscernibility facts, but this is, as Carrara and De Florio (2020) state, implausible.

Still, Carrara and De Florio define a role that identity criteria may play in metaphysics. Namely, they construe them as providing conceptual grounds for identity facts, thereby as being epistemic criteria of identity.

Finally, Fiocco (2021) argues that identity criteria cannot ground identity facts because the latter are primitive or basic, so there is no relation (nor any other ontological proxy) that is capable of explaining the former. Fiocco assumes that the essential aspect of the ontological criteria of identity is that they explain identity facts by means of referring to a relation in virtue of which they hold. And this sort of explanation cannot exist for identity facts:

In order for a thing to stand in any relation whatsoever, that thing must exist. It must exist as the very thing it is: it must exist as itself. If this is so, it cannot be by standing in some relation that a thing is itself. Being itself is required for it to be related at all. Therefore, there cannot be any such relation in virtue of which a is itself (i.e., a = b). (Fiocco, 2021, p. 5)

Thus that $\texttt{SameMembers}(ABC,BAC)$ (or, for that matter, any other relation between ABC and BAC) is, so to speak, ontically possible only if ABC (and BAC) exists. And if ABC exists, then it exists as itself, so the fact that ABC is itself (i.e., is identical to ABC or BAC) cannot explain the fact that $\texttt{SameMembers}(ABC,BAC)$. Although Fiocco (2021) does not claim that, one may add that if we must describe this situation in terms of grounding, then ABC’s being itself and BAC’s being itself would explain that $\texttt{SameMembers}(ABC,BAC)$ rather than vice versa.

If we agree that there are fundamental issues with identity criteria grounding identity facts, is there any other role that the former may play in the ontological discourse? Before attempting to address this question, let me make a quick detour focused on what kinds of things we mean when we speak about identity criteria and identity facts.

3 Criteria of identity—conceptual shift

As observed by (Fine, 2016, p. 3), when one makes reference to criteria of identity, one may refer to two, ontologically diverse, kinds of “things.” First, criteria of identity may be understood as statements or claims, so linguistic or mental “things”⁴: see, for example, Leitgeb (2013). In other cases they are simply facts, which, being contrasted with statements or claims, can be construed as ontic “things”—see, for example, the narrative in section 2 of Noonan and Curtis (2022). So, given that there exist sets ABC and BAC, then that $\texttt{SameMembers}(ABC, BAC)$ can be interpreted as referring to a sentence, which says that ABC and BAC have the same members or as referring to a fact, i.e., that ABC and BAC have the same members.

I cannot see any particular issue with the first interpretation, e.g., when ‘$\texttt{SameMembers}(ABC, BAC)$’ denotes a sentence—if a given language has the required expressivity, $\texttt{SameMembers}(ABC, BAC)$ is just one sentence among others within this language. Also uttering identity criteria sentences, one usually entertains identity criteria beliefs, so the notion of identity criteria as mental acts seems to be unproblematic. On the other hand, interpreting ‘$\texttt{SameMembers}(ABC, BAC)$’ as a fact may be tricky because if it is a fact that $\texttt{SameMembers}(ABC, BAC)$, then this fact involves only one set, which, rather unsurprisingly, has the same elements as itself, i.e., if the set in question contains a certain element, then it contains the same element (and vice versa). It is true that if one says that $\texttt{SameMembers}(ABC, BAC)$, one uses two terms, i.e., ‘ABC’ and ‘BAC’, to refer to this set, but this saying (a mental or linguistic event) does not ontologically duplicate the set in question. But what is this latter fact, actually? Again, at face value, it will probably be something like: a set has the same elements as itself. This fact is of a rather peculiar sort because it looks like an ontological referent of an ontological tautology, i.e., it holds in every possible world. Namely, one can reasonably maintain the view that everything, not just sets, has the same elements as itself: if something is not a set, then it has no elements, so it has every element it has, i.e., none. After all, the empty set shares the same plight: it has the same element as itself despite the fact that it has none.⁵ Even we restrict the domain of this relation to sets, then every object in this domain, i.e., every set, will be related by it (to itself.) Then these “identity criteria facts” seem to lack any ontological content, and one may wonder whether such “ontological tautologies” ground anything, let alone identity facts—they perhaps can be grounded, but it is unclear what and how they can ground.

My concern here is not that there cannot be facts that only involve one thing and a relation (or a property). The issue I am alluding to is that I find facts like that a set has the same elements as itself as abundant entities in the same sense as the property of being red or not red can be said to be abundant (cf. Lewis, 1986, pp. 61–62). If objects x and y share the property of being red or not red, this fact does not make them similar. Also, having this property does not contribute to any causal power of x or y. Similarly, the fact that x has the same members as itself does not bring any causal contribution to the world, e.g., it cannot be a cause or a result of some event. Also if x is identical to itself, then this fact is casually inert. As far as the relation of similarity is concerned, one can probably say that x’s being itself makes x’s being similar to itself, but the latter fact again is an ontological tautology because everything is similar to itself in any possible ontological scenario. So even if x’s being similar to itself, then this (allegedly abundant) fact seems to be in sharp contrast to such (less abundant) facts that this postbox is similar to that one because they both share the same shade of red. I believe that these observations can be generalised. Because any set has the same elements as itself in all possible worlds, the fact that a given set has the same elements as itself is of no relevance to any ontological theory of sets—and similarly, the fact that this set is identical to itself.

Let me elaborate on this a little further. Any statement of a fact that x is identical to x or a fact that x has the same elements as x is a logical tautology (or an instance thereof). Since you can get all logical tautologies ‘for free’ in any theory, they cannot play any inferential, and therefore any explanatory role, in an ontological theory. Suppose that you developed a theory by means of which some, sparse or not, facts are represented by a set of sentences. The goal of your theory is to describe these facts in some concise way, or to explain them, or to reduce them to other facts, or to $\phi$-y them. Now if, for some reason or other, you included in this set sentences like ‘a is identical to a’, then you can safely remove all such sentences from the set, and nothing will change as far as the prospects of achieving these goals are concerned: if your theory succeeded in explaining these facts beforehand, it still could explain them in exactly the same way; if your theory presupposed some epistemologically controversial view beforehand, it still presupposes the same view; etc. This is the sense in which all such facts as $x=x$ are ontologically abundant.

The issues aforementioned in the previous section and the current misgivings may lead to a doubt whether the notion of fact is really needed or is the most suitable concept to capture possible roles of identity criteria in ontology. We may, for example, reconceptualise the problem by focusing just on properties or relations, without any ontological commitments to facts. In the next section, where I attempt to outline an alternative ontological interpretation of identity criteria, I will drop the ‘fact-focused discourse’ for the sake of the ‘relation-focused discourse’, i.e., this alternative account assumes that criteria of identity are relations of a certain kind.

One may point to a similar issue regarding identity: ‘$ABC=BAC$’ may refer to a sentence or a fact—incidentally, as the reader probably already noticed, I follow here Carrara and De Florio (2020) in referring to facts of this kind as to identity facts. Again I cannot see any particular issue with interpreting ‘$ABC=BAC$’ as a sentence or as a mental act. On the other hand, interpreting ‘$ABC=BAC$’ as a fact is tricky because if it is a fact that $ABC=BAC$, then this fact involves only one set. This time, however, in contrast to “identity criteria facts”, it is not apparent what structure or content this fact exhibits—in any case the content or structure of the sentence ‘$ABC=BAC$’, which refers to this fact, does not seem to give us a satisfactory insight in this respect. Obviously, one must not say that ‘$ABC=BAC$’ refers to the fact that some set is identical to itself as this would be a clearly circular explanation—after all we attempt to understand what it is that a set is identical to itself. Following Fiocco (2021) we may say that $ABC=BAC$ is the fact that some set has the property of being itself, but this seems to be very close to saying that some set is identical to self, so just another label and not an explanation.

As we saw above, Carrara and De Florio discuss a possibility to the effect that identity facts are just indiscernibility facts. Unfortunately, if these indiscernibility facts are taken in the ontological sense, each such fact involves one object and amounts to this object having the same properties as itself. If, going back to our example, $ABC=BAC$, then there exists the fact that some set has the same properties as itself. So again we end up in the domain of ‘tautological facts’ (they hold in any possible circumstance)—this time probably one may justifiably insist that every object, not just sets, has the same properties as itself, so these facts are indeed void of ontological content.

It is unfortunate that the notion of identity property (or identity relation) appears to be ontologically as obscure as the notion of identity fact. Even the question of whether being itself is a (binary) relation or a (unary) property does not come with an obvious answer or, to put it more conspicuously, the appropriate answer to this question heavily depends on external ontological assumptions. For example, suppose that one claims that identity should be understood primarily as a relation and only derivatively as a property because being itself can be defined as being identical to some object, so it can be defined by or reduced to the relation of identity. But the realist about universals may protest saying that being itself does not have to be interpreted as being identical to some object, so the property of being itself is not easily reducible to the relation of being itself (in the sense of $\lambda x[x=A]$ being reducible to $\lambda x y [x=y]$). Suppose that $ABC=BAC$, $DEF=FED$, and $ABC \not = DEF$. From the realist point of view, the property of being itself that ABC exhibits may be considered as the same property as DEF’s property of being itself. But then being itself can be represented neither by $\lambda x[x=ABC]$ nor $\lambda x[x=DEF]$ because they are different properties (given the assumption that $ABC \not = DEF$).⁶

If identity criteria are grounds for identity facts and both are indeed facts, then the problem with the ontological interpretation of identity criteria and identity facts is pertinent to whether and how we should construe identity criteria as grounds for identity facts. Given that (i) both are ontologically dubious, (ii) the former arguably are not capable of grounding anything, (iii) the former do not seem to require any grounds, the conception of grounding identity criteria may seem obsolete. However this does not imply that identity criteria may play any other ontological role. After all, answering the question of what it is for one object to be identical to another, we may mention something else than just the ground for this fact.

4 Criteria of identity as identity modes

Let me start my exposition of an alternative ontological interpretation of identity criteria with a short and somewhat superficial comparison of the criterion of identity for sets to Davidson’s criterion of identity for events Davidson (1969). If you follow the formal pattern established by Formula 1, you are likely to render the latter criterion as follows:

$$\begin{aligned} \forall x, y [\texttt{event}(x) \wedge \texttt{event}(y) \rightarrow (\texttt{SameCauseEffect}(x,y) \rightarrow x=y)] \end{aligned}$$

(13)

where

$$\begin{aligned}{} & {} \forall x, y [\texttt{SameCauseEffect}(x,y) \nonumber \\{} & {} \equiv \nonumber \\{} & {} \forall z [(\texttt{cause}(z, x) \equiv \texttt{cause}(z, y)) \wedge (\texttt{cause}(x, z) \equiv \texttt{cause}(x, z))] \end{aligned}$$

(14)

and ‘$\texttt{cause}$’ means that x causes y.⁷

Clearly the criterion for sets is different from the criterion for events—despite the structural similarity of the respective conditions—because having the same members is different from having the same causes and effects. On the other hand, the identity facts for sets are the same or similar to the identity facts for events in the following sense. Suppose that there exist sets ABC and BAC such that $ABC=BAC$. And suppose that there exist events E and F such that $E=F$. It seems to me that if ‘$ABC=BAC$’ represents a fact, then ‘$E=F$’ also represents a fact. Given that one is prepared to acknowledge such facts, he or she is likely to admit that these facts are clearly different from one another but exhibit some kind of structural similarity. For instance, both involve only one object, respectively, a set and an event. And if they both involve a property or a relation, they apparently involve the same property or relation (or something) of being itself, whatever this property or relation might be. If we agree with Fiocco that existence facts are on a par with identity facts, we may probably compare this similarity to the similarity between the fact that ABC exists and the fact that E exists. That is to say, that $ABC=BAC$ is similar to that $E=F$ in the same or similar way to that ABC exists is similar to that E exists. One is tempted to say here that identity is univocal although this temptation needs to be resisted for the reasons to be explained shortly.

The view expressed in the previous paragraph should be further elaborated. Assume that such predicates as ‘$\texttt{SameMembers}$’ and ‘$\texttt{SameCauseEffect}$’ represent some “ontic things”, e.g., some (possibly higher-order) relations: having the same members and having the same causes and effects.⁸ Then the first point I am trying to make is that these relations are different because sets and events belong to two different ontological domains. Assume also that ‘$=$’ represents some “ontic thing”.⁹ Now my second point was that both the fact that $ABC=BAC$ and the fact that $E=F$ involve the same “thing”, i.e., being itself. This would make the ontological basis for the similarity therebetween while the other components of these facts, i.e., some set and some event, would account for their dissimilarity. The third point is that being itself is a different relationship than having the same members and than having the same causes and effects. Being itself cannot be the same relation as both having the same members and having the same causes and effects because if it were, then having the same members would become amount to having the same causes and effects. On the other hand, it cannot be identical to just one of them because none is a better candidate for this role than the other. Finally, although having the same members is different from having the same events and causes, the former is similar to the latter in two aspects: (i) they both are related to being itself in the same way (given by Formula 1) and that (ii) they are conceptually similar because they both exhibit the ‘having the same ...’ component—although the internal structure of these ‘same ...’ is different as one can easily verify comparing the right-hand sides of Formulae 3 and 14.

Having these observations in place, I propose that identity criteria play the following role in ontology. I put forward that principles like 2 and 13 can be seen not as ontological explanations of identity facts but rather as ontological descriptions or specifications thereof.¹⁰ According to this view, a criterion of identity is a mode or way in which things are identical. Given this account Formula 2 says that having the same members is the way in which sets are identical. Similarly for events: 13 says that having the same causes and effects is the way in which events are identical. In general, if $\phi$ is a sentence of the form given by 1, i.e., a criterion of identity for domain $\kappa$, then the relation denoted by $\rho$, which occurs in $\phi$, is the way in which entities from $\kappa$ are themselves.

I believe that this interpretation is valid for both kinds of identity criteria: one- and two-level, although in the latter case the interpretation looks awkward: lines being parallel is the way in which their directions are identical. In other words, the directions of lines exhibit a peculiar mode or way in which the relation of identity is exemplified because the identity of line directions amounts to the respective lines being parallel. Generally, a two-level identity criterion specifies the way in which certain objects are identical by referring to a relation between other objects and the difference between these objects can be traced down to the difference in ontological levels.

Conditions IC6 and IC9 imply that criteria of identity, being relations, are restricted to their domains, i.e., if $\rho$ is a criterion of identity for ontological domain $\kappa$, then its (logical) domain and range are restricted to objects from $\kappa$. So having the same members is a relation that relates only sets, and having the same causes and effects is a relation that relates only events. In other words, I propose to redefine definition 3 as follows:

$$\begin{aligned} \forall x, y [\texttt{SameMembers}(x,y) \equiv \texttt{Set}(x) \wedge \texttt{Set}(y) \wedge \forall z (z \in x \equiv z \in y)] \end{aligned}$$

(15)

Definition 14 should be adapted accordingly.

Also I need to acknowledge that since identity criteria are relations and are ways of identity, identity should be construed here as a (binary) relation and not a (unary) property.

This view on identity criteria may seem similar to the so-called ontological pluralism, which view has it that if x and y belong to different ontological categories, e.g., when x is a set and y is an event, the way in which x exists is different from the way in which y exists.¹¹ Following W. v. O. quine, the most popular version of ontological pluralism interprets different ways of existence by means of different existential quantifiers:

Put in those terms, pluralism holds that any metaphysically perspicuous representation of reality’s ultimate structure will represent it as including multiple ontological structures. [...] On the neo-quinean view, which our pluralist accepts, a fundamental language represents ontological structure with quantifiers. In this case, we need multiple quantifiers to represent multiple ontological structures. (Turner, 2010, pp. 9–10)

Contrary to the neo-Quinean version of ontological pluralism, I do not claim that we need to posit multiple identities even if we accept multiple identity criteria, i.e., different modes of identity. In fact my interpretation of identity criteria presupposes the claim that there is just one identity symbol or, better, one relation of being itself, and (possibly) multiple ways in which it can be realised, where each way of identity, so to speak, corresponds to a criterion of identity.

One may object to such a view pointing out that it hardly explains what it means that one relation describes another relation. In particular, one may point out that the conceptual link between identity criteria and the relation of identity is far too mysterious. How is it possible, one may ask, that a single relation may be realised in multiple ways? I suspect that a request like this may be addressed in more than one way. The explanation that seems most natural to me resorts to the well-known distinction between determinables and their determinates, which distinction goes back at least to Johnson (1921).

Originally, determinables and determinates were properties that stand in a distinctive relation: the ‘determinable–determinate’ relation, where, roughly speaking, a determinate is a more specific version of its determinable. Thus, the property of being sea green is a determinate for the property of being green, which is a determinate for the property of having colour. Conversely, the property of having colour is a determinable for multiple properties, like being red, being green, being sea green, etc.

Using this distinction and construing identity criteria as modes of identity I can explain the difference between identity criteria and identity facts saying that having the same members is the determinate of being itself (in the domain of sets). And having the same causes and effects is the determinate of being itself (in the domain of events). In general, an identity criterion is a determinate for the relation of identity or, conversely, the relation of identity is a determinable for any identity criterion.

Let me illustrate this application with an example. As I stipulated above $\texttt{SameMembers}$, i.e., a criterion of identity for sets, is the way or the mode in which sets are identical. So the former relation specifies or determines the particular way in which sets can be identical. This particular way is different from the way in which events can be identical, which way is, allegedly, given by $\texttt{SameCauseEffect}$. The two criteria thus determine two ways in which different kinds of things can be identical. It seems to me that this particular kind of determination is sufficiently similar to the determination whereby the property of being sea green determines the property of being green, i.e., the $\texttt{SameMembers}$relation determines the relation of identity in a similar manner as the property of being sea green determines the property of being green. In both cases the former makes the latter more specific or establishes how the latter is realised.

This intuition may be supported by an argument. In the remainder of this section, I will argue that the relation between identity criteria and identity can be seen as a special case of the relation of determination (between relations) by showing that the former satisfies the sufficient number of properties of the latter.

Obviously, this “application” of the relation of determination looks somewhat unorthodox. First, the usual scope of determinables and determinates is restricted to (unary) properties, so it does not include relations. Still, the idea of relations determining other relations is by no means new or alien—for instance (Johansson, 2014, pp. 116–117) notices:

I do not think there are any good reasons for the restriction. Obviously, just as there are determinate-determinable series between property concepts [...], so there are such series between relation concepts (e.g., “a little longer than $\rightarrow$ longer than in general $\rightarrow$ length relation” and “much brighter than $\rightarrow$ brighter than in general $\rightarrow$ intensity relation”). Relational concepts, just like property concepts, can be fitted into determinable-determinate trees.

Johansson is not particularly picky in giving examples of the relation of determination between relations: being little longer than determines being longer than, being much brighter than determines being brighter, etc. We may extend this list with such items as being a child of determinates being a descendant of, being an alumnus of determinates being a student of, being a charter member of determinates being a member of, being an improper function of (an artefact) determinates being a function of.

Secondly, and probably more importantly, it is easy to note that the usual examples of determinables and determinates may form long chains out of the determination relation, e.g., bright sea green determines sea green determines green determines colour determines property while the application in question restricts them to just one level, e.g., having the same members determines identity. This implies that there are formal properties of this determination relation when it is applied to, say, colours or shapes, that are not exhibited by this relation when we apply it to identity.

Let us investigate this issue in more detail. For the sake of brevity I will use the following notation:

• 1. ‘$\texttt{Received}$’ is to denote the “received” relation of determination, i.e., the one whose domain includes colours, shapes, etc.

• 2. ‘$\texttt{Revised}$’ is to denote the relation that holds between identity criteria and identity. The specific properties of this relation, including formal properties, are defined by Formula 1 accompanying by the usual additional constraints—see IC1—IC9 above. I will argue that because of these properties and constraints $\texttt{Revised}$ is sufficiently similar to $\texttt{Received}$ so that the latter can be seen as a particular species of the former in the domain of relations.

$\texttt{Received}$ is usually described by means of examples and certain characteristic properties it exhibits. Looking at the list of those collated in Wilson (2021)¹², one may notice that there are quite a few properties of $\texttt{Received}$ that are also satisfied by $\texttt{Revised}$:

• 1. By its very definition a criterion of identity describes the relation of identity in the sense given by 1, so the former is at least as specific as the latter and, given that there are multiple domain-specific criteria of identity, it is more specific than the latter because it details how the relation of identity is realised in a given domain. (Increased specificity)

• 2. $\texttt{Revised}$ is irreflexive: no mode of identity determines itself. $\texttt{Revised}$ is asymmetric: if x $\texttt{Revised}$-determines y, then y does not $\texttt{Revised}$-determine x. $\texttt{Revised}$ is transitive – but only because there are not such x, y, and z that x $\texttt{Revised}$-determines y and y $\texttt{Revised}$-determines z.(Irreflexivity, Asymmetry, Transitivity)

• 3. No non-circular identity criterion (see condition IC3 in Introduction) is a conjunction of identity and other properties or relations. (Non-conjunctive specification) And conversely the relation of identity is not a disjunction of any identity criterion and other properties or relations. (Non-disjunctive specification)

• 4. It follows from the definition of identity criteria that $\texttt{Revised}$ satisfies Determinable inheritance (‘For every determinable Q of a determinate P: if x has P at a time t then x must have Q at t.’), e.g., if set ABC has the same members as BAC (at a time), then $A=B$ (at that time).

• 5. The principle ‘Objects must have a determinate of every determinable they have’ is satisfied if the slogan ‘No entity without identity’ is true because every entity has its corresponding criterion of identity when the latter is true.(Requisite determination)

• 6. $\texttt{Revised}$ also satisfies the principle of multiple determinates provided that there are at least two valid instances of identity criteria, e.g., having the same members and having the same causes and effects. (Multiple determinates)

• 7. Consider now the principle of Unique determination, i.e., the principle that has it that if x has a determinable Q at a time, then x has a unique-one and only one-determinate P at any given level of specification at that time. Since all identity criteria are unique in the sense of conditions IC6 and IC9 aforementioned in Introduction, $\texttt{Revised}$ satisfies Unique determination.

• 8. $\texttt{Revised}$ seems to satisfy the following extrapolation of Asymmetric modal dependence principle: if relation $R_1$ is a determinate of relation $R_2$, then if $x R_1 y$ then $x R_2 y$, but for some $z_1$ and $z_2$, $z_1 R_2 z_2$ and not $z_1 R_1 z_2$. If R is a mode of identity, i.e., a determinate of identity, then if R, then $x=y$, but since R is domain-specific, there exists some $z_1$ and $z_2$ such that $z_1=z_2$ but not $R(z_1, z_2)$ provided that there exist such domains as sets and events with their own criteria of identity.

On the other hand, not all properties of $\texttt{Received}$ are exhibited by $\texttt{Revised}$. First, as opposed to $\texttt{Received}$, a relation’s characterisation as determinable or determinate is never relative for $\texttt{Revised}$–an identity criterion is always a determinate and identity is always a determinable. This respective principle in Wilson (2021) is labelled as Relative, leveled determination.¹³ Secondly, the principle of Determinate similarity/comparability is not satisfied by $\texttt{Revised}$. Having the same member does not look similar to such relations as the spatiotemporal continuity or the relation of parallelhood (which can be taken as the criterion of identity for line directions.) Even having the same causes and effects may be argued to be dissimilar to having the same members due to the structural dissimilarity between the definentia clauses in their definitions, i.e., the right-hand sides of Formulae 3 and 14. Finally, one-level identity criteria are also structurally different from two-level identity criteria, so they are not sufficiently similar to satisfy Determinate similarity/comparability.

Finally there are other principles that at first sight are not applicable to $\texttt{Revised}$ in the sense they one can maintain neither that they are satisfied nor that they are not satisfied by $\texttt{Revised}$. Consider, for instance, the principle of Causal compatibility, which states that determinables and determinates do not causally compete. Being red does not causally compete with being a colour, but arguably it makes little sense to say that being itself causally competes or not with having the same members. Similarly, it is not clear to me how one may make sense of Determination ‘in respect of’ determinables principle, which requires that if P determines Q, P is more specific than Q in respect of Q. One may say that red is more specific than colour in respect of colour. But if some relation $R_1$ $\texttt{Revised}$-determines the relation of identity, then I am not sure what it may mean that $R_1$ is more specific with than identity in respect of identity. One can probably say that an identity criterion is more specific than identity in respect of how the latter relation is realised, i.e., in respect of how two things are identical. But this does not go much beyond the initial claim that criteria of identity are ways of identity.

If one insists that all of the above conditions must be met by the relation of determination (i.e., the one which relates determinates and their determinables), then my interpretation of identity modes as domain-specific determinates for the identity determinable will fail. But if you allow for a more liberal approach or make room for imprecise concepts, and you think about this idea in terms of, say, family resemblance, then taking into account all these similarities and differences may lead to the conclusion that the above application of the ‘determinable–determinate’ pattern to identity and its modes is at worst a borderline case of the relation of determination in question. In fact Wilson (2021) explicitly states a caveat that there might be cases where some of the aforementioned properties do not hold in full generality and may require adaptation in the case of relations—I consider my analysis as an attempt at such adaptation.

Unfortunately, this interpretation of identity criteria as determinables of identity cannot be applied to two-level criteria due to the difference in ontological levels. Having the same causes and effects, which relation relates causes and effects, can be seen as a determinate of being identical, but being parallel is not a determinate of being identical because the former relates lines and the latter relates their directions. As a result, some of the crucial conditions that first-level identity criteria satisfy (if taken as determinates of identity) are false when applied to two-level criteria—most importantly: Increased specificity and Determinable inheritance.

In conclusion let me reiterate that my strategy to justify the claim that one-level criteria of identity are determinates for identity was to show that the relation between the former and the latter exhibits a (sufficient) number of formal properties of the relation of determination, which is the basis for the distinction between determinables and determinates.

5 Identity modes do not ground identity

Now some believe that the relation of determination, which defines the determinable–determinate distinction, is a special case of the relation of grounding—see, for example, Rosen (2010). If this were the case, my solution would be open to the objections from Sect. 2.

Prima facie, it cannot be the case because the relation of grounding is usually interpreted as relating facts or propositions—and this assumption is shared by both the so-called predicational and the operational view (Correia & Schnieder, 2012, pp. 10–11)—and my account concerns the relation of determination, which binds relations, not facts or propositions. Still, one can bridge these two notions adapting the principle called in (Rosen, 2010, p. 126) “Determinable–Determinate Link”: if property F determines property G and if F(x) is a fact, then F(x) grounds G(x). So in the case of relations we would get something like: if (binary) relation R determines (binary) relation S and if R(x, y) is a fact, then R(x, y) grounds S(x, y).¹⁴

Although a detailed discussion of the relevant similarities and dissimilarities between the two relations goes somewhat beyond the scope of this paper, let me mention one objection against the “Determinable–Determinate Link” principle and its use by Rosen (2010). The objection in question boils down to the claim that the relation of determination does not share certain essential features of the relation of grounding. So the latter cannot be a special case of the former.

Suppose that there are some properties F and G such that F determines G. Then “Determinable–Determinate Link” implies that F(x) grounds G(x) for any x such that F(x) is a fact. Many believe that if one fact grounds another, then the former is somehow more fundamental than the latter (e.g., Schnieder, 2020 or Werner, 2021). Still, something’s being sea green may be seen as equally fundamental as its being green or its having mass as equally fundamental as its having a mass of 1.3 kg.

First, intuitively the former seem to be on a par with the latter as far as their “ontological credentials" are concerned. So, for instance, you cannot explain away something’s having a mass by pointing to this thing’s having a mass of 1.3 kg and saying that the latter is somehow ontologically more basic than the former. Secondly, there are theoretical reasons, elaborated for example in Wilson (2012), for a determinable’s being equally fundamental as its determinates. In a nutshell, Wilson argues that because there are certain facts about determinables, e.g., that each of them has a specific range of determinates, which cannot be grounded in facts about determinates, determinate facts do not “fix” the determinable facts and, for that reason, the former are not more fundamental than the latter. As a matter of fact, Wilson is more careful. She maintains only that each determinable is of the type whose instances can be differently determinate (by its instances). For example, if the property of being red is determined by a particular shade of crimson, then the former property is of the type such some of its instances, not necessarily the former property itself, can be determined by another shade of crimson.

More generally, it is a constitutive modal fact about every determinable instance that it is of a type whose instances might be differently determined. But no specific determinate instance seems suited to ground this fact about its associated determinable instance. How could it, given that the ground for this modal fact involves reference to multiple determinates, all but one of which the specific determinate, whether instance or type, not just actually but necessarily excludes? At best, the determinate instance seems suited to ground certain non-modal facts about the determinable instance — most promisingly, that it is actually determined as it is. Left ungrounded by the determinate instance is the constitutive modal fact about the determinable instance, that it is of a type that might be differently determined. (Wilson, 2012, p. 12)

Thus, if one aims to explain both non-modal and modal facts about the world, both determinables and their determinates need to be parts of the (relative or absolute) base of fundamental entities, and we cannot maintain that any is more fundamental than the other just by using the “Determinable–Determinate Link” principle.

I believe that a similar argument can be reconstructed from Koslicki (2016)’s evaluation of J. Schaffer’s model of structural equations as applied to the relations causation and the relation of determination. Koslicki argues that these two “alleged cases of grounding” cannot be represented by means of this model unless one is willing to make severe modifications for each type. For my purposes the most relevant point she makes concerns the relation of dependency between “determinate facts” and “determinable facts”:

[...] there are thus dependence relations running in both directions in an alleged case of determinate/determinable grounding: a rigid dependence relation connecting particular states of affairs, running from the shirt’s exemplifying some determinate shade of color (e.g., maroon) to its exemplifying the corresponding determinable shade of color (viz., red); but also a generic dependence relation connecting the shirt’s exemplifying some determinable shade of color (e.g., red) to the shirt’s exemplifying some one particular determinate shade of color (e.g., maroon) which falls into the relevant range of admissible alternatives (viz., maroon or crimson or...) set by the determinable shade in question. (Koslicki, 2016, p. 108)

Being red rigidly depends on being maroon (because if something is maroon, then it is red). At the same time, being maroon generically depends on being red because the latter sets the scope of its possible determinates, which set happens to include the former. Although it is generally agreed that if one fact grounds another, then the latter depends on the former, it is not evident whether one can consistently maintain that the former also depends on the latter—even if each case involves a different type of dependency. Therefore something’s being maroon cannot ground this thing’s being red, pace Rosen’s principle.

To sum up there seem to be certain features of the relation of determination that set it apart from the relation of grounding, so the former cannot be seen as a special case of the latter. Thus if one construes identity criteria as specific ways of identity, one is not bound to represent the relation therebetween as the relation of grounding.

6 Identity criteria do not fail to specify

My interpretation of identity criteria as modes of identity may be in conflict with the more or less explicit established view of logical relations, which has it that the relation of identity cannot occur in various forms but is unitary. Let me take McGinn (2000) as an expositor of this position. Following Frege McGinn argues that the relation of identity, as opposed to other more mundane relations like being more intelligent, does not divide up into sub-varieties:

There is no equivocation or vagueness in the notion of identity, and it operates as a determinate property, not a determinable one. (McGinn, 2000, p. 1)

Although he states upfront that he does not enter into an elaborate defence of this view, he offers three arguments or rather the counter-arguments towards those who, as I do, perceive multiple varieties of identity. Out of these three only the last one is relevant here:

A third reason for denying that identity is unitary might issue from the recognition that objects come in many kinds and that the conditions of their persistence vary depending upon what kind of object is in question. Here we encounter the idea that objects have different ‘criteria of identity’, different conditions under which x may be said to be the same. Sets are the same iff they have all the same members as y. ; [...] But, however plausible this idea of varying ‘criteria of identity’ might be, it is confused to believe that it shows that the identity relation itself admits of such variation. Many different kinds of objects can be blue, too — solids, liquids, gases, rays of light, persons - but this does not show that blueness itself reflects the differences in these objects. Similarly, though many kinds of objects stand in the identity relation, it does not follow that this relation itself reflects or incorporates the kinds in question. We might indeed allow that identity can be supervenient on a variety of bases in objects, depending upon the kind of object in question, but this does not imply that what thus supervenes is not a unitary property. (McGinn, 2000, pp. 5–6)

An obvious reply may point out that this argument implicitly presupposes that identity pluralism posits multiple determinates of identity just because this relation occurs in multiple domains. Actually, identity pluralism assumes more, namely, it recognises varieties of identity not just because this relation occurs in every domain but (also) because there are domains in which different identity criteria are satisfied. Thus the relation of identity occurs in the domain of sets in a different way than it occurs in the domain of events, not just because sets are not events, but because different principles provide identity criteria for these domains. In other words, multiple determinates of identity are postulated because there are multiple criteria of identity (i.e., different domains that have different criteria of identity) and not just because there are multiple domains of objects.¹⁵

But McGinn seems to assume something else as well:

Compare the property of goodness: it supervenes on a variety of descriptive bases, but this does not compromise its conceptual integrity. Similarly, identity might be constituted by different things for different types of object, but it does not follow that identity itself shifts its identity from case to case (identity does not suffer from identity problems). (McGinn, 2000, footnote 11, p. 6)

It looks as if McGinn may also be worried about something else. Lack of conceptual integrity, identity shifts, identity problems with identity may indicate that he had in mind a position that would postulate multiple relations of identity.¹⁶ This interpretation is supported by his other two arguments for identity unitarianism, where he rejects the distinction between numeric vs qualitative identity and the distinction between absolute vs relative identity. This, however, is where my version of identity pluralism and his unitarianism on identity actually agree, as I tried to explain in the previous section since both positions maintain that there is one relation of identity. Alas, McGinn does not stop here: [identity] operates as a determinate property, not a determinable one.

Although I do not find McGinn’s argument conclusive, I do recognise that one may uphold a more general view to the effect that no logical notion can refer to a determinable property (or relation). Still, I am not aware of any arguments to this effect, so I cannot develop any reliable rejoinder to this view. Thus, I consider my interpretation of identity criteria as a tenable alternative to the “grounding interpretation.”

7 Conclusions

Recent attempts at interpreting identity criteria as grounds for identity facts have been met with serious criticism. The most substantial objections revolve around the claim that identity facts are sufficiently primitive so that they would not require grounds or explanations, at least if we think about such mundane objects like events, persons, or even sets, and not about theoretically sophisticated entities like arbitrary objects of Fine (2016).

However the notion of grounding is not the only way out for those who believe that identity criteria may play certain crucial theoretical roles in metaphysics. Another option, outlined above, interprets criteria of identity as means of elucidation of how identity is realised in different ontological domains. This view, if tenable, gives rise to a different form of ontological pluralism, which concerns identity and not existence. As opposed to its predecessor, this particular form of pluralism does not require multiple identities but interprets these modes of identity as different determinates for one and the same relation of identity. Still even this weak form of pluralism may be found objectionable.

Appendix—Wilson (2021)’s criteria for the determinables–determinates relation

• 1. Increased specificity: If P is a determinate of (‘determines’) Q, then to be P is to be Q, in a specific way. Determination relates properties that are more or less specific, relative to each other. For example, red is a determinate of color; scarlet is a determinate of red.

• 2. Irreflexivity, Asymmetry, Transitivity: For no P does P determine P If P determines Q, then P does not determine P if R determines P, and P determines Q, then R determines Q. These features, characteristic of strict partial orderings, follow just from Increased specificity. For example, scarlet does not determine itself, since no property can be more specific than itself. If scarlet determines red, then red does not determine scarlet, since red cannot be both less and more specific than scarlet. And if to be scarlet is to be red, in a specific way, and to be red is to be colored, in a specific way, then to be scarlet is to be a yet more specific way of being colored, such that scarlet determines color.

• 3. Relative, leveled determination: Except for minimally specific determinables and maximally specific determinates, a property’s characterization as determinable or determinate is relative. Relatedly, determinables can be determined at different ‘levels’ of specificity. For example, red is a determinate of color, but a determinable of scarlet; red, blue, and yellow determine color at one level of specification; scarlet, navy, and buttercup determine color at another (more finely grained) level of specification.

• 4. Determination ‘in respect of’ determinables: If P determines Q, P is more specific than P in respect of Q.

  The increase in specificity associated with determinates is, more specifically, ‘in respect of’ the determinable.[5] For example, if red determines color, red is more specific than color in respect of color. Determination in respect of a determinable is sometimes further explicated as involving specification along one or more ‘determination dimensions’ of the determinable-for example, the determination dimensions of color are commonly supposed to be hue, saturation, and brightness.

• 5. Determinate similarity/comparability: If P and R are different same-level determinates of a determinable Q, then P and R are similar, and moreover comparable, in respect of Q.

  That determination is ‘in respect of’ determinables gives rise to similarity and systematic comparability relations among determinates. For example, determinates of color-red, blue, orange, etc.-are all similar in being colors, and any two colors are comparable in this respect: red is more similar to orange than to blue in respect of color, and so on.

• 6. Non-conjunctive specification: if P determines Q, then P is not identical with any conjunctive property conjoining P with any property or properties independent of Q.

  The increase in specificity associated with determinates does not involve conjoining or otherwise combining the determinable with another property independent of the determinable. For example, red is not appropriately understood as the conjunction of color and another property independent of color.

• 7. Non-disjunctive specification: if P determines Q, then P is not identical with any disjunctive property disjoining P with any property or properties independent of P.

  The increase in specificity associated with determinates does not involve disjunctive elimination. For example, color is not a disjunction of red and some other property or properties independent of red.

• 8. Determinable inheritance: For every determinable P of a determinate P: if x has P at a time t then x must have P at t.

  Objects must have every determinable of any determinates they have. For example, if something is scarlet, then it must also be both red and colored.

• 9. Requisite determination: If x has P at a time t, then for every level L of determination of Q: x must have some L-level determinate P of P at t. Objects must have a determinate of every determinable they have. For example, if something is colored, then it must be some specific color (e.g., red).

• 10. Multiple determinates: For every determinate P of a determinable Q, there is a determinate R of P that is distinct from, but at the same level of specificity as, P. With determinate specificity comes multiplicity or diversity. For example, color has multiple determinates (red, blue, yellow, and so on); red has multiple determinates (scarlet, crimson, burgundy, and so on).

• 11. Determinate incompatibility: If x has determinate P of determinable P at time t, then x cannot have, at t, any other determinate R of P at the same level of specificity as P.

  For example, an object cannot be both red and blue all over, at a time. Determinate incompatibility is moreover ‘opponent’ in that same-level determinates are exclusive alternatives.

• 12. Unique determination: If x has a determinable P at a time, then x has a unique-one and only one-determinate P at any given level of specification at that time.

  Unique determination is entailed by Requisite determination and Determinate incompatibility.

• 13. Asymmetric modal dependence: if P is a determinate of Q, then if x has P then x must have Q, but for some y, y might have P without having P.

  For example, anything that is scarlet must be red, but something might be red without being scarlet (it might rather be, e.g., burgundy). Asymmetric dependence is entailed by Determinable inheritance, Requisite determination, and Multiple determinates.

• 14. Causal compatibility: Determinables and determinates do not causally compete.

  For example, if a given patch is both red and scarlet, there is no in-principle difficulty with both red and scarlet being causally efficacious vis-á-vis the pecking of a pigeon trained to peck at any red patch.