Paraconsistent Logic
Summary | In classical logic, every sentence is entailed by a contradiction: A and ¬A together entail B, for any sentences A and B whatsoever. This principle is often known as ex contradictione sequitur quodlibet (from a contradiction, everything follows), or the explosion principle. In paraconsistent logic, by contrast, this principle does not hold: arbitrary contradictions do not paraconsistently entail every sentence. Accordingly, paraconsistent logics are said to be contradiction tolerant. Semantics for paraconsistent logics can be given in a number of ways, but a common theme is that a sentence is allowed to be both true and false simultaneously. This can be achieved by introducing a third truth-value, thought of as both true and false; alternatively, it can be achieved (in the propositional case) be replacing the usual valuation function with a relation between sentences and the usual truth-values, true and false, so that a sentence may be related to either or both of these. Those who think there really are true contradictions are dialethists. Not all paraconsistent logicians are dialethists: some present paraconsistent logic as a better notion of what follows from what, or as a way to reason about inconsistent data. |
Key works | Asenjo 1966 and da Costa 1974 develop the Logic of Paradox (based on theor earlier work on paraconsistency in the 1950s). Priest et al 1989 is a classic early collection of papers. Priest 1987 is the classic philosophical defense of paraconsistent logic (and of dialethism). |
Introductions | da Costa & Bueno 2010 and Priest 2008 are good encyclopaedia entries on paraconsistent logic. The introduction to Priest 2005 is a clear statement of the case for paraconsistent logics; chapter 7 of Priest 2001 gives basic logical details of a few paraconsistent logics. |
- Adaptive Logic (4)
- Free Logic (35)
- Fuzzy Logic (416)
- Infinitary Logic (26)
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- Many-Valued Logic (398)
- Quantum Logic (556)
- Relevance Logic (639)
- Substructural Logic (250)
- Nonclassical Logic, Misc (37)
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