A motivation is given for expressing classical mechanics in terms of diagonal projection matrices and diagonal density matrices. Then quantum mechanics is seen to be a simple generalization in which one replaces the diagonal real matrices with suitable Hermitian matrices.

In a recent paper, a “distance” function, $\cal D$ , was defined which measures the distance between pure classical and quantum systems. In this work, we present a new definition of a “distance”, D, which measures the distance between either pure or impure classical and quantum states. We also compare the new distance formula with the previous formula, when the latter is applicable. To illustrate these distances, we have used 2 × 2 matrix examples and two-dimensional vectors for simplicity and (...) clarity. Several specific examples are calculated. (shrink)

After analyzing the difficulties for a local realistic interpretation of quantum theory, it is argued that such an interpretation might be possible if some new postulates are added to the standard ones. We propose a stochastic interpretation of quantum theory, which involves the need of joint probability distributions for all relevant observables. The well known problems for the existence of joint distributions are solved by assuming that neither all Hermitian operators correspond to observables nor all density matrices represent (...) physical states. A research program along these lines is presented studying in particular the Maxwell quantum field and the Dirac field. (shrink)

The two-component spinor theory of van der Waerden is put into a convenient matrix notation. The mathematical relations among various types of matrices and the rule for forming covariant expressions are developed. Relativistic equations of classical mechanics and electricity and magnetism are expressed in this notation. In this formulation the distinction between time and space coordinates in the four-dimensional space-time continuum falls out naturally from the assumption that a four-vector is represented by a Hermitian matrix. The indefinite metric (...) of tensor analysis is a derived result rather than an arbitrary ad hoc assumption. The relation to four-component spinor theory is also discussed. (shrink)

Verbs and nouns gear θ-dependencies, Case, agreement, or construal relations. Building on Chomsky’s 1974 decomposition of such categories into ±N, ±V features, by translating said features into ±1, ±i scalars that allow for the construction of a vector space, this paper studies the possibility of organizing said features into 2 × 2 square matrices. In the system proposed to explore “head-complement” relations, operating on nouns yields a measurable/observable (Hermitian matrix), which in turn limits other potential combinations with abstract (...) lexical categories. Functional/grammatical categories in the system deploy the same features, albeit organized differently in the matrix diagonal and off-diagonal. The algebraic result is a group with well-defined mathematical properties, which properly includes the Pauli group of standard use in quantum computation. In the system, the presumed difference between categories and interactions—here, in a context of the head-complement sort—reduces to whether the magnitude of the matrix eigenvalue is 1 or not, in the latter instance inducing asymmetric interactions. (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)

For several examples of Hermitian operators, the issues involved in their possible self-adjoint extension are shown to conform with recognizable properties in the solutions to the associated classical equations of motion. This result confirms the assertion made in an earlier paper (Ref. 1) that there are sufficient classical “symptoms” to diagnose any quantum “illness.”.

We relate so-called spin factors and generalized Hermitian (GH-) algebras, both of which are partially ordered special Jordan algebras. Our main theorem states that positive-definite spin factors of dimension greater than one are mathematically equivalent to generalized Hermitian algebras of rank two.

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.

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)

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.

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)

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)

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)

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

Logic and Philosophy of Logic

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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.

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)

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)

Viale introduced covering matrices in his proof that SCH follows from PFA. In the course of the proof and subsequent work with Sharon, he isolated two reflection principles, CP and S, which, under certain circumstances, are satisfied by all covering matrices of a certain shape. Using square sequences, we construct covering matrices for which CP and S fail. This leads naturally to an investigation of square principles intermediate between □κ and □ for a regular cardinal κ. We (...) provide a detailed picture of the implications between these square principles. (shrink)

“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)

This paper outlines a critique of the use of the genetic variance–covariance matrix (G), one of the central concepts in the modern study of natural selection and evolution. Specifically, I argue that for both conceptual and empirical reasons, studies of G cannot be used to elucidate so-called constraints on natural selection, nor can they be employed to detect or to measure past selection in natural populations – contrary to what assumed by most practicing biologists. I suggest that the search for (...) a general solution to the difficult problem of identifying causal structures given observed correlation’s has led evolutionary quantitative geneticists to substitute statistical modeling for the more difficult, but much more valuable, job of teasing apart the many possible causes underlying the action of natural selection. Hence, the entire evolutionary quantitative genetics research program may be in need of a fundamental reconsideration of its goals and how they correspond to the array of mathematical and experimental techniques normally employed by its practitioners. (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)

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)

The main result of the present article is the following: Let N be an infinite subset of,, and let be a matrix with infinitely many rows of completely Ramsey subsets of such that for every n,. Then there exist, a sequence of nonempty finite subsets of N, and an infinite subset T of such that for every infinite subset I of. We also give an application of this result to partitions of an uncountable analytic subset of a Polish space X (...) into sets belonging to the σ‐algebra generated by the analytic subsets of X. (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)

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.

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)

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)

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

We show that the physical principle, “the adjoint associates to each state a ‘test’ for that state”, fully characterises the Hermitian adjoint for pure quantum theory, therefore providing the adjoint with operational meaning beyond its standard mathematical definition. Moreover, we demonstrate that for general process theories, which all admit a diagrammatic representation, this physical principle induces a diagrammatic reflection operation.

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.