An Abstract Approach to Consequence Relations

We generalise the Blok-J\'onsson account of structural consequence relations, later developed by Galatos, Tsinakis and other authors, in such a way as to naturally accommodate multiset consequence. While Blok and J\'onsson admit, in place of sheer formulas, a wider range of syntactic units to be manipulated in deductions (including sequents or equations), these objects are invariably aggregated via set-theoretical union. Our approach is more general in that non-idempotent forms of premiss and conclusion aggregation, including multiset sum and fuzzy set union, are considered. In their abstract form, thus, deductive relations are defined as additional compatible preorderings over certain partially ordered monoids. We investigate these relations using categorical methods, and provide analogues of the main results obtained in the general theory of consequence relations. Then we focus on the driving example of multiset deductive relations, providing variations of the methods of matrix semantics and Hilbert systems in Abstract Algebraic Logic

Algebraic semantics for one-variable lattice-valued logics

The one-variable fragment of any first-order logic may be considered as a modal logic, where the universal and existential quantifiers are replaced by a box and diamond modality, respectively. In several cases, axiomatizations of algebraic semantics for these logics have been obtained: most notably, for the modal counterparts S5 and MIPC of the one-variable fragments of first-order classical logic and intuitionistic logic, respectively. Outside the setting of first-order intermediate logics, however, a general approach is lacking. This paper provides the basis for such an approach in the setting of first-order lattice-valued logics, where formulas are interpreted in algebraic structures with a lattice reduct. In particular, axiomatizations are obtained for modal counterparts of one-variable fragments of a broad family of these logics by generalizing a functional representation theorem of Bezhanishvili and Harding for monadic Heyting algebras. An alternative proof-theoretic proof is also provided for one-variable fragments of first-order substructural logics that have a cut-free sequent calculus and admit a certain bounded interpolation property

One-variable fragments of first-order logics

The one-variable fragment of a first-order logic may be viewed as an "S5-like" modal logic, where the universal and existential quantifiers are replaced by box and diamond modalities, respectively. Axiomatizations of these modal logics have been obtained for special cases -- notably, the modal counterparts S5 and MIPC of the one-variable fragments of first-order classical logic and intuitionistic logic -- but a general approach, extending beyond first-order intermediate logics, has been lacking. To this end, a sufficient criterion is given in this paper for the one-variable fragment of a semantically-defined first-order logic -- spanning families of intermediate, substructural, many-valued, and modal logics -- to admit a natural axiomatization. More precisely, such an axiomatization is obtained for the one-variable fragment of any first-order logic based on a variety of algebraic structures with a lattice reduct that has the superamalgamation property, building on a generalized version of a functional representation theorem for monadic Heyting algebras due to Bezhanishvili and Harding. An alternative proof-theoretic strategy for obtaining such axiomatization results is also developed for first-order substructural logics that have a cut-free sequent calculus and admit a certain interpolation property.Comment: arXiv admin note: text overlap with arXiv:2209.0856

Slabě implikační predikátové fuzzy logiky

There are two classes of propositional logics related to the area of mathematical fuzzy logics proposed in work of the author (see also joint paper by the author and Libor Běhounek where philosophical, methodological, and pragmatical reasons for introducing these two classes appear.) After we recall same basic definitions we turn our attention to the first-order variants of these two classes of logics. The results presented here are mainly from the author's thesis and his upcoming paper. Because of the lack of space we present the basic definitions and theorems only and we completely disregard the important concept of Baaz delta

Fuzzy teorie tříd: některá pokročilá témata

The goal of this paper is to push forward the development of the apparatus of the Fuzzy Class theory. We concentrate on three areas: strengthening the universal quantifier, formalizing the idea that `similar' fuzzy sets fulfill their properties to `similar' degrees, and embedding of classical crisp theories into Fuzzy Class theory