A Simple Paraconsistent Logic
One well-known system of paraconsistent logic is the simple system known as LP ("Logic of Paradox"), first proposed by the Argentinian logician F. G. Asenjo in 1966 and later popularized by Priest and others.
One way of presenting the semantics for LP is to replace the usual functional valuation with a relational one. The binary relation relates a formula to a truth value: means that is true, and means that is false. A formula must be assigned at least one truth value, but there is no requirement that it be assigned at most one truth value. The semantic clauses for negation and disjunction are given as follows:
(The other logical connectives are defined in terms of negation and disjunction as usual.) Or to put the same point less symbolically:
- not A is true if and only if A is false
- not A is false if and only if A is true
- A or B is true if and only if A is true or B is true
- A or B is false if and only if A is false and B is false
(Semantic) logical consequence is then defined as truth-preservation:
- if and only if is true whenever every element of is true.
Now consider a valuation such that and but it is not the case that . It is easy to check that this valuation constitutes a counterexample to both explosion and disjunctive syllogism. However, it is also a counterexample to modus ponens for the material conditional of LP. For this reason, proponents of LP usually advocate expanding the system to include a stronger conditional connective that is not definable in terms of negation and disjunction.
As one can verify, LP preserves most other inference patterns that one would expect to be valid, such as De Morgan's laws and the usual introduction and elimination rules for negation, conjunction, and disjunction. Surprisingly, the logical truths (or tautologies) of LP are precisely those of classical propositional logic. (LP and classical logic differ only in the inferences they deem valid.) Relaxing the requirement that every formula be either true or false yields the weaker paraconsistent logic commonly known as FDE ("First-Degree Entailment"). Unlike LP, FDE contains no logical truths.
It must be emphasized that LP is but one of many paraconsistent logics that have been proposed. It is presented here merely as an illustration of how a paraconsistent logic can work.
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