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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]

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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]

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]

Copyright of Cognitio - Revista de Filosofia is the property of Cognitio - Revista de Filosofia and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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]

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]

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]

Consider one of several things. Is the one thingnecessarilyone of the several? This key question in the modal logic of plurals is clarified. Some defenses of an affirmative answer are developed and compared. Various remarks are made about the broader philosophical significance of the question. [ABSTRACT FROM PUBLISHER]

The paper concerns Grzegorczyk's non-Fregean logics that are intended to be a formal representation of the equimeaning relation defined on descriptions. We argue that the main Grzegorczyk logics discussed in the literature are too strong and we propose a new logical system, $${\mathsf{MGL}}$$ , which satisfies Grzegorczyk's fundamental requirements. We present a sound and complete semantics for $${\mathsf{MGL}}$$ and we prove that it is decidable. Finally, we show that many non-classical logics are extensions of $${\mathsf{MGL}}$$ , which makes it a generic non-Fregean logic. [ABSTRACT FROM AUTHOR]

The paper presents a new argument supporting the ontological standpoint according to which there are no mathematical facts in any set theoretic model (world) of arithmetical theories. It may be interpreted as showing that it is impossible to construct fact-arithmetic. The importance of this conclusion arises in the context of cognitive science. In the paper, a new type of slingshot argument is presented, which is called hyper-slingshot. The difference between meta-theoretical hyper-slingshots and conventional slingshots consists in the fact that the former are formulated in the semantic meta-language of mathematical theories without the use of the iota-operator or the lambda-operator as the abstractor, whereas the latter require for their expression at least one of these non-standard term-operators. Hyper-slingshots implement simpler language tools in comparison with those used in conventional slingshots. [ABSTRACT FROM AUTHOR]

Those who accept the necessity of mereological universalism face what has come to be known as the ‘junk argument’ due to Bohn [2009], which proceeds from (i) the incompatibility of junk with universalism and (ii) the possibility of junk, to conclude that mereological universalism isn't metaphysically necessary. Most attention has focused on (ii); however, recent authors have cast doubt on (i). This paper undertakes a defence of premise (i) against three main objections. The first is a new objection to the effect that Bohn's defence of that premise presupposes far too much. I show that one can defend premise (i) from a much weaker set of assumptions. The second objection, due to Contessa [2012], is that those who accept unrestricted composition should only accept the existence of binary sums (which are compatible with junk) rather than infinitary fusions. I argue that this conception of unrestricted composition is problematic: it is in conflict with an intuitive remainder principle. The final objection is due to Spencer [2012]. His view is that there is no absolutely unrestricted plural universal quantifier; so any statement of the unrestricted fusion axiom will simply not rule out the existence of junky worlds. I argue that the failure of unrestricted quantification will not be enough by itself to establish the existence of junk. Furthermore, it is not clear whether this view counts as a form of mereological universalism. As a result, I suggest that if one wants to reject the junk argument, premise (ii) is the only viable option. [ABSTRACT FROM PUBLISHER]

This paper presents an extensional account of manyand few that explains data that have previously motivated intensional analyses of these quantifiers (cf. Fernando and Kamp, Proceedings of semantics and linguistic theory, ; Lappin, Linguist Philos, 23(6):599-620, ). 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 (J Semant, 16:43-65, ; Nat Lang Semant, 9:41-67, ), 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 (proportional, reverse, focus-affected, and cardinal) 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. [ABSTRACT FROM AUTHOR]

We propose a purely extensional semantics for higher-order logic programming. In this semantics program predicates denote sets of ordered tuples, and two predicates are equal iff they are equal as sets. Moreover, every program has a unique minimum Herbrand model which is the greatest lower bound of all Herbrand models of the program and the least fixed-point of an immediate consequence operator. We also propose an SLD-resolution proof system which is proven sound and complete with respect to the minimum Herbrand model semantics. In other words, we provide a purely extensional theoretical framework for higher-order logic programming which generalizes the familiar theory of classical (first-order) logic programming. [ABSTRACT FROM AUTHOR]

I argue that it is intuitive and useful to think about composition in the light of the familiar functionalist distinction between role and occupant. This involves factoring the standard notion of parthood into two related notions: being a parthood slot and occupying a parthood slot. One thing is part of another just in case it fills one of that thing's parthood slots. This move opens room to rethink mereology in various ways, and, in particular, to see the mereological structure of a composite as potentially outreaching the individual entities that are its parts. I sketch one formal system that allows things to have individual entities as parts multiple times over. This is particularly useful to David Armstrong, given Lewis's charge that his structural universals must do exactly that. I close by reflecting upon the nature and point of formal mereology. [ABSTRACT FROM PUBLISHER]

Framing effects occur when different descriptions of the same decision problem give rise to divergent decisions. They can be seen as a violation of the decision-theoretic version of the principle of extensionality (PE). The PE in logic means that two logically equivalent sentences can be substituted salva veritate. We explore what this notion of extensionality becomes in decision contexts. Violations of extensionality may have rational grounds. Based on some ideas proposed by the psychologist Craig McKenzie and colleagues, we contend that framing effects are justified when the selection of one particular frame conveys choice relevant information. We first discuss this idea from a philosophical point of view, and proceed next to formalize it first in the context of the Bolker–Jeffrey decision theory. Finally, we extend the previous analysis to non-expected utility theories using the Biseparable Preference model introduced by Ghirardato and Marinacci (2001) and therefore show that the analysis is independent of the assumptions of Bayesian decision theory. [ABSTRACT FROM AUTHOR]

It has often been remarked that the metatheory of strong reduction $$\succ$$, the combinatory analogue of βη-reduction $${\twoheadrightarrow_{\beta\eta}}$$ in λ-calculus, is rather complicated. In particular, although the confluence of $$\succ$$ is an easy consequence of $${\twoheadrightarrow_{\beta\eta}}$$ being confluent, no direct proof of this fact is known. Curry and Hindley’s problem, dating back to 1958, asks for a self-contained proof of the confluence of $$\succ$$ , one which makes no detour through λ-calculus. We answer positively to this question, by extending and exploiting the technique of transitivity elimination for ‘analytic’ combinatory proof systems, which has been introduced in previous papers of ours. Indeed, a very short proof of the confluence of $$\succ$$ immediately follows from the main result of the present paper, namely that a certain analytic proof system G e[ $$\mathbb {C}$$] , which is equivalent to the standard proof system CL ext of Combinatory Logic with extensionality, admits effective transitivity elimination. In turn, the proof of transitivity elimination—which, by the way, we are able to provide not only for G e[ $$\mathbb {C}$$] but also, in full generality, for arbitrary analytic combinatory systems with extensionality—employs purely proof-theoretical techniques, and is entirely contained within the theory of combinators. [ABSTRACT FROM AUTHOR]

We introduce new proof systems G[ β] and G ext[ β], which are equivalent to the standard equational calculi of λβ- and λβη- conversion, and which may be qualified as ‘analytic’ because it is possible to establish, by purely proof-theoretical methods, that in both of them the transitivity rule admits effective elimination. This key feature, besides its intrinsic conceptual significance, turns out to provide a common logical background to new and comparatively simple demonstrations—rooted in nice proof-theoretical properties of transitivity-free derivations—of a number of well-known and central results concerning β- and βη-reduction. The latter include the Church–Rosser theorem for both reductions, the Standardization theorem for β- reduction, as well as the Normalization ( Leftmost reduction) theorem and the Postponement of η-reduction theorem for βη-reduction [ABSTRACT FROM AUTHOR]

AGM-theory, named after its founders Carlos Alchourrón, Peter Gärdenfors and David Makinson, is the leading contemporary paradigm in the theory of belief-revision. The theory is reformulated here so as to deal with the central relational notions ‘ J is a contraction of K with respect to A’ and ‘ J is a revision of K with respect to A’. The new theory is based on a principal-case analysis of the domains of definition of the three main kinds of theory-change (expansion, contraction and revision). The new theory is stated by means of introduction and elimination rules for the relational notions. In this new setting one can re-examine the relationship between contraction and revision, using the appropriate versions of the so-called Levi and Harper identities. Among the positive results are the following. One can derive the extensionality of contraction and revision, rather than merely postulating it. Moreover, one can demonstrate the existence of revision-functions satisfying a principle of monotonicity. The full set of AGM-postulates for revision-functions allow for completely bizarre revisions. This motivates a Principle of Minimal Bloating, which needs to be stated as a separate postulate for revision. Moreover, contractions obtained in the usual way from the bizarre revisions, by using the Harper identity, satisfy Recovery. This provides a new reason (in addition to several others already adduced in the literature) for thinking that the contraction postulate of Recovery fails to capture the Principle of Minimal Mutilation. So the search is still on for a proper explication of the notion of minimal mutilation, to do service in both the theory of contraction and the theory of revision. The new relational formulation of AGM-theory, based on principal-case analysis, shares with the original, functional form of AGM-theory the idealizing assumption that the belief-sets of rational agents are to be modelled as consistent, logically closed sets of sentences. The upshot of the results presented here is that the new relational theory does a better job of making important matters clear than does the original functional theory. A new setting has been provided within which one can profitably address two pressing questions for AGM-theory: (1) how is the notion of minimal mutilation (by both contractions and revisions) best analyzed? and (2) how is one to rule out unnecessary bloating by revisions? [ABSTRACT FROM AUTHOR]

Gareth Evans proved that if two objects are indeterminately equal then they are different in reality. He insisted that this contradicts the assumption that there can be vague objects. However we show the consistency between Evans's proof and the existence of vague objects within classical logic. We formalize Evans's proof in a set theory without the axiom of extensionality, and we define a set to be vague if it violates extensionality with respect to some other set. There exist models of set theory where the axiom of extensionality does not hold, so this shows that there can be vague objects. [ABSTRACT FROM AUTHOR]

How natural is natural deduction? – Gentzen's system of natural deduction intends to fit logical rules to the effective mathematical reasoning in order to overcome the artificiality of deductions in axiomatic systems (¶ 2). In spite of this reform some of Gentzen's rules for natural deduction are criticised by psychologists and natural language philosophers for remaining unnatural. The criticism focuses on the principle of extensionality and on formalism of logic (¶ 3). After sketching the criticism relatively to the main rules, I argue that the criteria of economy, simplicity, pertinence etc., on which the objections are based, transcend the strict domain of logic and apply to arguments in general (¶ 4). (¶ 5) deals with Frege's critique of the concept of naturalness as regards logic. It is shown that this concept means a regression into psychologism and is exposed to the same difficulties as are: relativity, lack of precision, the error of arguing from `is' to `ought' (the naturalistic fallacy). Despite of these, the concept of naturalness plays the role of a diffuse ideal which favours the construction of alternative deductive systems in contrast to the platonic conception of logic (¶ 6). [ABSTRACT FROM AUTHOR]

It is well known that the extensional axiom of choice (ACext) implies the law of excluded middle (EM). We here prove that the converse holds as well if we have the intensional (‘type-theoretical’) axiom of choice ACint, which is provable in Martin-Löf's type theory, and a weak extensionality principle (Ext-), which is provable in Martin-Löf's extensional type theory. In particular, EM is equivalent to ACext in extensional type theory. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

We introduce, in a general setting, an ‘‘analytic’’ version of standard equational calculi of combinatory logic. Analyticity lies on the one side in the fact that these calculi are characterized by the presence of combinatory introduction rules in place of combinatory axioms, and on the other side in that the transitivity rule proves to be eliminable. Apart from consistency, which follows immediately, we discuss other almost direct consequences of analyticity and the main transitivity elimination theorem; in particular the Church-Rosser and the leftmostreduction theorems for the associated notions of reduction. The last two sections deal with analytic combinatory calculi with the extensionality rule added. Here, as far as the elimination of transitivity is concerned, we have only partial results, which unfortunately do not cover, at present, full CL + Ext. Yet, they are sufficient to prove the decidability of weaker combinatory calculi with extensionality, including e.g. BCK + Ext. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with the RCC8 composition table entailed by the Region Connection Calculus (RCC), a well-known formalism for Qualitative Spatial Reasoning. This table has been independently generated by Egenhofer in the context of Geographic Information Systems. It has been known for some time that the table is not extensional for each RCC model. This paper however shows that the Egenhofer model is indeed an extensional one for the RCC8 composition table. Moreover this model is the maximal extensional one for the RCC8 composition table in a sense. [ABSTRACT FROM AUTHOR]

Several properties of monotone functionals (MF) and monotone majorizable functionals (MMF) used in the earlier work by the author and van de Pol are proved. It turns out that the terms of the simply typed lambda-calculus define MF, but adding primitive recursion, and even monotonic primitive recursion changes the situation: already λZ.Z(1 — sg) is not MMF. It is proved that extensionality is not Dialectica-realizable by MMF, and a simple example of a MF which is not hereditarily majorizable is given. [ABSTRACT FROM AUTHOR]