Multi-valued logic and Gröner bases with applications to modal logic

Among non-monotonic systems of reasoning, non-monotonic modal logics, and autoepistemic logic in particular, have had considerable success. The presence of explicit modal operators allows flexibility in the embedding of other approaches. Also several theoretical results of interest have been established concerning these logics. In this paper we introduce non-monotonic modal logics based on many-valued logics, rather than on classical logic. This extends earlier work of ours on many-valued modal logics. Intended applications are to situations involving several reasoners, not just one as in the standard development.

Tractable reasoning via approximation

Boolean contact algebras constitute a suitable algebraic theory for qualitative spatial reasoning. They are Boolean algebras with an additional contact relation grasping the topological aspect of spatial entities. In this paper we present a logic that combines propositional logic, relevance logic, and modal logic to reason about Boolean contact algebras. This is done in two steps. First, we use the relevance logic operators to obtain a logic suitable for Boolean algebras. Then we add modal operators that are based on the contact relation. In both cases we present axioms that are equivalent to requirement that every frame for the logic is indeed a Boolean algebra resp. Boolean contact algebra. We also provide a natural deduction system for this logic by defining introduction and eliminations rules for each logical operator. The system is shown to be sound. Furthermore, we sketch an implementation of the natural deduction system in the functional programming language and interactive theorem prover Coq.

Sören Halldén. On the logic of ‘better.’ Library of Theoria, no. 2. C. W. K. Gleerup, Lund, and Ejnar Munksgaard, Copenhagen, 1957, 112 pp. - Lennart Åqvist. Deontic logic based on a logic of ‘better.’Proceedings of a Colloquium on Modal and Many-valued Logics, Helsinki, 23–26 August, 1962, Acta philosophica Fennica, no. 16, Helsinki1963, pp. 285–290. - Mark Fisher. A three-valued calculus for deontic logic. Theoria (Lund), vol. 27 (1961), pp. 107–118. - Lennart Åqvist. Postulate sets and decision procedures for some systems of deontic logic. Theoria (Lund), vol. 29 (1963), pp. 154–175. - Volume 31 Issue 2

(1978). Orthomodular Lattices and Quantum Physics. Mathematics Magazine: Vol. 51, No. 5, pp. 299-303.

Chapter 20 - Automated Deduction for Many-Valued Logics

To enhance the expressive power and the declarative ability of a deductive database, various CWA (Closed World Assumption) formalizations including the naive CWA, the generalized CWA and the careful CWA are extended to multi-valued logics. The basic idea is to embed logic formulas into some polynomial ring. The extensions can be applied in a uniform manner to any finitely multi-valued logics. Therefore they are also of computational significance.

1 Modal logic: a semantic perspective

We review, and in the process unify two techniques (due to Németi and Pigozzi), for proving results concerning amalgamation in several classes studied in algebraic logic. The logical counterpart of these results adress interpolation and definability properties in modal and algebraic logic. Presenting them in a functorial context as adjoint situations, we show that both techniques can indeed be seen as instances of the use of the Keisler-Shelah ultrapower Theorem in proving Robinson's Joint Consistency Theorem. Some new results are surveyed. The results of this paper are presented in such a way that further establishes the hitherto existing well developed duality theory between boolean algebras with operators and modal logic.

EQUATIONAL LOGIC

Typically, simultaneous Boolean equations are expressed by a set of relations Fi = fi, i = 1,2 ?, where the Fi and fi are switching functions of N binary variables. In many logical design problems, a special type of Boolean equation is often encountered; namely, F(A0, A1 ?, An?1, X0, X1 ?, Xm?1) = f(a1, a2 ?, ak, X0, X1 ?, Xm?1), in which the arguments Ai's and Xi's are binary variables and the a1's are implicit functions of the Ai's only. This paper presents a somewhat new approach to solving such special types of Boolean functional equations. The solution function (see reference 1) can be formulated and mapped on a Veitch chart, especially tailored to the present problem, and all possible sets of solutions of the implicit functions ai's which satisfy the original functional equation are obtained simultaneously. Other possible applications and extensions of this method are discussed also.

We are going to study, from the logical point of view, expressions as “usually ϕ”, “ϕ is probable”, “for many x, ϕ” and similar. We shall handle them both as generalized quantifiers and as modalities. This should not look unnatural since in general modalities can be viewed as hidden quantifiers. We shall try to show how the theory of generalized quantifiers and modal logic can be applied to the above expressions (stressed by Zadeh as items of the specific agenda of fuzzy logic in the narrow sense) and that they admit “classical” logical analysis. We shall offer a logical apparatus, precise definitions, some results and various problems: here much study and development is still necessary. To be able to use and generalize the results and approaches of Boolean logic, we shall have to review and summarize several notions and facts on generalized quantifiers, some modal logics, and also on measures of uncertainty (that will be used to define various quantifiers and modalities). Thus we shall have two preliminary sections in this chapter: Section 1 on generalized quantifiers (with an appendix on uncertainty measures) and Section 2 on modal logics, both sections in the frame of Boolean (two-valued) logic. Generalized quantifiers in fuzzy logic and fuzzy modal logics are studied in Sections 3, 4.

conceptual connectedness, which they cannot discern in the usual farraginous reading lists of anthologies and journal articles. To give some indication of the scope of the book, the chapter headings are: I. 'Philosophy of logics'; 2. Validity; 3. Sentence connectives; 4. Quantifiers; 5. Singular terms; 6. Sentences, statements, propositions; 7. Theories of truth; 8. Paradoxes; 9. Logic and logics; io. Modal logic; ii. Many-valued logic; 12. Some metaphysical and epistemological questions about logic. Anyone familiar with the literature will realise how much controversy has to be condensed under each heading. The impressive bibliography alone, if blended in the way one can expect from Haack's fair and even hand, would result in only the most anodyne discussion. Haack can serve as background reading at the very best. One must, as she herself does, point the student firmly in the direction of the classical sources. Otherwise Haack's tentativeness, and way of abruptly ending a discussion just when it starts to bear the fruits of deeper disagreement, might be taken as a model of philosophical style; and the student, in learning the price of every idea, will appreciate the value of none. At several points Haack's survey style renders her discussion too superficial:

In this article, an “algebra” (sometimes also called a “universal algebra”) is a nonempty set X together with a set of “basic operations”; each operation is a finitary total function on X , i.e., a function ƒ : X n → X for some n ∈ {1, 2, …}. While our operations always have finite arity, we allow infinitely many basic operations in our algebras. Clone theory investigates the structure of all algebras on a given set. It turns out that several questions originating in algebra can be solved (or at least attacked) with set-theoretic methods. This article collects a few examples of this approach, based on recent results of Pinsker, Shelah and the author. Background . A large part of universal algebra investigates the relations between the basic operations of an algebra; here we take a different point of view—all term operations are considered to live an an equal level; thus we will, for example, not distinguish between a Boolean algebra and the corresponding Boolean ring, because each ring operation can be written as composition of operations of the Boolean algebra, and conversely. This point of view is motivated by the observation that many properties of algebras depend only on the clone of its term functions or polynomials, without any need to distinguish basic operations; for example, an equivalence relation is a congruence relation iff it is closed under all unary polynomials.

In this chapter we will give an outline of applications of logic functions and equations that are not directly related to the design of hardware. The applications in propositional logics are surely the oldest and reach back to the developments of science in ancient Greece. The names ”Boolean Algebra” and ”Boolean Ring” have been selected in honour of George Boole who wanted to formalize the ideas expressed by texts in natural language and to calculate the truth of complex constructions (in order to be precise and more accurate) . Naturally we already had to use logics in this sense throughout the book, however, we will go back to an introductory or elementary point of view and show that the knowledge that has been prepared up to now is fully appropriate to solve these problems. There was an enormous development of logics (in a broad mathematical sense) during the last 150 years that resulted in a highly sophisticated and specialized level, however, the propositional calculus is still the foundation of all these developments, and it is still necessary and useful to have a good understanding of these concepts.