Abstract Valuation Semantics

Abstract

We define and study abstract valuation semantics for logics, an algebraically well-behaved version of valuation semantics. In the context of the behavioral approach to the algebraization of logics, we show, by means of meaningful bridge theorems and application examples, that abstract valuations are suited to play a role similar to the one played by logical matrices in the traditional approach to algebraization.

On meet-combination of logics

“…Second, so far we defined meet-combination only in the case of logics in the adopted universe (language of propositional nature, Hilbert calculus and matrix semantics). In order to widen the applicability of meet-combination, the work should be carried over to other kinds of semantics, such as non-deterministic matrices [1], possible-translations semantics [4], abstract valuations [3], and graph-theoretic interpretations [8], as well as to other kinds of deduction systems, such as sequent calculi. Furthermore, at some point the attempt should be made to leave the realm of propositional-based logics and address logics with variables and binding operators.…”

“…We use L and L(Ξ) for denoting the set of concrete formulas 3 and the set of all 2 That is, with no binding operators. 3 Formulas without schema variables.…”

On Combined Connectives

“…Even within the adopted setting (language of propositional nature, Hilbert calculus and matrix semantics) many other logics can be considered, namely modal and intuitionistic logics. However, in order to address other logics the work should be carried over to other kinds of semantics, such as non-deterministic matrices [1], possible-translations semantics [5], abstract valuations [4], and graphtheoretic interpretations [9], as well as to other kinds of deduction systems, such as sequent calculi. Furthermore, at some point the attempt should be made to come out of the realm of propositional-based logics and address logics with variables and binding operators.…”

Compositional Meaning in Logic

“…Overall, the impact and relevance of such a line of research is to be measured directly by its foundational character with respect to a better and deeper understanding of meaning in logics modeling complex phenomena and, of necessity, suitable general forms of compositional reasoning. Some highlights of the GeTFun project, covering topics such as truth-values, valuation semantics, bivalence, proof-systems and focusing, controlled nondeterminism, distance-based reasoning, information sources, paraconsistency, labeled and hybrid logic, inter alia, can be found in [2,10,12,20,19,21,24,25,29,32,45,44,55,56,58,74,50,7,3,27,26,59,11,47,57,43,49,60,54,1,66].…”