I provide a theory of the metaphysical foundations of identity: an account what grounds facts of the form a=b. In particular, I defend the claim that indiscernibility grounds identity. This is typically rejected because it is viciously circular; plausible assumptions about the logic of ground entail that the fact that a=b partially grounds itself. The theory I defend is immune to this circularity.

We investigate the theory Peano Arithmetic with Indiscernibles ( \(\textrm{PAI}\) ). Models of \(\textrm{PAI}\) are of the form \(({\mathcal {M}},I)\), where \({\mathcal {M}}\) is a model of \(\textrm{PA}\), _I_ is an unbounded set of order indiscernibles over \({\mathcal {M}}\), and \(({\mathcal {M}},I)\) satisfies the extended induction scheme for formulae mentioning _I_. Our main results are Theorems A and B following. _Theorem A._ _Let_ \({\mathcal {M}}\) _be a nonstandard model of_ \(\textrm{PA}\) _ of any cardinality_. \(\mathcal {M }\) _has an expansion (...) to a model of _ \(\textrm{PAI}\) _iff_ \( {\mathcal {M}}\) _has an inductive partial satisfaction class._ Theorem A yields the following corollary, which provides a new characterization of countable recursively saturated models of \(\textrm{PA}\) : _Corollary._ _A countable model_ \({\mathcal {M}}\) of \(\textrm{PA}\) _is recursively saturated iff _ \({\mathcal {M}}\) _has an expansion to a model of _ \(\textrm{PAI}\). _Theorem B._ _There is a sentence _ \(\alpha \) _ in the language obtained by adding a unary predicate_ _I_(_x_) _to the language of arithmetic such that given any nonstandard model _ \({\mathcal {M}}\) _of_ \(\textrm{PA}\) _ of any cardinality_, \({\mathcal {M}}\) _has an expansion to a model of _ \(\text {PAI}+\alpha \) _iff_ \({\mathcal {M}}\) _has a inductive full satisfaction class._. (shrink)

The author has previously shown that for a certain class of structures $\mathcal {I}$, $\mathcal {I}$-indexed indiscernible sets have the modeling property just in case the age of $\mathcal {I}$ is a Ramsey class. We expand this known class of structures from ordered structures in a finite relational language to ordered, locally finite structures which isolate quantifier-free types by way of quantifier-free formulas. This result is applied to give new proofs that certain classes of trees are Ramsey. To aid this (...) project we develop the logic of EM-types. (shrink)

Some authors have claimed that ante rem structuralism has problems with structures that have indiscernible places. In response, I argue that there is no requirement that mathematical objects be individuated in a non-trivial way. Metaphysical principles and intuitions to the contrary do not stand up to ordinary mathematical practice, which presupposes an identity relation that, in a sense, cannot be defined. In complex analysis, the two square roots of –1 are indiscernible: anything true of one of them is true of (...) the other. I suggest that ‘i’ functions like a parameter in natural deduction systems. I gave an early version of this paper at a workshop on structuralism in mathematics and science, held in the Autumn of 2006, at Bristol University. Thanks to the organizers, particularly Hannes Leitgeb, James Ladyman, and Øystein Linnebo, to my commentator Richard Pettigrew, and to the audience there. The paper also benefited considerably from a preliminary session at the Arché Research Centre at the University of St Andrews. I am indebted to my colleagues Craige Roberts, for help with the linguistics literature, and Ben Caplan and Gabriel Uzquiano, for help with the metaphysics. Thanks also to Hannes Leitgeb and Jeffrey Ketland for reading an earlier version of the manuscript and making helpful suggestions. I also benefited from conversations with Richard Heck, John Mayberry, Kevin Scharp, and Jason Stanley. CiteULike Connotea Del.icio.us What's this? (shrink)

The bundle theory is a theory about the internal constitution of individuals. It asserts that individuals are entirely composed of universals. Typically, bundle theorists augment their theory with a constitutional approach to individuation entailing the thesis ‘identity of constituents is a sufficient ground for numerical identity’ (CIT). But then the bundle theory runs afoul of Black’s duplication case—a world containing two indiscernible spheres. Here I propose and defend a new version of the bundle theory that denies ‘CIT’, and which instead (...) conjoins it with a structural diversity thesis , according to which being separated by distance is a sufficient ground for numerical diversity. This version accommodates Black’s world as well as the three-spheres world —a world containing three indiscernible spheres, arranged as the vertices of an equilateral triangle. In this paper, I also criticize Rodriguez-Pereyra’s alternative attempt to defend the bundle theory against Black’s case and the case of the three-spheres world. (shrink)

It is shown that typical arguments from intensionality against the Principle of Indiscernibility of Identicals (InI) misconstrue this principle, confusing it with the Principle of Substitution (PS). It has been proposed that Leibniz, in his statements like, "If A is the same as B, then A can be substituted for B, salva veritate, in any proposition", is not applying InI to objects nor PS to signs, but is talking about substitution of concepts in propositions, or applying InI to concepts. (...) It is shown in the paper that since Leibniz holds that there are exceptions to the principle thus stated, either the proposal in question is misguided, or else Leibniz is mistaken in thinking that there are such exceptions. (shrink)

Putative counterexamples to the Principle of Identity of Indiscernibles (PII) are notoriously inconclusive. I establish ground rules for debate in this area, offer a new response to such counterexamples for friends of the PII, but then argue that no response is entirely satisfactory. Finally, I undermine some positive arguments for PII.

According to strong composition as identity, the logical principles of one–one and plural identity can and should be extended to the relation between a whole and its parts. Otherwise, composition would not be legitimately regarded as an identity relation. In particular, several defenders of strong CAI have attempted to extend Leibniz’s Law to composition. However, much less attention has been paid to another, not less important feature of standard identity: a standard identity statement is true iff its terms are coreferential. (...) We contend that, if coreferentiality is dropped, indiscernibility is no help in making composition a genuine identity relation. To this aim, we analyse as a case study Cotnoir’s theory of general identity, in which indiscernibility is obtained thanks to a revisionary semantics and true identity statements are allowed to connect non-coreferential terms. We extend Cotnoir’s strategy for indiscernibility to the relation of comaternity, and we show that, neither in the case of composition nor in that of comaternity, indiscernibility contibutes to show that they are genuine identity relations. Finally, we compare Cotnoir’s approach with other versions of strong CAI endorsed by Wallace, Bøhn, and Hovda, and canvass the extent to which they violate coreferentiality. The comparative analysis shows that, in order to preserve coreferentiality, strong CAI is forced to adopt a non-standard semantic treatment of the singular/plural distinction. (shrink)

According to the indiscernibility characterization of the distinction between particulars and universals, only and all the former have possible numerically distinct indiscernible intrinsic qualitative duplicates. It is argued here that both the sufficiency and the necessity directions are defective and that indiscernibility thus does not distinguish particularity. Against sufficiency: universals may lack intrinsic qualitative character and thus be trivially indiscernible from one another. Against necessity: pluralities of duplicate-less entities are at once duplicate-less and particular.

The concept of indiscernibility in a structure is analysed with the aim of emphasizing that in asserting that two objects are indiscernible, it is useful to consider these objects as members of (the domain of) a structure. A case for this usefulness is presented by examining the consequences of this view to the philosophical discussion on identity and indiscernibility in quantum theory.

Structuralism about mathematical objects and structuralist accounts of physical computation both face indeterminacy objections. For the former, the problem arises for cases such as the complex roots i and \, for which a automorphism can be defined, thus establishing the structural identity of these importantly distinct mathematical objects. In the case of the latter, the problem arises for logical duals such as AND and OR, which have invertible structural profiles :369–400, 2001). This makes their physical implementations indeterminate, in the sense (...) that their structural profiles alone cannot establish whether a given physical component is an AND-gate or an OR-gate. Doherty has recently shown both problems to be analogous, and has argued that computational structuralism is threatened with the absurd conclusion that computational digits might be indiscernible, such that, if structural properties are all that we have to go on, the binary digit 0 must be treated as identical to the binary digit 1. However, we think that a solution to the indiscernibility problem for mathematical structuralists, drawing on the work of David Hilbert, can be adapted for the analogous problem in the computational case, thereby rescuing the structuralist approach to physical computation. (shrink)

Universals have traditionally thought to obey the identity of indiscernibles, that is, it has traditionally been thought that there can be no perfectly similar universals. But at least in the conception of universals as immanent, there is nothing that rules out there being indiscernible universals. In this paper, I shall argue that there is useful work indiscernible universals can do, and so there might be reason to postulate indiscernible universals. In particular, I shall argue that postulating indiscernible universals can allow (...) a theory of universals to identify particulars with bundles of universals, and that postulating indiscernible universals can allow a theory of universals to develop an account of the resemblance of quantitative universals that avoids the objections that Armstrong’s account faces. Finally, I shall respond to some objections and I shall undermine the criterion of distinction between particulars and universals that says that the distinction between particulars and universals lies in that while there can be indiscernible particulars, there cannot be indiscernible universals. (shrink)

This paper examines a variety of contexts in metaphysics which employ a strategy I consider to be suspect. In each of these contexts, ‘The Suspect Strategy’ (TSS) aims at excluding a series of troublesome contexts from a general principle whose truth the philosopher in question wishes to preserve. We see (TSS) implemented with respect to Leibniz’s Law (LL) in the context of Gibbard’s defense of contingent identity, Myro and Gallois’ defense of temporary identity, as well as Terence Parsons’ defense of (...) indeterminate identity. Our example of (TSS) as implemented with respect to the Existence Principle (EP) is Terence Parsons’ defense of non-existent objects. Finally, the coincidence-theorist’s analysis of the problem of constitution as given by Baker, Fine and Yablo, as well as Deutsch’s recent defense of the relative-identity theory provide examples of (TSS) as implemented with respect to restricted indiscernibility principles of the form in (RI). While we of course cannot conclude from our exposure to extant versions of (TSS) that no exclusion-procedure could ever overcome the troubling features we encountered, my remarks in this paper should, I think, at least give us reasons to be skeptical that any strategy which proceeds by means of purely formal (e.g., syntactic) individuation-criteria could achieve its intended purpose; for we have seen that such strategies are in general too coarse-grained to individuate contexts correctly into those that should and those that should not be excluded from the reaches of the general principle under discussion. I suspect, moreover, though I do not argue for this stronger claim, that any strategy which does not proceed by means of purely formal criteria would in some way succumb to the charge of circularity. As a result, I recommend that we should learn to become accustomed to a universe populated with a surprising multitude of numerically distinct yet almost indiscernible objects, such as statues and the lumps of clay that constitute them. Given the large amounts of overlap among these almost indiscernible objects, such a densely populated universe is actually quite easy to live with, but that’s a story for a separate occasion. (shrink)

We prove a theorem about models with indiscernibles that are cofinal in a given linear order. We apply this theorem to obtain new independence results for Quine's set theory New Foundations, thus solving two open problems in this field.

Universals have traditionally thought to obey the identity of indiscernibles, that is, it has traditionally been thought that there can be no perfectly similar universals. But at least in the conception of universals as immanent, there is nothing that rules out there being indiscernible universals. In this paper, I shall argue that there is useful work indiscernible universals can do, and so there might be reason to postulate indiscernible universals. In particular, I shall argue that postulating indiscernible universals can allow (...) a theory of universals to identify particulars with bundles of universals, and that postulating indiscernible universals can allow a theory of universals to develop an account of the resemblance of quantitative universals that avoids the objections that Armstrong’s account faces. Finally, I shall respond to some objections and I shall undermine the criterion of distinction between particulars and universals that says that the distinction between particulars and universals lies in that while there can be indiscernible particulars, there cannot be indiscernible universals. (shrink)

What is the meaning of general covariance? We learn something about it from the hole argument, due originally to Einstein. In his search for a theory of gravity, he noted that if the equations of motion are covariant under arbitrary coordinate transformations, then particle coordinates at a given time can be varied arbitrarily - they are underdetermined - even if their values at all earlier times are held fixed. It is the same for the values of fields. The argument can (...) also be made out in terms of transformations acting on the points of the manifold, rather than on the coordinates assigned to the points. So the equations of motion do not fix the particle positions, or the values of fields at manifold points, or particle coordinates, or fields as functions of the coordinates, even when they are specified at all earlier times. It is surely the business of physics to predict these sorts of quantities, given their values at earlier times. The principle of general covariance therefore seems untenable. (shrink)

We give definitions that distinguish between two notions of indiscernibility for a set {aη∣η∈ω>ω}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\{a_{\eta} \mid \eta \in ^{\omega>}\omega\}}$$\end{document} that saw original use in Shelah [Classification theory and the number of non-isomorphic models. North-Holland, Amsterdam, 1990], which we name s- and str−indiscernibility. Using these definitions and detailed proofs, we prove s- and str-modeling theorems and give applications of these theorems. In particular, we verify a step in the argument that TP (...) is equivalent to TP1 or TP2 that has not seen explication in the literature. In the Appendix, we exposit the proofs of Shelah [Classification theory and the number of non-isomorphic models. North-Holland, Amsterdam, 1990, App. 2.6, 2.7], expanding on the details. (shrink)

It is well known that the manner in which a definitely descriptive term contributes to the meaning of a sentence depends on the place the term occupies in the sentence. A distinction is accordingly drawn between ordinary contexts and contexts variously termed non-referential, intensional, oblique, or opaque. The aim of the present article is to offer a general account of the phenomenon, based on transparent intensional logic. It turns out that on this approach there is no need to say (as (...) Frege does) that descriptive terms are referentially ambiguous or to deny (as Russell does) that descriptive terms represent self-contained units of meaning. There is also no need to tolerate (as Montague does) exceptions to the Principle of Functionality. The notion of an ordinary (i.e., non-intensional) context is explicated exclusively in terms of logical structure and it is argued that two aspects of ordinariness (termed hospitality and exposure) must be distinguished. (shrink)

The literature on quantum logic emphasizes that the algebraic structures involved with orthodox quantum mechanics are non distributive. In this paper we develop a particular algebraic structure, the quasi-lattice ( $${\mathfrak{I}}$$ -lattice), which can be modeled by an algebraic structure built in quasi-set theory $${\mathfrak{Q}}$$. This structure is non distributive and involve indiscernible elements. Thus we show that in taking into account indiscernibility as a primitive concept, the quasi-lattice that ‘naturally’ arises is non distributive.

Universals have traditionally thought to obey the identity of indiscernibles, that is, it has traditionally been thought that there can be no perfectly similar universals. But at least in the conception of universals as immanent, there is nothing that rules out there being indiscernible universals. In this paper, I shall argue that there is useful work indiscernible universals can do, and so there might be reason to postulate indiscernible universals. In particular, I shall argue that postulating indiscernible universals can allow (...) a theory of universals to identify particulars with bundles of universals, and that postulating indiscernible universals can allow a theory of universals to develop an account of the resemblance of quantitative universals that avoids the objections that Armstrong’s account faces. Finally, I shall respond to some objections and I shall undermine the criterion of distinction between particulars and universals that says that the distinction between particulars and universals lies in that while there can be indiscernible particulars, there cannot be indiscernible universals. (shrink)

Ideally, our account of knowledge would help us to understand the appeal of (and flaws in) skepticism,2 while remaining consistent with our ‘intuitions,' and supporting epistemic principles that seem eminently plausible. Of course, we don't always get what we want; we may not be able to move from intuitions and principles to an account that fully squares with them. As a last resort, we may have to move in the other direction, and give up intuitions or principles that are undermined (...) by an otherwise compelling account of knowledge, so as to achieve ‘reflective equilibrium.'3 But last resorts come last. (shrink)

Cet article est un nouvel examen des objections soulevées par Leibniz dans la controverse avec Clarke contre la théorie absolutiste de l'espace et du temps. Or la plupart de ces objections sont fondées sur le principe de raison suffisante; mais Leibniz utilise aussi le principe de l'identité des indiscernables, qu'il prétend déduire du principe de raison suffisante . Ce qui m'intéresse c'est que Leibniz présente parfois deux versions de la même objection: l'une reposant uniquement sur le principe de raison suffisante, (...) l'autre uniquement sur le principe de l'identité des indiscernables. Se pose alors la question suivante: Leibniz aurait-il pu soulever toutes les objections qu'il voulait contre la théorie absolutiste, en utilisant seulement le principe de l'identité des indiscernables? Il me semble qu'il aurait certainement pu le faire. Et de plus, je crois que si Leibniz avait suivi cette voie, son raisonnement contre la théorie absolutiste aurait été beaucoup plus convaincant. Ainsi donc, je me propose de mener l'attaque de Leibniz contre la théorie absolutiste, en isolant, reformulant, et exposant systématiquement sept arguments dont il se servait ou dont il aurait pu se servir, qui sont tous fondés sur le principe de l'identité des indiscernables. (shrink)

We prove several results giving lower bounds for the large cardinal strength of a failure of the singular cardinal hypothesis. The main result is the following theorem: Theorem. Suppose κ is a singular strong limit cardinal and 2κ λ where λ is not the successor of a cardinal of cofinality at most κ. If cf > ω then it follows that o λ, and if cf = ωthen either o λ or {α: K o α+n} is confinal in κ for (...) each n ε ω.We also prove several results which extend or are related to this result, notably Theorem. If 2ω ω1 then there is a sharp for a model with a strong cardinal.In order to prove these theorems we give a detailed analysis of the sequences of indiscernibles which come from applying the covering lemma to nonoverlapping sequences of extenders. (shrink)

Assuming the position that takes properties to be tropes rather than universals and takes ordinary objects as bundles of tropes, the essay first argues that the Law of the Identity of Indiscernibles survives the challenge raised by Black's "two-sphere universe". It is because the Law of Indiscernibles becomes a trivialconsequence of the assumed trope ontology. The essay then considers four construals of the thesis of Uniqueness differing in strength. The construals are developed in terms of both the possibility that tropes (...) of an object may be transferred to another object and the possibility that objects may survive cross-worldproperty changes. Finally, the essay argues that a trilemma arises from three intuitively innocent claims: (1) that same world objects do not share tropes, (2) that cross world objects may share some (but not all) tropes, and (3) that objects may survive cross-world property changes. And, unfortunately, Indiscernibles plays a role in giving rise to one horn of the trilemma. The essay suggests "ultraessentialism" as a possible way out of the trilemma. (shrink)

There is a certain argument against the principle of the indiscernibility of identicals, or the thesis that whatever is true of a thing is true of anything identical with that thing. In this argument, PInI is used together with the self-evident principle of the necessity of self-identity to reach the conclusion, which is held to be paradoxical and, thus, fatal to PInI. My purpose is to show that the argument in question does not have this consequence. Further, I argue (...) that PInI is a universally valid principle which can be used to prove the necessity of identity. (shrink)

This paper is about the nature of God’s pre-creation knowledge of possible creatures. I distinguish three theories: non-qualitative singularism, qualitative singularism, and qualitative generalism, which differ in terms of whether the relevant knowledge is qualitative or non-qualitative, and whether God has singular or merely general knowledge of creatures. My main aim is to argue that qualitative singularism does not depend on a version of the Principle of Identity of Indiscernibles to the effect that, necessarily, qualitatively indiscernible individuals are identical. It (...) follows that qualitative singularism does not depend on the view that possible creatures categorically have qualitative individual essences. (shrink)

Arthur C. Danto’s The Transfiguration of the Commonplace is one of the most influential recent books on philosophy of art. It is noteworthy for both his method, which emphasizes indiscernible pairs and sets of objects, and his conclusion, which is that artworks are distinguished from non-artwork counterparts by a semantic and aesthetic transfiguration that depends on their relationship to art history. In numerous contexts, Danto has confirmed that the relevant concept of art is the concept of fine art. Examples of (...) music that are not fine art demonstrate that semantic and aesthetic transfiguration does not require a relationship to art history or art theory. Appropriate interpretation and individuation of a great deal of music can be achieved by listeners who do not grasp art theory and who do not guide their interpretation by reference to the concept of art. (shrink)

In this article I discuss identity and indiscernibility for person‐stages and persons. Identity through time is not an identity relation (it is a unity relation). Identity is carefully distinguished from persistence. Identity is timeless and necessary. Person‐stages are carefully distinguished from persons. Theories of personal persistence are not theories of identity for persons. I deal not with the persistence of persons through time but with the timeless and necessary identity and indiscernibility of persons. I argue that it is (...) possible that there are non‐identical but indiscernible temporally whole persons. I discuss the biographies of persons and develop the type or token distinction for persons. Twins in symmetrical or eternally recurrent universes are examples of indiscernible persons. I discuss temporal and modal branching, and I end with survival for person‐tokens and eternity for person‐types. (shrink)

In a recent paper, Sun Demirli (2010) proposes an allegedly new way of conceiving of individuation in the context of the bundle theory of object constitution. He suggests that allowing for distance relations to individuate objects solves the problems with worlds containing indiscernible objects that would otherwise affect the theory. The aim of the present paper is i) To show that Demirli’s proposal falls short of achieving this goal and ii) To carry out a more general critical assessment of the (...) issue by appraising the costs and benefits of Demirli’s view as well as of existing alternatives. (shrink)

Basic results on the model theory of substructures of a fixed model are presented. The main point is to avoid the use of the compactness theorem, so this work can easily be applied to the model theory of L ω 1 ,ω and its relatives. Among other things we prove the following theorem: Let M be a model, and let λ be a cardinal satisfying λ |L(M)| = λ. If M does not have the ω-order property, then for every $A (...) \subseteq M, |A| \leq \lambda$ , and every $\mathbf{I} \subseteq M$ of cardinality λ + there exists $\mathbf{J} \subseteq \mathbf{I}$ of cardinality λ + which is an indiscernible set over A. This is an improvement of a result of S. Shelah. (shrink)

In ‘Something to Do with Vagueness ...’, Linda Burns defends an analogy between the informational and the borderline-case variety of vagueness. She argues that the latter is in fact less extraordinary and less disastrous than people in the tradition of Michael Dummett and Crispin Wright have told us. However, her account involves presuppositions that cannot be taken for granted. Here is to take a closer look at some of these presuppositions and argue hat they may--when left unguarded--undermine much of Burns’ (...) general account. (shrink)

We associate with every first order structure [Formula: see text] a family of invariant, locally Noetherian topologies. The structure is almost determined by the topologies, and properties of the structure are reflected by topological properties. We study these topologies in particular for stable structures. In nice cases, we get a behaviour similar to the Zariski topology in algebraically closed fields.

We present a new proof of the existence of Morley sequences in simple theories. We avoid using the Erdős–Rado theorem and instead use only Ramsey’s theorem and compactness. The proof shows that the basic theory of forking in simple theories can be developed using only principles from “ordinary mathematics,” answering a question of Grossberg, Iovino, and Lessmann, as well as a question of Baldwin.

In ‘Something to Do with Vagueness ...’, Linda Burns defends an analogy between the informational and the borderline-case variety of vagueness. She argues that the latter is in fact less extraordinary and less disastrous than people in the tradition of Michael Dummett and Crispin Wright have told us. However, her account involves presuppositions that cannot be taken for granted. Here is to take a closer look at some of these presuppositions and argue hat they may—when left unguarded—undermine much of Burns’ (...) general account. (shrink)