Partiality and its dual

Studia Logica 66 (1):5-40 (2000)
This paper explores allowing truth value assignments to be undetermined or "partial" (no truth values) and overdetermined or "inconsistent" (both truth values), thus returning to an investigation of the four-valued semantics that I initiated in the sixties. I examine some natural consequence relations and show how they are related to existing logics, including ukasiewicz's three-valued logic, Kleene's three-valued logic, Anderson and Belnap's (first-degree) relevant entailments, Priest's "Logic of Paradox", and the first-degree fragment of the Dunn-McCall system "R-mingle". None of these systems have nested implications, and I investigate twelve natural extensions containing nested implications, all of which can be viewed as coming from natural variations on Kripke's semantics for intuitionistic logic. Many of these logics exist antecedently in the literature, in particular Nelson's "constructible falsity".
Keywords Philosophy   Logic   Mathematical Logic and Foundations   Computational Linguistics
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DOI 10.1023/A:1026740726955
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Heinrich Wansing (2002). Diamonds Are a Philosopher's Best Friends. Journal of Philosophical Logic 31 (6):591-612.
Heinrich Wansing (2008). Constructive Negation, Implication, and Co-Implication. Journal of Applied Non-Classical Logics 18 (2-3):341-364.

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