This is a contribution to the discussion on the role of truth degrees in manyvalued logics from the perspective of abstract algebraic logic. It starts with some thoughts on the so-called Suszko’s Thesis (that every logic is two-valued) and on the conception of semantics that underlies it, which includes the truth-preserving notion of consequence. The alternative usage of truth values in order to define logics that preserve degrees of truth is presented and discussed. Some recent works studying these in the (...) particular cases of Łukasiewicz’s many-valued logics and of logics associated with varieties of residuated lattices are also presented. Finally the extension of this paradigm to other, more general situations is discussed, highlighting the need for philosophical or applied motivations in the selection of the truth degrees, due both to the interpretation of the idea of truth degree and to some mathematical difficulties. (shrink)

Summary of the talk given to the 22nd Conference on the History of Logic, Cracow (Poland), July 5–9, 1976.

The monoidal t-norm based logic MTL is obtained from Hájek''s Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational completeness (...) (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL. Moreover, we investigate varieties of MTL algebras whose linearly ordered countable algebras embed into algebras whose lattice reduct is the real and/or the rational interval [0,1]. These embedding properties are used to investigate finite strong standard and/or rational completeness of the corresponding logics. (shrink)

Building on Wójcicki’s work on infinite-valued Łukasiewicz logic Ł ∞ , we give a self-contained proof of the deductive interpolation theorem for Ł ∞ . This paper aims at introducing the reader to the geometry of Łukasiewicz logic.

For decades Ryszard Wójcicki has been a highly influential scholar in the community of logicians and philosophers. Our aim is to outline and comment on some essential issues on logic, methodology of science and semantics as seen from the perspective of distinguished contributions of Wójcicki to these areas of philosophical investigations.

In the paper some consequence operations generated by ukasiewicz's matrices are examined.

The notion of local deduction theorem (which generalizes on the known instances of indeterminate deduction theorems, e.g. for the infinitely-valued ukasiewicz logic C ) is defined. It is then shown that a given finitary non-pathological logic C admits the local deduction theorem iff the class Matr(C) of all matrices validating C has the C-filter extension property (Theorem II.1).

In this paper the completeness theorems for the finite-valued ukasiewicz logics are proved with the use of the Lindenbaum algebra.

The paper is concerned with the problem of characterization of strengthenings of the so-called Łukasiewicz-like sentential calculi. The calculi under consideration are determined by n-valued Łukasiewicz matrices with superdesignated logical values. In general, Łukasiewicz-like sentential calculi are not implicative in the sense of [7]. Despite of this fact, in our considerations we use matrices analogous to S-algebras of Rasiowa. The main result of the paper says that the degree of maximality of any n-valued Łukasiewicz-like sentential calculus is finite and equal (...) to the degree of maximality of the corresponding n-valued Łukasiewicz calculus. (shrink)

First, we prove that the lattice of all structural strengthenings of a given strongly finite consequence operation is both atomic and coatomic, it has finitely many atoms and coatoms, each coatom is strongly finite but atoms are not of this kind — we settle this by constructing a suitable counterexample. Second, we deal with the notions of hereditary: algebraicness, strong finitisticity and finite approximability of a strongly finite consequence operation. Third, we formulate some conditions which tell us when the lattice (...) of all structural strengthenings of a given strongly finite consequence operation is finite, and subsequently we give some applications of them. (shrink)