Correspondence Analysis for Some Fragments of Classical Propositional Logic

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 ⋅ 4n 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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“…Key references include Kooi and Tamminga[14], Tamminga[24], Petrukhin[18], Leszczyńska-Jasion et al[15] and Petrukhin and Shangin[19].…”

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

Kooi

Tamminga

2022

J Philos Logic

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Non-transitive Correspondence Analysis

Petrukhin

Shangin

2022

J of Log Lang and Inf

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The paper’s novelty is in combining two comparatively new fields of research: non-transitive logic and the proof method of correspondence analysis. To be more detailed, in this paper the latter is adapted to Weir’s non-transitive trivalent logic $${\mathbf{NC}}_{\mathbf{3}}$$ NC 3 . As a result, for each binary extension of $${\mathbf{NC}}_{\mathbf{3}}$$ NC 3 , we present a sound and complete Lemmon-style natural deduction system. Last, but not least, we stress the fact that Avron and his co-authors’ general method of obtaining n-sequent proof systems for any n-valent logic with deterministic or non-deterministic matrices is not applicable to $${\mathbf{NC}}_{\mathbf{3}}$$ NC 3 and its binary extensions.

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Correspondence Analysis for Some Fragments of Classical Propositional Logic

Petrukhin

Shangin

2021

Preprint

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In the paper, we apply Kooi and Tamminga’s correspondence analysis (that has been previously applied to some notable three- and four-valued logics) to some conventional and functionally incomplete fragments of classical propositional logic. In particular, the paper deals with the implication, disjunction, and negation fragments. Additionally, we consider an application of correspondence analysis to some connectiveless fragment with certain basic properties of the logical consequence relation only. As a result of the application, one obtains a sound and complete natural deduction system for any binary extension of each fragment in question. With the focus on exclusive disjunction we comparatively study the proposed systems. Finally, we discuss Segerberg’s systems for connectiveless and negation fragments and compare them with our systems.

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Correspondence Analysis for Some Fragments of Classical Propositional Logic

Cited by 3 publications

References 18 publications

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

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

Non-transitive Correspondence Analysis

Correspondence Analysis for Some Fragments of Classical Propositional Logic

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