Sequent calculi for three-valued logics (Q1277163)

This paper presents complete sequent calculi for different 3-valued propositional logics, in a uniform way. Each calculus is built around the structural rules: thinning, contraction, interchange and cut, using so-called beginning sequents and logical rules of inference which are specific for that logic. Complete calculi are presented for the weakly intuitionistic logic of \textit{A. M. Sette} and \textit{W. A. Carnielli} [``Maximal weakly-intuitionistic logics'', Stud. Log. 55, No. 1, 181-203 (1995; Zbl 0841.03009)], the 3-valued paraconsistent logic of \textit{A. M. Sette} [``On the propositional calculus P\(^1\)'', Math. Japonicae 18, 173-180 (1973; Zbl 0289.02013)], \textit{A. Wronski}'s logic [``A three element matrix whose consequence operation is not finitely based'', Bull. Sect. Logic, Pol. Acad. Sci. 8, 68-71 (1979; Zbl 0419.03017)] and \textit{K. Palasińska'}s logics [``Three-element nonfinitely axiomatizable matrices'', Stud. Log. 53, No. 3, 361-372 (1994; Zbl 0808.03004)]. For the first two, also a cut-free version is considered. The completeness of those calculi refers to the specific validity notion of a sequent, based on the choice of the set of designated values, either \(\{ t \}\) or \(\{t,u\}\) depending on the logic. The paper concludes with a complete calculus (including a cut-free system) for the 3-valued conditional logic of \textit{F. Guzman} [``Gentzen system for conditional logic'', Stud. Log. 53, No. 2, 243-257 (1994; Zbl 0807.03011)], where the validity notion is based on the linear ordering \( f < u < t \) of the truth-values.