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)

Actualist counterpart theory replaces David Lewis’s concrete possible worlds and individuals with ersatz worlds and individuals, but retains counterpart theory about de re modality. While intuitively attractive, this view has been rejected for two main reasons: the problem of indiscernibles and the Humphrey objection. I argue that in insisting that ersatz individuals play the same role as Lewisian individuals, actualists commit the particularist fallacy. The actualist should not require stand-ins for every Lewisian individual. Ersatz individuals should instead be construed as (...) representations of actually existing qualitative ways for individuals to be, or qualitative properties individuals can instantiate. This necessitates changes elsewhere. Non-instrumental uses of Kripke semantics and standard counterpart semantics also require stand-ins for particular non-actual individuals. I argue that the actualist should instead adopt a non-standard counterpart semantics that more clearly illuminates the role that actual properties and relations play in explaining de re possibilities and necessities. The result is an intuitive and forceful reply to both the problem of indiscernibles and the Humphrey objection. (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)

the Mona Lisa, the Mondscheinsonate, the Chanson d’automne are works of art, the salt shaker on your table, the car in your garage, or the pijamas on your bed are not. the basic question of the metaphysics of works of art is this: what makes a thing a work of art? that is: what sort of property do works of art have in virtue of which they are works of art? or more simply: what sort of property being a work (...) of art is? In this paper we argue that things are works of art in virtue of what they are like, their intrinsic features, that is, in virtue of the fact that they have the perceptual (auditory, visual, etc.) properties they have. In other words: being a work of art supervenes on perceptual-intrinsic features. Currently, this metaphysical view is extremely unpopular within the philosophy of art. It is unpopular because there allegedly exists a knock-down objection to it, the well-known argument from indiscernible counterparts. our thesis implies, among other things, that every perceptual duplicate of a work of art is also a work of art. according to the argument from indiscernible counterparts, however, there could be (or even: there are) indiscernible counterparts such that one of them is a work of art while the other is not. hence things cannot be works of art solely in virtue of what they are like. Our paper divides into three parts. In the first part we state our views. In the second part we defend it against various versions of the argument from indiscernible counterparts. (In doing so our position will become more plausible, we hope). In the final part we provide some meta-reflections on the matter. (shrink)

In late 2014, the jazz combo Mostly Other People Do the Killing released Blue—an album that is a note-for-note remake of Miles Davis's 1959 landmark album Kind of Blue. This is a thought experiment made concrete, raising metaphysical puzzles familiar from discussion of indiscernible counterparts. It is an actual album, rather than merely a concept, and so poses the aesthetic puzzle of why one would ever actually listen to it.

Arthur Danto once stood next to Andy Warhol at a gallery opening while Warhol autographed his wife's brochure. For Danto, Warhol embodied the age and its aspirations: a philosophical artist in gel. Danto was also writing autobiographically in describing Warhol's before and after. The before and after made them alike: each became a modernist figure of unparalleled boldness, audacity, and brilliance. A number of critics understood that Warhol was pushing against the limits of art in a way that forced abstract (...) reflection on what art was, is, and could be. Danto reasoned that a cultural background puts in place a set of concepts allowing one thing to be schematized differently from a visually indiscernible counterpart. Warhol's work was to him more like declarative language than anything else: it spoke in virtue of a shared background of concepts, which he called a theory. (shrink)

This essay constructs philosophical defenses against criticisms of my theory of the end of art. These have to do with the definition of art; the concept of artistic quality; the role of aesthetics; the relationship between philosophy and art; how to answer the question "But is it art?"; the difference between the end of art and "the death of painting"; historical imagination and the future; the method of using indiscernible counterparts, like Warhol's Brillo Box and the Brillo cartons (...) it resembles; the logic of imitation-and the differences between Hegel's views on the end of art and mine. These defenses amplify and fortify the thesis of the end of art as set forth in my After the End of Art: Contemporary Art and the Pale of History (1997). (shrink)

Covert spatial attention alters the way things look. There is strong empirical evidence showing that objects situated at attended locations are described as appearing bigger, closer, if striped, stripier than qualitatively indiscernible counterparts whose locations are unattended. These results cannot be easily explained in terms of which properties of objects are perceived. Nor do they appear to be cases of visual illusions. Ned Block has argued that these results are best accounted for by invoking what he calls ‘mental (...) paint’. In this paper I argue, instead, in favour of an account of these phenomena in terms of the perceptual experience of affordances concerning saccadic eye movement. As part of the argument I draw connections with the empirical literature on the way in which performance efficiency also alters visual appearance. (shrink)

The difficulties encountered by attempts to treat identity as a relation between an object and itself are well-known: "...the sentence 'The morning star is...the morning star' is analytic and a truism, while...'The morning star is the evening star' is synthetic and represents a 'valuable extension of our knowledge'... But if {the morning star} and {the evening star} are the same object, and identity is taken as a relation holding between this object and itself, then it is impossible to explain how (...) the two sentences can differ in cognitive content... ". Russell's solution to these difficulties rejects the identification of logical with grammatical form, in effect denying that such sentences assert relations between the morning star and itself. The logical representations which the Russellian proposes, contain quantifiers, predicate letters, and individual variables, but no expressions standing for particulars. The Fregean solution, while admitting logical representations whose nominal expressions stand for particulars, insists that the meaning of such expressions is different from their reference. Frege-Russell analyses thus both deny that the morning star is involved qua particular in the meaning of the sentences in question. ;Rather than replace the morning star as ontological subject of these sentences, I suggest that their difference in meaning arises from the kind of particular that the identity-relation relates--a particular which, unlike its Frege-Russell counterpart, is two-sided and multi-faceted. Such a particular requires an ambiguous singular term. In Chapter 1, I utilize such a term to provide a surfacist account of belief-context ambiguity requiring neither differences in relative scope nor distinctions between sense and reference. In Chapter 2, I go on to provide an account of negative existentials, necessity- and identity-statements which resolves philosophical problems that Russell-Frege analyses only avoid. To solve these problems, I show that it is necessary to reject two canons of philosophical logic, the Law of Identity and the Indiscernibility of Identicals. (shrink)

This is a short response to Aaron Cotnoir's 'Composition as General Identity', in which I suggest some further applications of his ideas, and try to press the question of why we should think of his 'general identity relation' as a genuine identity relation.

The Logic of Metaphor uses techniques from possible worlds semantics to provide formal truth-conditions for many grammatical classes of metaphors. It gives logically precise and practically useful syntactic and semantic rules for generating and interpreting metaphors. These rules are implemented in a working computer program. The book treats the lexicon as a conceptual network with semantics provided by an intensional predicate calculus. It gives rules for finding analogies in such networks. It shows how to syntactically and semantically analyze texts containing (...) metaphors and how to use structural similarities between parts of possible worlds to provide truth-conditions for metaphors. Meanings for metaphors are linked to the modal logics of identity and indiscernibility. The book shows how to extend deductive and abductive inference systems to handle metaphors. It shows how to handle novel metaphorical word-senses. The Logic of Metaphor will be useful to philosophers, logicians, linguists, and computer scientists. (shrink)

In the debate about Composition as Identity (CI), a recurring pattern is to ask whether a certain feature of identity is also instantiated by composition. This recurring pattern is followed when, for example, the question is asked whether a whole and its parts are indiscernible. In following this pattern, it is methodologically desirable to assume the most standard account of the philosophical problems at stake. However, when the necessity of identity and the problem whether composition is as necessary as (...) identity are at stake, the literature about CI often violates this methodological principle, and resorts to non-standard views about modality, such as counterpart theory. In this paper, we purport to remedy this anomaly and to assess CI on the background of a standard, broadly Kripkean view of modality, and in particular of the contention that a single entity exists in more than one possible world. Given this contention, the backer of CI is forced to relativize composition and, as a consequence, identity to possible worlds, thereby introducing a non-standard kind of identity. We will discuss the charge of adhocness which might be raised against the resulting variety of CI. (shrink)

When the necessity of identity (NI) is combined with composition as identity (CAI), the contingency of composition (CC) is at risk. In the extant literature, either NI is seen as the basis for a refutation of CAI or CAI is associated with a theory of modality, such that: either NI is renounced (if counterpart theory is adopted); or CC is renounced (if the theory of modal parts is adopted). In this paper, we investigate the prospects of a new variety of (...) CAI, which aims to preserve both NI and CC. This new variety of CAI (CCAI, contingent composition as identity) is the quite natural product of the attempt to make sense of CAI on the background of a broadly Kripkean view of modality, such that one and the same entity is allowed to exist at more than one possible world. CCAI introduces a world-relative kind of identity, which is different from standard identity, and claims that composition is this kind of world-relative identity. CCAI manages to preserve NI and CC. We compare CCAI with Gibbard’s and Gallois’ doctrines of contingent identity and we show that CCAI can be sensibly interpreted as a form of Weak CAI, that is of the thesis that composition is not standard identity, yet is significantly similar to it. (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 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)

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)

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.

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 (...) class='Hi'>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)

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.

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.

David Lewis advised essentialists to judge his counterpart theory a false friend. He also argued that counterpart theory needs natural properties. This essay argues that natural properties are all essentialists need to find a true friend in counterpart theory. Section one explains why Lewis takes counterpart theory to be anti-essentialist and why he thinks it needs natural properties. Section two establishes the connection between the natural properties counterpart theory needs and the essentialist consequences Lewis disavows. Section three answers two objections: (...) the first attempts to block the consequences of adding natural properties to counterpart theory; the second grants the consequences, but denies that they amount to essentialism. –Correspondence to: [email protected]. (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)

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 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)

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 give a model-theoretic account for several results regarding sequences of random variables appearing in Berkes and Rosenthal [12]. In order to do this,•We study and compare three notions of convergence of types in a stable theory: logic convergence, i.e., formula by formula, metric convergence (both already well studied) and convergence of canonical bases. In particular, we characterise א0-categorical stable theories in which the last two agree.•We characterise sequences that admit almost indiscernible sub-sequences.•We apply these tools to the theory (...) of atomless random variables (ARV). We characterise types and notions of convergence of types as conditional distributions and weak/strong convergence thereof, and obtain, among other things, the Main Theorem of Berkes and Rosenthal. (shrink)

Counterpart theory is often advertised by its track record at solving metaphysical puzzles. Here I focus on puzzles of occasional identity, wherein distinct individuals at one world or time appear to be identical at another world or time. To solve these puzzles, the usual interpretation rules of counterpart theory must be extended beyond the simple language of quantified modal logic. I present a more comprehensive semantics that allows talking about specific times and worlds, that takes into account the multiplicity and (...) sortal-dependence of counterpart relations, and that does not require names to denote actual or present individuals. In addition, the semantics I defend does not identify ordinary individuals with world-bound or time-bound stages and thereby avoids the most controversial aspect of counterpart theory. Humphrey’s counterpart at other worlds or times is none other than Humphrey himself. (shrink)

We prove a property of generic homogeneity of tuples starting an infinite indiscernible sequence in a simple theory and we use it to give a shorter proof of the Independence Theorem for Lascar strong types. We also characterize the relation of starting an infinite indiscernible sequence in terms of coheirs.

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)

Counterpart theory has many benefits, but few are happy to accept the metaphysical setting in which this account of de re modality was developed by its architect, David Lewis. I argue here that counterpart theory can be made acceptable by the lights of those who repudiate the existence of merely possible objects. To the "ersatz" counterpart theorist I offer two stories: one about the relate of the counterpart relation and one about the relation itself. With these in place, I then (...) defend ersatz counterpart theory from a number of objections that can be found in the literature. (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, supposed as a theory concerning the internal constitution of individuals, is often conjoined with a constitutional approach to individuation entailing the thesis ‘no two individuals can share all their constituents’ (CIT). But then it runs afoul of Black’s duplication case. Here a new bundle theory, takingdistance relations between bundles to be a sufficient ground for their diversity, will be proposed. This version accommodates Black’s world. Nonetheless, there is a possible objection. Consider the ‘triplication case’—a world containing three (...) indistinguishable spheres, each 5-meters from each other. Since distance relations are dyadic, this version must fail to distinguish the threespheres world from Black’s world. In response to this objection, I maintain that we must construe distance relations as irreducibly multigrade and n-ary. Then these two worlds will be distinguished by appealing to a triadic relation—R3—that three things enter mutually. Aren’t all polyadic relations in principle reducible to dyadic relations? I won’t deny that. But I will aim lower and argue that R3 cannot be reduced to dyadic relations that obliterate the distinction between the three-spheres world and Black’s world. (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)

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)

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)

I formulate a theory of groups based on pluralities and counterparts: roughly put, a group is a plurality of entities at a time. This theory comes with counterpart-theoretic semantics for modal and temporal sentences about groups. So this theory of groups is akin to the stage theory of material objects: both take the items they analyze to exist at a single time, and both use counterparts to satisfy certain conditions relating to the modal properties, temporal properties, and coincidence (...) properties of those items. (shrink)

Counterpart theory affords an especially flexible form of essentialism. Gideon Rosen and the author suggested that by endowing the entire world with peculiar essences, they could likewise provide truth makers for negative existential propositions. Thus they avoid the need for states of affairs or non‐transferable tropes as truth makers. States of affairs are somehow constructed from particular things and the properties they instantiate. A legitimate counterpart relation can exist which validates only one direction of set‐theoretical essentialism, but under which any (...) counterpart of a state of affairs must be a state of affairs. States of affairs or tropes are not needed for purposes of truth making. But also, if we help ourselves to the flexibility of counterpart theory, the principal difficulties that stand in the way of using states of affairs or tropes as truth makers vanish. (shrink)

We analyze the notions of indiscernibility and indeterminacy in the light of the Galois theory of field extensions and the generalization to \(K\) -algebras proposed by Grothendieck. Grothendieck’s reformulation of Galois theory permits to recast the Galois correspondence between symmetry groups and invariants as a Galois–Grothendieck duality between \(G\) -spaces and the minimal observable algebras that discern (or separate) their points. According to the natural epistemic interpretation of the original Galois theory, the possible \(K\) -indiscernibilities between the roots of a (...) polynomial \(p(x)\in K[x]\) result from the limitations of the field \(K\) . We discuss the relation between this epistemic interpretation of the Galois–Grothendieck duality and Leibniz’s principle of the identity of indiscernibles. We then use the conceptual framework provided by Klein’s Erlangen program to propose an alternative ontologic interpretation of this duality. The Galoisian symmetries are now interpreted in terms of the automorphisms of the symmetric geometric figures that can be placed in a background Klein geometry. According to this interpretation, the Galois–Grothendieck duality encodes the compatibility condition between geometric figures endowed with groups of automorphisms and the ‘observables’ that can be consistently evaluated at such figures. In this conceptual framework, the Galoisian symmetries do not encode the epistemic indiscernibility between individuals, but rather the intrinsic indeterminacy in the pointwise localization of the figures with respect to the background Klein geometry. (shrink)

Kripke complained that counterpart theory makes modal claims be about counterparts and not about us, and that it is a misguided model of modality since we do not care about counterparts in the same way we care about ourselves. The first part of the complaint, I argue, has been met by Hazen and Lewis, while the second can be countered by observing that most of our modal concerns are about role‐fillers and that counterparts are well‐suited to such (...) concerns. The role‐filler analysis of modal concerns, however, does not seem to capture certain attitudes ‐ especially that of regret. Regret, however, I will argue, is not intrinsically modal and thus is no challenge to my general analysis of modal concerns. (shrink)

In Parmenides, the young Socrates defends several candidate forms against Parmenides, who makes five objections: the objection of forms of common things, the question of the part vs. the whole, the third man argument, infinite regress, and the greatest difficulty problem. I define forms in terms of Leibniz’s Principle of the Identity of Indiscernibles (PII) in an attempt to overcome Parmenides’ opposition. I show that the main force in Parmenides’ objections consists of absurdities that emerge in relations between forms and (...) particulars: absurdities that are avoided if the form and its instantiation in the particular are identical. (shrink)

Nonreductive physicalism faces serious problems regarding causal exclusion, causal heterogeneity, and the nature of realization. In this paper I advance solutions to each of those problems. The proposed solutions all depend crucially on embracing modal counterpart theory. Hence, the paper’s thesis: counterpart theory saves nonreductive physicalism. I take as my inspiration the view that mental tokens are constituted by physical tokens in the same way statues are constituted by lumps of clay. I break from other philosophers who have pursued this (...) line, however, in that I hold that constitution is identity. Much of the value of the comparison to statues and lumps is that it calls to mind the resources used to defend constitution-as-identity, most notably that of counterpart theory. Along the way, I discuss the virtues of a trope ontology, modal objections to token identity theories, the prospects of conditional analyses of causal powers, the subset account of realization, and the grounding problem. I also endorse a novel, empirical argument in favor of counterpart theory. (shrink)

Kant's argument from incongruent counterparts for substantival space is examined; it is concluded that the argument has no force against a relationist. The argument does suggest that a relationist cannot give an account of enantiomorphism, incongruent counterparts and orientability. The prospects for a relationist account of these notions are assessed, and it is found that they are good provided the relationist is some kind of modal relationist. An illustration and interpretation of these modal commitments is given.