We show by way of example how one can provide in a lot of cases simple modular semantics for rules of inference, so that the semantics of a system is obtained by joining the semantics of its rules in the most straightforward way. Our main tool for this task is the use of finite Nmatrices, which are multi-valued structures in which the value assigned by a valuation to a complex formula can be chosen non-deterministically out of a certain nonempty (...) set of options. The method is applied in the area of logics with a formal consistency operator (known as LFIs), allowing us to provide in a modular way effective, finite semantics for thousands of different LFIs. (shrink)

Dugundji proved in 1940 that most parts of standard modal systems cannot be characterized by a single finite deterministic matrix. In the eighties, Ivlev proposed a semantics of four-valued non-deterministic matrices (which he called quasi-matrices), in order to characterize a hierarchy of weak modal logics without the necessitation rule. In a previous paper, we extended some systems of Ivlev’s hierarchy, also proposing weaker six-valued systems in which the (T) axiom was replaced by the deontic (D) axiom. In (...) this paper, we propose even weaker systems, by eliminating both axioms, which are characterized by eight-valued non-deterministic matrices. In addition, we prove completeness for those new systems. It is natural to ask if a characterization by finite ordinary (deterministic) logical matrices would be possible for all those Ivlev-like systems. We will show that finite deterministic matrices do not characterize any of them. (shrink)

In [7] Wojtylak showed that the supremum of a finite collection of strongly finite logics is not necessarily strongly finite, putting to an end the dispute concerning the underlying algebraic structure of the set of all strongly finite logics defined on a propositional language. In this paper we prove that Wojtylak’s result extends to a wider class of propositional logics defined by finite sets of finite matrices with admissible valuations.

It is well known that every propositional logic which satisfies certain very natural conditions can be characterized semantically using a multi-valued matrix ([Los and Suszko, 1958; W´ ojcicki, 1988; Urquhart, 2001]). However, there are many important decidable logics whose characteristic matrices necessarily consist of an infinite number of truth values. In such a case it might be quite difficult to find any of these matrices, or to use one when it is found. Even in case a logic does (...) have a finite characteristic matrix it might be difficult to discover this fact, or to find such a matrix. The deep reason for these difficulties is that in an ordinary multi-valued semantics the rules and axioms of a system should be considered as a whole, and there is no method for separately determining the semantic effects of each rule alone. (shrink)

The resistance between two arbitrary nodes in an infinite square lattice of identical resistors is calculated. The method is generalized to infinite triangular and hexagonal lattices in two dimensions, and also to infinite cubic and hypercubic lattices in three and more dimensions. © 1999..

In the …rst part of this paper, the origins of modal-structuralism are traced from Hilary Putnam’s seminal article, "Mathematics without Foundations" (1967) to its transformation and development into the author’s modal-structural approach. The addition of a logic of plurals is highlighted for its recovery (in combination with the resources of mereology) of full, second-order logic, essential for articulating a good theory of mathematical structures. The second part concentrates on the motivation of large trans…nite cardinal numbers, arising naturally from the second-order (...) machinery combined with an extendability principle on structures for set theories due independently to Zermelo (1930) and Putnam (in the paper just cited). The power of this is enhanced by a novel modal re‡ection principle recently introduced by the author. This is illustrated in detail with the …rst axiom of in…nity: After reviewing some of the trouble this basic classical axiom has caused for previous foundational programs, we show how it is derived easily using the re‡ection principle and a weak form of extendability for …nite structures. We conclude with some comparative remarks on how this improves on the closest set-theoretic analogue to the present proposal. (shrink)

The plurivalent logics considered in Graham Priest's recent paper of that name can be thought of as logics determined by matrices whose underlying algebras are power algebras, where the power algebra of a given algebra has as elements textit{subsets} of the universe of the given algebra, and the power matrix of a given matrix has has the power algebra of the latter's algebra as its underlying algebra, with its designated elements being selected in a natural way on the basis (...) of those of the given matrix. The present discussion stresses the continuity of Priest's work on the question of which matrices determine consequence relations which remain unaffected on passage to the consequence relation determined by the power matrix of the given matrix with the corresponding question in equational logic as to which identities holding in an algebra continue to hold in its power algebra. Both questions are sensitive to a decision as to whether or not to include the empty set as an element of the power algebra, and our main focus will be on the contrast, when it is included, between the power matrix semantics and the four-valued Dunn--Belnap semantics for first-degree entailment a la Anderson and Belnap) in terms of sets of classical values, in which the empty set figures in a somewhat different way, as Priest had remarked his 1984 study, `Hyper-contradictions', in which what we are calling the power matrix construction first appeared. (shrink)

We examine the notion of primitive satisfaction in logical matrices. Theorem II. 1, being the matrix counterpart of Baker's well-known result for congruently distributive varieties of algebras (cf [1], Thm. 1.5), links the notions of primitive and standard satisfaction. As a corollary we give the matrix version of Jónsson's Lemma, proved earlier in [4]. Then we investigate propositional logics with disjunction. The main result, Theorem III. 2, states a necessary and sufficient condition for such logics to be finitely based.

Résumé L’article vise à souligner la signification politique de l’avènement de la Critique kantienne. Pour ce faire, une double thèse est défendue : il y a une matrice juridique sous-jacente au mouvement conceptuel de la philosophie critique, dont la puissance émancipatrice et révolutionnaire se manifeste à la lumière du contexte historique de sa formulation. L’argument se déploie de la façon suivante : une exploration des rapports entre la formule « révolution de la pensée » que l’on trouve dans la Critique (...) de la raison pure et le contexte historique de la publication l’œuvre ; ensuite un examen de la métaphore juridique et la façon dont elle incarne l’idée d’une philosophie critique ; et pour conclure la suggestion d’une hypothèse interprétative d’après laquelle le « contrat originaire » est un opérateur conceptuel accomplissant un mouvement analogue à celui qui est exigé de la philosophie critique : le passage du niveau individuel au niveau intersubjectif.The aim of the article is to highlight the political significance of the advent of Kant’s Critique. In order to establish it, a double thesis is defended: there is a juridical matrix underlying the conceptual movement of the critical philosophy, whose emancipatory and revolutionary strength is revealed in the light of the context of its formulation. The argument will be deployed in the following way: an exploration of the connections between the formula “a revolution in thought” articulated in the Critique and the historical context of its publication; followed by an exam of the juridical metaphor of the Tribunal and the way in which it embodies the idea of a critical philosophy and finally, as a conclusion, the suggestion of an interpretative hypothesis according to which the notion of contract is a conceptual operator accomplishing an analogous movement to the one required from the critical philosophy: the passage from the subjective level to the intersubjective one. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:American Journal of Philology 123.1 (2002) 119-122 [Access article in PDF] Mary Depew and Dirk Obbink, eds. Matrices of Genre: Authors, Canons, and Society. Center for Hellenic Studies Colloquia 4. Cambridge, Mass.: Harvard University Press, 2000. vi + 346 pp. Cloth, $50. The present collection of essays, which originated as a colloquium at the Center for Hellenic Studies, starts, in the words of editor and organizer Dirk Obbink, (...) from the premise that "genre is a conceptual orienting device that suggests to the hearer the sort of receptorial conditions in which a fictive discourse might have been delivered." Obbink goes on to note that, as the discussion at the colloquium developed, "a more text-based conception of authorial performance" began to emerge (6). This evolution in turn parallels the movement in the volume from an initial discussion of archaic and classical Greek genres to those of Hellenistic poetry and its Roman imitators. One can only applaud this attempt to provide a more sophisticated framework for the reading and reception of ancient literature. As befits the stature of the sponsoring institution and of the many distinguished participants, all the contributions are of an unquestionable competence. In the end, however, it is unclear that we have achieved a more complex, refined, or problematized concept of genre than that with which we began. Indeed, with a few exceptions, most of the modern debates about the concept of genre are either ignored or cited in passing rather than directly engaged. Nonetheless, there are a number of interesting and important individual papers in this volume and only a few that disappoint.Among the most praiseworthy is Eric Csapo's "From Aristophanes to Menander? Genre Transformation in Greek Comedy." Csapo takes dead aim at the traditional narrative of the evolution from a politically engaged Old Comedy of ribald invective to an apolitical plot-centered New Comedy typified by Menander and his Roman epigones. Csapo's argument is that this narrative was manufactured, not through the accidents of preservation, but through a deliberate selection and canonization of texts by early Peripatetic critics and scholars. When we characterize Old Comedy as being typified by Aristophanes' carnivalesque assaults on the Athenian political establishment and then point to his plays as proof, we are not only engaging in circular reasoning, we are also treading a path that has been deliberately and tendentiously drawn for us. Thus, while the traditional narrative depicts a linear evolution from iambic to Old Comedy to New Comedy, which is portrayed as a movement from the personal and the topical to the universal and the apolitical, Aristotle himself, as Csapo notes (117), admits that Crates, Aristophanes' predecessor, abandoned the iambic and began to compose generalizing plots (Po. 1449b7-8). In short, plots of the kind deemed [End Page 119] typical of New Comedy are to be found at the very beginning of the comic form. By the same token, we have evidence of political comedy surviving well past Aristophanes' supposed invention of the Middle Comic style in the Plutus. Indeed, the last example of an Athenian comedy of political invective is Demetrius' Areopagite, dating from circa 294 B.C.E.Csapo's argument, of course, is not a simpleminded refusal to acknowledge change but rather an attempt to rethink the historical evolution of the comic genre outside the bounds of a narrow, teleological narrative. Genres are always complex negotiations between competing conceptions of the same form by different producers and consumers. At various times, for various intrinsic and extrinsic reasons, different versions of the same form will be ascendant or fading into eclipse. But it is never a question of an essence that either unfolds its fullness over time or declines from a single ideal instantiation.Less revolutionary but equally persuasive is Joseph W. Day's "Epigram and Reader: Generic Force as (Re-)Activation of Ritual." Day examines the relation between dedicatory epigrams on cult statues and their ritual contexts by arguing that the dedicatory genre is essentially performative rather than constative. These epigrams do their social work by activating a performative context in the mind of the reader... (shrink)

We show that base matrices for P(ω)∕fin of regular height larger than h necessarily have maximal branches that are not cofinal. The same holds for base matrices of height h if tSpoiler

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We first relate the random matrix model to a Fokker-Planck Hamiltonian system, such that the correlation functions of the model are expressed as the vacuum expectation values of equal-time products of density operators. We then analyze the universality of the random matrix model by solving the Focker-Planck Hamiltonian system for large N. We use two equivalent methods to do this, namely the method of relating it to a system of interacting fermions in one space dimension and the method of collective (...) fields for large N matrix quantum mechanics. The final result using both these methods is the same Hamiltonian system of chiral bosons on a circle, which manifestly exhibits the universality of the random matrix model. (shrink)

In the present paper, we study some properties of matrices for non-structural consequence operators. These matrices were introduced in a former work (see [3]). In sections 1. and 2., general definitions and theorems are recalled; in section 3. a correspondence is studied, among our matrices and Wójcicki's ones for structural operators. In section 4. a theorem is given about operators, induced by submatrices or epimorphic images, or quotient matrices of a given one.Such matrices are used (...) to characterize lattices of non-structural consequence operators, by constructing lattices, antiisomorphic to them (see section 5.). In the last section, a sufficient condition is given for a non-structural operator to be finite. (shrink)

The main result of the present paper — Theorem 3 — establishes the equivalence of the interpolation and amalgamation properties for a large family of logics and their associated classes of matrices.

Across languages, we find indefinites that trigger modal inferences. This article contributes to a semantic typology of these items by contrasting Spanish alg´un with in-.

Across languages, we find epistemic indefinites, i.e. existential determiners that can convey information about the speaker’s epistemic state.1 One such indefinite is Spanish alg´un, which marks ignorance on the part of the speaker. By using alg´un in (1a) the speaker signals that he is unable (or unwilling) to identify the doctor that Mar´ıa married. Hence, it would be odd for him to add a namely continuation that explicitly identifies the doctor in question, as in (1b). From now on, we will (...) refer to the marking of the speaker’s lack of knowledge as an ‘epistemic effect.’ (Alonso-Ovalle and Men´endez-Benito, 2003). (shrink)

The argument here comes from consideration of a certain sort of linguistic expression called negative polarity items (NPIs). These are expressions such as “any,” “at all” and “ever.” NPIs are of particular interest for semantics because they can only be used in contexts with a certain rather abstract semantic feature. However, the precise characterization of the feature is itself a matter of some controversy. For those interested in the semantics of natural language it is worthwhile to figure out precisely what (...) this feature is. Once we have an account of it, we can use the account to infer facts about the meanings of various expressions that interact with NPIs. (shrink)

In this book, the authors introduce the linguistic set associated with a linguistic variable and the structure of matrices, which they define as linguistic matrices. The authors build linguistic matrices only for those linguistic variables which yield a linguistic continuum or an ordered linguistic set. This book is organised into three chapters. The first chapter is introductory, in which we introduce all the basic concepts of linguistic variables and the associated linguistic set to make this book self-contained. (...) Chapter two introduces linguistic matrices and develops basic properties associated with them, like types of matrices, transpose of matrices and diagonal matrices. Most of the properties enjoyed by real or complex matrices are satisfied by these linguistic matrices. Chapter three deals with operations on the linguistic matrices. (shrink)

As is well known, the variable-sharing property (vsp) is, according to Anderson and Belnap, a necessary property of any relevant logic. In this paper, we shall consider two versions of the vsp, what we label the "weak vsp" (wvsp) and the "strong vsp" (svsp). In addition, the "no loose pieces property," a property related to the wvsp and the svsp, will be defined. Each one of these properties shall generally be characterized by means of a class of logical matrices. (...) In this way, any logic verified by an actual matrix in one of these classes has the property the class generally represents. Particular matrices (and so, logics) in each class are provided. (shrink)

Two-dimensional semantics is a theory in the philosophy of language that provides an account of meaning which is sensitive to the distinction between necessity and apriority. Usually, this theory is presented in an informal manner. In this thesis, I take first steps in formalizing it, and use the formalization to present some considerations in favor of two-dimensional semantics. To do so, I define a semantics for a propositional modal logic with operators for the modalities of necessity, actuality, and apriority that (...) captures the relevant ideas of two-dimensional semantics. I use this to show that some criticisms of two-dimensional semantics that claim that the theory is incoherent are not justified. I also axiomatize the logic, and compare it to the most important proposals in the literature that define similar logics. To indicate that two-dimensional semantics is a plausible semantic theory, I give an argument that shows that all theorems of the logic can be philosophically justified independently of two-dimensional semantics. (shrink)

Ce volume est la première étape dans la question philosophique de l'existence de Dieu. Qu'est-ce qu'être intelligent sinon la capacité de rechercher et de recevoir la vérité? Il y a plus : la raison peut conduite à la fois, à Dieu. Et même, l'acte de foi est essentiellement un acte d'intelligence. La connaissance de Dieu vivifie la raison et la couronne. L'intelligence a été donnée par Dieu à l'homme et, avec elle, les idées dont l'être est l'âme et Dieu l'intelligence (...) suprême. (shrink)

In support of a recent conjecture by Nielsen (1999), we prove that the phenomena of ‘incomparable entanglement’— whereby, neither member of a pair of pure entangled states can be transformed into the other via local operations and classical communication (LOCC)—is a generic feature when the states at issue live in an infinite-dimensional Hilbert space.  2002 Elsevier Science B.V. All rights reserved.

Sections 1, 2 and 3 contain the main result, the strong finite axiomatizability of all 2-valued matrices. Since non-strongly finitely axiomatizable 3-element matrices are easily constructed the result reveals once again the gap between 2-valued and multiple-valued logic. Sec. 2 deals with the basic cases which include the important F i from Post's classification. The procedure in Sec. 3 reduces the general problem to these cases. Sec. 4 is a study of basic algebraic properties of 2-element algebras. (...) In particular, we show that equational completeness is equivalent to the Stone-property and that each 2-element algebra generates a minimal quasivariety. The results of Sec. 4 will be applied in Sec. 5 to maximality questions and to a matrix free characterization of 2-valued consequences in the lattice of structural consequences in any language. Sec. 6 takes a look at related axiomatization. problems for finite algebras and matrices. We study the notion of a propositional consequence with equality and, among other things, present explicit axiomatizations of 2-valued consequences with equality. (shrink)

Ordinary dialogical logic offers processing memory as a history of dialogues. Find a reference in this history is an ambiguous process, which can transform in complex or impossible issues the questions that compose a dialogue. Thus, two researches of justifications overlap in a dual problem of justification. In addition, this consideration of the memory uses a lot of technical resources.We propose a dialogical formalism that solves these problems: double justification, composition, memory saving. To do this, we will develop "special" rules (...) that will reconfigure the formalism and consider the metaphysical and philosophical concepts in this logical system and its interpretations.This article is divided into three parts:A presentation of the dialogical framework oriented towards the presentation of the problem of double justification.A re-interpretation of this framework in terms of matrices: the logical formulas modeled on the linguistic and mathematical language reading are replaced by a concept borrowed from the game semantical approach.A brief summary of the new approaches presented by this method: reaffirmation of the clear separation between different levels of analysis and economy on the amount of memory required. (shrink)

This paper explains what an infinite regress argument is. Part 1 contains some examples of infinite regress arguments. Part 2 presents a schema for all such arguments an defines an infinite regress argument as one that approximates to the schema. Part 3 tests the schema on the examples. Part 4 contrasts my account of infinite regress arguments with that given by Passmore and shows that Passmore's theory succumbs to objections. Part 5 distinguishes an infinite regress argument from an infinite regress (...) and defines an infinite regress. Part 6 explains the dyslogistic force of 'infinite regress'. (shrink)

In this paper we extend some techniques from the theory of conse- quences operations and logical matrices into the wider area of entailment relations.

I discuss the connection between the Hamiltonian and path integral approaches for fermionic fields. I show how the temporal Wilson projection operators appear naturally in a lattice action. I also carefully treat the insertion of a chemical potential term.

The paper is a continuation of the investigations of [3]. We assume that the reader is familiar with the notion of an !-saturated model and the notation applied in the theory of models. We refer to [2] for all model- theoretic terms which appear in this note.

(1) Somebody arrived late. (Guess who?/Namely Mary) a. Conventional meaning: Somebody arrived late b. Ignorance implicature: The speaker doesn’t know who..

In support of a recent conjecture by Nielsen (1999), we prove that the phenomena of ‘incomparable entanglement’— whereby, neither member of a pair of pure entangled states can be transformed into the other via local operations and classical communication (LOCC)—is a generic feature when the states at issue live in an infinite-dimensional Hilbert space.  2002 Elsevier Science B.V. All rights reserved.

ABSTRACTThe articles Maximality and Refutability Skura [. Maximality and refutability. Notre Dame Journal of Formal Logic, 45, 65–72] and Three-valued Maximal Paraconsistent Logics Skura and Tuziak [. Three-valued maximal paraconsistent logics. In Logika. Wydawnictwo Uniwersytetu Wrocławskiego] introduced a simple method of proving maximality of a given paraconsistent matrix. This method stemmed from the so-called refutation calculus, where the focus in on rejecting rather than accepting formulas. The article A Generalisation of a Refutation-related Method in Paraconsistent Logics Trybus [. A generalisation (...) of a refutation-related method in paraconsistent logics. Logic and Logical Philosophy, 27. doi:10.12775/llp.2018.002] was a first step towards generalising the method. In it, a number of 3-valued paraconsistent matrices were shown maximal. In this article we extend these results to cover a number of n-valued paraconsistent matrices using the same method. (shrink)

Languages that have determiners often have a rich inventory of them. In English, indefinite determiners include a(n), some, a certain, this, one, another, cardinals, partitives, the zero determiner of bare plurals (in some analyses), and, according to Horn 1999 and Giannakidou 2001, any. Despite the attention indefinites have received in the literature, characterizing what is common to all of them and what is specific to each is still an elusive task. This paper investigates the first three determiners in this list, (...) attempting to provide a semantic characterization that accounts for their distribution. (For pertinent discussion of issues that overlap to some extent with those taken up below, see Kamp and Bende-Farkas 2002, and for a discussion of the French versions of some and a certain, see Jayez and Tovena (this volume).) The Ds that concern us here share two characteristics: (i) they are indefinite, and (ii) they are existential when not in the scope of any operator or quantifier. As seen in [1], they may occur as pivots in existential there constructions, and therefore all three form weak DPs. (shrink)

Engineering science is a scientific discipline that from the point of view of epistemology and the philosophy of science has been somewhat neglected. When engineering science was under philosophical scrutiny it often just involved the question of whether engineering is a spin-off of pure and applied science and their methods. We, however, hold that engineering is a science governed by its own epistemology, methodology and ontology. This point is systematically argued by comparing the different sciences with respect to a particular (...) set of characterization criteria. (shrink)

We study the matrices, reduced matrices and algebras associated to the systems SAT of structural annotated logics. In previous papers, these systems were proven algebraizable in the finitary case and the class of matrices analyzed here was proven to be a matrix semantics for them.We prove that the equivalent algebraic semantics associated with the systems SAT are proper quasivarieties, we describe the reduced matrices, the subdirectly irreducible algebras and we give a general decomposition theorem. As a (...) consequence we obtain a decision procedure for these logics. (shrink)

The class Matr(C) of all matrices for a prepositional logic (, C) is investigated. The paper contains general results with no special reference to particular logics. The main theorem (Th. (5.1)) which gives the algebraic characterization of the class Matr(C) states the following. Assume C to be the consequence operation on a prepositional language induced by a class K of matrices. Let m be a regular cardinal not less than the cardinality of C. Then Matr (C) is the (...) least class of matrices containing K and closed under m-reduced products, submatrices, matrix homomorphisms, and matrix homomorphic counter-images. (shrink)

In this paper I give conditions under which a matrix characterisation of validity is correct for first order logics where quantifications are restricted by statements from a theory. Unfortunately the usual definition of path closure in a matrix is unsuitable and a less pleasant definition must be used. I derive the matrix theorem from syntactic analysis of a suitable tableau system, but by choosing a tableau system for restricted quantification I generalise Wallen's earlier work on modal logics. The tableau system (...) is only correct if a new condition I call alphabetical monotonicity holds. I sketch how the result can be applied to a wide range of logics such as first order variants of many standard modal logics, including non-serial modal logics. (shrink)

The concept of supermatrix for social scientists was first introduced by Paul Horst. The main purpose of his book was to introduce this concept to social scientists, students, teachers and research workers who lacked mathematical training. This book introduces the concept of fuzzy super matrices and operations on them. The author has provided only those operations on fuzzy supermatrices that are essential for developing super fuzzy multi expert models. This book will be highly useful to social scientists who wish (...) to work with multi expert models. (shrink)

In this book authors study neutrosophic super matrices. The concept of neutrosophy or indeterminacy happens to be one the powerful tools used in applications like FCMs and NCMs where the expert seeks for a neutral solution. Thus this concept has lots of applications in fuzzy neutrosophic models like NRE, NAM etc. These concepts will also find applications in image processing where the expert seeks for a neutral solution.

In our paper “A general characterization of the variable-sharing property by means of logical matrices”, a general class of so-called “Relevant logical matrices”, RMLs, is defined. The aim of this paper is to define a class of simpler Relevant logical matrices RMLs′serving the same purpose that RMLs, to wit: any logic verified by an RML′has the variable-sharing property and related properties predicable of the logic of entailment E and of the logic of relevance R.

It is shown that the class of reduced matrices of a logic is a 1 st order -class provided the variety associated with has the finite replacement property in the sense of [7]. This applies in particular to all 2-valued logics. For 3-valued logics the class of reduced matrices need not be 1 st order.

There are exactly two nonfinitely axiomatizable algebraic matrices with one binary connective o such thatx(yz) is a tautology of . This answers a question asked by W. Rautenberg in [2], P. Wojtylak in [8] and W. Dziobiak in [1]. Since every 2-element matrix can be finitely axiomatized ([3]), the matrices presented here are of the smallest possible size and in some sense are the simplest possible.

“Matrixes and Frames: Two Figures of Norm Functioning in the Work of Judith Butler”. The concepts of “intelligibility matrix” and “frame”, each corresponding to different periods in the thought of the feminist philosopher Judith Butler, seem to converge in so many aspects that one is tempted to hold that they may be used indistinctly. Nevertheless, the problem this paper deals with is the possible uniqueness the usage of the latter concept involves, thus pointing to the differences between both concepts. I (...) aim to solve the problem by studying the main sources that Butler herself uses in order to formulate the concept of frame, and by putting in evidence that which is gained by the use of that notion. I also aim to show more clearly the difference between the two concepts by appealing to their functions in the critique of speciffical cultural phenomenon: pornography as a frame and element of the heterosexual matrix. (shrink)