On finitely-valued inference systems (Q1580664)

In this paper the author looks for a proof-theoretical evidence of finite-valuedness within the class of cumulative inference systems. The notion of an inferentially many-valued inference system is defined and investigated. This definition tries to capture the idea of the proof-theoretical representation of truth-functional many-valued semantics. The notion of inferential many-valuedness is opposed to semantic (or referential) many-valuedness and a propositional inference system is defined to be many-valued if these two notions coincide. Cumulative propositional inference systems are defined as in \textit{D. Makinson}'s paper ``General theory of cumulative inference'' [Lect. Notes Comput. Sci. 346, 1-18 (1989; Zbl 0675.03007)] and a suitable definition of a set of formulas called verifiers allows to decide if those systems are inferentially finitely-valued. It is shown that every cumulative and inferentially finitely-valued inference system is structural and decidable. The model theory introduced by the author in ``Nonmonotonic theories and their axiomatic varieties'' [J. Logic Lang. Inf. 4, No. 4, 317-334 (1995; Zbl 0842.03020)] provides the semantic framework for a cumulative inference system. A cumulative inference system \({\mathcal P}\) is called inferentially \(k\)-valued if \(k\) is the cardinality of a smallest set of verifiers for \({\mathcal P}\). \({\mathcal P}\) is said to be referentially \(k\)-valued if there exists a model of cardinality \(k\). \({\mathcal P}\) is said to be \(k\)-valued if it is both inferentially and referentially \(k\)-valued. Classical logic, finite-valued logics of Post and the so-called Słupecki three-valued logic are considered.