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Disjointism is the view that co-located objects do not share any parts. A human-shaped statue is composed from a torso, head and limbs; the co-located lump of clay is only composed from chunks of clay. This essay discusses the tenability of this relatively neglected view, focusing on two objections. The first objection is that disjointism implies co-located copies of microphysical particles. I argue that it doesn't imply this and that there are more plausible disjointist views of tiny parts available. The second objection is that disjointism is at a loss to explain how material objects can be co-located and why the weights of co-located objects don't add up. The standard pluralist account appeals to the fact that co-located objects stand in mereological relations and this account is not available to the disjointist. I sketch an alternative account that appeals to a notion of 'material identity': the statue is taken to be the same matter as the lump of clay. The resort to a new theoretical primitive may seem to invite a quick rejection on grounds of unnecessary theoretical complexity but I argue that an abductive comparison with rival forms of pluralism shows that such a rejection is misguided. [ABSTRACT FROM AUTHOR]

This study uses a hermeneutic phenomenological approach to better understand therapist experiencing of metaphor in non-directive person-centered relationships. Metaphor is ubiquitous in language. Meanwhile, literature in the field of psychotherapy posits it as indicative of a range of therapeutic attitudes, goals and outcomes. However, outside of noting its uses for communicating empathic understanding, and its function in process direction and diagnosis, the person-centered approach has no coherent account of it. Analysis of interviews with three therapists revealed experiencing related to relinquishing certainty, risking curiosity, and connecting briefly. These findings led to a brief consideration of the dialogical contact made when meeting in metaphor, with meaning noticed as an emergent quality of this contact. Whilst making a case for research that broadens the notion of rationality and embraces the complex ineffability of human experience, this article suggests some further questions, derived from the results, that might be addressed in future reflection. [ABSTRACT FROM AUTHOR]

We investigate whether ordinary quantification over objects is an extensional phenomenon, or rather creates non-extensional contexts; each claim having been propounded by prominent philosophers. It turns out that the question only makes sense relative to a background theory of syntax and semantics (here called a grammar) that goes well beyond the inductive definition of formulas and the recursive definition of satisfaction. Two schemas for building quantificational grammars are developed, one that invariably constructs extensional grammars (in which quantification, in particular, thus behaves extensionally) and another that only generates non-extensional grammars (and in which quantification is responsible for the failure of extensionality). We then ask whether there are reasons to favor one of these grammar schemas over the other, and examine an argument according to which the proper formalization of deictic utterances requires adoption of non-extensional grammars. [ABSTRACT FROM AUTHOR]

The aim of this paper is to offer an interpretation, partially novel, of passage 5.54-5.5423 of the Tractatus in which Wittgenstein examines the propositions that represent intentional relations such as judgment. The fundamental idea will be that these propositions are thought of as propositions about complexes and, consequently, should be analyzed in accordance with paragraph 2.0201, which will lead me to develop step by step the proposed analysis. Furthermore, I will argue that Wittgenstein's theory, understood in this way, rules out the possibility of judging nonsense, unlike Russell's theory of judgment as a multiple relation. [ABSTRACT FROM AUTHOR]

This study explores the phenomenon of the so-called “third reading” of propositional attitude reports. This reading, which was originally explored in the dissertation of J. Fodor (1973) and has since become one of the significant problems in the formal semantics of natural languages, differs from the more well-known de re and de dicto readings by being an intermediate case. If the de re interpretation can be referred to as transparent specific, and the de dicto interpretation as opaque non-specific, then the third reading is transparent non-specific. Fodor's standard solution has been the subject of much discussion in the literature and has given rise to a series of widely accepted counterexamples that are thought to demonstrate the limitations of Fodor's solution. At the same time, alternative approaches to the explication of these readings also suffer from formal shortcomings (for example, some of them did not satisfy the requirement of meaning compositionality, which is a basic requirement in formal semantic literature). This study points out that the analysis of all complex cases of the third reading did not fully take into account their syntactic structure. This remained unnoticed due to the ellipsis in many of these cases. It is shown that a restoration of the ellided syntactic structure makes it possible to analyze all hard cases as basic ones using the classical standard analysis proposed by Fodor supplemented by the principles of admissibility of substitution of L-equivalent expressions in intensional contexts (known since G. Frege and R. Carnap). In the final part of the work, it is demonstrated how exactly the main complex cases of the “third reading” are explicated in terms of the standard approach.

The literature dealing with Gödel's legacy is largely preoccupied with challenging his philosophical views, regarding them as outdated. We believe that such an approach prevents us from seeing Gödel's views in the right light and understanding their rationale. In this article, his views are discussed in the philosophical realm in which he himself understood them. We explore the consequences of Gödel's incompleteness theorems for the question of the objectivity of mathematics and its epistemology. Taking set theory as the paradigm of formal mathematical theories, we examine the relationship between its incompleteness and extensionality. We argue, based on his philosophical views, that Gödel believed incompleteness can be overcome only by some intensional considerations about concepts from the basis of mathematical theories. These considerations should eventually lead to founding the so-called logic of concepts. [ABSTRACT FROM AUTHOR]

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 $\mathsf {IZF}$. [ABSTRACT FROM AUTHOR]

Mereological atomism is the thesis that everything is ultimately composed of atomic parts, i.e., parts without proper parts. Typically, this thesis is characterized by an axiom stating that everything has atomic parts. The present paper argues that the success of this standard characterization depends on how the notions of sum and composition are defined. In particular, we put forward a novel definition of mereological sum that: (i) is not equivalent to existing definitions in the literature, if no strong decomposition principle is assumed; (ii) can be used to argue that the standard characterization of atomism fails, because having atomic parts is not sufficient to be a sum of atoms; and (iii) provides a purely mereological distinction between structured and unstructured wholes, contributing to the ongoing debate on this crucial topic. [ABSTRACT FROM AUTHOR]

A model of Martin-Löf extensional type theory with universes is formalized in Agda, an interactive proof system based on Martin-Löf intensional type theory. This may be understood, we claim, as a solution to the old problem of modeling the full extensional theory in the intensional theory. Types are interpreted as setoids, and the model is therefore a setoid model.We solve the problem of interpreting type universes by utilizing Aczel's type of iterative sets and show how it can be made into a setoid of small setoids containing the necessary setoid constructions. In addition, we interpret the bracket types of Awodey and Bauer. Further quotient types should be interpretable. [ABSTRACT FROM AUTHOR]

In this paper we try to find a computational interpretation for a strong form of extensionality, which we call "converse extensionality". Converse extensionality principles, which arise as the Dialectica interpretation of the axiom of extensionality, were first studied by Howard. In order to give a computational interpretation to these principles, we reconsider Brouwer's apartness relation, a strong constructive form of inequality. Formally, we provide a categorical construction to endow every typed combinatory algebra with an apartness relation. We then exploit that functions reect apartness, in addition to preserving equality, to prove that the resulting categories of assemblies model a converse extensionality principle. [ABSTRACT FROM AUTHOR]

Generalized numberings are an extension of Ershov's notion of numbering, based on partial combinatory algebra (pca) instead of the natural numbers. We study various algebraic properties of generalized numberings, relating properties of the numbering to properties of the pca. As in the lambda calculus, extensionality is a key notion here. [ABSTRACT FROM AUTHOR]

The word problem for a finite set of ground identities is known to be decidable in polynomial time using congruence closure, and this is also the case if some of the function symbols are assumed to be commutative or defined by certain shallow identities, called strongly shallow. We show that decidability in P is preserved if we add the assumption that certain function symbols f are extensional in the sense that f (s 1 , ... , s n) ≈ f (t 1 , ... , t n) implies s 1 ≈ t 1 , ... , s n ≈ t n . In addition, we investigate a variant of extensionality that is more appropriate for commutative function symbols, but which raises the complexity of the word problem to coNP. [ABSTRACT FROM AUTHOR]

Appeal to standard set theory in (mainstream) minimalist syntax is shown to be in conflict with the goal of analyzing dependency formation, a.k.a. movement, as involving genuine constituent copies. The underlying tension is due to (the axiom of) extensionality, which—other things being equal—favors a perspective on dependencies in terms of multidominance. The above argument is developed against the backdrop of a recent exposition of minimalist syntax (Chomsky et al. in Catal J Linguist (Special Issue):229–261, 2019), which can be seen as exemplary. The resulting critical assessment should be taken as removing obstacles on the way toward proper minimalist foundations for syntactic theory. [ABSTRACT FROM AUTHOR]

On the occasion of the 50th anniversary of the publication of Zadeh's significant paper Similarity Relations and Fuzzy Orderings, an account of the development of similarity relations during this time will be given. Moreover, the main topics related to these fuzzy relations will be reviewed. [ABSTRACT FROM AUTHOR]

The current paper overviews of the notion of intensionality as it is presented in the work of Ullin Place, with the aim of characterising Place's somewhat neglected thinking about this topic. Ullin Place's work showed a development regarding his views concerning this topic, which, in themselves, illustrate a variety of possible stances that can be taken towards the concept of intensionality. Ultimately, Place suggested that 'intensional' statements are not necessarily connected with 'mentalistic' language, nor with 'mentalistic' explanations. Rather, Place came to the view that intensionality should be taken to be the mark of the 'conversational' - that is, it is a property of verbal behaviour that characterises nonscientific everyday discourse. This view has relevance to furthering the understanding of Place's work regarding intensionality, and also relevance for understanding the types of language that could be used in explanations given by behavioural science. [ABSTRACT FROM AUTHOR]

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The extensionality of ⊤-convergence spaces is verified for a complete residuated lattice L with the top element ⊤. And also the E -connectedness of ⊤-convergence spaces for a class E of ⊤-convergence spaces is proposed by generalizing Preuss's connectedness of topological spaces. Then we establish a necessary and sufficient condition that for a class K of ⊤-convergence spaces, there exists a class E of ⊤-convergence spaces such that each space of K is E -connected, where we stress the point that the conclusion benefits from the extensionality of the category of ⊤-convergence spaces. We further present a deep relationship between E -connectedness and T 1 -separation for ⊤-convergence spaces, that is, the E -connectedness of each subset in a ⊤-convergence space implies that of its closure if and only if E precisely is a class of ⊤-convergence spaces being T 1 -separated, and as a natural result, the product theorem for E -connected ⊤-convergence spaces is obtained. [ABSTRACT FROM AUTHOR]

There are two main ways in which the notion of mereological fusion is usually defined in the current literature in mereology which have been labelled 'Leśniewski fusion' and 'Goodman fusion'. It is well-known that, with Minimal Mereology as the background theory, every Leśniewski fusion also qualifies as a Goodman fusion. However, the converse does not hold unless stronger mereological principles are assumed. In this paper I will discuss how the gap between the two notions can be filled, focussing in particular on two specific sets of principles that appear to be of particular philosophical interest. The first way to make the two notions equivalent can be used to shed some interesting light on the kind of intuition both notions seem to articulate. The second shows the importance of a little-known mereological principle which I will call 'Mild Supplementation'. As I will show, the mereology obtained by adding Mild Supplementation to Minimal Mereology occupies an interesting position in the landscape of theories that are stronger than Minimal Mereology but weaker than what Achille Varzi and Roberto Casati have labelled 'Extensional Mereology'. [ABSTRACT FROM AUTHOR]

The search for the extensions of sentences can be guided by Frege's "principle of compositionality of extension", according to which the extension of a composed expression depends only on its logical form and the extensions of its parts capable of having extensions. By means of this principle, a strict criterion for the admissibility of objects as extensions of sentences can be derived: every object is admissible as the extension of a sentence that is preserved under the substitution of co-extensional expressions. The question is: what are the extensions of elementary sentences containing empty singular terms, like 'Vulcan rotates'. It can be demonstrated that in such sentences, states of affairs as structured objects (but not truth-values) are preserved under the substitution of co-extensional expressions. Hence, such states of affairs are admissible (while truth-values are not) as extensions of elementary sentences containing empty singular terms. [ABSTRACT FROM AUTHOR]

Leo-III is an automated theorem prover for extensional type theory with Henkin semantics and choice. Reasoning with primitive equality is enabled by adapting paramodulation-based proof search to higher-order logic. The prover may cooperate with multiple external specialist reasoning systems such as first-order provers and SMT solvers. Leo-III is compatible with the TPTP/TSTP framework for input formats, reporting results and proofs, and standardized communication between reasoning systems, enabling, e.g., proof reconstruction from within proof assistants such as Isabelle/HOL. Leo-III supports reasoning in polymorphic first-order and higher-order logic, in many quantified normal modal logics, as well as in different deontic logics. Its development had initiated the ongoing extension of the TPTP infrastructure to reasoning within non-classical logics. [ABSTRACT FROM AUTHOR]

There are three families of solutions to the traditional Amputation Paradox: Eliminativism, Contingent Identity Theories, and Theories of Coincident Entities. Theories of Coincident Entities challenge our common understanding of the relation between identity and parthood, since they accept that two things can be mereologically coincident without being identical. The contemporary discussion of the Amputation Paradox tends to mention only one theory of Coincident Entities, namely the Constitution View, which violates the mereological principle of Extensionality. But in fact, there is another theory, namely the Unique Part View, which violates another mereological principle (the Weak Supplementation Principle). In this paper, I argue that the contemporary focus on the Constitution View is unmotivated, at least when we are confronted with the Amputation Paradox, and that a balanced comparison of the two views (as solutions to this specific paradox) should favour the Unique Part View. [ABSTRACT FROM AUTHOR]

Mereological principles are often controversial; perhaps the most stark contrast is between those who claim that Weak Supplementation is analytic—constitutive of our notion of proper parthood—and those who argue that the principle is simply false, and subject to many counterexamples. The aim of this paper is to diagnose the source of this dispute. I'll suggest that the dispute has arisen by participants failing to be sensitive to two different conceptions of proper parthood: the outstripping conception and the non-identity conception. I'll argue that the outstripping conception (together with a specific set of definitions for other mereological notions), can deliver the analyticity of Weak Supplementation on at least one sense of 'analyticity'. I'll also suggest that the non-identity conception cannot do so independently of considerations to do with mereological extensionality. [ABSTRACT FROM AUTHOR]

Two mereological theories are presented based on a primitive apartness relation along with binary relations of mereological excess and weak excess, respectively. It is shown that both theories are acceptable from the standpoint of constructive reasoning while remaining faithful to the spirit of classical mereology. The two theories are then compared and assessed with regard to their extensional import. [ABSTRACT FROM AUTHOR]

One of the more persistent debates in mereology is whether distinct wholes can have the same parts. Extensional mereologists hold that if there is no part that makes the difference, then there is nothing to distinguish the wholes, so sameness of parts implies identity. Non-extensionalists, however, do think there are cases where distinct wholes share all their parts. This paper argues that the kind of argument non-extensionalists employ can also be levelled against a widely accepted extensionality principle of plural logic. Non-extensionalists thus face a dilemma: either give up the argument for non-extensional mereology, or deny both the extensionality principle of mereology and the analogous principle for plural logic. [ABSTRACT FROM AUTHOR]

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 (or, more accurately, of "isomorphism type"). 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. [ABSTRACT FROM AUTHOR]

In this work we provide an alternative, and equivalent, formulation of the concept of λ-theory without introducing the notion of substitution and the sets of all, free and bound variables occurring in a term. We call αβ-relations our alternative versions of λ-theories. We also clarify the actual role of α-renaming in the lambda calculus: it expresses a property of extensionality for a certain class of terms. To motivate the necessity of α-renaming, we construct an unusual denotational model of the lambda calculus that validates all structural and beta conditions but not α-renaming. The article also has a survey character. [ABSTRACT FROM AUTHOR]

If one takes seriously the idea that a scientific language must be extensional, and accepts Quine's notion of truth-value-related extensionality, and also recognizes that a scientific language must allow for singular terms that do not refer to existing objects, then there is a problem, since this combination of assumptions must be inconsistent. I will argue for a particular solution to the problem, namely, changing what is meant by the word 'extensionality', so that it would not be the truth-value that had to be preserved under the substitution of co-extensional expressions, but the state of affairs that the sentence described. The question is whether or not elementary sentences containing empty singular terms, such as 'Vulcan rotates', are extensional in the substitutivity sense. Five conditions are specified under which extensionality in the substitutivity sense of such sentences can be secured. It is demonstrated that such sentences are state-of-affairs-as-extension-related extensional. This implies (in accordance with the basic idea of state-of-affairs semantics) that such sentences are also truth-value-related extensional in Quine's sense, but not truth-value-as-extension-related extensional. [ABSTRACT FROM AUTHOR]

David Lewis famously endorsed Unrestricted Composition. His defense of such a controversial principle builds on the alleged innocence of mereology. This innocence defense has come under different attacks in the last decades. In this paper I pursue another line of defense, that stems from some early remarks by van Inwagen. I argue that Unrestricted Composition leads to a better metaphysics. In particular I provide new arguments for the following claims: Unrestricted Composition entails extensionality of composition, functionality of location and four-dimensionalism in the metaphysics of persistence. Its endorsement yields an impressively coherent and powerful metaphysical picture. This picture shows a universe that might not be innocent but it is certainly elegant. [ABSTRACT FROM AUTHOR]

This interventions essay deploys the notion of the (non)urban human to address the conundrums associated with identifying spaces, operations and entities outside of urbanization's planetary encompassment. In the piece, the objective of which is to call for a programme of prospective research collaborations, it seeks to explore domains of intersection among that which appears 'left out' of urbanization's purported advantages, that whose time has yet to come, and forms of the human that exceed the possibilities of self‐reflexive consciousness and free will. The essay draws upon the temporalities, rhythms, spatial arrangements and sensoria generated through histories of blackness and 'natural worlds' and their interactions, to posit extensionality—a dispersal of bodies and their capacities into more reciprocal and mutual enactments with the earthly surrounds—as a generative by‐product of extended urbanization. [ABSTRACT FROM AUTHOR]

We shall try to defend two non-standard views that run counter to two well-entrenched familiar views. The standard views are (1) the universal and existential quantifiers of first-order logic are not modal operators, and (2) the quantifiers are extensional. If that is correct then the counterclaims create genuine problems for some traditional philosophical doctrines. [ABSTRACT FROM AUTHOR]

This paper revisits Daniel Kahneman and Amos Tversky’s work on framing. It shows how various conventions involved in economic theory allow for the establishment of different equivalence relations between pairs of problems in framing experiments. Following on from this, an exegesis of their comments on these experiments is conducted. This concerns the relation between their theoretical explanation using prospect theory and the positive/normative distinction involved in models of individual behaviors. Throughout, a methodological framework is developed with a distinction between identity, equivalence and equality (borrowed from philosopher Craig Dilworth) for a critical analysis of the relation between external frames (the empirical structure of a decision problem) and internal frames (the psychological representation of the decision problem by decision makers).

An elementary theory of concatenation, QT+, is introduced and used to establish mutual interpretability of Robinson arithmetic, Minimal Predicative Set Theory, quantifier-free part of Kirby’s finitary set theory, and Adjunctive Set Theory, with or without extensionality. The most basic arithmetic and simplest set theory thus turn out to be variants of string theory. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]