Many-valued Propositional Logic Research Articles

Multi-adjoint lattice logic and truth-stressing hedges

This paper presents the multi-adjoint lattice logic (MLL) and its completeness and soundness. Specifically, the proposed many-valued propositional logic framework is defined on a multi-adjoint algebra whose underlying algebraic structure is a bounded lattice, which embeds the well-known basic logic (BL) given by Hájek on residuated lattices. The consideration of truth-stressing hedges in the multi-adjoint algebra is also studied and, as a consequence, two new logics extensions of MLL arise: the multi-adjoint lattice logic very true intensified (MLLvt) and the multi-adjoint lattice logic ∨-very true intensified (MLL∨−vt). Finally, the soundness and completeness of the aforementioned logics are also proven.

On the Rosser–Turquette method of constructing axiom systems for finitely many-valued propositional logics of Łukasiewicz

A method of constructing Hilbert-type axiom systems for standard many-valued propositional logics was offered by Rosser and Turquette. Although this method is considered to be a solution of the problem of axiomatisability of a wide class of many-valued logics, the article demonstrates that it fails to produce adequate axiom systems. The article concerns finitely many-valued propositional logics of Łukasiewicz. It proves that if standard propositional connectives of the Rosser–Turquette axiom systems are definable in terms of the propositional connectives of Łukasiewicz’s logics, and thus, they are normal ones, then every Rosser–Turquette axiom system for a finite-valued Łukasiewicz’s logic is semantically incomplete.

Membership function based rough set

In this paper the notion of a kind of clusters of subsets of a set based on rough membership function is introduced. The algebraic structure emerged thereby is studied. A comparison with classical rough sets with respect to the algebraic properties has been made. A many-valued propositional logic for such entities is proposed. Representation theorems in the style of Obtułowicz have been established.

Lewis Dichotomies in Many-Valued Logics

In 1979, H. Lewis shows that the computational complexity of the Boolean satisfiability problem dichotomizes, depending on the Boolean operations available to formulate instances: intractable (NP-complete) if negation of implication is definable, and tractable (in P) otherwise [21]. Recently, an investigation in the same spirit has been extended to nonclassical propositional logics, modal logics in particular [2, 3]. In this note, we pursue this line in the realm of many-valued propositional logics, and obtain complexity classifications for the parameterized satisfiability problem of two pertinent samples, Kleene and Godel logics.

The Logics’ Explorer: a Maple package for exploring finite many-valued propositional logics

The big computer algebra systems like Maple are no longer restricted to symbolic computations, but are becoming general purpose tools for engineers, mathematicians, and scientists instead. We have worked for a long time with many-valued logics and we believe that a flexible and comfortable tool that allowed to perform logical computations (for instance, to explore properties) in any existing or proposed many-valued logic with any notation for the truth values of the logic, could be very useful and time-saving. Therefore we have implemented such an extension of Maple that goes beyond Maple’s logic package and also beyond some previous implementations of these authors included with the Derive and Macsyma distributions some years ago. The logical connectives can be defined both using truth tables or in a functional way.

Finite RDP-algebras: duality, coproducts and logic

The variety of RDP-algebras forms the algebraic semantics of RDP-logic, the many-valued propositional logic of the revised drastic product left-continuous triangular norm and its residual. We prove a Priestley duality for finite RDP-algebras, and obtain an explicit description of coproducts of finite RDP-algebras. In this light, we give a combinatorial representation of free finitely generated RDP-algebras, which we exploit to construct normal forms, strongest deductive interpolants and most general unifiers.We prove that RDP-unification is unitary, and that the tautology problem for RDP-logic is coNP-complete.

Very true on CBA fuzzy logic

Abstract CBA logic was introduced as a non-associative generalization of the Łukasiewicz many-valued propositional logic. Its algebraic semantic is just the variety of commutative basic algebras. Petr Hájek introduced vt-operators as models for the “very true” connective on fuzzy logics. The aim of the paper is to show possibilities of using vt-operators on commutative basic algebras, especially we show that CBA logic endowed with very true connective is still fuzzy.

Quantitative logic

In this paper, we introduce the concept of the degree of the truth in the framework of so called many-valued propositional logic systems with the intention of measuring to what extent a given formula is true. Such a concept can indeed induce, in a very natural way, the degree of the similarity as well as to induce a pseudo-metric among formulas as the ‘graded’ version of the notion of the logical equivalence. The basic properties of such induced logic metric space hence are investigated in this paper. We further introduce different concepts of the degree of the divergence and the degree of the consistency in order to grade the extent of the consistency of a logic theory. To the end, all the basic logic notions are graded. Lastly, we propose in this paper three patterns of ‘approximate reasoning’ so far as the many-valued propositional logic systems are concerned. Hence, we can claim that we have laid the foundation for the “quantitative logic”. Since this theory has touched upon several key concepts of similarity, comparison, measurements, etc. in information theory, we are convinced that the theory may have profound impact in contemporary information technology. In addition, quantitative logic can be regarded as a highly representative example of mathematics of uncertainty capable of handling vagueness.

Basic fuzzy logic and BL -algebras II

The many-valued propositional logic BL (basic fuzzy logic) is investigated. It is known to be complete for tautologies over BL-algebras (particular residuated lattices). Each continuous t-norm on [0,1] determines a BL-algebra; such algebras are called t-algebras. Two additional axioms B1, B2 are found such that BL+(B1,B2) is complete for tautologies over t-algebras. It remains open whether B1, B2 are provable in BL.

Systems of ordinal fuzzy logic with application to preference modelling

In this paper, we survey several many-valued propositional logics in which the truth-functions (in the real unit interval [0,1]) of their connectives are definable only from the natural ordering of the scale, without using any richer algebraic structure. In particular we describe a complete notion of proof for weighted formulas of the implication-free fragment of Gödel logic with involution. The usefulness of this logical framework for the formalization of ordinal preference modelling is also shown.

Interpolation in fuzzy logic

We investigate interpolation properties of many-valued propositional logics related to continuous t-norms. In case of failure of interpolation, we characterize the minimal interpolating extensions of the languages. For finite-valued logics, we count the number of interpolating extensions by Fibonacci sequences.

Basic fuzzy logic and BL-algebras

The many-valued propositional logic BL (basic fuzzy logic) is investigated. It is known to be complete for tautologies over BL-algebras (particular residuated lattices). Each continuous t-norm on [0,1] determines a BL-algebra; such algebras are called t-algebras. Two additional axioms B1, B2 are found such that BL+(B1,B2) is complete for tautologies over t-algebras. It remains open whether B1, B2 are provable in BL.

A complete many-valued logic with product-conjunction

A simple complete axiomatic system is presented for the many-valued propositional logic based on the conjunction interpreted as product, the coresponding implication (Goguen's implication) and the corresponding negation (Godel's negation). Algebraic proof methods are used. The meaning for fuzzy logic (in the narrow sense) is shortly discussed.

The Complete Axiomatisation of any Many‐Valued Propositional Logic

Mathematical Logic QuarterlyVolume 22, Issue 1 p. 299-306 Article The Complete Axiomatisation of any Many-Valued Propositional Logic John Williamson, John Williamson Liverpool (Great Britain)Search for more papers by this author John Williamson, John Williamson Liverpool (Great Britain)Search for more papers by this author First published: 1976 https://doi.org/10.1002/malq.19760220140AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Volume22, Issue11976Pages 299-306 RelatedInformation

On Słupecki T-functions

In [2] E. L. Post defined a many-valued propositional logic to be functionally complete if, for every function on the set of truth-values, there exists a formula of the logic having that function as its associated truth-value function. He proved that the logic with truth-values 1, 2, …, m and (i) a unary connective ∼ such that ∼p has truth-value i+1 (mod m) when p has truth-value i, (ii) a binary connective ∨ such that p ∨ q has truth-value min(i, j) when p, q have truth-values i, j respectively, is functionally complete.The many-valued logics described by Łukasiewicz and Tarski [1] are not functionally complete. These logics have truth-values 1, 2, …, m and (i) a unary connective ~ such that ~p has truth-value m−i+1 when p has truth-value i, (ii) a binary connective → such that if p, q have truth-values i, j respectively, then p → q has truth-value 1 for i ≧ j, and truth-value 1 for i ≧ j. The functional incompleteness of these logics is immediate, since there exists no formula in p having truth-value i (≠ 1 or m) when p has truth-value 1.In [4] Słupecki showed that if a new unary connective T, such that T(p) has truth-value 2 for all truth-values assigned to p, is added to the 3-valued Łukasiewicz-Tarski logic, then the resulting logic is functionally complete. In [3] Rosser and Turquette proved this result for the m-valued (m ≧ 3) logic.