Principle of explosion

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The principle of explosion is the law of classical logic and a few other systems (e.g., intuitionistic logic) according to which "anything follows from a contradiction." In symbolic terms, the principle of explosion can be expressed in the following way (where "Failed to parse (Missing texvc executable; please see math/README to configure.): \vdash " symbolizes the relation of logical consequence):

Failed to parse (Missing texvc executable; please see math/README to configure.): \{ \phi , \lnot \phi \} \vdash \psi.

This can be read as, "If one claims something is both true (Failed to parse (Missing texvc executable; please see math/README to configure.): \phi ) and not true (Failed to parse (Missing texvc executable; please see math/README to configure.): \lnot \phi ), one can logically derive any conclusion (Failed to parse (Missing texvc executable; please see math/README to configure.): \psi )."

The principle of explosion is also known as ex falso quodlibet, ex falso sequitur quodlibet (EFSQ for short), ex contradictione (sequitur) quodlibet (ECQ for short), and ex falso/contradictione (sequitur).

Arguments for explosion

There are two basic kinds of argument for the principle of explosion.

The semantic argument

The first argument is semantic or model-theoretic in nature. A sentence Failed to parse (Missing texvc executable; please see math/README to configure.): \psi

is a semantic consequence of a set of sentences Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma
only if every model of Failed to parse (Missing texvc executable; please see math/README to configure.): \Gamma
is a model of Failed to parse (Missing texvc executable; please see math/README to configure.): \psi

. But there is no model of the contradictory set Failed to parse (Missing texvc executable; please see math/README to configure.): \{\phi , \lnot \phi \} . A fortiori, there is no model of Failed to parse (Missing texvc executable; please see math/README to configure.): \{\phi , \lnot \phi \}

that is not a model of Failed to parse (Missing texvc executable; please see math/README to configure.): \psi

. Thus, vacuously, every model of Failed to parse (Missing texvc executable; please see math/README to configure.): \{\phi , \lnot \phi \}

is a model of Failed to parse (Missing texvc executable; please see math/README to configure.): \psi

. Thus Failed to parse (Missing texvc executable; please see math/README to configure.): \psi

is a semantic consequence of Failed to parse (Missing texvc executable; please see math/README to configure.): \{\phi , \lnot \phi \}

.

The proof-theoretic argument

The second type of argument is proof-theoretic in nature. Consider the following derivations:

1. Failed to parse (Missing texvc executable; please see math/README to configure.): \phi \wedge \neg \phi\,

1. assumption

2. Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\,

1. from (1) by conjunction elimination

2. Failed to parse (Missing texvc executable; please see math/README to configure.): \neg \phi\,

1. from (1) by conjunction elimination

2. Failed to parse (Missing texvc executable; please see math/README to configure.): \phi \vee \psi\,

1. from (2) by disjunction introduction

2. Failed to parse (Missing texvc executable; please see math/README to configure.): \psi\,

1. from (3) and (4) by disjunctive syllogism

2. Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \wedge \neg \phi) \to \psi

1. from (5) by conditional proof (discharging assumption 1)

Or:

1. Failed to parse (Missing texvc executable; please see math/README to configure.): \phi \wedge \neg \phi\,

1. assumption

2. Failed to parse (Missing texvc executable; please see math/README to configure.): \neg \psi\,

1. assumption

2. Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\,

1. from (1) by conjunction elimination

2. Failed to parse (Missing texvc executable; please see math/README to configure.): \neg \phi\,

1. from (1) by conjunction elimination

2. Failed to parse (Missing texvc executable; please see math/README to configure.): \neg \neg \psi\,

1. from (3) and (4) by reductio ad absurdum (discharging assumption 2)

2. Failed to parse (Missing texvc executable; please see math/README to configure.): \psi\,

1. from (5) by double negation elimination

2. Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \wedge \neg \phi) \to \psi

1. from (6) by conditional proof (discharging assumption 1)

Or:

1. Failed to parse (Missing texvc executable; please see math/README to configure.): \phi \wedge \neg \phi\,

1. hypothesis

2. Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\,

1. from (1) by conjunction elimination

2. Failed to parse (Missing texvc executable; please see math/README to configure.): \neg \phi\,

1. from (1) by conjunction elimination

2. Failed to parse (Missing texvc executable; please see math/README to configure.): \neg \psi\,

1. hypothesis

2. Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\,

1. reiteration of (2)

2. Failed to parse (Missing texvc executable; please see math/README to configure.): \neg \psi \to \phi

1. from (4) to (5) by deduction theorem

2. Failed to parse (Missing texvc executable; please see math/README to configure.): ( \neg \phi \to \neg \neg \psi)

1. from (6) by contraposition

2. Failed to parse (Missing texvc executable; please see math/README to configure.): \neg \neg \psi

1. from (3) and (6) by modus ponens

2. Failed to parse (Missing texvc executable; please see math/README to configure.): \psi\,

1. from (8) by double negation elimination

2. Failed to parse (Missing texvc executable; please see math/README to configure.): (\phi \wedge \neg \phi) \to \psi

1. from (1) to (9) by deduction theorem

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 Failed to parse (Missing texvc executable; please see math/README to configure.): \{\phi , \lnot \phi \}

and devise semantical systems in which there are such models.  As for the proof-theoretic arguments, they commonly reject disjunctive syllogism on the ground that it does not hold when applied to inconsistent situations. As well is common to deny the use of reductio ad absurdum in this way, on the grounds that even though a contradiction was derived while assuming a certain proposition, if that proposition was not used in the derivation, it is still not valid to derive its negation.

See also

• Dialetheism - belief in the existence of true contradictions
• Law of excluded middle - every proposition is either true or not true
• Law of noncontradiction - no proposition can be both true and not true
• Paraconsistent logic - the view that a contradiction does not allow absolutely every conclusion
• Paradox of entailment - a seeming paradox derived from the principle of explosion
• Reductio ad absurdum - concluding that a proposition is false because it produces a contradiction
• Trivialism - the belief that all statements of the form "P and not-P" are true

External links

• Ex Falso Quodlibet - explanation from Everything2de:Ex falso quodlibet

es:Ex falso quodlibet uk:Принцип вибухуit:ex falso quodlibet