Hilbert-style axiomatization of first-degree entailment and a family of its extensions
The present paper investigates the well-known \textit{first-degree entailment logic} (FDE) in the \textsc{Fmla-Fmla}\ framework. The logic is given a Hilbert-style formulation (actually, a Fregean one -- cf. \S 1, \S2), dubbed FDE\(_{\text{H}}\). It is shown (which is not immediate) that FDE\(_{\text{H}}\) is sound and complete w.r.t. an obvious restriction to the \textsc{Fmla-Fmla} case of the standard definable consequence relation in Belnap-Dunn 4-valued semantics (BD-semantics). The author then notes that the deductive equivalence of two given proof-systems does not entail the equivalence of the set of extensions of each one of them. Next, he proceeds to prove this claim by defining the lattice of a family of extensions of FDE\(_{\text{H}}\) built up by adding to it one or more of a set of enough plausible rules absent, of course, in FDE\(_{\text{H}}\). Belnap-Dunn 4-valued type semantics are provided for the more important of these extensions. The results are compared with previous work by other authors on FDE when regarded in the \textsc{Set-Fmla} framework. In the concluding remarks to the paper it is noted that it will be worth-while to carry out the ensuing two tasks: (1) to provide similar BD-semantics for all the extensions of FDE\(_{\text{H}}\) defined in the paper (not only for the most prominent ones); (2) to provide an unified algebraic semantics for all the extensions defined, given the direct correspondence between the \textsc{Fmla-Fmla} logical framework and the one underlying ordered algebras.
scientific article; zbMATH DE number 7374871
scientific article