Editing Paraconsistent logic (section)

== {{anchor|Example}}Logic of Paradox ==
One example of paraconsistent logic is the system known as LP ("'''Logic of Paradox'''"), first proposed by the [[Argentina|Argentinian]] logician [[Florencio González Asenjo]] in 1966 and later popularized by [[Graham Priest|Priest]] and others.<ref>Priest (2002), p. 306.</ref>

One way of presenting the semantics for LP is to replace the usual [[function (mathematics)|functional]] valuation with a [[relation (mathematics)|relational]] one.<ref>LP is also commonly presented as a [[many-valued logic]] with three truth values (''true'', ''false'', and ''both'').</ref>  The binary relation <math>V\,</math> relates a [[Well-formed formula|formula]] to a [[truth value]]: <math>V(A,1)\,</math> means that <math>A\,</math> is true, and <math>V(A,0)\,</math> means that <math>A\,</math> 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:
* <math>V( \neg A,1) \Leftrightarrow V(A,0)</math>
* <math>V( \neg A,0) \Leftrightarrow V(A,1)</math>
* <math>V(A \lor B,1) \Leftrightarrow V(A,1) \text{ or } V(B,1)</math>
* <math>V(A \lor B,0) \Leftrightarrow V(A,0) \text{ and } V(B,0)</math>
(The other [[logical connective]]s 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:
:  <math>\Gamma\vDash A</math> if and only if <math>A\,</math> is true whenever every element of <math>\Gamma\,</math> is true.
Now consider a valuation <math>V\,</math> such that <math>V(A,1)\,</math> and <math>V(A,0)\,</math> but it is not the case that <math>V(B,1)\,</math>. 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.<ref>See, for example, Priest (2002), §5.</ref>

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 [[Natural deduction|introduction and elimination rules]] for negation, [[Logical conjunction|conjunction]], and disjunction. Surprisingly, the [[logical truth]]s (or [[Tautology (logic)|tautologies]]) of LP are precisely those of classical propositional logic.<ref>See Priest (2002), p. 310.</ref>  (LP and classical logic differ only in the ''[[inference]]s'' they deem valid.)  Relaxing the requirement that every formula be either true or false yields the weaker paraconsistent logic commonly known as [[first-degree entailment]] (FDE). Unlike LP, FDE contains no logical truths.

LP is only one of ''many'' paraconsistent logics that have been proposed.<ref>Surveys of various approaches to paraconsistent logic can be found in Bremer (2005) and Priest (2002), and a large family of paraconsistent logics is developed in detail in Carnielli, Congilio and Marcos (2007).</ref> It is presented here merely as an illustration of how a paraconsistent logic can work.