Principle of explosion
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The principle of explosion (also known as ex falso sequitur quodlibet, ex contradictione sequitur quodlibet, and ex falso sequitur aliquot) is the law of classical logic and a few other systems, for example, intuitionistic logic, according to which "anything follows from a contradiction". In symbolic terms, the principle of explosion can be expressed in the following way:
- <math>\{ \phi , \lnot \phi \} \vdash \psi.</math>
Here, "<math>\vdash</math>" symbolizes the relation of logical consequence.
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Arguments for explosion
There are two basic arguments for the principle of explosion.
The semantic argument
The first argument is semantic or model-theoretic in nature. A sentence <math>\psi</math> is a semantic consequence of a set of sentences <math>\Gamma</math> just in case every model of <math>\Gamma</math> is a model of <math>\psi</math>. But there is no model of the contradictory set <math>\{\phi , \lnot \phi \}</math>. A fortiori, there is no model of <math>\{\phi , \lnot \phi \}</math> that is not a model of <math>\psi</math>. Thus, vacuously, every model of <math>\{\phi , \lnot \phi \}</math> is a model of <math>\psi</math>. Thus <math>\psi</math> is a semantic consequence of <math>\{\phi , \lnot \phi \}</math>.
The proof-theoretic argument
The second argument is proof-theoretic in nature. Consider the following derivation:
| (1) <math>\phi\,</math> | by assumption |
| (2) <math>\lnot \phi\,</math> | by assumption |
| (3) <math>\phi \lor \psi\,</math> | from (1) by disjunction introduction |
| (4) <math>\psi\,</math> | from (2) and (3) by disjunctive syllogism |
Rejecting the principle
Proponents of paraconsistent logic reject the principle of explosion, and thus must find flaw with both of the arguments above. As for the semantic argument, paraconsistent logicians often deny the assumption that there can be no model of <math>\{\phi , \lnot \phi \}</math> and devise semantical systems in which there are such models. As for the proof-theoretic argument, they commonly reject disjunctive syllogism on the ground that it does not hold when applied to inconsistent situations.