Hyperboolean algebras are Boolean algebras with operators, constructed as algebras of complexes (or, power structures) of Boolean algebras. They provide an algebraic semantics for a modal logic (called here a {\em hyperboolean modal logic}) with a Kripke semantics accordingly based on frames in which the worlds are elements of Boolean algebras and the relations correspond to the Boolean operations. We introduce the hyperboolean modal logic, give a complete axiomatization of it, and show (...) that it lacks the finite model property. The method of axiomatization hinges upon the fact that a "difference" operator is definable in hyperboolean algebras, and makes use of additional non-Hilbert-style rules. Finally, we discuss a number of open questions and directions for further research. (shrink)

Monoidal t-norm based logic $\mathbf {MTL}$ is the weakest t-norm based residuated fuzzy logic, which is a $[0,1]$ -valued propositional logical system having a t-norm and its residuum as truth function for conjunction and implication. Monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ that consists of the formulas with unary predicates and just one object variable, is the monadic fragment of fuzzy predicate logic $\mathbf {MTL\forall }$, which is indeed the predicate version of monoidal t-norm based logic $\mathbf {MTL}$. The main (...) aim of this paper is to give an algebraic proof of the completeness theorem for monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and some of its axiomatic extensions. Firstly, we survey the axiomatic system of monadic algebras for t-norm based residuated fuzzy logic and amend some of them, thus showing that the relationships for these monadic algebras completely inherit those for corresponding algebras. Subsequently, using the equivalence between monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and S5-like fuzzy modal logic $\mathbf {S5(MTL)}$, we prove that the variety of monadic MTL-algebras is actually the equivalent algebraic semantics of the logic $\mathbf {mMTL\forall }$, giving an algebraic proof of the completeness theorem for this logic via functional monadic MTL-algebras. Finally, we further obtain the completeness theorem of some axiomatic extensions for the logic $\mathbf {mMTL\forall }$, and thus give a major application, namely, proving the strong completeness theorem for monadic fuzzy predicate logic based on involutive monoidal t-norm logic $\mathbf {mIMTL\forall }$ via functional representation of finitely subdirectly irreducible monadic IMTL-algebras. (shrink)

Volume I provides a detailed analysis of cylindric algebras, starting with a formulation of their axioms and a development of their elementary properties, and proceeding to a deeper study of their interrelationships by means of general algebraic notions such as subalgebras, homomorphisms, direct products, free algebras, reducts and relativized algebras.

This comprehensive text shows how various notions of logic can be viewed as notions of universal algebra providing more advanced concepts for those who have an introductory knowledge of algebraic logic, as well as those wishing to delve into more theoretical aspects.

In classical logic, the proposition expressed by a sentence is construed as a set of possible worlds, capturing the informative content of the sentence. However, sentences in natural language are not only used to provide information, but also to request information. Thus, natural language semantics requires a logical framework whose notion of meaning does not only embody informative content, but also inquisitive content. This paper develops the algebraic foundations for such a framework. We argue that propositions, in order to embody (...) both informative and inquisitive content in a satisfactory way, should be defined as non-empty, downward closed sets of possibilities, where each possibility in turn is a set of possible worlds. We define a natural entailment order over such propositions, capturing when one proposition is at least as informative and inquisitive as another, and we show that this entailment order gives rise to a complete Heyting algebra, with meet, join, and relative pseudo-complement operators. Just as in classical logic, these semantic operators are then associated with the logical constants in a first-order language. We explore the logical properties of the resulting system and discuss its significance for natural language semantics. We show that the system essentially coincides with the simplest and most well-understood existing implementation of inquisitive semantics, and that its treatment of disjunction and existentials also concurs with recent work in alternative semantics. Thus, our algebraic considerations do not lead to a wholly new treatment of the logical constants, but rather provide more solid foundations for some of the existing proposals. (shrink)

This book presents an English translation of a classic Russian text on duality theory for Heyting algebras. Written by Georgian mathematician Leo Esakia, the text proved popular among Russian-speaking logicians. This translation helps make the ideas accessible to a wider audience and pays tribute to an influential mind in mathematical logic. The book discusses the theory of Heyting algebras and closure algebras, as well as the corresponding intuitionistic and modal logics. The author introduces the key notion of (...) a hybrid that “crossbreeds” topology and order, resulting in the structures now known as Esakia spaces. The main theorems include a duality between the categories of closure algebras and of hybrids, and a duality between the categories of Heyting algebras and of so-called strict hybrids. Esakia’s book was originally published in 1985. It was the first of a planned two-volume monograph on Heyting algebras. But after the collapse of the Soviet Union, the publishing house closed and the project died with it. Fortunately, this important work now lives on in this accessible translation. The Appendix of the book discusses the planned contents of the lost second volume. (shrink)

In the paper, pre-Hilbert algebras are defined as a generalization of Hilbert algebras (namely, a Hilbert algebra is just a pre-Hilbert algebra satisfying the property of antisymmetry). Pre-Hilbert algebras have been inspired by Henkin's Positive Implicative Logic. Their properties and characterizations are investigated. Some important results and examples are given. Moreover, positive implicative pre-Hilbert algebras are introduced and studied, their connections with some algebras of logic are presented. The hierarchies existing between the classes of (...) class='Hi'>algebras considered here are shown. (shrink)

This paper investigates a generalization of Boolean algebras which I call agglomerative algebras. It also outlines two conceptions of propositions according to which they form an agglomerative algebra but not a Boolean algebra with respect to conjunction and negation.

Algebraic quantum field theory provides a general, mathematically precise description of the structure of quantum field theories, and then draws out consequences of this structure by means of various mathematical tools -- the theory of operator algebras, category theory, etc.. Given the rigor and generality of AQFT, it is a particularly apt tool for studying the foundations of QFT. This paper is a survey of AQFT, with an orientation towards foundational topics. In addition to covering the basics of the (...) theory, we discuss issues related to nonlocality, the particle concept, the field concept, and inequivalent representations. We also provide a detailed account of the analysis of superselection rules by Doplicher, Haag, and Roberts (DHR); and we give an alternative proof of Doplicher and Robert's reconstruction of fields and gauge group from the category of physical representations of the observable algebra. The latter is based on unpublished ideas due to J. E. Roberts and the abstract duality theorem for symmetric tensor *-categories, a self-contained proof of which is given in the appendix. (shrink)

A sequent calculus S for the variety tqBa of all topological quasi-Boolean algebras is established. Using a construction of syntactic finite algebraic model, the finite model property of S is shown, and thus the decidability of S is obtained. We also introduce two non-distributive variants of topological quasi-Boolean algebras. For the variety TDM5 of all topological De Morgan lattices with the axiom 5, we establish a sequent calculus S5 and prove that the cut elimination holds for it. Consequently (...) the decidability of S5 is established. Furthermore, this proof-theoretic method is applied to some subvarieties of TDM5. (shrink)

The Belnap–Dunn logic is a well-known and well-studied four-valued logic, but until recently little has been known about its extensions, i.e. stronger logics in the same language, called super-Belnap logics here. We give an overview of several results on these logics which have been proved in recent works by Přenosil and Rivieccio. We present Hilbert-style axiomatizations, describe reduced matrix models, and give a description of the lattice of super-Belnap logics and its connections with graph theory. We adopt the point of (...) view ofAlgebraic Logic, exploring applications of the general theory of algebraization of logics to the super-Belnap family. In this respect we establish a number of new results, including a description of the algebraic counterparts, Leibniz filters, and strong versions of super-Belnap logics, as well as the classification of these logics within the Leibniz and Frege hierarchies. (shrink)

This volume is not restricted to papers presented at the 1988 Colloquium, but instead aims to provide the reader with a (relatively) coherent reading on Algebraic Logic, with an emphasis on current research. To help the non-specialist reader, the book contains an introduction to cylindric and relation algebras by Roger D. Maddux and an introduction to Boolean Algebras by Bjarni Joacute;nsson.

Provability, Computability and Reflection.

We prove two results intended to streamline proofs about cellularity that pass through mutual algebraicity. First, we show that a countable structure M is cellular if and only if M is \-categorical and mutually algebraic. Second, if a countable structure M in a finite relational language is mutually algebraic non-cellular, we show it admits an elementary extension adding infinitely many infinite MA-connected components. Towards these results, we introduce MA-presentations of a mutually algebraic structure, in which every atomic formula is mutually (...) algebraic. This allows for an improved quantifier elimination and a decomposition of the structure into independent pieces. We also show this decomposition is largely independent of the MA-presentation chosen. (shrink)

A new structure, called equality algebras, will be introduced. It has two connectives, a meet operation and an equivalence, and a constant. A closure operator will be defined in the class of equality algebras, and we call the closed algebras equivalential. We show that equivalential equality algebras are term equivalent with BCK-algebras with meet. As a by-product, we obtain a quite general result, which is analogous to a result of Kabziński and Wroński: we provide an (...) equational characterization for the equivalential fragment of BCK-algebras with meet. (shrink)

This paper deals with one kind of Kripke-style semantics, which we shall call algebraic Kripke-style semantics, for relevance logics. We first recall the logic R of relevant implication and some closely related systems, their corresponding algebraic structures, and algebraic completeness results. We provide simpler algebraic completeness proofs. We then introduce various types of algebraic Kripke-style semantics for these systems and connect them with algebraic semantics.

We show that the variety of equivalential algebras with regularization gives the algebraic semantics for the -fragment of intuitionistic propositional logic. We also prove that this fragment is hereditarily structurally complete.

The Algebra of Revolution is the first book to study Marxist method as it has been developed by the main representatives of the classical Marxist tradition, namely Marx and Engels, Luxembourg, Lenin, Lukacs, Gramsci, and Trotsky. This book provides the only single volume study of major Marxist thinkers' views on the crucial question of the dialectic, connecting them with pressing contemporary, political and theoretical questions. This title available in eBook format. Click here for more information . Visit our eBookstore at: (...) www.ebookstore.tandf.co.uk. (shrink)

We generalize the double negation construction of Boolean algebras in Heyting algebras to a double negation construction of the same in Visser algebras. This result allows us to generalize Glivenko’s theorem from intuitionistic propositional logic and Heyting algebras to Visser’s basic propositional logic and Visser algebras.

_The Algebra of Revolution_ is the first book to study Marxist method as it has been developed by the main representatives of the classical Marxist tradition, namely Marx and Engels, Luxembourg, Lenin, Lukacs, Gramsci and Trotsky. This book provides the only single volume study of major Marxist thinkers' views on the crucial question of the dialectic, connecting them with pressing contemporary, political and theoretical questions. John Rees's _The Algebra of Revolution_ is vital reading for anyone interested in gaining a new (...) and fresh perspective on Marxist thought and on the notion of the dialectic. (shrink)

This unique textbook states and proves all the major theorems of many-valued propositional logic and provides the reader with the most recent developments and trends, including applications to adaptive error-correcting binary search. The book is suitable for self-study, making the basic tools of many-valued logic accessible to students and scientists with a basic mathematical knowledge who are interested in the mathematical treatment of uncertain information. Stressing the interplay between algebra and logic, the book contains material never before published, such as (...) a simple proof of the completeness theorem and of the equivalence between Chang's MV algebras and Abelian lattice-ordered groups with unit - a necessary prerequisite for the incorporation of a genuine addition operation into fuzzy logic. Readers interested in fuzzy control are provided with a rich deductive system in which one can define fuzzy partitions, just as Boolean partitions can be defined and computed in classical logic. Detailed bibliographic remarks at the end of each chapter and an extensive bibliography lead the reader on to further specialised topics. (shrink)

An analysis of the structural properties of collections or pluralities, homogeneous objects like water, and the semantics and philosophy of events.

1 Cognitive Architectures 2 Multilayer Perceptrons 3 Relations between Variables 4 Structured Representations 5 Individuals 6 Where does the Machinery of Symbol Manipulation Come From? 7 Conclusions.

We investigate the variety corresponding to a logic, which is the combination of ukasiewicz Logic and Product Logic, and in which Gödel Logic is interpretable. We present an alternative axiomatization of such variety. We also investigate the variety, called the variety of algebras, corresponding to the logic obtained from by the adding of a constant and of a defining axiom for one half. We also connect algebras with structures, called f-semifields, arising from the theory of lattice-ordered rings, and (...) prove that every algebra can be regarded as a structure whose domain is the interval [0, 1] of an f-semifield, and whose operations are the truncations of the operations of to [0, 1]. We prove that such a structure is uniquely determined by up to isomorphism, and we establish an equivalence between the category of algebras and that of f-semifields. (shrink)

This article provides an algebraic study of the propositional system $\mathtt {InqB}$ of inquisitive logic. We also investigate the wider class of $\mathtt {DNA}$ -logics, which are negative variants of intermediate logics, and the corresponding algebraic structures, $\mathtt {DNA}$ -varieties. We prove that the lattice of $\mathtt {DNA}$ -logics is dually isomorphic to the lattice of $\mathtt {DNA}$ -varieties. We characterise maximal and minimal intermediate logics with the same negative variant, and we prove a suitable version of Birkhoff’s classic variety (...) theorems. We also introduce locally finite $\mathtt {DNA}$ -varieties and show that these varieties are axiomatised by the analogues of Jankov formulas. Finally, we prove that the lattice of extensions of $\mathtt {InqB}$ is dually isomorphic to the ordinal $\omega +1$ and give an axiomatisation of these logics via Jankov $\mathtt {DNA}$ -formulas. This shows that these extensions coincide with the so-called inquisitive hierarchy of [9].1. (shrink)

In the recent years, problem-solving become a central topic that discussed by educators or researchers in mathematics education. it’s not only as the ability or as a method of teaching. but also, it is a little in reviewing about the components of the support to succeed in problem-solving, such as student's belief and attitude towards mathematics, algebraic thinking skills, resources and teaching materials. In this paper, examines the algebraic thinking skills as a foundation for problem-solving, and learning cycle as a (...) breath of continuous learning. In this paper, learning cycle to be used is a modified type of 5E based on beliefs. (shrink)

This paper is an introduction: in particular, to algebras of relations of various ranks, and in general, to the part of algebraic logic algebraizing quantifier logics. The paper has a survey character, too. The most frequently used algebras like cylindric-, relation-, polyadic-, and quasi-polyadic algebras are carefully introduced and intuitively explained for the nonspecialist. Their variants, connections with logic, abstract model theory, and further algebraic logics are also reviewed. Efforts were made to make the review part relatively (...) comprehensive. In some directions we tried to give an overview of the most recent results and research trends, too. (shrink)

In this paper we show how the dynamics of the Schrödinger, Pauli and Dirac particles can be described in a hierarchy of Clifford algebras, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal{C}}_{1,3}, {\mathcal{C}}_{3,0}$\end{document}, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal{C}}_{0,1}$\end{document}. Information normally carried by the wave function is encoded in elements of a minimal left ideal, so that all the physical information appears within the algebra itself. The state of the quantum process can (...) be completely characterised by algebraic invariants of the first and second kind. The latter enables us to show that the Bohm energy and momentum emerge from the energy-momentum tensor of standard quantum field theory. Our approach provides a new mathematical setting for quantum mechanics that enables us to obtain a complete relativistic version of the Bohm model for the Dirac particle, deriving expressions for the Bohm energy-momentum, the quantum potential and the relativistic time evolution of its spin for the first time. (shrink)

The algebra of embeddings at the I3 level has been deeply analyzed, but nothing is known algebra-wise for embeddings above I3. In this article, we introduce an operation for embeddings at the level of I0 and above, and prove that they generate an LD-algebra that can be quite different from the one implied by I3.

The logic DAI of demodalised analytic implication has been introduced by J.M. Dunn as a variation on a time-honoured logical system by C.I. Lewis’ student W.T. Parry. The main tenet underlying this logic is that no implication can be valid unless its consequent is “analytically contained” in its antecedent. DAI has been investigated both proof-theoretically and model-theoretically, but no study so far has focussed on DAI from the viewpoint of abstract algebraic logic. We provide several different algebraic semantics for DAI, (...) showing their equivalence with the known semantics by Dunn and Epstein. We also show that DAI is algebraisable and we identify its equivalent quasivariety semantics. This class turns out to be a linguistic and axiomatic expansion of involutive bisemilattices, a subquasivariety of which forms the algebraic counterpart of Paraconsistent Weak Kleene logic. This fact sheds further light on the relationship between containment logics and logics of nonsense. (shrink)

Grishin algebras are a generalisation of Boolean algebras that provide algebraic models for classical bilinear logic with two mutually cancelling negation connectives. We show how to build complete Grishin algebras as algebras of certain subsets (“propositions”) of cover systems that use an orthogonality relation to interpret the negations. The variety of Grishin algebras is shown to be closed under MacNeille completion, and this is applied to embed an arbitrary Grishin algebra into the algebra of all (...) propositions of some cover system, by a map that preserves all existing joins and meets. This representation is then used to give a cover system semantics for a version of classical bilinear logic that has first-order quantifiers and infinitary conjunctions and disjunctions. (shrink)

In this paper we define the polyadic Pavelka algebras as algebraic structures for Rational Pavelka predicate calculus . We prove two representation theorems which are the algebraic counterpart of the completness theorem for RPL∀.

A Hilbert algebra with supremum is a Hilbert algebra where the associated order is a join-semilattice. This class of algebras is a variety and was studied in Celani and Montangie . In this paper we shall introduce and study the variety of $${H_{\Diamond}^{\vee}}$$ H ◊ ∨ -algebras, which are Hilbert algebras with supremum endowed with a modal operator $${\Diamond}$$ ◊ . We give a topological representation for these algebras using the topological spectral-like representation for Hilbert (...) class='Hi'>algebras with supremum given in Celani and Montangie . We will consider some particular varieties of $${H_{\Diamond}^{\vee}}$$ H ◊ ∨ -algebras. These varieties are the algebraic counterpart of extensions of the implicative fragment of the intuitionistic modal logic $${\mathbf{IntK}_{\Diamond}}$$ IntK ◊ . We also determine the congruences of $${H_{\Diamond}^{\vee}}$$ H ◊ ∨ -algebras in terms of certain closed subsets of the associated space, and in terms of a particular class of deductive systems. These results enable us to characterize the simple and subdirectly irreducible $${H_{\Diamond}^{\vee }}$$ H ◊ ∨ -algebras. (shrink)

The aim of this paper is to give a description of the free algebras in some varieties of Glivenko MTL-algebras having the Boolean retraction property. This description is given (generalizing the results of [9]) in terms of weak Boolean products over Cantor spaces. We prove that in some cases the stalks can be obtained in a constructive way from free kernel DL-algebras, which are the maximal radical of directly indecomposable Glivenko MTL-algebras satisfying the equation in the (...) title. We include examples to show how we can apply the results to describe free algebras in some well known varieties of involutive MTL-algebras and of pseudocomplemented MTL-algebras. (shrink)

The effects in a quantum-mechanical system form a partial algebra and a partially ordered set which is the prototypical example of the effect algebras discussed in this paper. The relationships among effect algebras and such structures as orthoalgebras and orthomodular posets are investigated, as are morphisms and group- valued measures (or charges) on effect algebras. It is proved that there is a universal group for every effect algebra, as well as a universal vector space over an arbitrary (...) field. (shrink)

The main aim of the present paper is to explain a nature of relationships exist between Nelson and Heyting algebras. In the realization, a topological duality theory of Heyting and Nelson algebras based on the topological duality theory of Priestley for bounded distributive lattices are applied. The general method of construction of spaces dual to Nelson algebras from a given dual space to Heyting algebra is described. The algebraic counterpart of this construction being a generalization of the (...) Fidel-Vakarelov construction is also given. These results are applied to compare the equational category N of Nelson algebras and some its subcategories with the equational category H of Heyting algebras. It is proved that the category N is topological over the category H. (shrink)

We study the variety generated by the three-element equivalential algebra with conjunction on the dense elements. We prove the representation theorem which let us construct the free algebras in this variety.

The first known statements of the deduction theorems for the first-order predicate calculus and the classical sentential logic are due to Herbrand [8] and Tarski [14], respectively. The present paper contains an analysis of closure spaces associated with those sentential logics which admit various deduction theorems. For purely algebraic reasons it is convenient to view deduction theorems in a more general form: given a sentential logic C (identified with a structural consequence operation) in a sentential language I, a quite arbitrary (...) set P of formulas of I built up with at most two distinct sentential variables p and q is called a uniform deduction theorem scheme for C if it satisfies the following condition: for every set X of formulas of I and for any formulas and , C(X{{a}}) iff P(, ) AC(X). [P(, ) denotes the set of formulas which result by the simultaneous substitution of for p and for q in all formulas in P]. The above definition encompasses many particular formulations of theorems considered in the literature to be deduction theorems. Theorem 1.3 gives necessary and sufficient conditions for a logic to have a uniform deduction theorem scheme. Then, given a sentential logic C with a uniform deduction theorem scheme, the lattices of deductive filters on the algebras A similar to the language of C are investigated. It is shown that the join-semilattice of finitely generated (= compact) deductive filters on each algebra A is dually Brouwerian. (shrink)

Algebraic equations in the tradition of Descartes and Frans Van Schooten accompany Christiaan Huygens’s early work on collision, which later would be reorganized and presented as De motu corporum ex percussione. Huygens produced the equations at the same time as his announcement of his rejection of Descartes’s rules of collision. Never intended for publication, the equations appear to have been used as preliminary scaffolding on which to build his critiques of Descartes’s physics. Additionally, Huygens used algebraic equations of this form (...) to accurately predict the speeds of bodies after collision in experiments carried out at the Royal Society. Despite their deceptive simplicity, Huygens’s algebraic equations pose significant conceptual problems both mathematically and for their physical interpretation especially for negative speeds; they may very well have been the source of a new principle, the conservation of quantity of motion with direction. (shrink)

We introduce a generalization of MV algebras motivated by the investigations into the structure of quantum logical gates. After laying down the foundations of the structure theory for such quasi-MV algebras, we show that every quasi-MV algebra is embeddable into the direct product of an MV algebra and a “flat” quasi-MV algebra, and prove a completeness result w.r.t. a standard quasi-MV algebra over the complex numbers.

Einstein algebras have been suggested (Earman 1989) and rejected (Rynasiewicz 1992) as a way to avoid the hole argument against spacetime substantivalism. In this article, I debate their merits and faults. In particular, I suggest that a gauge‐invariant interpretation of Einstein algebras that avoids the hole argument can be associated with one approach to quantizing gravity, and, for this reason, is at least as well motivated as sophisticated substantivalist and relationalist interpretations of the standard tensor formalism.

In this paper we present a general theory of normal forms, based on a categorial result for the free monoid construction. We shall use the theory mainly for proposictional modal logic, although it seems to have a wider range of applications. We shall formally represent normal forms as combinatorial objects, basically labelled trees and forests. This geometric conceptualization is implicit in and our approach will extend it to other cases and make it more direct: operations of a purely geometric and (...) combinatorial nature will be introduced in order to give a mathematical description of the basic logical/algebraic constructions .We begin by recalling the above-mentioned categorial construction: we need a careful inspection of it because in the various examples considered later we plan to deduce from it in a uniform way the normal forms and the description of finitely generated free algebras. This method always works whenever we can describe the category of algebras corresponding to the logic under consideration as a T-objects category. When this simple description seems not to be available, still the general theory might be of some interest, because a description of the category of algebras as a T-objects category plus equation is possible .The central part of the paper is more advanced and specific: we show how the general approach presented here can provide some insights even in the basic case of the modal system K. Section 4 contains a contribution to the theory of normal forms, namely the description of the uniform substitution. This result will enable us to give a duality theorem for the category of finitely generated free modal algebras and in Section 5 to provide a characterization of the collections of normal forms which happen to be normal forms for a logic, thus giving a description of the lattice of modal logics.Section 6 deals with some applications: we shall show how to use normal forms in order to prove for the modal system K the definability of higher-order propositional quantifiers and of the tense operator F .As to the prerequisites, the paper is almost self-contained. The reader is only assumed to have familiarity with standard techniques in algebraic logic ). Knowledge of the basic facts about adjoint functors is required too, see e.g. McLane or the appendix. (shrink)

We introduce new algebraic and topological semantics for inquisitive logic. The algebraic semantics is based on special Heyting algebras, which we call inquisitive algebras, with propositional valuations ranging over only the ¬¬-fixpoints of the algebra. We show how inquisitive algebras arise from Boolean algebras: for a given Boolean algebra B, we define its inquisitive extension H(B) and prove that H(B) is the unique inquisitive algebra having B as its algebra of ¬¬-fixpoints. We also show that inquisitive (...) algebras determine Medvedev’s logic of finite problems. In addition to the algebraic characterization of H(B), we give a topological characterization of H(B) in terms of the recently introduced choice-free duality for Boolean algebras using so-called upper Vietoris spaces (UV-spaces). In particular, while a Boolean algebra B is realized as the Boolean algebra of compact regular open elements of a UV-space dual to B, we show that H(B) is realized as the algebra of compact open elements of this space. This connection yields a new topological semantics for inquisitive logic. (shrink)

Brown and Pooley’s ‘dynamical approach’ to physical theories asserts, in opposition to the orthodox position on physical geometry, that facts about physical geometry are grounded in, or explained by, facts about dynamical fields, not the other way round. John Norton has claimed that the proponent of the dynamical approach is illicitly committed to spatiotemporal presumptions in ‘constructing’ space-time from facts about dynamical symmetries. In this article, I present an abstract, algebraic formulation of field theories and demonstrate that the proponent of (...) the dynamical approach is not committed, in special relativity, to the illicit presumptions to which Norton refers. (shrink)

We present an algebraic account of the Tongan kinship terminology (TKT) that provides an insightful journey into the fabric of Tongan culture. We begin with the ethnographic account of a social event. The account provides us with the activities of that day and the centrality of kin relations in the event, but it does not inform us of the conceptual system that the participants bring with them. Rather, it is a slice in time of an ongoing dynamic process that links (...) behavior with a conceptual system of kin relations and vice versa. To understand this interplay, we need an account of the underlying conceptual system that is being activated during the event. Thus, we introduce a formal, algebraically based account of TKT. This account brings to the fore the underlying logic of TKT and allows us to distinguish between features of the kinship system that arise from the logic of TKT as a generative structure and features that must have arisen through cultural intervention. (shrink)

The admissibility of Ackermann's rule γ is one of the most important problems in relevant logics. The admissibility of γ was first proved by an algebraic method. However, the development of Routley-Meyer semantics and metavaluational techniques makes it possible to prove the admissibility of γ using the method of normal models or the method using metavaluations, and the use of such methods is preferred. This paper discusses an algebraic proof of the admissibility of γ in relevant modal logics based on (...) modern algebraic models. (shrink)