Proof-theoretic semantics (P-tS) is the paradigm of semantics in which meaning in logic is based on proof (as opposed to truth). A particular instance of P-tS for intuitionistic propositional logic (IPL) is its base-extension semantics (B-eS). This semantics is given by a relation called support, explaining the meaning of the logical constants, which is parameterized by systems of rules called bases that provide the semantics of atomic propositions. In this paper, we interpret bases as collections of definite formulae and (...) use the operational view of them as provided by uniform proof-search—the proof-theoretic foundation of logic programming (LP)—to establish the completeness of IPL for the B-eS. This perspective allows negation, a subtle issue in P-tS, to be understood in terms of the negation-as-failure protocol in LP. Specifically, while the denial of a proposition is traditionally understood as the assertion of its negation, in B-eS we may understand the denial of a proposition as the failure to find a proof of it. In this way, assertion and denial are both prime concepts in P-tS. (shrink)

Russell argued, famously, that definite descriptions are not logical constituents of the sentences in which they appear. In neither of the following should we suppose that the definite description picks anything out: The King of France is bald The Prince of Wales is bald Since France is a republic, nothing could be picked out by the first; and if the semantic structures of each are the same, it cannot be the function of the second to pick anything out either. On (...) the alternative semantics developed in his 1905 article 'On Denoting', definite descriptions do not have meaning in isolation; they have meaning only in the context of a whole sentence. Andrew Botterell and Robert Stainton have pointed out that this conclusion appears to be at odds with the phenomenon of unembedded definite descriptions, in which definite descriptions are uttered, meaningfully, without accompanying predicates. For example, it is possible to utter 'The last temptation' on its own and in doing so express a proposition. Since definite descriptions can be used in this way, how can it be right to claim, with Russell, that they lack meaning in isolation? The present paper seeks to show how a Russellian semantics for definite descriptions is entirely compatible with the phenomenon of unembedded definite descriptions. In particular, Botterell and Stainton are wrong to think that generalized quantifier semantics is better able to cope with the phenomenon than a more authentically Russellian syncategorematic semantics. (shrink)

Implementation of formal specifications is very important in formal software development and can be described in terms of simple logical concepts. Formal specifications are presentations of theories in many-sorted first-order logic, and an implementation of a formal specification on another formal specification amounts to an interpretation of the former into a conservative extension of the latter. Here we present and analyse some sort introducing constructs akin to those found in many programming languages. This is of importance because it occurs (...) often in implementing formal specifications, when new sorts are ‘constructed’ from the concrete ones. We specify these constructs and show that these specifications provide extensions with behaviour close to that of the familiar extensions by definitions. (shrink)

ABSTRACT Joseph Halpern and Judea Pearl draw upon structural equation models to develop an attractive analysis of ‘actual cause’. Their analysis is designed for the case of deterministic causation. I show that their account can be naturally extended to provide an elegant treatment of probabilistic causation. 1Introduction 2Preemption 3Structural Equation Models 4The Halpern and Pearl Definition of ‘Actual Cause’ 5Preemption Again 6The Probabilistic Case 7Probabilistic Causal Models 8A Proposed Probabilistic Extension of Halpern and Pearl’s Definition 9Twardy and Korb’s Account (...) 10Probabilistic Fizzling 11Conclusion. (shrink)

In their article 'Causes and Explanations: A Structural-Model Approach. Part I: Causes', Joseph Halpern and Judea Pearl draw upon structural equation models to develop an attractive analysis of 'actual cause'. Their analysis is designed for the case of deterministic causation. I show that their account can be naturally extended to provide an elegant treatment of probabilistic causation.

In the domain in which the definition of property energy is available, a rigorous and general proof of additivity is provided. Then, both the definition of energy and the proof of additivity are extended to broader domains.

Extended simple regions have been the focus of recent developments in philosophical logic, metaphysics, and philosophy of physics. However, only a handful of works provides a rigorous characterization of an extended simple region. In particular, a recent paper in this journal defends a definition based on an extrinsic notion of least distance. Call it the Least Distance proposal. This paper provides the first assessment of it. It argues that Least Distance faces difficulties and drawbacks. The paper then goes on to (...) suggest a different proposal, the Diameter proposal that is able to handle such drawbacks and difficulties. (shrink)

This paper distinguishes between definiteness and determinacy. Definiteness is seen as a morphological category which, in English, marks a uniqueness presupposition, while determinacy consists in denoting an individual. Definite descriptions are argued to be fundamentally predicative, presupposing uniqueness but not existence, and to acquire existential import through general type-shifting operations that apply not only to definites, but also indefinites and possessives. Through these shifts, argumental definite descriptions may become either determinate or indeterminate. The latter option is observed in examples like (...) ‘Anna didn’t give the only invited talk at the conference’, which, on its indeterminate reading, implies that there is nothing in the extension of ‘only invited talk at the conference’. The paper also offers a resolution of the issue of whether possessives are inherently indefinite or definite, suggesting that, like indefinites, they do not mark definiteness lexically, but like definites, they typically yield determinate readings due to a general preference for the shifting operation that produces them. (shrink)

David Lewis, following in the tradition of Broad, Quine and Goodman, says that change in an object X consists in X's being temporally extended and having qualitatively different temporal parts. Analogously, change in a spatially extended object such as a road consists in its having different spatial parts . The alternative to this view is that ordinary objects undergo temporal change in virtue of having different intrinsic non-relational properties at different times. They endure, remaining the same object throughout change, whereas (...) Lewis's temporally extended 4D objects perdure.It is impossible to come down on one side or the other of the endurance/perdurance debate without a clear and unambiguous understanding of what ‘endurance’ is. ‘Perdurance’ is clear enough. Every 4D object of non-zero temporal thickness perdures simply by having temporal extension, i.e. by having different temporal parts, or stages, at different times. But Lewis muddies the waters by giving a confused and ultimately untenable definition of endurance.His first step is to cover both perduring and enduring by offering the neutral word persist. ‘Let us say that something persists iff, somehow or other, it exists at various times’ . Something then perdures if it persists by having different temporal parts at different times, no one part being wholly present at more than one time. In contrast, a thing endures if it persists by being wholly present at more than one …. (shrink)

Euclid’s definition of a point as “that which has no part” has been a major source of controversy in relation to the epistemological and ontological presuppositions of classical geometry, from the medieval and modern disputes on indivisibilism to the full development of point-free geometries in the 20th century. Such theories stem from the general idea that all talk of points as putative lower-dimensional entities must and can be recovered in terms of suitable higher-order constructs involving only extended regions (or bodies). (...) Here I focus on what is arguably the first thorough proposal of this sort, Whitehead’s theory of “extensive abstraction”, offering a critical reconstruction of the theory through its successive installments: from the purely mereological version of ‘La théorie relationniste de l’espace’ (1916) to the refined versions presented in An Enquiry Concerning the Principles of Natural Knowledge (1919) and in The Concept of Nature (1920) to the last, mereotopological version of Process and Reality (1929). (shrink)

Graph normal form, introduced earlier for propositional logic, is shown to be a normal form also for first-order logic. It allows to view syntax of theories as digraphs, while their semantics as kernels of these digraphs. Graphs are particularly well suited for studying circularity, and we provide some general means for verifying that circular or apparently circular extensions are conservative. Traditional syntactic means of ensuring conservativity, like definitional extensions or positive occurrences guaranteeing exsitence of fixed points, emerge as special (...) cases. (shrink)

Extensive and diverse conceptual work towards developing a definition of ‘mental disorder’ was motivated by the declassification of homosexuality from the Diagnostic and Statistical Manual in 1973. This highly politicised event was understood as a call for psychiatry to provide assurances against further misclassification on the basis of discrimination or socio-political deviance. Today, if a definition of mental disorder fails to exclude homosexuality, then it fails to provide this safeguard against potential abuses and therefore fails to do an important part (...) of the work it was intended to do. We argue that fact-based definitions of mental disorder, relying on scientific theory, fail to offer a robust definition of mental disorder that excludes homosexuality. Definitions of mental disorder based on values do not fare better: these definitions are silent on questions about the diagnostic status of individuals in oppressive societies and over-inclusive of mental or behavioural states that happen to be negatively valued in the individual's social context. We consider the latest definition proposed for the Diagnostic and Statistical Manual-5 in light of these observations. We argue that definition fails to improve on these earlier deficiencies. Its inclusion in the manual may offer false reassurance against repetition of past misclassifications. We conclude with a provocation that if candidate definitions of mental disorder are unable to exclude homosexuality, it might perhaps be preferable not to attempt a definition at all. (shrink)

The paper considers certain extensions of the system LC introduced in Dunn & Meyer 1997. LC is a structurally free system , but it has combinators as formulas in the place of structural rules. We consider two ways to extend LC with conjunction and disjunction depending on whether they distribute over each other or not. We prove the elimination theorem for the systems. At the end of the paper we give a Routley-Meyer style semantics for the distributive extension, including (...) some new definitions and an algorithm which are used in proving a representation theorem in general, i.e., for arbitrary combinators. (shrink)

In proof-theoretic semantics, meaning is based on inference. It may be seen as the mathematical expression of the inferentialist interpretation of logic. Much recent work has focused on base-extension semantics, in which the validity of formulas is given by an inductive definition generated by provability in a ‘base’ of atomic rules. Base-extension semantics for classical and intuitionistic propositional logic have been explored by several authors. In this paper, we develop base-extension semantics for the classical propositional modal systems (...) K, KT , K4, and S4, with □ as the primary modal operator. We establish appropriate soundness and completeness theorems and establish the duality between □ and a natural presentation of ♢. We also show that our semantics is in its current form not complete with respect to euclidean modal logics. Our formulation makes essential use of relational structures on bases. (shrink)

In this article, I use Edmund Gettier’s Ten Coins hypothetical scenario to illustrate some reasoning errors in the use of definite descriptions. The Gettier problem, central as it is to modern epistemology, is first and foremost an argument, which Gettier (Analysis 23(6):121–123, 1963) constructs to prove a contrary conclusion to a widely held view in epistemology. Whereas the epistemological claims in the case have been extensively analysed conceptually, the strategies and tools from other philosophical disciplines such as analytic philosophy of (...) language, logic and argumentation that Gettier deploys in the case have scarcely received any attention. This work abstracts from the epistemological content and examines Gettier’s handling of the definite description involved, and how that affects the cogency of his argument. (shrink)

In proof-theoretic semantics, meaning is based on inference. It may seen as the mathematical expression of the inferentialist interpretation of logic. Much recent work has focused on base-extension semantics, in which the validity of formulas is given by an inductive definition generated by provability in a ‘base’ of atomic rules. Base-extension semantics for classical and intuitionistic propositional logic have been explored by several authors. In this paper, we develop base-extension semantics for the classical propositional modal systems |$K$|⁠, (...) |$KT$|⁠, |$K4$| and |$S4$|⁠, with |$\square $| as the primary modal operator. We establish appropriate soundness and completeness theorems and establish the duality between |$\square $| and a natural presentation of |$\lozenge $|⁠. We also show that our semantics is in its current form not complete with respect to euclidean modal logics. Our formulation makes essential use of relational structures on bases. (shrink)

Frege’s famous definition of number famously uses the concept of “extension”. Extensions, in the Fregean framework, are susceptible to bringing many difficulties, and, some say, even paradoxes. Therefore, neo-logicist programs want to avoid the problems and to replace the classical Fregean definition of number with Hume’s Principle. I argue that this move, even if it makes sense from a computational point of view, is at odds with Frege’s larger philosophical project. For Frege, I claim, extensions were an important part (...) of his philosophical program of logic-as-an-universal-language. This is why Frege places his project in line with Leibniz’ philosophical project of finding a lingua characterica universalis. (shrink)

Frege’s famous definition of number famously uses the concept of “extension”. Extensions, in the Fregean framework, are susceptible to bringing many difficulties, and, some say, even paradoxes. Therefore, neo-logicist programs want to avoid the problems and to replace the classical Fregean definition of number with Hume’s Principle. I argue that this move, even if it makes sense from a computational point of view, is at odds with Frege’s larger philosophical project. For Frege, I claim, extensions were an important part (...) of his philosophical program of logic-as-an-universal-language. This is why Frege places his project in line with Leibniz’ philosophical project of finding a lingua characterica universalis. (shrink)

Frege’s famous definition of number famously uses the concept of “extension”. Extensions, in the Fregean framework, are susceptible to bringing many difficulties, and, some say, even paradoxes. Therefore, neo-logicist programs want to avoid the problems and to replace the classical Fregean definition of number with Hume’s Principle. I argue that this move, even if it makes sense from a computational point of view, is at odds with Frege’s larger philosophical project. For Frege, I claim, extensions were an important part (...) of his philosophical program of logic-as-an-universal-language. This is why Frege places his project in line with Leibniz’ philosophical project of finding a lingua characterica universalis. (shrink)

This paper proposes an extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system. Chaitin's Ω is defined as the probability that the universal self-delimiting Turing machine U halts, and plays a central role in the development of algorithmic information theory. In the theory, there are two equivalent ways to define the program-size complexity H of a given finite binary string s. In the standard way, H is defined as the length of the (...) shortest input string for U to output s. In the other way, the so-called universal probability m is introduced first, and then H is defined as –log2m without reference to the concept of program-size.Mathematically, the statistics of outcomes in a quantum measurement are described by a positive operator-valued measure in the most general setting. Based on the theory of computability structures on a Banach space developed by Pour-El and Richards, we extend the universal probability to an analogue of POVM in an infinite dimensional quantum system, called a universal semi-POVM. We also give another characterization of Chaitin's Ω numbers by universal probabilities. Then, based on this characterization, we propose to define an extension of Ω as a sum of the POVM elements of a universal semi-POVM. The validity of this definition is discussed.In what follows, we introduce an operator version equation image of H in a Hilbert space of infinite dimension using a universal semi-POVM, and study its properties. (shrink)

A rigorous extension of the full Lorentz group is found which is parameterized by interframe velocities v(t) and which reduces to Special Relativity for acceleration-free cases and to Galilean relativity for low velocity cases. Full group properties are exhibited. Four-momentum is defined and particle masses are shown to be invariants. Four-force is introduced and pseudoforces are shown to enter the equations of particle dynamics. Maxwell's equations are shown to take on pseudocurrent terms in accelerating frames. A four-vector Green function (...) solution to the modified Maxwell equations is presented. Finally, a discussion is offered concerning philosophical questions such as the operational definition of time. (shrink)

We introduce a new and general notion of canonical extension for algebras in the algebraic counterpart of any finitary and congruential logic . This definition is logic-based rather than purely order-theoretic and is in general different from the definition of canonical extensions for monotone poset expansions, but the two definitions agree whenever the algebras in are based on lattices. As a case study on logics purely based on implication, we prove that the varieties of Hilbert and Tarski algebras are (...) canonical in this new sense. (shrink)

This volume contains finalized versions of papers presented at an international workshop on extensions of logic programming, held at the Seminar for Natural Language Systems at the University of Tübingen in December 1989. Several recent extensions of definite Horn clause programming, especially those with a proof-theoretic background, have much in common. One common thread is a new emphasis on hypothetical reasoning, which is typically inspired by Gentzen-style sequent or natural deduction systems. This is not only of theoretical significance, but also (...) bears upon computational issues. It was one purpose of the workshop to bring some of these recent developments together. The volume covers topics such as the languages Lambda-Prolog, N-Prolog, and GCLA, the relationship between logic programming and functional programming, and the relationship between extensions of logic programming and automated theorem proving. It contains the results of the first conference concentrating on proof-theoretic approaches to logic programming. (shrink)

In this essay, Alain Caillé reconstructs the “singular history of the MAUSS (Anti-Utilitarian Movement in the Social Sciences)” from when in early 1980 a group of friends from different disciplines (sociologists, economists, philosophers, etc.) decided to found the “Bulletin du MAUSS” to counter the growing hegemony of utilitarianism and economism in the human sciences and in the philosophical disciplines themselves. The Bulletin would initially become the “Revue du MAUSS trimestrielle” from 1988 to 1992 and from 1993 to 2022 the “Revue (...) du MAUSS semestrielle” until its current anglophone extension “MAUSS International”. The author takes Marcel Mauss’ definition of the gift as a “total social fact” and explains that the gift paradigm is a “translation operator” in the sense that it is continuously enriched in interaction with existing discourses. The theoretical operation that Caillé proposes is to explicate Mauss’s discovery by moving from a simple gift paradigm, based on the perspective of simple reciprocity between giver and recipient, to an expanded gift paradigm. In this way, it can dialogue with other paradigms and other currents of thought (theory of recognition, theory of care, theory of human development, etc.). And from here to a conception of the gift as “adonnement”, i.e. the commitment of human subjects to bring something new and unprecedented into the world. (shrink)

In his “The Foundations of Mathematics”, Ramsey attempted to marry the Tractarian idea that all logical truths are tautologies and vice versa, and the logicism of the Principia. In order to complete his project, Ramsey was forced to introduce propositional functions in extension (PFEs): given Ramsey's definitions of 1 and 2, without PFEs even the quantifier-free arithmetical truth that 1 ≠ 2 is not a tautology. However, a number of commentators have argued that the notion of PFEs is incoherent. (...) This response was first given by Wittgenstein but has been best developed by Sullivan. While I agree with Wittgenstein and Sullivan's common conclusion, I believe that even the most compelling of Sullivan's arguments is importantly mistaken and that Wittgenstein's remarks are too opaque to be left as the end of the matter. In this article I uncover the fault in Sullivan's argument and present an alternative criticism of PFEs which is Wittgensteinian in spirit without being too mystifying. (shrink)

7.3.1 Ostension 7.3.2 Extensional Definition by Naming 7.3.3 Extensional Definition by Unique Description 7.4 Two Case Studies in the Application of the Intension/Extension Distinction 7.4.1 "God exists, by definition" 7.4.2 The 'Width' of an Intensional Definition..

We provide a version of first-order hybrid tense logic with predicate abstracts and definite descriptions as the only non-rigid terms. It is formalised by means of a tableau calculus working on sat-formulas. A particular theory of DD exploited here is essentially based on the approach of Russell, but with descriptions treated as genuine terms. However, the reductionist aspect of the Russellian approach is retained in several ways. Moreover, a special form of tense definite descriptions is formally developed. A constructive proof (...) of the interpolation theorem for this calculus is given, which is an extension of the result provided by Blackburn and Marx. (shrink)

In this paper, we generalize the definition of Neutrosophic sets and present a method for extending crisp functions on Neutrosophic sets and study some properties of such extended functions.

In this paper, we address an infamous argument against divisibility that dates back to Zeno. There has been an incredible amount of discussion on how to understand the critical notions of divisibility, extension, and infinite divisibility that are crucial for the very formulation of the argument. The paper provides new and rigorous definitions of those notions using the formal theories of parthood and location. Also, it provides a new solution to the paradox of divisibility which does not face some (...) threats that can possibly undermine the standard Lebesgue measure solution to such a paradox. (shrink)

Many opinions and ideas about aging exist. Biological theories have taken hold of the popular and scientific imagination as potential answers to a “cure” for aging. However, it is not clear what exactly is being cured or whether aging could be classified as a disease. Some scientists are convinced that aging will be biologically alterable and that the human lifespan will be vastly extendable. Other investigators believe that aging is an elusive target that may only be “statistically” manipulatable through a (...) better understanding of the operational principles of systems situated within complex environments. Not only is there confusion over definitions but also as to the safety of any potential intervention. Curing cell death, for example, may lead to cell cancer. The search for a cure for aging is not a clearly beneficial endeavour. This paper will first, describe contemporary ideas about aging processes and second, describe several current life extension technologies. Third, it analyses these theories and technologies, focusing on two representative and differing scientific points of view. The paper also considers the public health dilemma that arises from life extension research and examines two issues, risk/benefit ratio and informed consent, that are key to developing ethical guidelines for life extension technologies. (shrink)

Dummctt argues that classical quantification is illegitimate when the domain is given as the objects which fall under an indefinitely extensible concept, since in such cases the objects are not the required definite totality. The chief problem in understanding this complex argument is the crucial but unexplained phrase 'definite totality' and the associated claim that it follows from the intuitive notion of set that the objects over which a classical quantifier ranges form a set. 'Definite totality' is best understood as (...) disguised plural talk like Cantor's 'consistent multiplicity', although this does not help in understanding how a totality could be anything other than definite. Moreover, contrary to his claims, Dummett's own notion of set is not intuitive and he does not demystify the set-theoretic paradoxes. In conclusion, it is argued that Dummett's context principle is responsible for the incoherent projection of the haziness of a conception of some objects onto reality. (shrink)

Dummctt argues that classical quantification is illegitimate when the domain is given as the objects which fall under an indefinitely extensible concept, since in such cases the objects are not the required definite totality. The chief problem in understanding this complex argument is the crucial but unexplained phrase 'definite totality' and the associated claim that it follows from the intuitive notion of set that the objects over which a classical quantifier ranges form a set. 'Definite totality' is best understood as (...) disguised plural talk like Cantor's 'consistent multiplicity', although this does not help in understanding how a totality could be anything other than definite. Moreover, contrary to his claims, Dummett's own notion of set is not intuitive and he does not demystify the set-theoretic paradoxes. In conclusion, it is argued that Dummett's context principle is responsible for the incoherent projection of the haziness of a conception of some objects onto reality. (shrink)

As Adorno’s posthumous writings were released, a notable turn in his phi- losophy of music became noteworthy: by revealing an ambitious project on Beethoven, a quasi-axiomatic theory of musical time emerged, which seems to guide Adornian thought at very significant works and texts, as Philosophy of New Music. Sketched in three configuration types of musical time, this theory surprises with the so-called “ex- tensive type”. Adorno would then admit kinds of formal consistency (Stimmigkeit) that are not restricted to the ideal (...) of development (Entwicklung) and assumes the possibility of a non-regressive spatialization, which contrasts with his own ontological definition of the medium. This paper discusses the contradictions of extensive type by presenting an analysis of the Archduke Trio op. 97, in the light of the well-known originary affinity between Beethoven and Hegel. At last, the text presents by which means and for what reasons the extensive type would lead Adorno to the last and most decisive modality of musical time’s configuration, the late style. (shrink)

In this book Robert Stecker sets out to answer three basic questions in the philosophy of art: What is art? What is it to understand a work of art? And what is the value of art? Stecker addresses each question in turn and delivers what he claims to be “a unified, if incomplete, philosophy of art—a theory of the nature and functions of art and of the practice of interpreting and appreciating it”. His strategy is to discuss thoroughly recent contributions (...) to the debate concerning each of these questions and to put forward his own alternative answers. This leads to an extensive and detailed discussion of the journal literature on each of these subjects and considerable effort is spent both presenting and evaluating this literature. (shrink)

Quantum interference, manifest in the two slit experiment, lies at the heart of several quantum computational speed-ups and provides a striking example of a quantum phenomenon with no classical counterpart. An intriguing feature of quantum interference arises in a variant of the standard two slit experiment, in which there are three, rather than two, slits. The interference pattern in this set-up can be written in terms of the two and one slit patterns obtained by blocking one, or more, of the (...) slits. This is in stark contrast with the standard two slit experiment, where the interference pattern cannot be written as a sum of the one slit patterns. This was first noted by Rafael Sorkin, who raised the question of why quantum theory only exhibits irreducible interference in the two slit experiment. One approach to this problem is to compare the predictions of quantum theory to those of operationally-defined ‘foil’ theories, in the hope of determining whether theories that do exhibit higher-order interference suffer from pathological—or at least undesirable—features. In this paper two proposed extensions of quantum theory are considered: the theory of Density Cubes proposed by Dakić, Paterek and Brukner, which has been shown to exhibit irreducible interference in the three slit set-up, and the Quartic Quantum Theory of Życzkowski. The theory of Density Cubes will be shown to provide an advantage over quantum theory in a certain computational task and to posses a well-defined mechanism which leads to the emergence of quantum theory—analogous to the emergence of classical physics from quantum theory via decoherence. Despite this, the axioms used to define Density Cubes will be shown to be insufficient to uniquely characterise the theory. In comparison, Quartic Quantum Theory is a well-defined theory and we demonstrate that it exhibits irreducible interference to all orders. This feature of Życzkowski’s theory is argued not to be a genuine phenomenon, but to arise from an ambiguity in the current definition of higher-order interference in operationally-defined theories. Thus, to begin to understand why quantum theory is limited to a certain kind of interference, a new definition of higher-order interference is needed that is applicable to, and makes good operational sense in, arbitrary operationally-defined theories. (shrink)

According to the math tea argument, there must be real numbers that we cannot describe or define, because there are uncountably many real numbers, but only countably many definitions. And yet, the existence of pointwise-definable models of set theory, in which every individual is definable without parameters, challenges this conclusion. In this article, I introduce a flexible new method for constructing pointwise-definable models of arithmetic and set theory, showing furthermore that every countable model of Zermelo-Fraenkel ZF set theory and of (...) Peano arithmetic PA has a pointwise-definable end extension. In the arithmetic case, I use the universal algorithm and its Σ n generalizations to build a progressively elementary tower making any desired individual a n definable at each stage n, while preserving these definitions through to the limit model, which can thus be arranged to be pointwise definable. A similar method works in set theory, and one can moreover achieve V = L in the extension or indeed any other suitable theory holding in an inner model of the original model, thereby fulfilling the resurrection phenomenon. For example, every countable model of ZF with an inner model with a measurable cardinal has an end extension to a pointwise-definable model of ZFC + V = L[μ]. (shrink)

Three systems of double extension set theory have been proposed by Andrzej Kisielewicz in two papers. In this paper, it is shown that the two stronger systems are inconsistent, and that the third, weakest system does not admit extensionality for general sets or the use of general sets as parameters in its comprehension scheme. The parameter-free version of the comprehension principle of double extension set theory is also shown to be inconsistent with extensionality. The definitions of the systems (...) and a self-contained exposition of their properties is given, sufficient to develop the inconsistency proofs. (shrink)

This paper suggests a way of formalizing the amount of information that can be conveyed to each player along every possible play of an extensive game. The information given to each player i when the play of the game reaches node x is expressed as a subset of the set of terminal nodes. Two definitions are put forward, one expressing the minimum amount of information and the other the maximum amount of information that can be conveyed without violating the constraint (...) represented by the information sets. Our definitions provide intuitive characterizations of such notions as perfect recall, perfect information and simultanetty. (shrink)

Although delusion is one of the central concepts of psychopathology, it stills eludes precise conceptualization. In this paper, I present certain basic issues concerning the classification and definition of delusion, as well as its ontological status. By examining these issues, I aim to shed light on the ambiguity of the clinical term ‘delusion’ and its extension, as well as provide clues as to why philosophers are increasingly joining the ranks of psychiatrists, psychologists, and neuroscientists in the effort to come (...) to a comprehensive understanding of delusion. (shrink)

The study proposes a new approach towards a social phenomenon called convention and submits a minimalistic definition of convention, which provides a promising basis for future analysis unburdened by contra-Lewisian objections. The definition itself, based on the insights of Ruth Millikan in the study Language Conventions Made Simple, represents a simple and efficient means of delimiting essential components of conventional behaviour (stripped of most of the controversial issues from previous debates on Lewis’s notion) solely by means of the role of (...) precedent and its ability to reproduce. Yet, it is argued that a few additional conditions are required for a valid and distinct notion of conventionality: namely, the inclusion of a coordination aspect and an extension of the concept of precedent. The final version of the definition, thereafter, meets intuitive requirements of conventionality (e.g., arbitrariness) and has the generality to embrace different types of conventions. (shrink)

"The papers in this volume are extended versions of presentations at the fourth International Workshop on Extensions of Logic Programming, held at the University of St Andrews, March/April 1993. Among the topics covered in the volume are: defintional reflection and completion, modules in lambda-Prolog, representation of logics as partial inductive definitions, non-procedural logic programming, knowledge representation, contradiction avoidance, disjunctive databases, strong negation, linear logic programming, proof theory and regular search spaces, finite sets and constraint logic programming, search-space pruning and universal (...) algebra, and implementation on transputer networks."--PUBLISHER'S WEBSITE. (shrink)

In this paper, as an extension of CI-algebras, we discuss the new notions of Neutro-CI-algebras and Anti-CI-algebras. First, some examples are given to show that these definitions are different. We prove that any proper CI-algebra is a Neutro-BE-algebra or Anti-BE-algebra. Also, we show that any NeutroSelf-distributive and Anti-Commutative CI-algebras are not BE-algebras.

L’examen du « canon des Ecritures », après la consécration de l’expression par Athanase au IVe siècle, révèle d’abord une époque où une certaine fluidité caractérisait la réflexion sur l’extension des Ecritures et sur leurs limites. A partir du IVe siècle, une mutation décisive se produit, à laquelle il faut adjoindre une autre évolution sémantique préparée de longue date à travers l’emploi de diatèkè, traduit en latin par testamentum. Ainsi apparaissent deux temps forts dans l’histoire des Ecritures aux premiers (...) siècles du christianisme. Le premier est celui de l’extension du concept de livre saint à d’autres textes que les écrits hérités du judaïsme ; le second, dans la seconde moitié du IVe siècle, est celui des efforts de délimitation de l’Ancien et du Nouveau Testament, des décisions prises pour trancher les débats, commencés deux siècles plus tôt. L’examen de quelques-uns des problèmes que pose l’extension du canon et qui s’offrent ici à l’historien confirme qu’ils ne sont pas résolus définitivement par les Eglises.An examination of the ³canon of the Scriptures,² after the consecration of Athanasius¹ expression in the 4th century, first of all reveals a period when a certain fluidity characterized reflection on the extension of the Scriptures and their limits. From the 4th century a decisive change came about, to which must be added another semantic evolution, which was prepared for a long time through the use of diatéké, translated in Latin by testamentum. Thus, two important periods in the history of the Scriptures appear in the first centuries of Christianity. In the first there was an extension of the concept of the sacred book to texts other than the inherited writings of Judaism. The second period, in the second half of the 4th century, concerns efforts to demarcate the Old and New Testaments, and decisions taken to settle the debates begun two centuries earlier. The examination of some of these problems about the extension of the canon is offered here to the historian and confirms that they are not definitively resolved by the Churches. (shrink)

It will come as no surprise that I have a different interpretation of the four passages in which, McRae claims, Descartes “definitely includes extension and its modes in what is given through the senses”. In the first, Descartes includes extension, etc., among his ideas of corporeal bodies. This is not to say that he includes them among his adventitious ideas, though. All adventitious ideas are ideas of external bodies. But the converse is not true. Not all ideas of (...) corporeal bodies are ipso facto adventitious ideas, for, as I see it, the idea of the true and immutable nature of body is non-sensible and innate. McRae slides from “all adventitious ideas seem to be ideas of external bodies” to “all ideas of external bodies are adventitious”. (shrink)

Frege believes (1) that his definition of number is (partly) arbitrary; (2) that it "makes" numbers of certain extensions; (3) that without such a definition we cannot even think or understand arithmetical propositions. this position is part of a view according to which mathematics in general involves the free construction of objects, their properties, and the very contents of mathematical propositions. frege tries to avoid excess subjectivism by the kantian device of treating alternative systems of arithmetic (e.g.) as different appearances (...) of a single realm of unstructured truths. (shrink)

Decades of empirical and theoretical research has produced an extensive literature on the ethical judgments construct. Given its importance to understanding people’s ethical choices, future research should explore the psychological processes that produce ethical judgments. In this paper, the authors discuss two steps needed to advance this effort. First, they note that the business ethics literature lacks a single, generally accepted definition of ethical judgments. After reviewing several extant definitions, the authors offer a definition of the construct and discuss its (...) advantages. Second, future ethical judgment research would benefit from greater integration between theories of ethical decision making and theories of social cognition. Drawing upon the Hunt–Vitell ( Journal of Macromarketing 6 (Spring), 5–15, 1986 ; In: N. C. Smith and J. A. Quelch (eds.), Ethics in Marketing . Irwin, Homewood, IL, pp. 775–784, 1992 ) model and the heuristic-systematic model (Chaiken, Journal of Personality and Social Psychology 39 (November), 752–766, 1980 ), the authors present a brief research agenda intended to stimulate research on the psychological processes behind ethical judgments. (shrink)