Paraconsistent Logic (Substantive Revision)
Paraconsistent logic is defined by a non-explosive consequence relation, meaning that contradictions do not entail every possible statement, distinguishing it from classical logic. While paraconsistency allows for reasoning in the presence of contradictions, it does not necessarily imply dialetheism, the view that true contradictions exist. There is significant diversity among paraconsistent logics, and they vary in how they handle principles like the Law of Non-Contradiction and historical interpretations of logical consequence.
- ▪Paraconsistency is a property of a logical consequence relation that avoids explosion, meaning that from a contradiction (A and ¬A), not every statement (B) follows.
- ▪Dialetheism, the belief that some contradictions are true, is distinct from paraconsistency, which is solely about the structure of logical consequence.
- ▪Historically, the principle ex contradictione quodlibet (ECQ) was not widely accepted in antiquity or the medieval period, only becoming standard with the development of classical logic in the late 19th century.
- ▪Some paraconsistent logics still validate the Law of Non-Contradiction, showing that rejecting ECQ does not require rejecting classical principles outright.
- ▪Scholars such as Barrio and Da Ré, Carnielli and Rodrigues, and Asmus have contributed to clarifying the relationship between paraconsistency and dialetheism.
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1. Paraconsistency A logic is paraconsistent iff its logical consequence relation \((\vDash\), either semantic or proof theoretic) is not explosive. Paraconsistency is a property of a consequence relation. The argument ex contradictione quodlibet (ECQ) is paraconsistently invalid: in general, it is not the case that \(A\), \(\neg A \vDash B\). The role often played by the notion of consistency in logics, namely, the most basic requirement that any theory must meet, is relaxed to the notion of coherence: no theory can include every sentence whatsoever if it is to be considered tenable. Simple consistency of a theory (no contradictions) is a special case of absolute consistency, or non-triviality (not every sentence is a part of the theory).
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