Extensional scientific realism is the view that each believable scientific theory is supported by the unique first-order evidence for it and that if we want to believe that it is true, we should rely on its unique first-order evidence. In contrast, intensional scientific realism is the view that all believable scientific theories have a common feature and that we should rely on it to determine whether a theory is believable or not. Fitzpatrick argues that extensional realism is immune, while intensional (...) realism is not, to the pessimistic induction. I reply that if extensional realism overcomes the pessimistic induction at all, that is because it implicitly relies on the theoretical resource of intensional realism. I also argue that extensional realism, by nature, cannot embed a criterion for distinguishing between believable and unbelievable theories. (shrink)

I focus on three mereological principles: the Extensionality of Parthood (EP), the Uniqueness of Composition (UC), and the Extensionality of Composition (EC). These principles are not equivalent. Nonetheless, they are closely related (and often equated) as they all reflect the basic nominalistic dictum, No difference without a difference maker. And each one of them—individually or collectively—has been challenged on philosophical grounds. In the first part I argue that such challenges do not quite threaten EP insofar as they are (...) either self-defeating or unsupported. In the second part I argue that they hardly undermine the tenability of EC and UC as well. (shrink)

The paper addresses various questions about the logical and metaphysical relations between notions of parthood, location and persistence. In particular it argues that the conjunction of mereological extensionalism and multilocation, is highly problematic, if not utterly inconsistent. It thus provides an alternate route to reject multilocation, one that does not rely on Barker and Dowe's well known argument, at least for those who endorse extensionality of parthood. It then argues that other major metaphysical theses such as three-dimensionalism turn out (...) to be at odds with extensionalism. (shrink)

This paper presents an extensional account of manyand few that explains data that have previously motivated intensional analyses of these quantifiers :599–620, 2000). The key insight is that their semantic arguments are themselves set intersections: the restrictor is the intersection of the predicates denoted by the N’ or the V’ and the restricted universe, U, and the scope is the intersection of the N’ and V’. Following Cohen, I assume that the universe consists of the union of alternatives to the (...) nominal and verbal predicates, where an alternative to a property ψ is one that shares a pragmatic presupposition with ψ, and a pragmatic presupposition is one that is selected by context from a set of potential presuppositions associated with the sentence. A many/few-quantified sentence is then true iff the proportion of the scope to the restrictor is greater/less than some threshold, n. In addition to explaining various problematic cases from the literature, the analysis shows how the readings of a many/few-quantified sentence can be derived from the same syntactico-semantic structure, it being unnecessary to claim lexical or structural ambiguity. The analysis also provides strong support for the idea that natural language quantification is always purely extensional. (shrink)

A logic \ is called self-extensional if it allows to replace occurrences of a formula by occurrences of an \-equivalent one in the context of claims about logical consequence and logical validity. It is known that no three-valued paraconsistent logic which has an implication can be self-extensional. In this paper we show that in contrast, there is exactly one self-extensional three-valued paraconsistent logic in the language of \ for which \ is a disjunction, and \ is a conjunction. We also (...) investigate the main properties of this logic, determine the expressive power of its language, and provide a cut-free Gentzen-type proof system for it. (shrink)

Tarski characterized logical notions as invariant under permutations of the domain. The outcome, according to Tarski, is that our logic, which is commonly said to be a logic of extension rather than intension, is not even a logic of extension—it is a logic of cardinality. In this paper, I make this idea precise. We look at a scale inspired by Ruth Barcan Marcus of various levels of meaning: extensions, intensions and hyperintensions. On this scale, the lower the level of meaning, (...) the more coarse-grained and less “intensional” it is. I propose to extend this scale to accommodate a level of meaning appropriate for logic. Thus, below the level of extension, we will have a more coarse-grained level of form. I employ a semantic conception of form, adopted from Sher, where forms are features of things “in the world”. Each expression in the language embodies a form, and by the definition we give, forms will be invariant under permutations and thus Tarskian logical notions. I then define the logical terms of a language as those terms whose extension can be determined by their form. Logicality will be shown to be a lower level analogue of rigidity. Using Barcan Marcus’s principles of explicit and implicit extensionality, we are able to characterize purely logical languages as “sub-extensional”, namely, as concerned only with form, and we thus obtain a wider perspective on both logicality and extensionality. (shrink)

Ethical extensionism is a common argument pattern in environmental and animal ethics, which takes a morally valuable trait already recognized in us and argues that we should recognize that value in other entities such as nonhuman animals. I exposit ethical extensionism’s core argument, argue for its validity and soundness, and trace its history to 18th century progressivist calls to expand the moral community and legal franchise. However, ethical extensionism has its critics. The bulk of the paper responds to recent criticisms, (...) including (1) environmental ethicists’ objection against its austere conception of moral value (2), feminist ethicists’ claim that extensionism fails to account for the moral significance of difference, (3) disability theorists’ concern that ethical extensionist arguments are offensive, and (4) animal rights theorists’ lament that extensionism is a practical failure. While something is to be gained from each criticism, I argue that they ultimately fail and that extensionism remains compelling. (shrink)

A proof is presented showing that there is no paraconsistent logics with a standard implication which have a three-valued characteristic matrix, and in which the replacement principle holds.

Mereology is the theory of the relation “being a part of”. The first exact formulation of mereology is due to the Polish logician Stanisław Leśniewski. But Leśniewski’s mereology is not first-order axiomatizable, for it requires every subset of the domain to have a fusion. In recent literature, a first-order theory named General Extensional Mereology can be thought of as a first-order approximation of Leśniewski’s theory, in the sense that GEM guarantees that every definable subset of the domain has a fusion, (...) and this has been achieved by positing an axiom schema which in effect defines infinitely many axioms. Intuitively, in order to range over every definable subset, such an axiom schema seems unavoidable. But this paper will show that GEM is finitely axiomatizable and that all we need are just finitely many instances of the said axiom schema. (shrink)

The naive set theory problem is to begin with a full comprehension axiom, and to find a logic strong enough to prove theorems, but weak enough not to prove everything. This paper considers the sub-problem of expressing extensional identity and the subset relation in paraconsistent, relevant solutions, in light of a recent proposal from Beall, Brady, Hazen, Priest and Restall [4]. The main result is that the proposal, in the context of an independently motivated formalization of naive set theory, leads (...) to triviality. (shrink)

In this paper, we present an extensional system of mereology suitable to account for the intuitive distinction between heaplike and non-heaplike entities. Since the need to capture this distinction has been a key motivation for non-extensional mereologies, we first assess the main non-extensional systems advanced in the last years and highlight some mereological and metaphysical difficulties they involve. We then advance a novel program, according to which the distinction between heaplike and non-heaplike entities can be accounted for by bringing together (...) the parthood relation characterized by classical extensional mereology and an Aristotelian extensional notion of potential parthood. Thus, while rejecting the thesis of mereological monism, our proposal is consistent with the thesis of mereological extensionalism. We show that within this framework it is possible to characterize the above-mentioned distinction, to define the relation of material constitution, and to capture three fundamental standpoints in metaphysics. (shrink)

Quine’s “New Foundations” (NF) was first presented in Quine [1937] and later on in Quine [1963]. Ernst Specker [1958, 1962], building upon a previous result of Ehrenfeucht and Mostowski [1956], showed that NF is consistent if and only if there is a model of the Theory of Negative (and positive) Types (TNT) with full extensionality that admits of a “shifting automorphism,” but the existence of a such a model remains an open problem.

Philosophers of modern logic have cherished no project more dearly than that of extensional reduction. Despite occasional protests that this project was ill-conceived from the start, or that it fails to account for important areas of experience and thought, the extensionalist mills have been grinding away anyhow. Their grinding has brought with it a number of important technical successes, replete with philosophical claims that light has finally been shed on areas hitherto buried in incomprehensible darkness.

We analyze some extensions of Martin-Löf 's constructive type theory by means of extensional set constructors and we show that often the most natural requirements over them lead to classical logic or even to inconsistency.

I examine the link between extensionality principles of classical mereology and the anti‐symmetry of parthood. Varzi's most recent defence of extensionality depends crucially on assuming anti‐symmetry. I examine the notions of proper parthood, weak supplementation and non‐well‐foundedness. By rejecting anti‐symmetry, the anti‐extensionalist has a unified, independently grounded response to Varzi's arguments. I give a formal construction of a non‐extensional mereology in which anti‐symmetry fails. If the notion of ‘mereological equivalence’ is made explicit, this non‐anti‐symmetric mereology recaptures all of (...) the structure of classical mereology. (shrink)

The charge that the limit-frequency theory of probability is inconsistent due to incompatibility between the required features of randomness and limit convergence is inapplicable when probability sequences are taken to be empirically (i.e., extensionally) generated, as they must be on a strictly empirical conception of probability. All past attempts to meet this charge by formulating constructive definitions of randomness that would still allow for a demonstrable limit-convergence have, in their exclusive concern with logically (i.e., intensionally) prescribed sequences, left the logic (...) of extensional classes essentially untouched. In the light of a strict differentiation between intensional and extensional classes a generalized approach is possible under which several closely connected senses of randomness, i.e., the formal, material, restricted and unrestricted senses may be easily distinguished and related to the notion of relevance. (shrink)

I argue that 'c' occurs extensionally in 'c caused e' and 'D' occurs extensionally in 'c caused e because c is D'. I claim that this has been insufficiently appreciated because the two contexts are often run together and because it has not been clear that the description D of c is among the referents of an explanatory argument. I argue as well that Hume's analysis of causation is consistent with taking causation to be a relation between single events, and (...) that even if events are eliminated as virtual entities, extensionality holds for all terms in the resulting context. (shrink)

Semantic composition in language must be closely related to semantic composition in thought. But the way the two processes are explained differs considerably. Focusing primarily on propositional content, language theorists generally take semantic composition to be a truth‐conditional process. Focusing more on extensional content, cognitive theorists take it to be a form of concept combination. But though deep, this disconnect is not irreconcilable. Both areas of theory assume that extensional (i.e., denotational) meanings must play a role. As this article demonstrates, (...) they also have the potential to fulfill a mediative function. What is shown is that extensional meanings are themselves inherently compositional. On this basis, it becomes possible to model semantic composition without assuming the existence of any specifically linguistic/conceptual apparatus. Examples are presented to demonstrate this direct style of modeling. Abstract connections between composition in thought and language can then be made, raising the prospect of a more unified, theoretical account of semantic composition. (shrink)

Ethical extensionism generally involves drawing one or more lines of moral standing. I argue for all living organisms, there is a non- zero probability of sentience and consciousness, and we cannot justify excluding beings from consideration on the basis of uncertainty of their sentience, etc., and rather we should incorporate this uncertainty into the strength of our moral responsibilities. This use of probabilities differs critically from multi-criteria theories of moral standing and those that assign benefit of the doubt, which involve (...) unjustified exclusions and dilutions of duties. From uncertainty rises certainty : we have duties to non-human organisms, although they may often be minor. This modification of extensionist ethics provides foundation for an environmental ethic that parallels interpersonal and animal welfare ethics, and it suggests that we owe much greater concern to 'lower' organisms than they are typically given. (shrink)

It has been argued that prototypes cannot compose, and that for this reason concepts cannot be prototypes (Osherson and Smith in Cognition 9:35–58, 1981; Fodor and Lepore in Cognition 58:253–270, 1996; Connolly et al. in Cognition 103:1–22, 2007). In this paper I examine the intensional and extensional approaches to prototype compositionality, arguing that neither succeeds in their present formulations. I then propose a hybrid extensional theory of prototype compositionality, according to which the extension of a complex concept is determined as (...) a function of what triggers its constituent prototypes. I argue that the theory escapes the problems traditionally raised against extensional theories of compositionality. (shrink)

In [17], we introduced an extensional variant of generic realizability [22], where realizers act extensionally on realizers, and showed that this form of realizability provides inner models of $\mathsf {CZF}$ (constructive Zermelo–Fraenkel set theory) and $\mathsf {IZF}$ (intuitionistic Zermelo–Fraenkel set theory), that further validate $\mathsf {AC}_{\mathsf {FT}}$ (the axiom of choice in all finite types). In this paper, we show that extensional generic realizability validates several choice principles for dependent types, all exceeding $\mathsf {AC}_{\mathsf {FT}}$. We then show that adding (...) such choice principles does not change the arithmetic part of either $\mathsf {CZF}$ or $\mathsf {IZF}$. (shrink)

A development of Quine's views took place between the denial of analyticity (in "Two Dogmas") and the doctrine of indeterminacy (in Word and Object). Quine argues for the inscrutability of extensional as well as intensional content. The debate with Carnap in the mid-fifties pushes Quine to argue for full indeterminacy. Quine initially resists arguing for indeterminacy because the doctrine seems to lead to general skepticism, not just to skepticism about meanings. Quine draws on Tarski's work on truth to dispel the (...) skepticism, and only then argues for indeterminacy. This shows why Tarski's work is especially important for Quine. (shrink)

Neo-Fregean approaches to set theory, following Frege, have it that sets are the extensions of concepts, where concepts are the values of second-order variables. The idea is that, given a second-order entity $X$, there may be an object $\varepsilon X$, which is the extension of X. Other writers have also claimed a similar relationship between second-order logic and set theory, where sets arise from pluralities. This paper considers two interpretations of second-order logic—as being either extensional or intensional—and whether either is (...) more appropriate for this approach to the foundations of set theory. Although there seems to be a case for the extensional interpretation resulting from modal considerations, I show how there is no obstacle to starting with an intensional second-order logic. I do so by showing how the $\varepsilon$ operator can have the effect of “extensionalizing” intensional second-order entities. (shrink)

The signature of the formal language of mereology contains only one binary predicate P which stands for the relation “being a part of”. Traditionally, P must be a partial ordering, that is, ${\forall{x}Pxx, \forall{x}\forall{y}((Pxy\land Pyx)\to x=y)}$ and ${\forall{x}\forall{y}\forall{z}((Pxy\land Pyz)\to Pxz))}$ are three basic mereological axioms. The best-known mereological theory is “general extensional mereology”, which is axiomatized by the three basic axioms plus the following axiom and axiom schema: (Strong Supplementation) ${\forall{x}\forall{y}(\neg Pyx\to \exists z(Pzy\land \neg Ozx))}$ , where Oxy means ${\exists (...) z(Pzx\land Pzy)}$ , and (Fusion) ${\exists x\alpha \to \exists z\forall y(Oyz\leftrightarrow \exists x(\alpha \land Oyx))}$ , for any formula α where z and y do not occur free. In this paper, I will show that general extensional mereology is decidable, and will also point out that the decidability of the first-order approximation of the theory of complete Boolean algebras can be shown in the same way. (shrink)

Two straightforward “extensionalisations” of Kleene's realizability are considered; denoted re and e. It is shown that these realizabilities are not equivalent. While the re-notion is a subset of Kleene's realizability, the e-notion is not. The problem of an axiomatization of e-realizability is attacked and one arrives at an axiomatization over a conservative extension of arithmetic, in a language with variables for finite sets. A derived rule for arithmetic is obtained by the use of a q-variant of e-realizability; this rule subsumes (...) the well-known Extended Church's Rule. The second part of the paper focuses on toposes for these realizabilities. By a relaxation of the notion of partial combinatory algebra, a new class of realizability toposes emerges. Relationships between the various realizability toposes are given, and results analogous to Robinson and Rosolini's characterization of the effective topos, are obtained for a topos generalizing e-realizability. (shrink)

The classical theory of types in question is essentially the theory of Martin-Löf [1] but with the law of double negation elimination. I am ultimately interested in the theory of types as a framework for the foundations of mathematics and, for this purpose, we need to consider extensions of the theory obtained by adding ‘well-ordered types,’ for example the type N of the finite ordinals; but the unextended theory will suffice to illustrate the treatment of extensional equality.

I discuss Frege's argument - later called the slingshot - that if a construction is extensional and preserves logical equivalence then it is truth-functional. I consider some simple apparent counterexamples and conclude that they are not sentence-embedding in the required way.

A logic \ is called self-extensional if it allows to replace occurrences of a formula by occurrences of an \-equivalent one in the context of claims about logical consequence and logical validity. It is known that no three-valued paraconsistent logic which has an implication can be self-extensional. In this paper we show that in contrast, the famous Dunn–Belnap four-valued logic has exactly one self-extensional four-valued extension which has an implication. We also investigate the main properties of this logic, determine the (...) expressive power of its language, and provide a cut-free Gentzen-type proof system for it. (shrink)

In his 2000 book Logical Properties Colin McGinn argues that predicates denote properties rather than sets or individuals. I support the thesis, but show that it is vulnerable to a type-incongruity objection, if properties are (modelled as) functions, unless a device for extensionalizing properties is added. Alternatively, properties may be construed as primitive intensional entities, as in George Bealer. However, I object to Bealer’s construal of predication as a primitive operation inputting two primitive entities and outputting a third primitive entity. (...) Instead I recommend we follow Pavel Tichý in construing both predication and extensionalization as instances of the primitive operation of functional application. (shrink)

In this paper we re-examine the semantics of classical higher-order logic with the purpose of clarifying the role of extensionality. To reach this goal, we distinguish nine classes of higher-order models with respect to various combinations of Boolean extensionality and three forms of functional extensionality. Furthermore, we develop a methodology of abstract consistency methods (by providing the necessary model existence theorems) needed to analyze completeness of (machine-oriented) higher-order calculi with respect to these model classes.

The problems that Millikan addresses in theories of concepts arise from an extensional view of concepts and word meaning. If instead one assumes that concepts are psychological entities intended to explain human behavior and thought, many of these problems dissolve.

Achille Varzi has shown that it is harder to deny Extensionality than we may have thought: he's argued that if we define proper parthood as parthood with distinctness, cases we take to violate Extensionality don’t really involve sharing of all proper parts. Aaron Cotnoir has responded by showing that, if we instead define proper parthood as asymmetric parthood, we can take Extensionality to be violated in these cases. I will offer a new response to this argument. Even (...) granting Varzi's definition of proper parthood, there are versions of Extensionality very similar to the one Varzi and Cotnoir discuss. One of these is motivated by the same intuitions as Varzi's Extensionality, and is violated in cases traditionally taken to be anti-Extensional. Thus, even granting Varzi's definition of proper parthood, anti-Extensionalists can capture a sense in which their cases violate Extensionality, without this merely amounting to a violation of Uniqueness of Composition. (shrink)