Identity Criteria and Meta-Ontology

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Identity Criteria and Meta-ontology

MSc Thesis (Afstudeerscriptie)

written by Jacopo Guzzon

(born May 2nd, 1996 in Milan, Italy)

under the supervision of Dr Bahram Assadian, and submitted to the Examinations Board in partial fulfillment of the requirements for the degree

of

MSc in Logic

at the Universiteit van Amsterdam.

Date of the public defense: Members of the Thesis Committee:

June 10, 2021 Dr Maria Aloni (Chair)

Dr Bahram Assadian (Supervisor) Dr Paul Dekker

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Abstract

The notion of identity criterion is crucial in the field of ontology. The goal of the thesis is to explore the theoretical consequences that arise from includ-ing a demand for identity criteria in one’s meta-ontological principles. We develop a precise formal account of identity criteria that is able to capture all their prominent features. In addition, we enumerate exhaustively all re-quirements that criteria of identity must meet to qualify as legitimate, and all functions they can fulfil in a meta-ontological framework.

Analysis of the logical form of identity criteria will produce two results: the first is the formulation of the theory of basic sortals, which states that every ontology that employs identity criteria is committed to the existence of a class of sortals for which an identity criterion cannot be provided. The availability of an identity criterion has been used to establish whether a certain class of things is included in the realm of being or not: it is called the strong ontological function. A notable consequence of the theory of basic sortals is the defeat of such function; the second result is the commitment of any theory employing identity criteria to the strong versions of Leibniz’s principle of the identity of indiscernibles, and to the negation of the weak versions of the same principle.

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Introduction

The subject of this thesis is the notion of identity criteria, as employed in contemporary metaphysics and analytic ontology. Identity criteria relative to a certain kind of entities Φ can be defined as propositions that provide necessary and sufficient conditions to settle the identity or difference of Φs. Given arbitrary entities a and b belonging to Φ, an identity criterion states what conditions a and b must meet in order for them to be numerically identical. The paradigmatic recurring example of identity criteria is the Axiom of Extensionality from set theory, stated here in its informal version (informal identity criteria are marked by the asterisk):

(AE*) If a and b are sets, then a = b if and only if a and b have the same elements.

The condition to be met in order for sets a and b to be identical, i.e. to be the same set, is the inclusion of exactly the same elements.

Criteria similar to (AE) can be found for different kinds of objects: nat-ural numbers, directions, material objects, objects of geometry and events are among the most popular kinds of entities for which identity criteria have been provided. In the ontological debate, identity criteria have been accepted by a majority, but not the totality, of authors. They have sometimes been deemed as useless at best or nonsensical at worst (see Pollard 1986, Jubien 1996 or Strawson 1997). On the other hand, authors who accepted the gen-eral legitimacy of the notion debated whether an adequate identity criterion for specific kinds of objects can be formulated or not. For instance, many maintain that it is impossible to find a criterion of identity for kinds like propositions or properties (Quine 1975; 1986, 66-7; Carnap 1988).

Although many tend to attribute the introduction of identity criteria to Frege (1950), some philosophers, first and foremost E.J. Lowe (1989a, 2), claimed the paternity of the concept for John Locke (1690, Book II, Chapter XXVII):

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When we see anything [material] to be in any place in any in-stant of time, we are sure (be it what it will) that it is that very thing, and not another which at that same time exists in another place, how like and undistinguishable soever it may be in all other respects: and in this consists identity (ibid.,311, our emphasis). This may be considered the first formulation of an identity criterion for ma-terial objects, which in more recent terms will assume the form:

(MO*) If a and b are material objects, then a = if and only if a and b occupy the same spatio-temporal region.

Locke also posited that what identity consists in for a certain class of entities may not be the same for another, laying the foundation for the differentiation of identity criteria on the basis of the nature of the kind they are provided for.

If some merit has to be attributed to Frege, it is the merit of having framed the notion in terms that are more familiar to contemporary metaphysics. He also provided some of the most paradigmatic examples of identity criteria, like Hume’s principle for natural numbers. Another major contribution from Frege is the insistence on the theoretical necessity of having identity criteria for whatever class of objects we refer to:

If we are to use the symbol a to signify an object, we must have a criterion for deciding in all cases whether b is the same as a (Frege 1950, §73).

A second notion, strictly connected to identity criteria, is necessary for the understanding of the thesis: sortal concepts. Now, the definition of sortal can be tricky, since different philosophers used the term with different meanings. The main aspect that is generally agreed upon is the following: a sortal concept captures the essence of the entities falling under it (Grandy and Freund 2020). ‘Red thing’ is not a sortal, since for whatever entity may fall under it, whether it is a red book, a scarlet frog or a crimson crystal, being red is not its essence (Lowe 2009, 34). In other words, a sortal is supposed to tell us what an entity falling under it is. Now, in addition to this understanding, some authors imposed other restrictions on the concept. Many claimed that a sortal necessarily provides a criterion of identity for the entities falling under it (Wiggins 2001, 69). Some require a sortal to

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62). The discussion on the definition of sortal is not our focus, for now; the question will be tackled in section 3.2.1. For the time being, we will accept the definition based on the understanding that sortals encapsulate what its instantiations are: a concept Φ is a sortal if and only if it captures the essence of the entities falling under it. We will argue for the superiority of this definition in comparison with the definition that connects sortals and identity criteria in chapter 3. We agree, however, that if for a group of entities an identity criterion can be provided, then they all fall under the same sortal. In other words, it is impossible to produce an adequate identity criterion for {n1, n2, ..., nn}if there is no sortal Φ such that Φ(n1) ∧ Φ(n2) ∧ ... ∧ Φ(nn).

Meta-ontology

Since Quine (1948), the main Ontological Question to which ontologists are supposed to find the answer has been:

(OQ) What is there?

The goal of ontology, then, is to compile an exhaustive and inclusive cata-logue of the furniture of the world, of everything that has being. Thus, the elegant Ontological Question fractured into countless different sub-questions regarding the existence of specific kinds of entities. Questions like ‘Do num-bers exist?’, or ‘Are properties entities?’ and so forth. However, the main reason behind the disagreement on how to answer these questions was that many authors were interpreting words like ‘existence’, ‘being’ or ‘entity’ in different ways.

Peter van Inwagen (1975) found a more effective way to frame the dis-agreements on the interpretations of these concepts. Meta-ontology was de-fined precisely as the discipline which looks for the answer to the Meta-ontological Question:

(MQ) What are we asking when we ask the Ontological Question?

While looking for the answer to (MQ), ontologists tackled meta-ontological sub-questions like ‘Is there a difference between existence and being?’ and ‘Are there degrees of existence?’.

Every ontology is founded on a certain meta-ontological framework, even when such a framework is not explicitly stated. In order to argue about what exists, one necessarily needs an account of what ‘exist’ means, and, thus, one

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needs some meta-ontological assumptions. The meta-ontological framework can be formally captured as a set of sentences. For instance, van Inwagen elaborated what he considered to be the four tenets that were crucial to the meta-ontological framework employed by Quine when discussing ontological questions (van Inwagen 1975):

(I) Being is not an activity.

(II) Being is the same as existence. III) Being is univocal.

(IV) The single sense of being or existence is adequately captured by the existential quantifier of formal logic.

This is by no means an exhaustive list. Quine’s strong opinion on identity criteria and on the role that they are supposed to fulfil has been summarized in the motto ‘No Entity without Identity’. The motto has been considered as another meta-ontological tenet (Berto and Plebani 2015, 41), best captured by these two principles:

(T) If a class of entities falling under a single sortal concept has being/exists, then it can be provided with an identity criterion.

(O) If a class of entities falling under a single sortal concept cannot be provided with an identity criterion, then it does not have being/does not exist.

Thus, in light of this analysis of Quine’s assumptions, his meta-ontology is a set of sentences MQ that includes as elements all the propositions listed

above. Any ontological theory that operates, explicitly or not, according to these six theses is to be considered a Quinean ontology.

Quine’s meta-ontological framework gained extreme popularity in the field to the point where the set of principles MQ has been referred to as

the Standard View in ontology (Berto and Plebani 2015, 34). It is, by far, the most influential framework in which ontologists operate to this day. The relevance of this thesis in the field of ontology lies precisely in the fact that our claims apply heavily to the meta-ontological design of the Standard View, and suggest how it could be amended and improved.

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We interpret meta-ontologies as closed under entailment, i.e. given a meta-ontological framework M, if any subset of M entails proposition (X), then (X)∈ M. For instance, it would be included in Quine’s meta-ontology any proposition that can be logically implied from any subset of {(I), (II), (III), (IV), (T), (O)}. Our findings regard the way in which meta-ontological principles interact with each other and what consequences arise from these interactions. In particular, we will be concerned with a principle that is weaker than (O) and (T):

(*) Some subsets of the entities in any ontology must have an identity criterion.

By virtue of the closure under entailment of meta-ontologies, we can generalize our claims, which hence apply not only to meta-ontologies that directly include (*), but also to any meta-ontology that includes principles that entail (*). Thus, our conclusions will also apply to the Standard View, which is something we consider important in order for them to be relevant to the current debate in the field.

Research question and results

The primary goal of this thesis is to evaluate the theoretical consequences that arise from the inclusion of identity criteria in one’s meta-ontology.

Identity criteria are not themselves meta-ontological principles, but there can be meta-ontological principles that refer to them. In the Standard View, principles (T) and (O) explicitly prescribe identity criteria. Our main objec-tive is to analyse what kind of position the inclusion of a principle along the lines of (T) or (O) in one’s meta-ontology commits one to. The principle in question does not need to be as strong as these two; it can also require, for instance, identity criteria only for a particular subset of the entities included in one’s ontology, not for all of them. Yet, as soon as one meta-ontological framework includes a principle that demands a criterion of identity for at least one subset of the ontology’s objects, then the findings of this thesis will apply to that framework too.

In order to achieve our goal, the thesis will not address the general ques-tion concerning the legitimacy or usefulness of the noques-tion of identity criteria. On the contrary, we will remain eminently neutral on the topic. No argument in favour or against the inclusion of a principle demanding identity criteria

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for one or more groups of entities will be issued. Our main interest lies in what happens to a meta-ontological framework when one such principle is included, not the merit of the inclusion itself.

The main results, achieved by virtue of this method, will be two. First, we will argue that no meta-ontology can include a principle that prescribes the need for identity criteria for at least one sortal, without also including some sortal for which identity determinate, even if no criterion of identity can be provided: basic sortals. In other words, if a meta-ontology includes a principle like (*), or any principle stronger than (*) such that it implies (*), then it must also include a meta-ontological principle specifying that there must be a class of basic sortals.

Second, we will argue that including in one’s meta-ontology any demand for identity criteria for at least some sortal, i.e. including in any meta-ontology (*) or any principle that entails (*), amounts to a commitment to the truth of the stronger, more trivial versions of the principle of the identity of indiscernibles, and to the rejection of weaker, most controversial versions of the same principle. The principle of the identity of indiscernibles states that for all F , if F (x) if and only if F (y), then x = y, i.e. if arbitrary x and y share all their properties, then they are identical. The principle can be interpreted in different ways, according to the restrictions that one imposes on the second-order quantification over properties. If there is no restriction, one has a trivial version of the principle. If the only restriction is that the properties of identity and difference (for example, the property of x to be identical to x, or the property of x to be different from any other object that is not itself) are not included in the scope of the universal quantification, one has a strong version of the principle. If the restriction goes further, and excludes from the scope of the quantification also every property that refers to any other particular (for example, the property of being three miles from a specific tree or the property of having a specific radius), then one has the proper version of the identity of indiscernibles. Any additional restriction results in weak versions of the principle. A meta-ontology which includes some demand for identity criteria is committed, as we will show, to the truth of the trivial and strong versions, and to the falsehood of the proper version and of any other version weaker than that.

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Structure

The thesis is structured as follows: the first chapter analyses the debate concerning the best formal account that can be given of the notion of identity criterion. In light of the abundant technical literature on the topic, we will formulate a formal scheme for any identity criterion. This scheme will be called (IC) and it will have consequences in the next chapters.

The second chapter will present, once again on the basis of debates in metaphysics and ontology, the list of requirements that an identity criterion must meet to be considered legitimate. For instance, a criterion of identity for squares that states that for x and y, which are squares, x = y iff x and y have the same sides and angles, would violate the minimality requirement, since the identity of squares can be settled effectively by appealing exclu-sively to the identity of sides alone. The former principle is not minimal, since it requires sameness of sides and angles, while sameness of sides alone gives us a necessary and sufficient criterion. The other requirements will be: non-tautology, informativity, non-circularity and uniqueness. Furthermore, the second chapter will also present the different possible functions that an identity criterion can fulfil. Out of the five functions we encountered in the literature, one will be immediately discarded. This is the definitory function, according to which a criterion of identity for Φ provides the definition of identity concerning Φs or, at least, the definition of the concept of Φ. The four remaining functions are: the metaphysical function, according to which identity criteria for Φ tell us what identity for Φs consists in; the ontological function, according to which identity criteria for Φ tells whether Φs have being or not; the semantic function, according to which identity criteria tells us whether two names have the same reference; and the epistemic function, according to which identity criteria for Φ tells us how we can know if Φs are identical.

The first two chapters can be considered, to some extent, a presentation of the state of the art in the literature on identity criteria. Their conclusions are functional to the other two chapters, which bring forth some original claims and criticisms of currently held positions in metaphysics, in particular towards the Standard View. The aim of the third chapter is to disprove the meta-ontological principle that we called (T), the totality assumption. The defeat of such assumption will prove that the notion of identity criteria cannot be applied unrestrictedly and that there must be, in any ontology employing identity criteria, a group of sortals, whose instantiations are included in the

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list of the furniture of the world, but that do not possess identity criteria. We call it the theory of basic sortals, by virtue of which we will also show how the other Quinean assumption on identity criteria, (O), is untenable and must be rejected by any meta-ontology.

The fourth chapter will highlight the theoretically meaningful connec-tions between identity criteria and the Leibnizian principle of the identity of indiscernibles. We will draw distinctions between the different version of the principle that we already described and how they relate in different ways to identity criteria. It will be shown that the inclusion of identity criteria in one’s ontology necessarily leads to two meta-ontological consequences: a commitment to the trivial and strong version of the identity of indiscernibles and also to a rejection of the proper version and of any version weaker than that. These are two separate consequences since accepting the trivial and strong version of the identity of indiscernibles does not entail the rejection of the proper and weak versions. In fact, it is entirely possible for one theory to hold both the trivial or strong versions and the proper or weaker versions as true simultaneously. One can, for example, claim that if x and y share all their properties (identity and difference included), then they are identical (thus claiming the truth of the trivial version of the principle), and simulta-neously one can hold that even in the case where x and y share all and only their qualitative properties, x is identical to y (which amounts to claiming the truth of the proper version of the principle).

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Chapter 1 Formal structure

In order to inaugurate a fruitful discussion on the role, function and legit-imacy of identity criteria, as well as on the consequences of supplementing one’s meta-ontology with such concept, some agreement is needed on the ba-sic formal structure that they embody, since contra principia negantem non est disputandum. The issue nevertheless prompted a great deal of philosoph-ical and logphilosoph-ical debate, so that essentially every single glyph in the standard formulation we are about to present has been questioned or deemed unsuit-able at some point. The following scheme, presented by E.J. Lowe as the ‘B-form’ identity criterion (Lowe 1989a, 6), can claim the highest degree of widespread acceptance in the technical literature on the topic:

(ICp) (∀x)(∀y)(Φ(x) ∧ Φ(y)) → (x = y ↔ R(x, y))

The formula reads: for every entity x and y, if they belong to the same sortal, then they are numerically identical if and only if they stand in a specific relation R, which must be an equivalence relation, i.e. it must be transitive, symmetric and reflexive. The only scheme that can rival (ICp) with regard to consensus would be the two-level formulation of identity criteria, which will be explained in section 1.2.

The main objective of this chapter is a review of the disparate arguments made in favour of or opposed to (ICp), along with some proposed amend-ments to it informed by an extensive overview of the literature in the field. The final outcome of the chapter is the replacement of (ICp), where the ‘p’ signifies its provisional status, with the definitive (IC). The resulting formal account will inform the discussion in the next chapters.

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1.1 Sortal specification

The present section will exhibit evidence for the necessity of the restriction of the range of values of the variables in (ICp) to a specific sortal. We call this sortal specification: formally it consists in the attribution to variables x and y in (ICp) of a sortal predicate Φ, in the form Φ(x) ∧ Φ(b). The idea is grounded in the understanding that, for any identity statement a = b, there is some sortal concept Φ such that both a and b are Φs. Eminent and influential authors in the field of ontology have considered this a relatively uncontroversial claim, and proceeded to implent it into their logical systems. Lowe (2009, 72) names it ‘Sortal Expandability Thesis’:

(S) a = b ↔ (∃Φ)(a =Φ b)

where a =Φ b is to be understood as a = b ∧ Φ(a) ∧ Φ(b) (we slightly altered

Lowe’s notation to match ours). David Wiggins (2001, 17) came to the same conclusion before Lowe. His reasoning is derived from Quine’s treatment of the sentence ‘a is the same donkey as b’, which is considered to be a condensation of ‘a is a donkey and a is the same as b’ (Quine 1964). Any identity statement ‘a = b’, in conjunction with the proposition that a is something, i.e. a is the instantiation of a sortal, is equivalent to the claim that a is the same something as b.

Thus, the only assumption needed to obtain a =Φ b from a = b is Φ(a).

Therefore, to argue against (S), one is committed to the claim that there are some particulars that do not instantiate any sortal at all. This view is sometimes referred to as ‘theory of bare particulars’ (Lowe 1998, 141; Wiggins 2001, 8). However, bare particulars are not a popular notion in ontology anymore, the theory concerning them is virtually dead in the field and therefore it cannot be the backbone of a solid rebuttal of Wiggins’s argument.

An additional argument for the necessity of sortal specification in (ICp), albeit not a definitive one, is that individuating an identity criterion which is not supplied with a sortal specification seems like a hopeless enterprise. Without restriction on the values of the variables, one would have to identify a suitable relation R that holds for the identity of every conceivable sortal, from ‘material object’ to ‘chiliagon’, from ‘event’ to ‘personality’, from ‘elementary particle’ to ‘property’. Although it cannot be shown in a conclusive way, it is almost self-evident that whatever R can be found to account for the

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identity of any two entities in general, it will violate one of more of the requirements for identity criteria which will be the topic of the next chapter, and, in particular, it would struggle with the conditions of non-circularity and informativity.

These considerations make a compelling case for the inclusion of sortal specification in (ICp).

1.2 One-level and two-level identity criteria

The second clause of (ICp), i.e. x = y ↔ R(x, y) has been so intensely discussed that both formulae flanking the biconditional symbol require a separate analysis. The identity statement x = y alone is indeed tied to a debate between two prominent figures in the identity criteria literature: the aforementioned E.J. Lowe (1991) and Timothy Williamson (1991, 2013). Their respective stances will be presented and the preeminence of Lowe’s proposal will be defended.

It is appropriate to comment here on the biconditional sign in itself. This glyph embeds the necessity and sufficiency of the identity criterion. Most authors agreed that both these conditions are essential in order to obtain a legitimate criterion of identity. Consider these two examples of possible identity criteria with respect to object-identity and personal identity:

(1) If x and y are material objects, then, if x = y, x is perceptually indis-criminable from y

(2) If x and y are stages of personhood, then, if they are in a relation of overwhelming and well-founded psychological resemblance, x = y (1) employs a relation like R1 (perceptual indistinguishability) as a

neces-sary condition for the identity of material objects, that, however, fails to be sufficient. It can very well be the case that two objects are perceptually in-discriminable without being numerically identical. In the same fashion, the relation R2 (extreme psychological resemblance) is sufficient to establish the

identity of consciousness, but it is not necessary: it might be the case that a sudden event, like amnesia or a head injury leading to massive and abrupt personality shifts, allows for x = y to hold while R2(x, y)does not.

In formal terms, (1) and (2) can be captured by this two modified versions of (ICp):

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(ICp1) ... (x = y → R(x, y))

(ICp2) ... (R(x, y) → x = y)

(1) and (2) are examples provided by Williamson (1986, 383-4). (ICp1) and

(ICp2) have also been referred to as ‘weak identity conditions’ (Guarino and

Welty 2000). We will not consider such schemes as acceptable amendments to (ICp), and therefore we will not accept only necessary or only sufficient iden-tity criteria as legitimate, since they lack one of the few formal constraints we imposed on R: transitivity. A scenario in which object a is indistinguishable from b and the same holds for b and c, but not for a and c defeats (1), while a scenario in which a person undergoes a series of gradual changes defeats (2) (Williamson 1986, 381). We argue Williamson would have agreed with this choice since he spent a great deal of theoretical work in an attempt to approximate R1 and R2 into proper candidates for a genuine equivalence

re-lation (Williamson 1986). Simply put, in order for an identity criterion to be valid, one needs a condition that is both necessary and sufficient, on pain of violating the transitivity requirement imposed on R.

1.2.1 The distinction

The theoretical core of this section is the distinction between one-level and two-level identity criteria, advocated by Williamson (2013, 145), implicitly drawn by Frege (1950, 75) in presenting his criterion for the identity of di-rections, and made explicit by Dummett (1973, 580-1). The main difference between the two flavours of criteria lies in the logical form that they instan-tiate. The main feature that qualifies a two-level identity criterion is the fact that the identity statement is not flanked by the entities that the criterion quantifies over, but by another class of entities functionally related to them. Consider the aforementioned Fregean criterion for directions, where P is the parallel predicate:

(Di) (∀x)(∀y)(L(x) ∧ L(y)) → ((d(x) = d(y)) ↔ P (x, y))

It reads: for every x and y, if they are lines, then the direction of x is identical to the direction of y if and only if x and y are parallel. One can clearly see how in (Di) the initial quantification ranges over lines (L is the predicate of linehood), while the identity statement deals with entities functionally

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related to lines, i.e. their directions (d being a function that maps from lines to their directions).

Another example of two-level identity criterion can be the criterion for properties, as expressed by Jackson (1982). In his model, predicats, such as ‘redness’, ascribe properties, such as ‘redhood’ to objects. Thus, two proper-ties are identical if and only if the predicates they are ascribed by are synony-mous, i.e. they have identical meaning (an example of predicate synonymy would be the perfect identity of meanings of the predicate of bachelorhood and of the predicate of unmarried-manhood) In formal terms:

(Prop) (∀x)(∀y)(Pred(x)∧Pred(y)) → ((Prop(x) =Prop(y)) ↔ S(x, y))

(Prop) reads: for every x and y, if x and y are predicates, then the properties they ascribe are identical if and only if x and y are synonymous. Here Pred(x) is a predicate while Prop(x) is a function mapping from predicates to the properties they ascribe. Without discussing the actual merit of this criterion, we can evaluate its structure. Once again, the initial quantification ranges over predicates, while the identity statements concerns entities that are functionally related to predicates: properties. It is another example of a two-level criterion.

In order to further elucidate the distinction between one-level and two-level, consider the Axiom of Extensionality, the identity criterion for sets: (AE) (∀x)(∀y)(S(x) ∧ S(y)) → (x = y ↔ (∀z)(z ∈ x ↔ z ∈ y))

In (AE) the initial quantification ranges over sets (S is the predicate of sethood), and the identity statement is flanked by variables ranging over sets. There is no reference to a different kind of entities related to sets via a function, therefore this is an example of a one-level criterion.

The introduction of this distinction between these two kinds of identity criteria paves the way for questions like: what is legitimacy of two-level identity criteria? Is it possible to reduce one to the other, i.e. to successfully achieve a method of translation for two-level identity criteria in the one-level scheme, or vice versa? Are both schemes needed? The next sections will address the point of view of Williamson and Lowe.

1.2.2 Williamson’s argument

Williamson (1991; 2013, 144 ff) argues for two main points: the first is the irreducibility of two-level identity criteria to one-level identity criteria,

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i.e. the impossibility of finding a reliable method to effectively translate the former to the latter, without any significant loss of meaning (we can call this irreducibility claim); second, the superiority of two-level identity criteria to their one-level companions, in virtue of two arguments (superiority claim).

The first argument to support the superiority claim goes as follows: iden-tity is, by overwhelming consensus, considered to be logically primitive, i.e. it cannot be explained in logically more basic terms. No formula can be more fundamental than x = y, yet one-level identity criteria claim to explain precisely that formula in terms of another formula, R(x, y). Williamson sees this as incompatible with the primitiveness of identity. On the other hand, two-level identity criteria do not run into the same problem. Consider the cri-terion for directions (Di): the identity statement is not x = y but d(x) = d(y), for which a more basic formula, specifically P (x, y), can be provided without violating the primitiveness of identity (Williamson 2013, 145).

The second argument deals with the much-discussed requirement of non-circularity for identity criteria. Informally speaking, an identity criterion is circular if it appeals to the identity it is supposed to settle. In section 2.1.5 we will analyse more precisely what the requirement consists in, nevertheless it will not align with Williamson’s understanding of it. Williamson claims, in order to argue for the superiority of two-level criteria, that the non-circularity requirement, which was, to some extent, unclear and lacking a formal defini-tion for one-level criteria, is rather easy to capture formally when it comes to two-level criteria. A two-level criterion is circular iff the function symbol on the left-hand side (in the case of the criterion for direction, the function symbol is d) also appears on the right side, (ibid., 144). Since non-circularity cannot be expressed formally in a similarly straightforward fashion for one-level criteria, two-one-level criteria are superior.

The argument from primitiveness and the argument from non-circularity are enough, according to Williamson, to substantiate the superiority claim.

1.2.3 The irreducibility claim

This section will present the replies formulated by Lowe in response to the irreducibility claim and Williamson’s rebuttals. Lowe’s comments on the arguments for the superiority claim will be addressed in the next section.

Lowe argues for the reducibility of two-level identity criteria to the one-level class. The reducibility claim is two-folded: (a) two-one-level criteria can be

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the converse operation cannot be successfully accomplished.

Claim (b) is supported by the following argument: the main formal dif-ference between the two classes of criteria is that, for two-level criteria, there is a structural necessity of reference to another class of entities. This means that in order to find a second-level identity criterion for a class of entities Φ, there must be a class of entities Φ0 _{to which Φs are functionally related to.}

For instance, a two-level criterion for directions structurally needs to include reference to a group of entities functionally related to directions, i.e. lines. One-level criteria completely lack this requirement, and therefore claiming that any one-level criterion can be translated into two-level form amounts to claiming that for any object for which a one-level identity criterion is given, there necessarily is a class of entities in a suitable functional relation to it. It is obvious that directions are always directions of lines, but what about more problematic cases, like human beings or material objects, or polygons? In or-der for their one-level identity criterion to be translated into two-level form, one needs an uncontroversial, well-founded understanding of what these enti-ties are entienti-ties of. Lacking this feature for most kinds of entienti-ties an ontology can be interested in, with the exception of lines and numbers, translation in this direction seems hopeless.

With regard to claim (a), the reformulation of two-level identity criteria in one-level form, Lowe’s claim is that, for instance, (Di) can be effectively rendered as a one-level identity criterion as follows:

(Di0_{) (∀x)(∀y)(D(x) ∧ D(y)) →}

(x = y ↔ (∀w)(∀z)(L(w) ∧ L(z) ∧ Of (x, w) ∧ Of (y, z) → P (w, z)) where D is directionhood, L is, again, linehood and Of ness is the relation of a direction to the line that has it. (Di0_{) expresses the idea that if directions}

x and y are such that the lines which they are directions of are parallel, they are identical. The initial quantification ranges over directions, and so do the variables flanking the identity sign. This looks prima facie as a legitimate equivalent expression to (Di). Nevertheless, Williamson raised a number of doubts on the appropriacy of such operation (Williamson 1991):

Firstly, Williamson argues, Lowe’s proposal can be reformulated as fol-lows:

(Di00_{) (∀x)(∀y)(D(x) ∧ D(y)) → (x = y ↔ (∃z)(L(z) ∧ Of (x, z) ∧ Of (y, z)))}

It reads: for every x and y, if they are directions, then x = y if and only if there is a line z such that z has x as its direction and z has y as its direction.

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Williamson claims that (Di00_{), on one hand, violates our immediate}

under-standing of the nature of directions, since it does not mention the property of parallelity, which is supposed to be crucial for the essence of directions; on the other hand, (Di00_{) is unable to differentiate between directions and}

other qualitatively different properties of a line, e.g. length, since it holds indiscriminately for both these entities. Consider how, if there is a line z that has x as its length and y as its length, then x = y: (Di00_{) holds with}

equal efficacy for two kinds of entities whose different natures seem to beg for different criteria! To summarize: in Williamson’s view, Lowe’s proposed translation (Di0_{) can be rendered into the equivalent form (Di}00_{), which is}

theoretically undesirable for the two reasons mentioned: it does not seem to deal with the nature of directions and it can easily be applied to entities that are not directions, e.g. lengths.

Lowe (1991) replied that Williamson was not addressing his original for-mula, but another formula that can be derived from it. Lowe’s (Di0_{) explicitly}

mentions parallelism and it explicitly not applies to lengths. The flaws high-lighted by Williamson are not present in (Di0_{), but only in (Di}00_{). If this was}

sound reasoning, Lowe argues, one could accuse the Axiom of Extensionality (AE) of being inadequate, since from (AE) and the definition of set-inclusion one can obtain (Se0_):

(Se0_{) (∀x)(∀y)(S(x) ∧ S(y)) → (x = y ↔ (∀j)(j ⊆ x ↔ j ⊆ y))}

which reads: for all x and y, if they are sets, then x = y if and only if every subset of x is a subset of y and vice versa. It is clear that (Se0_{) displays one}

of the flaws of (Di00_{): it does not do justice to the nature of sets, since it does}

not mention elements. Lowe agrees that (Se0_{) is erroneous, but he disagrees}

on the fact that, since (Se0_{) is to be rejected, (AE) must suffer the same fate.}

(Se0_{)’s flaws are not reflected in (AE), which is a perfectly valid criterion,}

and the same applies to (Di0_).

Williamson’s second criticism of Lowe’s (Di0_{) deals with the fact that}

(Di0_{) employs a relation, Of ness, that is beyond the conceptual resources in}

Frege’s original (Di), i.e. there is no such relation in the language used to express the two-level criterion for directions. Lowe’s reply is that (Di0_{)’s aim}

was never perfect synonymy, but functional synonymy, which allows him to use in his translation relations and predicates that were absent in the original formulation, if the outcome is a translation that is operationally identical to the original. Furthermore, Lowe adds, one might argue that the meaning of

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the Of ness relation must already be grasped, albeit implicitly, in order to understand d(x), the function that maps from each line to its direction (Lowe 1997, 999).

It is precisely on this last point that Williamson’s last criticism is nested. The relation of Of ness in (Di0_{) is defined in terms already included in (Di):}

Of (x, w) (the relation between direction x and the line that instantiates it, w) would be a paraphrase for d(w) = x (the function that maps from line w to its direction x). Thus, the goal of disposing of the functional term d, which was the main aim of Lowe’s translation, would not be achieved: the functional term d would still be included in (Di0_{), just disguised in the form of}

the Ofness relation. In response, Lowe highlights the fact that paraphrasing do not establish priority, since it is a symmetrical relation and therefore the same criticism could be applied to Williamson’s (Di). There is nothing more primitive in (Di) with respect to (Di0_).

1.2.4 The superiority claim

The previous section thoroughly addressed Williamson’s first original point, the irreducibility claim, and the various arguments are rebuttals between the two philosophers. The present section will address the superiority claim, which Williamsons defended via the argument from primitiveness and the argument from non-circularity.

The primitiveness point stems from a misconception: Lowe’s understand-ing of identity criteria is not that they are, in any sense, definitions, not of the identity relation, nor of such relation in a restricted sense, nor of the sortals involved. On this point, not surprisingly, Williamson and Lowe are in perfect agreement. Identity is a primitive notion, but this does not imply that the identity between two entities belonging to a sortal, e.g. triangles, cannot be expressed in ‘more basic terms’ (without this violating the primitiveness of identity), e.g. in terms of the identity of two sides and the included angle.

Lastly, the non-circularity argument can be approached acknowledging that the non-circularity constraint is controversial in its structure. Several formulations have been proposed to capture its meaning and Williamson’s claim that it makes more sense when applied to the two-level class is, quite ironically, circular. A couple of pages before stating his argument, he asserts that he lacks a precise definition for non-circularity and then he puts forward a proposal explicitly modelled around two-level identity criteria (Williamson 2013, 146): ‘an identity criterion is circular iff the function symbol on the

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right-hand side also appears on the left’. In other words: it is obvious that Williamson’s definition of non-circularity perfectly fits two-level identity cri-teria precisely because the formulation of Williamson’s definition was based on that class of criteria. It is not surprising that a definition of circular-ity tailored to two-level identcircular-ity criteria is a better fit for two-level criteria, rather than one-level ones. The whole argument is circular and, thus, inef-fective: non-circularity can be understood in ways that, unlike Williamson’s definition of circularity, do not apply exclusively to two-level identity criteria. In light of the unfolding of this debate, we consider one-level identity criteria the only basic and fundamental formal structure of identity criteria in general.

1.3 Amending (ICp)

In the provisional formal scheme (ICp)

(ICp) (∀x)(∀y)(Φ(x) ∧ Φ(y)) → (x = y ↔ R(x, y))

the R relation is, in some capacity, the core of the debate on identity cri-teria. Generally speaking, when philosophers argue about the adequacy of the criteria for the identity of some sortal, the discussion revolves around the adequacy of the interpretation given to the R relation. Is the specifi-cation of R for material objects, which consists in the relation of sameness of spatio-temporal coordinates, acceptable to settle the issue of their iden-tity? Is there any legitimate interpretation of R for properties or sentences? These discussions, as relevant as they may be, do not impact in any way the suitability of R as the best formal scheme to capture the essence of identity criteria. Quite interestingly, this assumption is very rarely disputed and even more rarely advocated, to the extent that a potential critic would face serious trouble in finding arguments to rebut. The only formal point with regard to R that almost every author seems to agree upon is that R must be an equivalence relation, and therefore transitive, symmetric and reflexive, since it is in connected through a biconditional to an identity statement.

Despite being effective in capturing some features of the concept, R is not the only one, and, we argue, not even the most effective way, to model the left-hand clause of the identity condition. An alternative formal scheme could discard the problem by disposing of the relation between x and y, in favour of

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another formulation that does not directly associate the two variables. This is the topic of the next section.

1.3.1 Identifying properties and characteristic relations

Consider the following examples of identity criteria (in their informal version, marked by an asterisk):

(NN*) For all x and y, if they are natural numbers, then x = y if and only if they have the same predecessor

(MO*) For all x and y, if they are material objects, then x = y if and only if they have the same spatio-temporal location

(Ev*) For all x and y, if they are events, then x = y if and only if they have the same causes and effects

(Ci*) For all x and y, if they are circles, then x = y if and only if they have the same radius.

(NN*) and (Ci*) are unanimously accepted in the field, (MO*) is widely recognized as our best option for the identity of material objects (Strawson (1959, 36-8) and Lowe (1998, 165) being two notable exceptions), (Ev*) has been heavily discussed in virtue of its supposed violation of the non-circularity requirement (a proper treatment of the topic in 2.1.5). We picked random identity criteria for all over the spectrum of accepted and rejected accounts. Yet, every instance of identity criteria displays the same feature, namely the reference to sameness of other entities. This holds for every other identity criterion that we encountered in the field (regarding polygons, properties, meanings, organisms, lengths, heaps and so forth, as well as the already mentioned sets). Our intuition is that the equivalence relation R ultimately amounts to a relation between the sortal to which x and y belong and a second sortal whose entities are related to the first sortal. In other words, a criterion of identity for Φs seems to rely on the reference to another sortal Ψ, and Φs are connected to Ψs via specific relations.

Now, the sheer fact that every identity criterion so far conforms to this impression is not sufficient to establish conclusively that no identity criteria can defy our intuition. Induction can only lead us so far. Moreover, one could object that (Di0_{) or (Prop) do not adhere to our idea, since they do}

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not openly mention sameness of different entities. Nevertheless, parallelity is nothing but sameness of the slope for straight lines, and synonymy is tantamount to sameness of meaning; (Di0_{) can be expressed without any}

semantic loss as:

(Di000_{) (∀x)(∀y)(D(x)∧D(y)) → (x = y ↔ (∀w)(∀j)(L(w)∧L(j)∧Of (x, w)∧}

Of (y, j)) → (∀(z))(Q(w, z) ↔ Q(j, z))

(Di000_{) reads: for every x and y, if they are directions, then x = y if and only}

if, for every line w and z such that w has direction x and z has direction y, for every slope j, w has j as slope if and only if z has j as slope. Parallelity is thus conceived as a relation that both lines instantiating direction x and y entertain with a different entity, the slope j. Along the same lines, (Prop) can be translated as:

(Prop0_{) (∀x)(∀y)P (x) ∧ P (y)) → (x = y ↔ (∀m)(∀n)(Pred(m)∧Pred(n)}

∧ Of (x, m) ∧ Of (y, n)) → (∀(z))(S(m, z) ↔ S(n, z))

which reads: for every x and y, if they are properties, then x = y if and only if, for every predicate m and n such that m ascribes property x and n ascribes property y, for every meaning z, m has z as meaning if and only if n has z as meaning.

It seems that our intuition that identity criteria settle identity in terms of identity of other entities can be applied also to criteria where this aspect is not prima facie evident. Criteria like (Prop) and (D0_{) can be succesfully}

translated into (Prop0_{) and (D}000_{), that do settle identity in terms of the}

identity of other entities.

Guarino and Welty (2000) reached roughly the same conclusion: Suppose we stipulate, e.g., that two persons are the same iff they have the same fingerprint: the reason why this relation can be used as an IC [identity criterion] for persons lies in the fact that a property like ‘having this particular fingerprint’ is an identify-ing property, since it holds exactly for one person. Fidentify-ingerprints are then identifying characteristics of persons. Identifying prop-erties can be seen as relational propprop-erties, involving a character-istic relation between a class of individuals and their identifying characteristics (Guarino and Welty 2000, 3, our emphasis).

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The identity of the entities falling under sortal Φ, for instance persons, is settled in terms of the identity of another class of entities (fingerprints) that are connected to the original entities via a specific relation (the relation ‘... has as fingerprints...’). In other words, the correct way to formalize our previous intuition regarding identity criteria as relying on the identity of another class of related entities is the following: every object a within a specific sortal Φ is in a specific relation R (characteristic relation) with an object z (i.e. a’s identifying characteristic) of another sortal Ψ such that it is the case that a, and only a, is R-related to b and such that there is no other entity in Ψ that a is R-related to. Thus, a is identical to another Φ, b, if and only if both a and b are R-related to the same z.

These new formal concepts introduced by Guarino and Welty need a proper nomenclature and definition:

1. Identifying property: a property that holds exactly for one entity of kind Φ (e.g. the property of having a precise slope for lines, or the property of having a specific totality of members for sets). We refer to it as property Ω.

2. Identifying characteristics: a class of entities such that each of them is uniquely paired to entities of sortal Φ (e.g. slopes for lines, totality of members for sets, spatio-temporal regions for material objects). We refer to it as the class of zs.

3. Characteristic relation: the relation held by entities in Φ with their identifying characteristics (e.g. the relation between lines and slopes, sets and totality of members, circles and radii). We refer to it via the symbol χ.

Others have tapped into this insight. Noonan (2009) notably argued that identity criteria in their standard form (ICp), are equivalent to the conjunction of two clauses:

(a) (∀x)(Φ(x) → R(x, x))

(b) (∀x)(Φ(x) → (∀y)(Φ(y) ∧ R(x, y) → (x = y)))

(a) says that if x is a Φ, then there is a certain relation that x bears with x itself. (b) adds that if any other entity y is in an R relation with x, then x = y.

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This analysis shows how ultimately R amounts to the relation that holds exclusively for one entity belonging to the sortal Φ with itself. R, when dealing with sets, is interpreted as the relation of having the same elements; when dealing with material objects, it is the relation of having the same spatio-temporal coordinates. It is not difficult to see how we can extract from this a property that only one single entity of kind Φ can have the identifying property Ω. If the relation R for sets is the relation of having the same elements, the Ω property is the property of having that exact totality of elements, so that for any x and y both having that property, x = y. And, with equal ease, we can extract the characteristic relation and the identifying characteristics. If Ω(x) for objects is the property of having some specific spatio-temporal coordinates, then it can be modelled as a relation χ between the material objects, x and that specific spatio-temporal coordinates a: χ(x, a).

With relatively strong confidence on the legitimacy of this understanding, we can let the intuitive reformulation of (D0_{) into (D}000_{) and of (Prop) into}

(Prop0_{), and the accepted logical form of the axiom of extensionality (AE)}

guide the amendment of (ICp), that constitutes the prime achievement of this chapter:

(IC) (∀x)(∀y)(Φ(x) ∧ Φ(y)) → (x = y ↔ (∀z)(χ(x, z) ↔ χ(y, z)))

It reads: for every x and y, if they are Φs, then x = y if and only if for every identifying characteristic z, x is in a specific characteristic relation χ with z if and only if y is in a specific characteristic relation χ with z. Note that this is not the only possible amendment of (ICp) that fits the new model based on Guarino and Welty’s insight. In Chapter 4 we will present a version of (IC), labelled (ICΩ) which privileges Ω, the identifying property over the

combination of χ and zs. The two formulations are, however, equivalent. (IC) is our desired point of arrival for several reasons: it effectively cap-tures our intuition that the identity of Φs is always settled by reference to another kind of entities in a specific relation with Φs; it also employs the con-cepts provided by Guarino and Welty, since zs are identifying characteristics and χ is the characteristic relation. In light of all these reasons, we assume (IC) to be the definitive formal account of identity criteria which will inform our research from the next chapter on.

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Chapter 2 Requirements and functions

Once we have settled the formal part of the debate surrounding identity cri-teria, the notion must be further analysed and specified, in order to eliminate from the class of genuine identity criteria some very questionable candidates whose inclusion would violate our intuition on what identity criteria ought to be. Not any statement that can be formalized according to our scheme (IC) is to be regarded as legitimate. Consider, for example, the defective criterion for sets adduced by Lowe (1991) that we presented in section 1.2.3. This criterion seems to abide by the formal structure outlined in the previous chapter:

(Se0_{) (∀x)(∀y)(S(x) ∧ S(y)) → (x = y ↔ (∀j)((j ⊆ x) ↔ (j ⊆ y))).}

(Se0_{) reads: if x and y are sets, then x = y if and only if for every subset z, z is}

a subset of x if and only if z is a subset of y. (Se0_{) complies with our formal}

requirements: it is a one-level criterion supplied with sortal specification (x and y are restricted to sets), with an identifying characteristic (subsets) and a characteristic relation (χ here is the relation of set-inclusion). It also happens to be an exact criterion, meaning that sets a and b are numerically identical iff it cannot be the case that one includes a set that the other does not. Nevertheless, something is immediately problematic with (Se0_).

This criterion is formally adequate, yet intuitively unsatisfying. Therefore there must be some supplementary condition a criterion must meet, alongside formal adequacy. The enumeration of these conditions will be the subject of section 2.1.

The second avenue to offer supplementary clarification on identity criteria is the analysis of the most prevalent uses of the notion, of the function

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iden-tity criteria can fulfil. Is an ideniden-tity criterion to be considered a statement regarding the very nature of the objects it concerns, their ontological essence, or a method to assess our knowledge of their identity, or the central concept in fixing the meaning of the word, or a definition of identity as restricted to its sortal? Section 2.2 will address these points.

2.1 Requirements for adequacy

This section will present an overview of proposed conditions that identity criteria must meet in order to qualify as such. Unlike the first chapter, most of the entries in the catalogue were not object of fierce debating, and the variations between authors only consist of what number of the following req-uisites they explicitly endorse (arguably most authors have an understanding of the totality of them), with one notable exception: the non-circularity con-dition. The discussion upon the most suitable way to formalize the need for an identity criterion in order to avoid circularity involved Davidson (1969, 2001), Quine (1985), Horsten (2010) and, once again, Lowe (1989b). Some additional requirements are present in the literature, however, we argue, de-spite being labelled with disparate names, they are perfectly tantamount to criteria we did include.

The requirements are marked with the exclamation point and are all formulated as an implication whose antecedent is: ‘If a criterion of identity is adequate...’. The only exception being the two initial conditions, who are sufficient to grant the label of ‘identity criterion’ to any proposition fulfilling them (hence their antecedent is: ‘a proposition is a criterion of identity if and only if...’). On the other hand, no single condition is single-handedly sufficient to ensure adequacy; however, the conjunction of all requirements is to be considered as such in our view.

2.1.1 Formal and material adequacy

The most self-evident, and consequently the least interesting conditions to be met are formal and material adequacy. The names for the criteria have been borrowed from Horsten (2010, 421).

(FA!) A proposition is a criterion of identity if and only if it is structured according to the scheme (IC).

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(MA!) A proposition is a criterion of identity if and only if it is correct, i.e. it can’t be the case that it turns out false for some Φs.

(FA!) and (MA!) are essentially our starting point. They are so general that our paradigm of intuitively unacceptable identity criterion (Se0_{) actually}

qualifies for both. Two-level identity criteria can, as shown in 1.2.3, be amended in order to meet (FA!). (MA!) is relatively unproblematic while dealing with abstract objects, since deduction offers necessary and universal conclusions: it cannot be the case that in a hundred years from now humanity discovers a new triangle for which the criterion of the identity of two sides and the included angle does not hold. The same necessity cannot be found on identity criteria regarding the natural world. Induction cannot prevent us from considering the possibility of human beings x and y that are numerically different but have the exact same DNA sequence at t, or the chance of the discovery of material objects that can happily overlap. For this reason, one can consider (MA!) as restricted to the current scientific understanding of the world.

2.1.2 Non-tautology

The non-tautology condition is rather straightforward. Simply put, this re-quirement states:

(NT!) If a criterion of identity is adequate, then it cannot employ identity as characteristic relation χ.

(NT!) is necessary to prevent something like:

(Se00_{*) For all x and y, if they are sets, then x = y if and only if for every set}

z, z = x iff z = y

from being considered a legitimate identity criterion for sets. Note that (Se00₎

satisfies (FA!) and (MA!) and, in contrast to what one might think, it does not violate the non-circularity condition, as we will illustrate. Thus, a specific condition is required, in order to exclude all kind of tautological identity criteria. Horsten is essentially agreeing with this understanding when he says: “An identity criterion must not be a truth of logic” (Horsten 2010, 419). This requirement, however, is connected with informativity (for which we have a different definition), and not non-tautology.

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We borrowed (NT!) from Jackson (1982, 292). Nevertheless, in the liter-ature the term non-tautology condition denotes a different kind of require-ment, namely the condition that the clause (∀z)(χ(x, z) ↔ χ(y, z)) does not hold for every couple of Φs; it must be the case that for some a and some b (a and b being Φs), (∀z)(χ(a, z) ↔ χ(b, z)) is false.

Such is the definition provided by Brand (1977, 331) and by Carrara and Gaio (2010, 7). We consider this formulation of (NT!) virtually useless: if it was the case that the clause can be satisfied by any a and b included in sortal Φ, it would also be the case that the identity criterion embedding it would violate the material adequacy condition (MA!).

If any arbitrary Φ a and b can make (∀z)(χ(a, z) ↔ χ(b, z)) true, then (∀z)(χ(a, z) ↔ χ(b, z)) would be true also in the case in which a and b are not identical. Therefore it would be possible to have a = b as false while (∀z)(χ(a, z) ↔ χ(b, z)) is true. Hence, (a = b ↔ (∀z)(χ(a, z) ↔ χ(b, z))) would be false.

Thus, the criterion:

(IC) (∀x)(∀y)(Φ(x) ∧ Φ(y)) → (x = y ↔ (∀z)(χ(x, z) ↔ χ(y, z))

would turn out false, since Φs a and b would make a = b false and (∀z)(χ(a, z) ↔ χ(b, z)) true.

2.1.3 Informativity

Various accounts of the informativity requirement have been produced. Horsten (2010, 419) states that for a criterion of identity to be formally in-formative, it must not be a truth of logic. Carrara and Gaio (2009), following the path outlined by Brand (1976, 137), make the informativity requirement lie on the assumption that an identity criterion needs to furnish a specific un-derstanding of the sortal it relates to. This entails that an identity criterion must contribute to the specification of the nature of the sortal.

Another possible way to tackle the issue is advanced by Lowe. According to his definition, an informative identity criterion is “at least non-trivial and non circular” (Lowe 1998, 59). We can assume non-triviality to be equivalent to (NT!) and non-circularity will be discussed in the upcoming sections. Nev-ertheless, Lowe states that his understanding of the informativity condition can effectively disqualify the fallacious identity criterion for sets (Se0_):

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iden-explicitly or implicitly, in the statement of the relevant criterial condition Cφ (Lowe 2009, 23).

Since (Se0_{) explains the identity of any two sets in the identity of other sets,}

specifically subsets of them, it cannot be considered informative. The next section on non-circularity will show why such concept of informativity cannot be accepted in our framework, and, ironically, how Lowe himself delivers the arguments in favour of the rejection of his definition.

We opt for the formulation by Carrara and Gaio (2009) and Brand (1977, 137) and define the informativity condition as:

(I!) If a criterion of identity is adequate, then it must specify or contribute to specifying characteristics essential (i.e. non-contingent) to the cor-responding sortal Φ.

To some extent Lowe is right: (I!), albeit employing a different definition of the informativity condition, can dispose of (Se0_{), since the latter does not}

contribute in any way to the specification of the nature of sets. Given that every set is a subset of itself, (Se0_{) is stating, in a way that avoids violating}

(NT!), that x = y if and only if x = y, which is eminently uninformative on the nature of sets.

2.1.4 Non-circularity

The history of the non-circularity condition is of theoretical interest and it deserves a thorough analysis. First, one needs an understanding of the intuition underlying this requirement, which is: a criterion of identity cannot appeal to what it is trying to elucidate, i.e. it cannot employ identity of Φs to state the condition of identity for Φs. When it comes to actually defining the constraint that this intuition entails, opinions differ widely. The casus belli for the debate we want to analyse was Davidson’s criterion for the identity of events (Davidson 1969), which we already stated in informal terms in 1.3.1: two events are numerically identical if and only if they have the same effects and causes, or:

(Ev) (∀x)(∀y)(E(x) ∧ E(y)) →

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where E is the predicate of eventhood and C is a predicate that takes as arguments a cause and its effect.

Quine (1985) notoriously commented on this finding, criticizing its al-leged circularity. Quine concedes, that (Ev) is not formally circular, since the right-hand side of the biconditional does not contain any identity sign, which was Davidson’s preemptive arguments against possible objections to his criterion (Davidson 2001, 150). Nevertheless, (Ev) is informally circu-lar, or ‘impredicative’, since it does presuppose an account of the identity of events in order to assess the identity of events. If x and y are events they are numerically identical if and only if they have the same causes and effects, which are events themselves! The criterion seems to presuppose a grasp on what it is trying to express: the identity of events. If one has to establish the identity of events x and y, in order to do so according to (Ev), one needs to establish the identity of the causes of x and y, call them respectively z and w, which are events themselves. What the criterion is supposed to provide is already presupposed.

Quine’s account of impredicativity amounts to the prohibition on having the scope of variable z be the same as, or a subset of, the scope of variables x and y. In our terminology, introduced in 1.3.1: the set of identifying characteristics cannot be the same as or a subset of the set including all instantiations of sortal Φ. In (Ev), the identifying characteristics (zs) are causes and effects, and they are definitely a subset of the sortal E (events) for which the criterion is provided. After Quine’s powerful criticism, Davidson publicly renounced to defend (Ev) and acknowledged its impredicative nature (2001, 244).

Lowe (1989b) shows how Quine’s definition is not sharp enough. In his view, the core of the problem does not lie solely on the range of the variable z. Consider (AE), the criterion for sets. Lowe claims that such criterion would resist any claim of impredicativity, even when applied to ‘pure’ Zermelo-Fraenkel, or ZF, set theory, in which sets can only be empty or have other sets as elements. In such theoretical framework, the formulation:

(AE) (∀x)(∀y)(S(x) ∧ S(y)) → (x = y ↔ (∀z)(z ∈ x ↔ z ∈ y))

is Quine-impredicative. The scope of x and y is that of sets, and the same goes for z, since the universe of discourse is restricted to sets only. However, Lowe still maintains that (AE) is predicative, even when embedded in pure ZF. His reasoning is the following: the root of impredicativity lies in the fact

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this is achieved by appealing to the identity of a different kind of entities. For instance, material-object-identity is settled in terms of the identity of spatio-temporal regions, circle-identity is settled in terms of radius-identity. However, when a criterion appeals to the identity of more entities from the original sortal Φ, the question is inevitable: how can the identity of those entities be settled, and so forth?

In the case of (Ev), how can the identity of events be settled in terms of the identity of other events? Suppose one has to establish the identity of events a and b such that C(a, c) and C(b, d), where C is the relation between a cause and its effect. So, a is the cause of c, and b is the cause of d. By virtue of (Ev), a and b are the same event if they have the same effect. Thus, are c and d the same event? By applying (Ev), they are if and only if they have the same causes and effect. If C(c, e) and C(d, f), then c = d if e = f. But since e and f are also events to establish their identity we need to apply (Ev). This can go on indefinitely and event-identity will never be settled since it will always rely on other event-identities. Lowe claims impredicativity applies to (Ev), but not to (AE): the difference is that in the case of sets in pure ZF, there is an element, i.e. the empty set, that acts as bedrock to stop the reiteration ad infinitum.

In order to clarify Lowe’s point, consider sets A and B in ZF. According to (AE), they are one and the same set if and only if they have the same elements. Their elements are also sets, therefore the same criterion must be applied to them, and to their elements too, and to the elements of their elements and so on, until one of the sets in the sequence is the empty sets. This stops the infinite iteration of identity questions and settles set-identity for good. If A : {C} and B : {D}, A = B iff C = D. Suppose C : {E} and D : {F }, so that C = D iff E = F . However: E : {∅} and F : {G}. By repeating the application of (AE), at the nth application of the criterion one will reach the empty set, which stops the iterations and settles the identity of all the sets involved. In our example, E 6= F , therefore C 6= D, therefore A 6= B.

According to this view, the main difference between (Ev) and (AE) lies outside the identity criteria in themselves, in the theoretical framework in which sets and events are embedded. The requirement that must be met to avoid impredicativity is the existence, within the theory that shapes the concept of the sortal in question, of a ground-notion able to stop the infinite iteration of the identity question in virtue of its being at the base of the hierarchy. Davidson’s theory of events is not, and arguably never will be,

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able to provide a concept of this kind to ground the succession of events. The final act of this saga is Horsten’s contribution (2010). He claims that predicativity is an epistemic virtue, which some criteria possess and some do not. He argues that, when identity criteria are employed in their metaphys-ical function (see section 2.2.2), impredicativity ceases to be theoretically troublesome:

If a criterion would somehow create the objects of the sort under investigation, then circular criteria would be problematic. But since the objects exist independently of the criterion, some crite-ria may well have to be circular (Horsten 2010, 426).

In other words: the fact that we may never be able to know if event e1

is identical with e2 is not relevant if the criterion is supposed to tell us in

what the identity of events consists in (metaphysical function). (Ev) does exactly that and the fact that human agents are unable to know whether e1

and e2 are different does not change the fact that the answer to the question

is genuinely settled by (Ev). It just happens to fall outside of the grasp of our knowledge. Impredicativity is therefore a matter of applicability of a certain identity criterion and thus it is bothering only for those who employ identity criteria in a function that requires effective applicability (i.e. the epistemic function that we will analyse in 2.2.5). An impredicative criterion, like Davidson’s, can still meet all the other requirements we enumerated, informativity included.

Lowe’s stance (as expressed in Lowe 2009, 18) does fall prey to Horsten criticism: he claims to be employing identity criteria in their metaphysical and semantic function, neither of which is supposed to be bothered by a difficulty in the applicability of the criterion. We feel confident in restricting the following version of the non-circularity condition to the epistemic use of identity criteria:

(NC!) If a criterion of identity for Φs is adequate in its epistemic function, it must not quantify in the right-hand side clause of the biconditional over Φs, unless the theory of Φ informing the concept is supplied with an element which can ground the hierarchy of Φs.

Identity Criteria and Meta-Ontology