Many Faces of Logic

The algebra of logic, as an explicit algebraic system showing the underlying mathematical structure of logic, was introduced by George Boole (1815-1864) in his book The Mathematical Analysis of Logic (1847). The methodology initiated by Boole was successfully continued in the 19th century in the work of William Stanley Jevons (1835-1882), Charles Sanders Peirce (1839-1914), Ernst Schröder (1841-1902), among many others, thereby establishing a tradition in (mathematical) logic. Furthermore, this tradition motivated the investigations of Leopold Löwenheim (1878-1957) that eventually gave rise to model theory. The tradition of the algebra of logic played a key role in the notion of Logic as Calculus as opposed to the notion of Logic as Universal Language. This entry is divided into 10 sections: 0. Introduction 1. 1847—The Beginnings of the Modern Versions of the Algebra of Logic. 2. 1854—Boole's Final Presentation of his Algebra of Logic. 3. Jevons: An Algebra of Logic Based on Total Operations. 4. Peirce: Basing the Algebra of Logic on Subsumption. 5. De Morgan and Peirce: Relations and Quantifiers in the Algebra of Logic. 6. Schröder’s systematization of the Algebra of Logic. 7. Huntington: Axiomatic Investigations of the Algebra of Logic. 8. Stone: Models for the Algebra of Logic. 9. Skolem: Quantifier Elimination and Decidability.

2007

Since all the algebras connected to logic have, more or less explicitely, an associated order relation, it follows that they have two presentations, dual to each other. We classify these dual presentations in "left" and "right" ones and we consider that, when dealing with several algebras in the same research, it is useful to present them unitarily, either as "left" algebras or as "right" algebras. In some circumstances, this choice is essential, for instance if we want to build the ordinal sum (product) between a BL algebra and an MV algebra. We have chosen the "left" presentation and several algebras of logic have been redefined as particular cases of BCK algebras. We introduce several new properties of algebras of logic, besides those usually existing in the literature, which generate a more refined classification, depending on the pro- perties satisfied. In this work (Parts I-V) we make an exhaustive study of these algebras - wit...

We propose a generalization of first-order logic originating in a neglected work by C.C. Chang: a natural and generic correspondence language for any types of structures which can be recast as Set-coalgebras. We discuss axiomatization and completeness results for several natural classes of such logics. Moreover, we show that an entirely general completeness result is not possible. We study the expressive power of our language, both in comparison with coalgebraic hybrid logics and with existing first-order proposals for special classes of Set-coalgebras (apart from relational structures, also neighbourhood frames and topological spaces). Basic model-theoretic constructions and results, in particular ultraproducts, obtain for the two classes that allow completeness---and in some cases beyond that. Finally, we discuss a basic sequent system, for which we establish a syntactic cut-elimination result.

2009

In this chapter we determine the classes of S-algebras and of full models for several logics, especially for some which do not fit into the classical approaches to the algebraization of logic. We classify them according to several of the criteria we have been considering, i.e., the properties of the Leibniz, Tarski and Frege operators, which determine the classes of selfextensional logics, Fregean logics, strongly selfextensional logics, protoalgebraic logics, etc. We also study the counterexamples promised in the preceding chapters of this monograph. It goes without saying that the number of cases we have examined is limited, and that many more are waiting to be studied 32. In our view this is an interesting program, especially for non-algebraizable logics. Among those already proven in Blok and Pigozzi [1989a] not to be algebraizable we find many quasi-normal and other modal logics like Lewis' S1, S2 and S3, entailment system E, several purely implicational logics like BCI, the system R → of relevant implication, the "pure entailment" system E → , the implicative fragment S5 → of the Wajsbergstyle version of S5, etc. Other non-algebraizable logics not treated in the present monograph are Da Costa's paraconsistent logics C n (see Lewin, Mikenberg, and Schwarze [1991]), and the "logic of paradox" of Priest [1979] (see Pynko [1995]). This program is also interesting for some algebraizable logics whose class of Salgebras is already known, but whose full models have not yet been investigated; this includes Łukasiewicz many-valued logics (see Rodríguez, Torrens, and Verdú [1990]), BCK logic and some of its neighbours (see Blok and Pigozzi [1989a] Theorem 5.10), the equivalential fragments of classical and intuitionistic logics 32 The full models of several subintuitionistic logics have been determined in Bou [2001]; those of

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2014

We study the residuated basic logic ($\mathsf{RBL}$) of residuated basic algebra in which the basic implication of Visser's basic propositional logic ($\mathsf{BPL}$) is interpreted as the right residual of a non-associative binary operator $\cdot$ (product). We develop an algebraic system $\mathsf{S_{RBL}}$ of residuated basic algebra by which we show that $\mathsf{RBL}$ is a conservative extension of $\mathsf{BPL}$. We present the sequent formalization $\mathsf{L_{RBL}}$ of $\mathsf{S_{RBL}}$ which is an extension of distributive full non-associative Lambek calculus ($\mathsf{DFNL}$), and show that the cut elimination and subformula property hold for it.

Studies in Fuzziness and Soft Computing, 2003

Algebraic Logic is a general theory of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum-Tarski process. The notions of logical matrix and of Leibniz congruence are among its main building blocks. Its most successful part has been developed mainly by BLOK, PIGOZZI and CZELAKOWSKI, and obtains a deep theory and very nice and powerful results for the so-called protoalgebraic logics. I will show how the idea (already explored by W ÓJCKICI and NOWAK) of deÞning logics using a scheme of "preservation of degrees of truth" (as opposed to the more usual one of "preservation of truth") characterizes a wide class of logics which are not necessarily protoalgebraic and provide another fairly general framework where recent methods in Abstract Algebraic Logic (developed mainly by JANSANA and myself) can give some interesting results. After the general theory is explained, I apply it to an inÞnite family of logics deÞned in this way from subalgebras of the real unit interval taken as an MV-algebra. The general theory determines the algebraic counterpart of each of these logics without having to perform any computations for each particular case, and proves some interesting properties common to all of them. Moreover, in the Þnite case the logics so obtained are protoalgebraic, which implies they have a "strong version" deÞned from their Leibniz Þlters; again, the general theory helps in showing that it is the logic deÞned from the same subalgebra by the truth-preserving scheme, that is, the corresponding Þnite-valued logic in the most usual sense. However, for inÞnite subalgebras the obtained logic turns out to be the same for all such subalgebras and is not protoalgebraic, thus the ordinary methods do not apply. After introducing some (new) more general abstract notions for non-protoalgebraic logics I can Þnally show that this logic too has a strong version, and that it coincides with the ordinary inÞnite-valued logic of Łukasiewicz. * The term "(semi)lattice-based" has been used in , in a non-technical way, to describe a large class of logics

2011

Algebraic Logic and Computer Science Jacinta Poças, Manuel Martins, and Carlos Caleiro 1 Dep. Mathematics, IST TU Lisbon, Portugal 2 Dep. Mathematics, U Aveiro, Portugal 3 SQIG Instituto de Telecomunicações Abstract. Abstract algebraic logic (AAL) is a branch of logic that Abstract algebraic logic (AAL) is a branch of logic that uses universal algebra to study the properties of logical systems by associating them with representative classes of algebras, thus generalizing the Lindenbaum-Tarski process that leads to linking Boolean algebras with classical propositional logic. The theory of AAL then classifies logical systems and their metaproperties, systematically, along bridge theorems that relate them with properties of the associated algebras. Recently, concepts of behavioral algebraic specification have influenced the development of the behavioral approach to AAL. Namely, the notion of behavioral equivalence, imported from computer science, has been used to weaken the traditional...

2000

Connections between Algebraic Logic and (ordinary) Logic. Algebraic co- unterpart of model theoretic semantics, algebraic counterpart of proof theory, and their connections. The class Alg(L) of algebras associated to any logic L. Equivalence theorems stating that L has a certain logical property iff Alg(L) has a certain algebraic property. (E.g. L admits a strongly complete Hilbert- style inference system iff

Arxiv preprint cs/0501039, 2005

This paper is the second part of an introduction to linear logic and ludics, both due to Girard. It is devoted to proof nets, in the limited, yet central, framework of multiplicative linear logic (section 1) and to ludics, which has been recently developped in an aim of further ...