In this revolutionary work, the author sets the stage for the science of the 21st Century, pursuing an unprecedented synthesis of fields previously considered unrelated. Beginning with simple classical concepts, he ends with a complex multidisciplinary theory requiring a high level of abstraction. The work progresses across the sciences in several multidisciplinary directions: Mathematical logic, fundamental physics, computer science and the theory of intelligence. Extraordinarily enough, the author breaks new ground in all these fields. In the field of fundamental (...) physics the author reaches the revolutionary conclusion that physics can be viewed and studied as logic in a fundamental sense, as compared with Einstein's view of physics as space-time geometry. This opens new, exciting prospects for the study of fundamental interactions. A formulation of logic in terms of matrix operators and logic vector spaces allows the author to tackle for the first time the intractable problem of cognition in a scientific manner. In the same way as the findings of Heisenberg and Dirac in the 1930s provided a conceptual and mathematical foundation for quantum physics, matrix operator logic supports an important breakthrough in the study of the physics of the mind, which is interpreted as a fractal of quantum mechanics. Introducing a concept of logic quantum numbers, the author concludes that the problem of logic and the intelligence code in general can be effectively formulated as eigenvalue problems similar to those of theoretical physics. With this important leap forward in the study of the mechanism of mind, the author concludes that the latter cannot be fully understood either within classical or quantum notions. A higher-order covariant theory is required to accommodate the fundamental effect of high-level intelligence. The landmark results obtained by the author will have implications and repercussions for the very foundations of science as a whole. Moreover, Stern's Matrix Logic is suitable for a broad spectrum of practical applications in contemporary technologies. (shrink)

In this pioneering work, the author develops a fundamental formulation of logic in terms of theory of matrices and vector spaces. The discovery of matrix logic represents a landmark in the further formalization of logic. For the first time the power of direct mathematical computation is applied to the whole set of logic operations, allowing the derivation of both the classical and modal logics from the same formal base. The new formalism allows the author to (...) enlarge the alphabet of the truth-values with negative logic antivalues and to link matrix logic descriptions with the Dirac formulation of quantum theory - a result having fundamental implications and repercussions for science as a whole. As a unified language which permits a logical examination of the underlying phenomena of quantum field theory and vice versa, matrix logic opens new avenues for the study of fundamental interactions and gives rise to a revolutionary conclusion that physics as such can be viewed and studied as a logic in the fundamental sense. Finally, modelling itself on exact sciences, matrix logic does not refute the classical logic but instead incorporates it as a special deterministic limit. The book requires multidisciplinary knowledge and will be of interest to physicists, mathematicians, computer scientists and engineers. (shrink)

Resolution Matrix Semantics (RMS) introduces the alternative truth-value-based framework for modal logic, providing a substantive alternative to Kripke’s relational semantics of possible worlds. Drawing inspiration from Y. Ivlev’s substantive semantics, RMS utilizes a 4-valued structure—necessary truth (tn), contingent truth (tc), contingent false (fc), and necessary false (fn)—augmented by indeterminate values (t, f, t/f) to define modal systems Km, KDm, KTm, S4m, and S5m, analogous to Kripke’s K, KD, T, S4, and S5. By directly assigning determined and indeterminate truth (...) values via an interpretation function, RMS validates formulas without relying on accessibility relations, as demonstrated through soundness and completeness proofs and a tailored tableau method. The framework’s versatility extends to applications in deontic logic, artificial intelligence, and quantum computing, enabling context-dependent reasoning with computational efficiency. RMS thus bridges philosophical logic and practical domains, offering a truth-centric perspective on modal reasoning’s complexities. (shrink)

The aim of this paper is to define the logical system Sm4 characterised by the degree of truth-preserving consequence relation defined on the ordered set of values of Smiley’s four-element matrix MSm4. The matrix MSm4 has been of considerable importance in the development of relevant logics and it is at the origin of bilattice logics. It will be shown that Sm4 is a most interesting paraconsistent logic which encloses a sound theory of logical necessity similar to that (...) of Anderson and Belnap’s logic of entailment E. Intuitively, Sm4 can be described as a four-valued expansion of the positive fragment of Lewis’ S5 or, alternatively, as a four-valued version of S5. (shrink)

It is well established that classical propositional logic is Boolean. However, this view has recently been challenged. In their paper Non-Orthomodular Models for Both Standard Quantum Logic and Standard Classical Logic: Repercussions for Quantum Computers, Mladen Pavic̆ić and Norman Megill present a non-distributive, non-orthomodular model for both classical and quantum logic based on lattice O6, and argue that classical propositional logic is non-distributive. In this paper, we examine this claim. Pavic̆ić and Megill’s model is formulated (...) within unital matrix semantics rather than as an algebraic model in the sense of Abstract Algebraic Logic. An analysis of the lattice O6 in the framework of matrix semantics reveals that the matrix (O6,{1,a,b}) is adequate for CL, but not reduced, and induces the same consequence relation as the two-element Boolean matrix B2. Similarly, the unital matrix (O6,{1}) is adequate for CL through reduction to the four-element Boolean matrix B4. Furthermore, we present two lattice constructions that yield matrix models for CL lacking nontrivial lattice-theoretic properties. These results show that the adequacy of O6 is not intrinsic to its algebraic structure, but is inherited from its reducibility to Boolean matrices, and more generally that classical logic admits models with highly unconstrained lattice structure. Consequently, the existence of such non-distributive models does not undermine the distributive character of classical propositional logic. (shrink)

The aim of this paper is to show that the operations of forming direct products and submatrices suffice to construct exhaustive semantics for all structural strengthenings of the consequence determined by a given class of logical matrices.

The paper offers a matrix-based logic (relevant matrix quantum physics) for propositions which seems suitable as an underlying logic for empirical sciences and especially for quantum physics. This logic is motivated by two criteria which serve to clean derivations of classical logic from superfluous redundancies and uninformative complexities. It distinguishes those valid derivations (inferences) of classical logic which contain superfluous redundancies and complexities and are in this sense from those which are or in (...) the sense of allowing only the most informative consequences in the derivations. The latter derivations are strictly valid in RMQ, whereas the former are only materially valid. RMQ is a decidable matrix calculus which possesses a semantics and has the finite model property. It is shown in the paper how RMQ by its strictly valid derivations can avoid the difficulties with commensurability, distributivity, and Bell's inequalities when it is applied to quantum physics. (shrink)

We provide a logical matrix semantics and a Gentzen-style sequent calculus for the first-degree entailments valid in W. T. Parry’s logic of Analytic Implication. We achieve the former by introducing a logical matrix closely related to that inducing paracomplete weak Kleene logic, and the latter by presenting a calculus where the initial sequents and the left and right rules for negation are subject to linguistic constraints.

Implicational tonoid logics and their extensions with abstract Galois properties have been introduced by Yang and Dunn. They introduced matrix semantics for the implicational tonoid logics but did not do for the extensions. Here we provide such semantics for implicational partial gaggle logics as one sort of such extensions. To this end, first we discuss implicational partial gaggle logics in Hilbert-style. We next introduce one kind of matrix semantics based on Lindenbaum– Tarski matrices for the logics and show (...) that those logics are complete with respect to the matrix semantics. Finally, we further introduce a slightly different kind of matrix semantics based on reduced models for the logics and show that those logics are complete with respect to this matrix semantics. (shrink)

In this paper, we define the family FiCn:={FiCn}n∈ω of logics by means of semantics of bivaluations. This family is a variant of the family of paraconsistent bivaluated Ciun-logics, defined and studied by J. Ciuciura. We will show that every logic in FiCn can be alternatively defined by means of finite matrices. This result arises from the characterisation of the truth-values of the involved matrices (relative to each FiCn-logic) as being specific finite sequences of elements of the set 2 (...) := {0,1}. Moreover, we will show in this paper that such characterisation is related to the well-known standard Fibonacci Sequence, which is presented here by means of its binary expansion. (shrink)

The present article shows that there are consistent and decidable manyvalued systems of propositional logic which satisfy two or all the three criteria for non-trivial inconsistent theories by da Costa . The weaker one of these paraconsistent system is also able to avoid a series of paradoxes which come up when classical logic is applied to empirical sciences. These paraconsistent systems are based on a 6-valued system of propositional logic for avoiding difficulties in several domains of empirical (...) science ). (shrink)

A recent rethinking of the early history of Quantum Mechanics deemed the late 1920s agreement on the equivalence of Matrix Mechanics and Wave Mechanics, prompted by Schrödinger’s 1926 proof, a myth. Schrödinger supposedly failed to achieve the goal of proving isomorphism of the mathematical structures of the two theories, while only later developments in the early 1930s, especially the work of mathematician John von Neumman (1932) provided sound proof of equivalence. The alleged agreement about the Copenhagen Interpretation, predicated to (...) a large extent on this equivalence, was deemed a myth as well. If such analysis is correct, it provides considerable evidence that, in its critical moments, the foundations of scientific practice might not live up to the minimal standards of rigor, as such standards are established in the practice of logic, mathematics, and mathematical physics, thereby prompting one to question the rationality of the practice of physics. In response, I argue that Schrödinger’s proof concerned primarily a domain-specific ontological equivalence, rather than the isomorphism. It stemmed initially from the agreement of the eigenvalues of Wave Mechanics and energy-states of Bohr’s Model that was discovered and published by Schrödinger in his First and Second Communications of 1926. Schrödinger demonstrated in this proof that the laws of motion arrived at by the method of Matrix Mechanics could be derived successfully from eigenfunctions as well (while he only outlined the reversed derivation of eigenfunctions from Matrix Mechanics, which was necessary for the proof of isomorphism of the two theories). This result was intended to demonstrate the domain-specific ontological equivalence of Matrix Mechanics and Wave Mechanics, with respect to the domain of Bohr’s atom. And although the full-fledged mathematico-logical equivalence of the theories did not seem out of the reach of existing theories and methods, Schrödinger never intended to fully explore such a possibility in his proof paper. In a further development of Quantum Mechanics, Bohr’s complementarity and Copenhagen Interpretation captured a more substantial convergence of the subsequently revised (in light of the experimental results) Wave and Matrix Mechanics. I argue that both the equivalence and Copenhagen Interpretation can be deemed myths if one predicates the philosophical and historical analysis on a narrow model of physical theory which disregards its historical context, and focuses exclusively on its formal aspects and the exploration of the logical models supposedly implicit in it. (shrink)

Referential matrix semantics of R. W´ojcicki [5] and [4] is extended to cover the class of all structural propositional calculi.

Originating in the framework of philosophical preoccupations with paradoxes and determinism, the many-valued logics are today essential, finding abundant applications in STEM fields from electronics to knowledge bases. Essentially characterized by many-valuedness, or the property of having more than the two classical truth values, the many-valued logical systems are often presented only axiomatically and/or truth-functionally, but this presentation largely obliterates the intended meanings of many-truthfulness, the property that in these logics more than a single truth value can be distinguished as (...) true. Matrix semantics, a type of algebraic semantics, handles many-truthfulness in a set-theoretic way that typically allows for a meaning account. In this review article, the core aspects of many-valued logics with matrix semantics are elaborated on from the viewpoint of logical consequence. (shrink)

Using matrix iterations of ccc posets, we prove the consistency with ZFC of some cases where the cardinals on the right hand side of Cichon’s diagram take two or three arbitrary values (two regular values, the third one with uncountable cofinality). Also, mixing this with the techniques in J Symb Log 56(3):795–810, 1991, we can prove that it is consistent with ZFC to assign, at the same time, several arbitrary regular values on the left hand side of Cichon’s diagram.

This work adapts techniques and results first developed by Malinowski and by Marek in the context of referential semantics of sentential logics to the context of logics formalized as π-institutions. More precisely, the notion of a pseudoreferential matrix system is introduced and it is shown how this construct generalizes that of a referential matrix system. It is then shown that every π–institution has a pseudo-referential matrix system semantics. This contrasts with referential matrix system semantics which is (...) only available for self-extensional π-institutions by a previous result of the author obtained as an extension of a classical result of Wójcicki. Finally, it is shown that it is possible to replace an arbitrary pseudoreferential matrix system semantics by a discrete pseudo-referential matrix system semantics. (shrink)

In this paper we define n+1-valued matrix logic Kn+1 whose class of tautologies is non-empty iff n is a prime number. This result amounts to a new definition of a prime number. We prove that if n is prime, then the functional properties of Kn+1 are the same as those of ukasiewicz's n +1-valued matrix logic n+1. In an indirect way, the proof we provide reflects the complexity of the distribution of prime numbers in the natural (...) series. Further, we introduce a generalization K n+1 * of Kn+1 such that the set of tautologies of Kn+1 is not empty iff n is of the form p , where p is prime and is natural. Also in this case we prove the equivalence of functional properties of the introduced logic and those of n+1. In the concluding part, we discuss briefly a partition of the natural series into equivalence classes such that each class contains exactly one prime number. We conjecture that for each prime number the corresponding equivalence class is finite. (shrink)

The aim of this paper is to provide a reduced Routley–Meyer semantics for the logics characterized by all natural implicative expansions of Kleene’s strong 3-valued matrix (with two designated values, as well as with only one) susceptible to be interpreted in Routley–Meyer semantics.

This paper deals with the axiomatizability problem for the matrix-based logics RMQ − and RMQ *. We present a Hilbert-style axiom system for RMQ −, and a quasi-axiomatization based on it for RMQ *. We further compare these logics to different well-known modal logics, and assess its status as relevance logics.

A recent rethinking of the early history of Quantum Mechanics deemed the late 1920s agreement on the equivalence of Matrix Mechanics and Wave Mechanics, prompted by Schrödinger's 1926 proof, a myth. Schrödinger supposedly failed to prove isomorphism, or even a weaker equivalence (“Schrödinger-equivalence”) of the mathematical structures of the two theories; developments in the early 1930s, especially the work of mathematician von Neumann provided sound proof of mathematical equivalence. The alleged agreement about the Copenhagen Interpretation, predicated to a large (...) extent on this equivalence, was deemed a myth as well. In response, I argue that Schrödinger's proof concerned primarily a domain-specific ontological equivalence, rather than the isomorphism or a weaker mathematical equivalence. It stemmed initially from the agreement of the eigenvalues of Wave Mechanics and energy-states of Bohr's Model that was discovered and published by Schrödinger in his first and second communications of 1926. Schrödinger demonstrated in this proof that the laws of motion arrived at by the method of Matrix Mechanics are satisfied by assigning the auxiliary role to eigenfunctions in the derivation of matrices (while he only outlined the reversed derivation of eigenfunctions from Matrix Mechanics, which was necessary for the proof of both isomorphism and Schrödinger-equivalence of the two theories). This result was intended to demonstrate the domain-specific ontological equivalence of Matrix Mechanics and Wave Mechanics, with respect to the domain of Bohr's atom. And although the mathematical equivalence of the theories did not seem out of the reach of existing theories and methods, Schrödinger never intended to fully explore such a possibility in his proof paper. In a further development of Quantum Mechanics, Bohr's complementarity and Copenhagen Interpretation captured a more substantial convergence of the subsequently revised (in light of the experimental results) Wave and Matrix Mechanics. -/- I argue that both the equivalence and Copenhagen Interpretation can be deemed myths if one predicates the philosophical and historical analysis on a narrow model of physical theory which disregards its historical context, and focuses exclusively on its formal aspects and the exploration of the logical models supposedly implicit in it. -/- . (shrink)

Motivated by a question of W. Rautenberg, we prove that any matrix that is term-equivalent to the well-known nonfinitely based matrix of A. Wroski is itself also nonfinitely based.

Referential semantics importantly subscribes to the programme of theory of logical calculi. Defined by Wójcicki in [8], it has been subsequently studied in a series of papers of the author, till the full exposition of the framework in [9] and its intuitive characterisation in [10]. The aim of the article is to present several generalizations of referential semantics as compared and related to the matrix semantics for propositional logics. We show, in a uniform way, some own generalizations of referentiality: (...) the first, directed to unrestricted cluster referential semantics, [4], its "discrete" version, a counterpart of algebraic semantics and a many-valued referentiality based on matrices, whose elements are functions from the set of indices to a finite n-element set of values, n≥2, [3]. Next to this we outline pragmatic matrices introduced by Tokarz in [6] as an alternative for cluster referential approach and discuss together all presented versions of referential semantics. (shrink)

“The Matrix is a computer-generated dreamworld built to keep us under control” Morpheus, early in The Matrix. “ In dreaming, you are not only out of control, you don’t even know it…I was completely duped again and again the minute my pons, my amygdala, my perihippocampal cortex, my anterior cingulate, my visual association and parietal opercular cortices were revved up and my dorsolateral prefrontal cortex was muffled” ” J. Allan Hobson, The Dream Drugstore, p.64 The Matrix is (...) an exercise in ambivalence, and at the very heart of that ambivalence lies the Dream. In our dreams, we are not in control. Real dreaming, unlike many popular philosophers’ fictions, is an altered state, closely related to the states induced by chemical manipulations such as the use of (certain) medical or recreational drugs. The dreaming brain is not like the wakeful brain. Normal sensory input is blocked, attentional capacities are impaired or lost, memory is distorted, reasoning and logic are weakened, narratives run wild, self-reflection is dampened or destroyed, emotion and instinct are hyperstimulated, and forms of ‘top-down’ willed control and decision-making diluted and easily overwhelmed. (shrink)

We show that short bounded-depth Frege proofs of matrix identities, such as PQ=I⊃QP=I (over the field of two elements), imply short bounded-depth Frege proofs of the pigeonhole principle. Since the latter principle is known to require exponential-size bounded-depth Frege proofs, it follows that the propositional version of the matrix principle also requires bounded-depth Frege proofs of exponential size.

Vector logic is a mathematical model of the propositional calculus in which the logical variables are represented by vectors and the logical operations by matrices. In this framework, many tautologies of classical logic are intrinsic identities between operators and, consequently, they are valid beyond the bivalued domain. The operators can be expressed as Kronecker polynomials. These polynomials allow us to show that many important tautologies of classical logic are generated from basic operators via the operations called Type (...) I and Type II products. Finally, it is described a matrix version of the Fredkin gate that extends its properties to the many-valued domain, and it is proved that the filtered Fredkin operators are second degree Kronecker polynomials that cannot be generated by Type I or Type II products. (shrink)

ABSTRACTA conditional is natural if it fulfils the three following conditions. It coincides with the classical conditional when restricted to the classical values T and F; it satisfies the Modus Ponens; and it is assigned a designated value whenever the value assigned to its antecedent is less than or equal to the value assigned to its consequent. The aim of this paper is to provide a ‘bivalent’ Belnap-Dunn semantics for all natural implicative expansions of Kleene's strong 3-valued matrix with (...) two designated elements. (shrink)

The objective of this chapter is to study abstract sentential calculi (alternatively, abstract propositional logics) construed as pairs (ℒ,⊢)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathcal {L}, \vdash )$$\end{document}. Here, ℒ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}$$\end{document} is the sentential language (absolutely free algebra) generated by a countably infinite set of propositional variables and a finite set of finitary connectives, and ⊢\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vdash $$\end{document} is a multiple-conclusion (...) consequence relation. To call a binary relation between arbitrary sets of formulas of ℒ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}$$\end{document} a (multiple-conclusion) consequence relation, it is assumed that it fulfills the three laws—(R), (M), and (C): (R)If X∩Y≠∅\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\cap Y\neq \varnothing $$\end{document}, then X⊢Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\vdash Y$$\end{document}. (Reflexivity)(M)If X1⊢Y1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_1\vdash Y_1$$\end{document}, where X1⊆X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_1\subseteq X$$\end{document} and Y1⊆Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_1\subseteq Y$$\end{document}, then X⊢Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\vdash Y$$\end{document}. (Monotonicity or Dilution)(C)If X,S⊢Y,T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X, S\vdash Y, T$$\end{document} for all S and T such that S∩T=∅\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S\cap T=\varnothing $$\end{document} and S∪T=L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S\cup T = L$$\end{document}, then X⊢Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\vdash Y$$\end{document}. (Cut)Some relations may be additionally structural in the sense that they satisfy the following condition, which takes into account the algebraic structure of ℒ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}$$\end{document}: (S)If X⊢Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\vdash Y$$\end{document}, then {e(α):α∈X}⊢{e(β):β∈Y}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{e(\alpha ): \alpha \in X\} \vdash \{e(\beta ): \beta \in Y\}$$\end{document} for every endomorphism e of ℒ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}$$\end{document}.Section 3 reviews the basic properties of consequence relations from a lattice-theoretical point of view. In particular, a description is given of the least consequence relation extending a fixed set of pairs (X, Y ), where X and Y are finite sets of formulas.The main thread of discussion appears in Sect. 4, extends to the rest of the work, and concerns consequence relations determined by (logical) matrices, i.e., the relations of the form ⊢M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vdash _M$$\end{document}, where M=(A,D)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M = (\mathcal {A}, D)$$\end{document} is a matrix for ℒ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}$$\end{document}. One defines X⊢MY\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\vdash _{M} Y$$\end{document} to mean that, for every homomorphism h from ℒ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}$$\end{document} to A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document}, either h(α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(\alpha )$$\end{document} is undesignated (does not belong to D) for some α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} in X or h(β)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(\beta )$$\end{document} is designated for some β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} in Y. Since every such a relation ⊢M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vdash _M$$\end{document} is structural, in discussing the problem of representing a consequence relation ⊢\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vdash $$\end{document} by a class of matrices, one naturally has to assume that ⊢\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vdash $$\end{document} is structural. Much attention is devoted to the study of strongly finite logics, i.e., logics whose consequence relations may be determined by a finite set of finite matrices. In Sect. 4.3, a criterion, expressed in terms of Lindenbaum-type matrices, for ⊢\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vdash $$\end{document} to be strongly finite is given, and in Sect. 4.4 by the well-known method of ultraproducts it is proved that every strongly finite consequence relation is compact. The subject of strengthenings of logics is treated in Sects. 4.5 and 4.6, where the so-called (A, B)-strengthenings are examined in detail. Intuitively, given two arbitrary sets of formulas A and B, the (A, B)-strengthening of consequence relation ⊢\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vdash $$\end{document} is obtained by treating formulas in A as new axiom schemas, while those from B as counter-axiom schemas. It is shown (by two methods) that any (A, B)-strengthening of a strongly finite ⊢\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vdash $$\end{document} is strongly finite, and the number of such strengthenings is finite.The remaining part of our work (Sect. 5) is devoted to a purely model-theoretical proof of Shoesmith and Smiley’s finite basis theorem, which states that every strongly finite logic is finitely based, i.e., generated by a finite set of pairs (X, Y ) with X and Y being finite sets of formulas.In the Appendix, we abandon multiple-conclusion consequence relations and turn briefly to single-conclusion ones (i.e., consequence operations in Tarski’s sense) and furnish other proofs of two important theorems: Czelakowski’s characterization theorem for the class Matr(CnK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Cn_K$$\end{document}) and Wojtylak’s representation theorem for consequence operations that are ≥\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\geq $$\end{document} than CnK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Cn_K$$\end{document}. (shrink)

This paper is a sequel to ‘Belnap-Dunn semantics for natural implicative expansions of Kleene's strong three-valued matrix with two designated values’, where a ‘bivalent’ Belnap-Dunn semantics is provided for all the expansions referred to in its title. The aim of the present paper is to carry out a parallel investigation for all natural implicative expansions of Kleene's strong 3-valued matrix now with only one designated value.

Phenomenology, Sociology, Hermeneutics and Psychoanalysis have accumulated different methods and knowledge on the imaginary. Nevertheless, the crucial connection between the social imagining and communication has not always been truly examined. In this article, we take the imaginary (seen as a symbolic thought of images) and communication (seen as a process of symbolic reproduction) as correlated notions, and work upon a communicational matrix of the imaginary. We emphasize three key elements of the imaginary: by pointing to the verbal-iconic, technical and (...) mediatized dimensions we will be paying special attention to the media imaginary and the effects of its commemorative logic to the propagation of public imaginaries. (shrink)

Drawing on the analytic of the “colonial matrix of power” developed by Aníbal Quijano within the Latin American modernity/coloniality research program, this article theorises how a system of coloniality underpins the structuring logic of artificial intelligence (AI) systems. We develop a framework for critiquing the regimes of global labour exploitation and knowledge extraction that are rendered invisible through discourses of the purported universality and objectivity of AI. ​​Through bringing the political economy literature on AI production into conversation with (...) scholarly work on decolonial AI and the modernity/coloniality research program, we advance three main arguments. First, the global economic and political power imbalances in AI production are inextricably linked to the continuities of historical colonialism, constituting the colonial supply chain of AI. Second, this is produced through an international division of digital labour that extracts value from majority world labour for the benefit of Western technology companies. Third, this perpetuates hegemonic knowledge production through Western values and knowledge that marginalises non-Western alternatives within AI’s production and limits the possibilities for decolonising AI. By locating the production of AI systems within the colonial matrix of power, we contribute to critical and decolonial literature on the legacies of colonialism in AI and the hierarchies of power and extraction that shape the development of AI today. (shrink)

Indian Philosophy has reached across the globe. It is popular for its practical way towards life. Study of Indian philosophy should be part of all streams of education. Film is effective tool of communication. It attracts all generations and makes strong impression in the mind. Film is always considered as an effective tool in Pedagogy. Philosophy deals with abstract concepts, their correlation and logical reasoning. It deals with the complex problem of reality. People have notion that philosophy is a dry (...) and theoretical subject. Students can understand complex philosophical ideas if they are demonstrated through films. There are films based on biographies of philosophers, life picture of great people, incidences in life of philosophers. But it can attract only limited number of audience. Some popular Hollywood films have amalgamation of science fiction, super natural elements, fantastic story guided by some philosophical principles. Such films attract audience from different strata. Viewer is totally captivated by extra ordinary super natural and fictional elements in the film. As a result, certain philosophical principles are taught to audience without their conscious effort. (Kantāsammita upadeśa) Teaching philosophy with the means of such films can be definitely effective tool in Pedagogy. Such practise will not only create liking for Indian Philosophy but will result in effective understanding of complex ideas. It can create awareness that Indian Philosophy is embraced across the globe, there are no boundaries for philosophical deliberations and there are certain common threads in different philosophical schools. Paper will take example of popular movie ‘Matrix’ to explain the principles of Advaita Vedānta. The Matrix is a 1999 science fiction film written and directed by The Wachowskis, starring Keanu Reeves, Laurence Fishburne, Carrie-Anne Moss, Hugo Weaving, and Joe Pantoliano. It states the journey of hero of the film ‘Neo’ who is aspiring for reality. He is struggling between illusory world and the real world which is presented through the means of science fiction. This paper will analyse Matrix from the perspective of Advaita Vedānta. Māyā projects the illusory world (vikṣepa) in place of Brahman (āvaraṇa). Film portrays digital world as illusory world. One sees this world as real due to limitations of body, mind and intellect. Neo goes to the real world and realises his true nature. True reality is beyond senses and sense objects (atindrīya). Concepts of Advaita Vedānta can be explained with this film. Such movies can be shown to students along with Vedanta discourses. Paper will explore ‘Film and Philosophy’ as an effective pedagogical model. (shrink)

There exist important deductive systems, such as the non-normal modal logics, that are not proper subjects of classical algebraic logic in the sense that their metatheory cannot be reduced to the equational metatheory of any particular class of algebras. Nevertheless, most of these systems are amenable to the methods of universal algebra when applied to the matrix models of the system. In the present paper we consider a wide class of deductive systems of this kind called protoalgebraic logics. (...) These include almost all (non-pathological) systems of prepositional logic that have occurred in the literature. The relationship between the metatheory of a protoalgebraic logic and its matrix models is studied. The following results are obtained for any finite matrix model U of a filter-distributive protoalgebraic logic : (I) The extension U of is finitely axiomatized (provided has only finitely many inference rules); (II) U has only finitely many extensions. (shrink)