Two-sided Sequent Calculi for FDE-like Four-valued Logics

Journal of Philosophical Logic 52 (2):495-518 (2023)
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Abstract

We present a method that generates two-sided sequent calculi for four-valued logics like "first degree entailment" (FDE). (We say that a logic is FDE-like if it has finitely many operators of finite arity, including negation, and if all of its operators are truth-functional over the four truth-values 'none', 'false', 'true', and 'both', where 'true' and 'both' are designated.) First, we show that for every n-ary operator * every truth table entry f*(x1,...,xn) = y can be characterized in terms of a pair of sequent rules. Secondly, we use these sequent rules to build sequent calculi and prove their completeness. With the help of two simplification procedures we then show that the 2⋅4^n sequent rules that characterize an n-ary operator can be systematically reduced to at most four sequent rules. Thirdly, we use our method to investigate the proof-theoretical consequences of including intuitive truth-functional implications in FDE-like logics.

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Author Profiles

Barteld Kooi
University of Groningen
Allard Tamminga
University of Greifswald

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References found in this work

An Introduction to Non-Classical Logic: From If to Is.Graham Priest - 2008 - Bulletin of Symbolic Logic 14 (4):544-545.
Partiality and its dual.J. Michael Dunn - 2000 - Studia Logica 66 (1):5-40.
Nothing but the Truth.Andreas Pietz & Umberto Rivieccio - 2013 - Journal of Philosophical Logic 42 (1):125-135.
The value of the four values.Ofer Arieli & Arnon Avron - 1998 - Artificial Intelligence 102 (1):97-141.

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