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arXiv · 2312.16035

On three-valued presentations of classical logic

Bruno da Ré·Damian Szmuc·Emmanuel Chemla·Paul Égré
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Abstract

Given a three-valued definition of validity, which choice of three-valued truth tables for the connectives can ensure that the resulting logic coincides exactly with classical logic? We give an answer to this question for the five monotonic consequence relations $st$, $ss$, $tt$, $ss\cap tt$, and $ts$, when the connectives are negation, conjunction, and disjunction. For $ts$ and $ss\cap tt$ the answer is trivial (no scheme works), and for $ss$ and $tt$ it is straightforward (they are the collapsible schemes, in which the middle value acts like one of the classical values). For $st$, the schemes in question are the Boolean normal schemes that are either monotonic or collapsible.

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Bruno da Ré, Damian Szmuc, Emmanuel Chemla, Paul Égré. 2023-12-26. On three-valued presentations of classical logic. https://doi.org/10.1017/s1755020323000114

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