Abstract -/- The indistinction between repose and uniform movement exposed in the principle of inertia marks one of the most famous dissertations: the discussion between Newton and Leibniz. Through their respective conceptions of space, both seek to solve the problem of indistinction. The relational space of Leibniz, supported by the principle of sufficient reason and the identity of indiscernibles leads to the kinematic solution of the problem of inertia. The objective of this paper is to show the contribution of Leibniz, (...) in terms of relations of order, by distinguishing between equality by identity and equality by equivalence. Under a philosophical approach we link identity, equivalence and equality as relations of order with the indiscernibility between different states of movement. We expose first the principle of sufficient reason and the identity of indiscernibles. Under Bernoulli's principle of indifference, we argue that we distinguish between equality in abstract and equality in concrete. As a third part, we present the relationship between the identity of the indiscernibles and the predicate of equality under logical analysis. Finally we present the conclusions. -/- . (shrink)

After sketching some aspects of truthmaker doctrines and "truthmaker projects", and canvassing some prima facie objections to the latter, I turn to an issue which might seem to involve confusion about the nature of character of truthmakers if such there be, viz for statements of identity and (specially) distinctness. The real issue here is versions of the Identity of Indiscernibles. I discuss ways of discriminating versions, which are almost certainly true but trivial, which almost certainly substantive but false, and explore (...) an interesting intermediate possibility which might if developed yield a plausibly true yet not-entirely-trivial version of the doctrine: it is equivalent to what I call "the Denial of Bare Distinctness". (shrink)

In this note I discuss some issues around the law of Identity of Indiscernibles and, above all, its difference with the so-called law of indiscernibilty of identicals. In this way I distinguish between the notions identity, sameness and equality, through a phenomenological discussion and using the key idea of intentionality. In order to formulate the Leibnizian law of Identity of Indiscernibles, and examine its validity, we need higher order logic. I will give semantic rules for a second-order logic with identity (...) in the framework of the dialogical logic, introduced by P. Lorenzen. Then I will demonstrate the validity of the law of Identity of Indiscernibles by means of the introduced logic. (shrink)

ObjectiveThis paper explores the interrelation among the concepts of identity, individuality and indiscernibilty, primarily from the standpoint of contemporary western analytic ontology and logic.MethodI review, compare and evaluate the classical and the alternative approaches to identity. In this regard, I focus on the issue whether these purportedly alternative approaches do really provide us with alternative conceptions of identity, or they are considering some other forms of equivalence relations weaker than the relation of identity. Arguments for and against the Principles of (...) Indiscernibility of Identicals and Identity of Indiscernibles are delineated to unravel the connection between identity and indiscernibility. Various attempts to provide the definition of identity in terms of indiscernibility have also been evaluated.FindingsI argue that it is a verbal issue whether we use the term ‘identity’, when analysed in terms of indiscernibility, or ‘congruence’, but the underlying metaphysical issues are not simple. If identity is analysed in terms of indiscernibility, stipulation of one’s philosophical understanding of property and possession of properties in the concerned logic and ontology should be made in clear terms to avoid confusion. But there can also be ontological and logical frameworks where numerical identity and difference are not qualitatively explicable.ConclusionA pluralist stance, at least at the phenomenal level, is proposed to be maintained with regard to the explanation of the phenomenon of identity. This pluralist stance has also been proposed as a plausible metatheoretical framework of metaphysical debates. (shrink)

The present paper is a continuation of a previous one by the same title, the content of which faced the issue concerning the relations of coreference and qualification in compliance with the Navya-Nyāya theoretical framework, although prompted by the Advaita-Vedānta enquiry regarding non-difference. In a complementary manner, by means of a formal analysis of equivalence, equality, and identity, this section closes the loop by assessing the extent to which non-difference, the main issue here, cannot be reduced to any of the (...) former. The following sections of this study will focus on the assessment of the eventual possibility of causation and transformation in non-difference. (shrink)

Equality and identity. Bulletin of Symbolic Logic. 19 (2013) 255-6. (Coauthor: Anthony Ramnauth) Also see https://www.academia.edu/s/a6bf02aaab This article uses ‘equals’ [‘is equal to’] and ‘is’ [‘is identical to’, ‘is one and the same as’] as they are used in ordinary exact English. In a logically perfect language the oxymoron ‘the numbers 3 and 2+1 are the same number’ could not be said. Likewise, ‘the number 3 and the number 2+1 are one number’ is just as bad from a logical point (...) of view. In normal English these two sentences are idiomatically taken to express the true proposition that ‘the number 3 is the number 2+1’. Another idiomatic convention that interferes with clarity about equality and identity occurs in discussion of numbers: it is usual to write ‘3 equals 2+1’ when “3 is 2+1” is meant. When ‘3 equals 2+1’ is written there is a suggestion that 3 is not exactly the same number as 2+1 but that they merely have the same value. This becomes clear when we say that two of the sides of a triangle are equal if the two angles they subtend are equal or have the same measure. -/- Acknowledgements: Robert Barnes, Mark Brown, Jack Foran, Ivor Grattan-Guinness, Forest Hansen, David Hitchcock, Spaulding Hoffman, Calvin Jongsma, Justin Legault, Joaquin Miller, Tania Miller, and Wyman Park. -/- ► JOHN CORCORAN AND ANTHONY RAMNAUTH, Equality and identity. Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA E-mail: [email protected] The two halves of one line are equal but not identical [one and the same]. Otherwise the line would have only one half! Every line equals infinitely many other lines, but no line is [identical to] any other line—taking ‘identical’ strictly here and below. Knowing that two lines equaling a third are equal is useful; the condition “two lines equaling a third” often holds. In fact any two sides of an equilateral triangle is equal to the remaining side! But could knowing that two lines being [identical to] a third are identical be useful? The antecedent condition “two things identical to a third” never holds, nor does the consequent condition “two things being identical”. If two things were identical to a third, they would be the third and thus not be two things but only one. The plural predicate ‘are equal’ as in ‘All diameters of a given circle are equal’ is useful and natural. ‘Are identical’ as in ‘All centers of a given circle are identical’ is awkward or worse; it suggests that a circle has multiple centers. Substituting equals for equals [replacing one of two equals by the other] makes sense. Substituting identicals for identicals is empty—a thing is identical only to itself; substituting one thing for itself leaves that thing alone, does nothing. There are as many types of equality as magnitudes: angles, lines, planes, solids, times, etc. Each admits unit magnitudes. And each such equality analyzes as identity of magnitude: two lines are equal [in length] if the one’s length is identical to the other’s. Tarski [1] hardly mentioned equality-identity distinctions (pp. 54-63). His discussion begins: -/- Among the logical concepts […], the concept of IDENTITY or EQUALITY […] has the greatest importance. -/- Not until page 62 is there an equality-identity distinction. His only “notion of equality”, if such it is, is geometrical congruence—having the same size and shape—an equivalence relation not admitting any unit. Does anyone but Tarski ever say ‘this triangle is equal to that’ to mean that the first is congruent to that? What would motivate him to say such a thing? This lecture treats the history and philosophy of equality-identity distinctions. [1] ALFRED TARSKI, Introduction to Logic, Dover, New York, 1995. [This is expanded from the printed abstract.] . (shrink)

We present a finitary regularly algebraizable logic not finitely equivalential, for every similarity type. We associate to each of these logics a class of algebras with an equivalence relation, with the property that in this class, the identity is atomatically definable but not finitely atomatically definable.

There are several relations which may fall short of genuine identity, but which behave like identity in important respects. Such grades of discrimination have recently been the subject of much philosophical and technical discussion. This paper aims to complete their technical investigation. Grades of indiscernibility are defined in terms of satisfaction of certain first-order formulas. Grades of symmetry are defined in terms of symmetries on a structure. Both of these families of grades of discrimination have been studied in some (...) detail. However, this paper also introduces grades of relativity, defined in terms of relativeness correspondences. This paper explores the relationships between all the grades of discrimination, exhaustively answering several natural questions that have so far received only partial answers. It also establishes which grades can be captured in terms of satisfaction of object-language formulas and draws connections with definability theory. (shrink)

The indiscernibility of identicals is incompatible with geach's theory of 'relative' identity, But consistent with the view that x is identical with y iff x is the same a as y, For some count-Noun 'a'. 'x is the same a as y' expresses identity only if x is an a, Otherwise it is merely an equivalence relation.

We study the coherence, that is the equality of canonical natural transformations in non-free symmetric monoidal closed categories . To this aim, we use proof theory for intuitionistic multiplicative linear logic with unit. The study of coherence in non-free smccs is reduced to the study of equivalences on terms in the free category, which include the equivalences induced by the smcc structure. The free category is reformulated as the sequent calculus for imll with unit so that only equivalences on derivations (...) in this system are to be considered. We establish that any equivalence induced by the equality of canonical natural transformations over a model can be axiomatized by some set of “critical” pairs of derivations. From this, we derive certain sufficient conditions for full coherence, and establish that the system of identities defining smccs is not Post-complete: extending this system with an identity that does not hold in the free smcc does not in general cause the free smcc to collapse into a preorder. In order to give a larger context to these results, we study the equality of canonical morphisms in non-free symmetric monoidal categories, and establish that w.r.t. a broad subclass of smccs, the equivalences induced by the equality of canonical natural transformations over a model coincide with the equivalences induced by the equality of canonical morphisms for all interpretations in that model. (shrink)

ABSTRACT Vague concepts are represented by L-fuzzy sets. It is argued that any vague concept carries with it an approximate identity which is a fuzzy equivalence relation. The relation also fulfills the criterion of ? indiscernibility of Identicals ?, which is called ? saturatedness ? in this context. An application in knowledge representation is indicated.

This paper investigates the metaphysics in higher-order counterfactual logic. I establish the necessity of identity and distinctness and show that the logic is committed to vacuism, which entails that all counteridenticals are true. I prove the Barcan, Converse Barcan, Being Constraint and Necessitism. I then show how to derive the Identity of Indiscernibles in counterfactual logic. I study a form of maximalist ontology which has been claimed to be so expansive as to be inconsistent. I show that it is equivalent (...) to the collapse of the counterfactual into the material conditional---which is itself equivalent to the modal logic TRIV. TRIV is consistent, from which it follows that maximalism is, surprisingly, consistent. I close by arguing that stating the limit assumption requires a higher-order logic. (shrink)

Theories of structured meanings are designed to generate fine-grained meanings, but they are also liable to overgenerate structures, thus drawing structural distinctions without a semantic difference. I recommend the proliferation of very fine-grained structures, so that we are able to draw any semantic distinctions we think we might need. But, in order to contain overgeneration, I argue we should insert some degree of individuation between logical equivalence and structural identity based on structural isomorphism. The idea amounts to forming an equivalence (...) class of different structures according to one or more formal criteria and designating a privileged element as a representative of all the elements, i.e., a first among equals. The proposed method helps us to a cluster of notions of co-hyperintensionality. As a test case, I consider a recent objection levelled against the act theory of structured propositions. I also respond to an objection against my methodology. (shrink)

This is a copy of my DPhil thesis, the abstract for which is as follows: The first third of this thesis argues for a B-theoretic conception of time according to which all times exist equally and the present is in no way privileged. I distinguish "ontological" A-theories from "non-ontological" ones, arguing that the latter are experientially unmotivated and barely coherent. With regard to the former, I focus mainly on presentism. After some remarks on how to formulate this (and eternalism) non-trivially, (...) I review the non-relativistic case against presentism. I then consider the impact of Special Relativity on the debate, and attempt to deepen this impact by supplying a modal variation on the standard arguments. The middle third of the thesis investigates persistence, contending that both endurance and perdurance are consonant with the eternalism already endorsed. After introducing these theories of persistence, and discussing in particular how best to formulate an eternalist endurance, I proceed to defend the coherence of this combination. The Problem of Change is addressed here. I then respond in some detail to recent allegations of relativistic threats to endurance. The final third of the thesis questions the validity of the endurantist-perdurantist dispute. I criticize two recently proposed translation schemes that aim to show this dispute to be non-substantive. However, the second scheme suggests a doctrine of "Ontological Equivalence" which I develop and consider. I then address the Rotating Discs Argument, using this to launch a discussion of identity, genidentity, and the relationship between them. (shrink)

There are logics where necessity is defined by means of a given identity connective: \ is a tautology). On the other hand, in many standard modal logics the concept of propositional identity \ can be defined by strict equivalence \}\). All these approaches to modality involve a principle that we call the Collapse Axiom : “There is only one necessary proposition.” In this paper, we consider a notion of PI which relies on the identity axioms of Suszko’s non-Fregean logic SCI. (...) Then S3 proves to be the smallest Lewis modal system where PI can be defined as SE. We extend S3 to a non-Fregean logic with propositional quantifiers such that necessity and PI are integrated as non-interdefinable concepts. CA is not valid and PI refines SE. Models are expansions of SCI-models. We show that SCI-models are Boolean prealgebras, and vice-versa. This associates non-Fregean logic with research on Hyperintensional Semantics. PI equals SE iff models are Boolean algebras and CA holds. A representation result establishes a connection to Fine’s approach to propositional quantifiers and shows that our theories are conservative extensions of S3–S5, respectively. If we exclude the Barcan formula and a related axiom, then the resulting systems are still complete w.r.t. a simpler denotational semantics. (shrink)

In his latest paper, Loland1 tackles the question of whether athletes with differences of sexual development may compete in the women’s division. The topic is one of the most complex in sport and, as such, is fraught with debate. On one hand, the higher testosterone levels of athletes with DSD means they have an unfair performance advantage over their female competitors. On the other hand, it is argued that women with DSD should be able to compete in the gender division (...) with which they identify, especially as many will have been raised with a clear gender identity. Loland uses the Rawlsian notion of ‘fair equality of opportunity’ in which individuals with similar talents and ambitions should have roughly equivalent prospects for competitive success. For example, in most situations in society, one’s age, biological sex, ethnic and religious background, colour of skin, or sexual orientation are irrelevant to whether an individual should be given a job or is eligible for social welfare. In addition to being irrelevant, these factors are also stable—they are inequalities that ‘individuals cannot control or impact in any significant way’.1 However, Loland recognises that some stable inequalities, such as biological sex and body size, are relevant to FEO in sport. For example, a featherweight boxer may have similar talents and …. (shrink)

The problem of the meaning of a reasonable natural language expression is solved. First, traditional ”denotational” approach is criticized. The meaning of a sentence is not its truth value, similarly the meaning of, eg, ”The president of U.S.A.” is not Bill Clinton, etc. Frege met this problem when analyzing the so called propositional attitudes in which ”denotational” approach has lead to the paradox of analysis. His well-known solution consists in splitting the meaning into sense and reference. But this is rejected (...) in the paper as well, for its radical contextualism. In the first attempt, meaning is defined as an intension — mapping from possible worlds and time points — or extension , respectively. The problem of the propositional attitudes is seemingly solved. The proposition that Morning Star = Evening Star is diffe_rent than that of Morning Star = Morning Star. But, alas, in the case of analytical expressions we get the paradox of omniscience. A fine-grained solution is, therefore, proposed: the meaning is a closed construction denoted by the respective expression. Yet this solution is still not precise enough. Eventually the meaning is a concept which is the equivalence class of quasi-identical constructions indiscernible from the conceptual point of view. Finally, homonymous, synonymous and equi_valent expressions are precisely defined. Concluding we state that only synonymous expressions, having exactly the same meaning, ie. representing one and the same concept, can be mutually substituted in propositional attitudes without lea_ding to paradoxes. (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.

This essay examines an argument of perennial importance against naive Leibnizian absolute identity theory, originating with Ruth Barcan in 1947 (Barcan, R. 1947. ?The identity of individuals in a strict functional 3 calculus of second order?, Journal of Symbolic Logic, 12, 12?15), and developed by Arthur Prior in 1962 (Prior, A.N. 1962. Formal Logic. Oxford: The Clarendon Press), presented here in the form offered by Nicholas Griffin in his 1977 book, Relative Identity (Griffin, N. 1977. Relative Identity. Oxford: The Clarendon (...) Press). The objection considers the property of being necessarily identical to a specific object a as a counterexample to Leibnizian identity conditions, and more particularly to the indiscernibility of identicals, when it is only contingently true that a=b. The inquiry eliminates necessity and reference to a specifically designated object as responsible for the counterexample, leaving only identity. The requirements for an exact reinterpretation of Leibniz's Law in light of counterexamples involving converse intentional properties and the family of properties suggested by the property of being necessarily identical to a where a=b (or an equivalent definite descriptor variation) is only logically contingently true, are formalized in a more powerful, counterexample-resistant version of Leibniz's Law that does not succumb to relative identity as a necessary alternative motivated by desperation over the failure of the absolute identity principle. (shrink)

Is identity fundamental to formal systems? Even if a system have no the identity relation, is that concept is not assumed in any way – whether in a metalinguistic or intuitive level? In this paper we shall discuss this issue. Otávio Bueno (2014, 2016) argues against the elimination of identity, holding that this concept is fundamental and non-eliminable (even in does systems that claim to do so). Décio Arenhart Krause and Jonas (2015), by the other hand, have a number of (...) objections to Bueno’s thesis. Firstly, we look at how the concept of identity was accounted in the philosophical tradition, as well as part of its formal account. We will also examine the notion of indiscernibility that, according to the traditional approach to identity, is equivalent to the notion of identity, but (arguably) is not it logically equivalent. Thus, we shall state Krause and Arenhart’s strategy, according to, in a formal system we can eliminate identity in favor of the indiscernibility notion. Finally, we shall expose Bueno’s criticism to such strategy, and then to put forward some objections against it. (shrink)

In this paper, I explore several versions of the bundle theory and the substratum theory and compare them, with the surprising result that it seems to be true that they are equivalent (in a sense of 'equivalent' to be specified). In order to see whether this is correct or not, I go through several steps : first, I examine different versions of the bundle theory with tropes and compare them to the substratum theory with tropes by going through various standard (...) objections and arguing for a tu quoque in all cases. Emphasizing the theoretical role of the substratum and of the relation of compresence, I defend the claim that these views are equivalent for all theoretical purposes. I then examine two different versions of the bundle theory with universals, and show that one of them is, here again, equivalent to the substratum theory with universals, by examining how both views face the famous objection from Identity of Indiscernibles in a completely parallel way. It is only the second, quite extreme and puzzling, version of the bundle theory with universals that is not be equivalent to any other view; and the diagnosis of why this is so will show just how unpalatable the view is. Similarly, only a not-so-palatable version of the substratum theory is genuinely different from the other views; and here again it's precisely what makes it different that makes it less appealing. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Leibniz on Purely Extrinsic DenominationsGary L. CesarzDennis Plaisted. Leibniz on Purely Extrinsic Denominations. Rochester: University of Rochester Press, 2002. Pp. viii + 128 Cloth, $70.00.Interpreting Leibniz is like trying to flatten a balloon without deflating it; press it here and it bulges there. Press one of his controversial principles and problems result for another. Leibniz on contingent and necessary truths exemplifies this difficulty. So does Leibniz's denial that (...) there are "extrinsic denominations which have no foundationin the denominated thing" (NPE).Plaisted questions the meaning and challenges the current interpretations of this principle. Does "foundation" mean "included in a thing," or "reducible to an intrinsic denomination"? Furthermore, what is the thing denominated? Is it the individual concept of a substance, or an intrinsic denomination in the concept of the substance? Leibniz holds, "for all truths the predicate is contained in the subject" (PS) which, in some sense, includes both denominations (Leroy E. Loemker, ed., Gottfried Wilhelm Liebniz: Philosophical Papers and Letters [Dordrecht: Reidel, 1976], 268). This suggests that both denominations have the same status as predicates. However, such statements as, "there is no denomination so extrinsic that it does not have an intrinsic denomination as its basis" (Loemker, 527) and "all extrinsic denominations are founded in intrinsic denominations" (Benson Mates, The Philosophy of Leibniz: Metaphysics and Language [New York: Oxford University Press, 1989], 225) suggest that the former reduce to the latter. The last two quotes seem to trump all hands. But textual citations are seldom enough to settle disputes of this sort.This has inspired two approaches to the question, one reductionistic, the other non-reductionistic. Plaisted adopts a non-reductionistic approach but regards those prior to his as implausible and unsatisfactory (1, 113 n.34). As for reductionism, he claims to "stand [End Page 494] [it] on its head" (1) and denies its "textual merit" (14). Given Plaisted's broad grasp of Leibniz's writings, this is not an oversight of the passages quoted above. Rather, his case depends on reinterpreting the same passages in light of his version of NPE.A reduction, according to some, requires that a relational proposition be logically equivalent (or at least truth-functionallyequivalent) to a set of non-relational propositions. For others, implication is sufficient, i.e., for a given relational proposition z, and two non-relational propositions x and y, it is impossible that (x & y) is true but z false (Mates, 218). In response, Plaisted examines NPE from four angles, including Leibniz's predicate-in-subject notion of truth, the identity of indiscernibles, and two others. However, since all that follows rides on his understanding of PS, let us focus on it.Plaisted's interprets NPE to mean "There are no extrinsic denominations of an individual... that are not included in that individual's concept" (11). To support this, he observes that Leibniz applies PS to all truths and often derives NPE from PS. But to justify substituting "included in" for Leibniz's use of "foundation in," Plaisted elects to interpret PS in a troubling way. Now, Leibniz insists that necessary truths state an identity between a subject and predicate, either explicit or implicit, and that the latter are reducible to the former by an a priori proof of finite length. Leibniz further says that identity, as explained above, is the reason that PS holds for necessary truths. Thus, Plaisted infers that "identity" and "having a reason" are equivalent, that this sense of PS generalizes to all truths (24-25), and claims support for his version of NPE. This is a positive result for Plaisted, for from it he infers that extrinsic denominations are included in the individual's concept as directly as intrinsic denominations, hence the latter cannot serve as the foundation for the former (25ff.). But this is problematic.First, it leads him to affirm that some denominations are both extrinsic and intrinsic (75). This is a strange conclusion to consider positive, for it implies that there are two types of intrinsic plus extrinsic denominations. Surely this strains the boundaries of "textual merit" at least as much as Plaisted charges against reductionists. More importantly, even if the equivalence Plaisted presses holds for necessary... (shrink)

Abstract: There are two traditionally rival views about the nature of time: substantivalism that takes time to be a substance that exists independently of events located in it, and relationism that takes time to be constructed out of events. In this paper, first, I want to make some progress with respect to the debate between these two views, and I do this mainly by examining the strategies they use to face the possibilities of ‘empty time’ and ‘time without change’. As (...) we shall see, the two allegedly very different rival views are much less different than has been thought: their structure is extremely similar, their strategies are extremely similar, and they can both face the possibilities of ‘empty time’ and ‘time without change’ in the same way. Thus, I argue in favour of a certain kind of equivalence between the two views; I discuss a Strong and a Weak version of this claim; and I provide reasons for endorsing the former. I also discuss the parallel between this pair of views about the nature of time and another analogous pair of views: the bundle theory and the substratum theory about the nature of material objects, with respect to the problem with Identity of Indiscernibles. (shrink)

Leibniz notoriously insisted that no two individuals differ solo numero, that is, by being primitively distinct, without differing in some property. The details of Leibniz’s own way of understanding and defending the principle –known as the principle of identity of indiscernibles (henceforth ‘the Principle’)—is a matter of much debate. However, in contemporary metaphysics an equally notorious and discussed issue relates to a case put forward by Max Black (1952) as a counter-example to any necessary and non-trivial version of the principle. (...) Black asks us to imagine, via one of the fictional characters of his dialogue, a world consisting solely of two completely resembling spheres, in a relational space. The supporter of the principle is then forced to admit that although there are ex hypothesi two objects in that universe, there is no property (except trivial ones), not even relational ones, to distinguish them, and hence the necessary version of the principle is falsified. In this essay I will argue that Black’s possible world, together with the dialectic between the potential friends and foes of the Principle as expounded by Black himself.. (shrink)

The notion of strict identity is sometimes given an explicit second-order definition: objects with all the same properties are identical. Here, a somewhat different problem is raised: Under what conditions is the identity relation on the domain of a structure first-order definable? A structure may have objects that are distinct, but indiscernible by the strongest means of discerning them given the language (the indiscernibility formula). Here a number of results concerning the indiscernibility formula, and the definability of identity, (...) are collected and a number of applications discussed. (shrink)

Schrödinger logics are logical systems in which the principle of identity is not true in general. The intuitive motivation for these logics is both Erwin Schrödinger's thesis (which has been advanced by other authors) that identity lacks sense for elementary particles of modern physics, and the way which physicists deal with this concept; normally, they understandidentity as meaningindistinguishability (agreemment with respect to attributes). Observing that these concepts are equivalent in classical logic and mathematics, which underly the usual physical theories, we (...) present a higher-order logical system in which these concepts are systematically separated. A classical semantics for the system is presented and some philosophical related questions are mentioned. One of the main characteristics of our system is that Leibniz' Principle of the Identity of Indiscernibles cannot be derived. This fact is in accordance with some authors who maintain that quantum mechanics violates this principle. Furthermore, our system may be viewed as a way of making sense some of Schrödinger's logical intuitions about the nature of elementary particles. (shrink)

In his well-known 1952 dialogue Max Black describes a counterexample to the Principle of the Identity of Indiscernibles (PII). The counterexample is a world containing nothing but two purportedly indiscernible iron spheres. Reflecting on Black's example, Robert Adams uses the possibility of a world containing two almost indiscernible spheres to argue for the possibility of the indiscernible spheres world. One of Adams's almost indiscernible spheres has a small impurity, and, Adams writes, "Surely... the absence of the impurity would not make (...) such a universe impossible." The appeal to "surely" constitutes a gap in Adams's argument. This paper bridges the gap with a premise that exploits the counterfactual conditional and the related notion of causal independence. The paper then argues that causal independence bears in a similar way on an issue Nathan Salmon raises concerning Kripke's argument for the Essentiality of Origins. (shrink)

Gonzalo Rodriguez-Pereyra presents an original study of the place and role of the Identity of Indiscernibles in Leibniz's philosophy. The Principle of the Identity of Indiscernibles rules out numerically distinct but perfectly similar things; Leibniz derived it from more basic principles and used it to establish important philosophical theses. Rodriguez-Pereyra aims to establish what Leibniz meant by the Principle of Identity of Indiscernibles, what his arguments for and from it were, and to assess those arguments and Leibniz's claims about the (...) Principle of Identity of Indiscernibles. He argues that Leibniz had a very strong version of the principle, according to which no possibilia are intrinsically perfectly similar, where this excludes things that differ in magnitude alone. The book discusses Leibniz's arguments for the Identity of Indiscernibles in the Meditation on the Principle of the Individual, the Discourse on Metaphysics, Notationes Generales, Primary Truths, the letter to Casati of 1689, the correspondence with Clarke, as well as the use of the Identity of Indiscernibles in Leibniz's arguments against the Cartesian conception of the material world, atoms, absolute space and time, the Lockean conception of the mind as a tabula rasa, and freedom of indifference. Rodriguez-Pereyra argues that the Identity of Indiscernibles was a central but inessential principle of Leibniz's philosophy. (shrink)

A and B continue their conversation concerning the Identity of Indiscernibles. Both are aware of recent critiques of the principle that haven’t received replies; B summarizes those critiques, and A offers the replies that are due. B then raises a new worry.

The identity of indiscernibles states that indiscernible objects must be identical. Many philosophers have held that the PII turns out to be either true but trivial, or non-trivial but false, depending on how the notion of discernibility is spelled out. In this paper, I propose and defend an account of this notion which aims to yield a minimally non-trivial and yet plausible version of the PII. I argue moreover that this version of the principle is immune to a number of (...) well-known and recent objections to the PII. (shrink)

The Identity of Indiscernibles seems like a good enough way to define identity. Roughly it simply says that if x and y have all and only the same properties, these will be the same object. However the principle has come under attack using a series of thought experiments employing the idea of radical symmetry. I follow the history of the debate including its theological origins to assess the contemporary arguments against the Identity of Indiscernibles. I argue that the principle is (...) viable as a logical truth, and so can be put to work in our idea of objects. (shrink)

This paper is a contribution to the study of equality-free logic, that is, first-order logic without equality. We mainly devote ourselves to the study of algebraic characterizations of its relation of elementary equivalence by providing some Keisler-Shelah type ultrapower theorems and an Ehrenfeucht-Fraïssé type theorem. We also give characterizations of elementary classes in equality-free logic. As a by-product we characterize the sentences that are logically equivalent to an equality-free one.

A number of commentators (especially French and Redhead, 1988, and Butterfield, 1993) have investigated the status of the principle of the identity of indiscernibles (PII) for bosons and fermions. In this paper I extend that investigation to the full range of quantum particles of any allowed kind of statistics -- `quarticles', that is. I show that for any kind (except bosons and fermions) there are states in which PII is violated by every pair of particles, some pairs and not others, (...) and by no pairs. (shrink)

The paper formulates and defends a version of the Identity of Indiscernibles and demonstrates that it entails a non-trivial version of the doctrine of essentialism.

: Hegel is commonly thought to affirm Leibniz’s principle of the identity of indiscernibles, which states that no two things are exactly alike. I argue that this interpretation is mistaken: it cannot accommodate passages in which Hegel rejects PII, and the texts cited in favor of this interpretation admit of another reading, which I provide. On my view, Hegel distinguishes between different senses of PII, and the sense of PII he accepts only entails that determinacy is immanent to individuals qua (...) concrete particulars. (shrink)

Paul (Noûs 36:578–596, 2002; Noûs 40:623–659, 2006, The Handbook of Mereology, forthcoming) has argued for a bundle theory of objects that analyzes the bundling relation between properties and objects in terms of parthood relations. In this paper I argue that any mereological bundle theory with the explanatory power of Paul’s theory will entail the principle of the identity of indiscernibles (PII). This is problematic, since similar bundle theories seem to fall to Max Black’s two sphere counterexample to (PII). I argue, (...) however, that a fully developed mereological bundle theory provides a new way of interpreting Black’s two sphere universe that dispels the counterexample. I argue that this solution to Black’s puzzle is superior to other solutions on offer, and consequently that mereological bundle theory is an attractive ontological strategy for friends of (PII). (shrink)

SummaryOne form of the principle of the identity of indiscernibles asserts that for any two distinct entities it is necessarily the case that there is a description which distinguishes between them, in that it applies to one but not to the other. In this paper it is argued that this form of the principle is false. The argument is based upon the nature of language, in particular the need for ostensive definition of terms which are not verbally definable. It is (...) concluded that the identity of indiscernibles, in its linguistic form, requires a power of discrimination which no language could possess. Other versions of the principle, however, formulated in terms of properties rather than descriptions, are unaffected by the argument and may well be true. (shrink)

Arthur Danto has made important contributions to both aesthetics and philosophy of history. Furthermore, as I shall try to show in this essay, his aesthetics is of great relevance to his philosophy of history, while his philosophy of history is of no less interest for his aesthetics.By focusing on the notions of representation, identity, and the identity of indiscernibles we shall discover how fruitful this cooperation of aesthetics and philosophy of history may be. Crucial to all historical writing and, hence, (...) to all philosophy of history, is the notion of identity through time and change. How could the historian write the history of x if x cannot be said to remain the same in the course of its history? It will become clear that aesthetics will provide us with a satisfactory solution for the problem, for the aestheticist notion of representation will enable us to define the notion of identity that the historian needs.Nevertheless, a certain friction can be observed between Danto's aesthetics and his philosophy of history. At the end of this essay I hope to show that Danto's philosophy of history will be our best guide to dealing adequately with this friction. (shrink)

This article is an extended promenade strolling along the winding roads of identity, equality, nameability and completeness, looking for places where they converge. We have distinguished between identity and equality; the first is a binary relation between objects while the second is a symbolic relation between terms. Owing to the central role the notion of identity plays in logic, you can be interested either in how to define it using other logical concepts or in the opposite scheme. In the first (...) case, one investigates what kind of logic is required. In the second case, one is interested in the definition of the other logical concepts in terms of the identity relation, using also abstraction. The present paper investigates whether identity can be introduced by definition arriving to the conclusion that only in full higher-order logic a reliable definition of identity is possible. However, the definition needs the standard semantics and we know that with this semantics completeness is lost. We have also studied the relationship of equality with comprehension and extensionality and pointed out the relevant role played by these two axioms in Henkin’s completeness method. We finish our paper with a section devoted to general semantics, where the role played by the nameable hierarchy of types is the key in Henkin’s completeness method. (shrink)

Recently, several authors have claimed to have found graph-theoretic counterexamples to the Principle of the Identity of Indiscernibles. In this paper, I argue that their counterexamples presuppose a certain view of what unlabeled graphs are, and that this view is optional at best.

The mathematical field of graph theory has recently been used to provide counterexamples to the Principle of the Identity of Indiscernibles. In response to this, it has been argued that appeal to relations between graphs allows the Principle to survive the counterexamples. In this paper, I aim to show why that proposal does not succeed.

Leibniz was a philosopher of principles: the principles of Contradiction, of Sufficient Reason, of Identity of Indiscernibles, of Plenitude, of the Best, and of Continuity are among the most famous Leibnizian principles. In this article I shall focus on the first three principles; I shall discuss various formulations of the principles (sect. 1), what it means for these theses to have the status of principles or axioms in Leibniz’s philosophy (sect. 2), the fundamental character of the Principles of Contradiction and (...) Sufficient Reason (sect. 3), some attempts to demonstrate the Principles of Contradiction and Sufficient Reason (sect. 4), and one attempt to demonstrate the Principle of Identity of Indiscernibles (sect. 5). The main results of the chapter are summarized in a short conclusion (sect. 6). (shrink)

It is argued that recent discussion of the principle of the identity of indiscernibles (PII) and quantum mechanics has lost sight of the broader philosophical motivation and significance of PII and that the `received view' of the status of PII in the light of quantum mechanics survives recent criticisms of it by Muller, Saunders, and Seevinck.

It is surprisingly difficult to determine what modal strength Leibniz wants to ascribe to his principle of the identity of indiscernibles (PII). I consider this question by examining (i) some direct textual evidence, (ii) Leibniz's main arguments for PII, (iii) Leibniz's presumable response to a prominent contemporary defense of the necessity of PII against Max Black style counterexamples, and (iv) Leibniz's views about the possibility of primitive haecceities. I conclude that Leibniz probably takes PII to be necessary.