Elsevier

Fuzzy Sets and Systems

Volume 407, 1 March 2021, Pages 161-174
Fuzzy Sets and Systems

Completeness for monadic fuzzy logics via functional algebras

https://doi.org/10.1016/j.fss.2020.02.002Get rights and content

Abstract

We study S5-modal (monadic) expansions of extensions of Hájek's basic logic BL. Hájek proposed Hilbert-style systems axiomatizing these logics and we prove that completeness theorems for these logics follow from algebraic representation results, namely, functional representations of finitely subdirectly irreducible algebras. We prove a general theorem linking these concepts and give two major applications, namely, for the S5-modal expansions of Łukasiewicz and Gödel logics.

Section snippets

Introduction and preliminaries

We denote by BL the basic logic defined by Hájek in [18]. BL is an example of an implicative logic, so it is an algebraizable logic whose equivalent algebraic semantics AlgBL is the variety of BL-algebras that we usually denote by BL. As a result of the general theory of algebraizability, every axiomatic extension of BL is also algebraizable and its equivalent algebraic semantics is a subvariety of BL. More precisely, if C is an axiomatic extension of BL, the equivalent algebraic semantics Alg

Completeness via functional algebras

In this section we define C-functional algebras and prove that, given an axiomatic extension C of BL, the completeness theorem holds, that is, S5(C)=S5(C), if and only if the variety MBLC is generated as a quasivariety by its C-functional algebras.

To define C-functional algebras, we need to recall how to define monadic BL-algebras based on m-relatively complete subalgebras of a BL-algebra. In [10] we give a characterization of those subalgebras of a given BL-algebra that may be the range of

Monadic Łukasiewicz logic

One of the main axiomatic extensions of basic logic is Łukasiewicz logic L, which may be obtained by adding the axiom schema ¬¬pp to BL. Its corresponding equivalent algebraic semantics is the variety MV of MV-algebras, which is precisely the subvariety of BL determined by the identity ¬¬xx. We refer the reader to [11] for all basic properties of MV-algebras.

We recall here some useful notions of BL-algebras and MV-algebras that will be needed. Given a BL-algebra A, the congruences on A are in

Monadic Gödel logic

Another important axiomatic extension of BL is the one obtained by adding the axiom pp2. This is usually called Gödel logic and denoted by G. The corresponding equivalent algebraic semantics is the variety of Gödel algebras, that is, the subvariety of BL determined by the identity x2x (or, equivalently, by the identity xyxy). We denote this variety by G. In addition, Gödel algebras may also be defined as prelinear Heyting algebras, that is, G is the variety generated by totally ordered

Concluding remarks and future work

In this article we presented a way of proving the completeness of certain monadic logical calculi by means of algebraic representation theorems. We would like to pursue this route even deeper in order to prove the completeness for monadic basic logic. The key fact we used to prove the representation theorems in the cases addressed in this article was the amalgamation property. However, this is not enough in general. We already have some partial results that point in the right direction, but

Acknowledgements

We would like to thank the anonymous referees for their useful remarks that improved the clarity of this article. We would also like to acknowledge the funding received by Universidad Nacional del Sur (grant number 24/L108) and CONICET (grant number 11220170101010CO).

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