Principle of explosion

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The principle of explosion (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) 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:

$\{ \phi , \lnot \phi \} \vdash \psi.$

Here, " $\vdash$ " symbolizes the relation of logical consequence.

1 Arguments for explosion
- 1.1 The semantic argument
- 1.2 The proof-theoretic argument
2 Rejecting the principle
3 See also
4 External link

[edit] Arguments for explosion

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

[edit] The semantic argument

The first argument is semantic or model-theoretic in nature. A sentence $ψ$ is a semantic consequence of a set of sentences $Γ$ just in case every model of $Γ$ is a model of $ψ$ . But there is no model of the contradictory set $\{\phi , \lnot \phi \}$ . A fortiori, there is no model of $\{\phi , \lnot \phi \}$ that is not a model of $ψ$ . Thus, vacuously, every model of $\{\phi , \lnot \phi \}$ is a model of $ψ$ . Thus $ψ$ is a semantic consequence of $\{\phi , \lnot \phi \}$ .

[edit] The proof-theoretic argument

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

$\phi \wedge \neg \phi\,$

assumption
$\phi\,$

from (1) by conjunction elimination
$\neg \phi\,$

from (1) by conjunction elimination
$\phi \vee \psi\,$

from (2) by disjunction introduction
$\psi\,$

from (3) and (4) by disjunctive syllogism
$(\phi \wedge \neg \phi) \to \psi$

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

Or:

$\phi \wedge \neg \phi\,$

assumption
$\neg \psi\,$

assumption
$\phi\,$

from (1) by conjunction elimination
$\neg \phi\,$

from (1) by conjunction elimination
$\neg \neg \psi\,$

from (3) and (4) by reductio ad absurdum (discharging assumption 2)
$\psi\,$

from (5) by double negation elimination
$(\phi \wedge \neg \phi) \to \psi$

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

Or:

$\phi \wedge \neg \phi\,$

hypothesis
$\phi\,$

from (1) by conjunction elimination
$\neg \phi\,$

from (1) by conjunction elimination
$\neg \psi\,$

hypothesis
$\phi\,$

reiteration of 1
$\neg \psi \to \phi$

from (4) to (5) by deduction theorem
$( \neg \phi \to \neg \neg \psi)$

from (6) by contraposition
$\neg \neg \psi$

from (3) and (6) by modus ponens
$\psi\,$

from (8) by double negation elimination
$(\phi \wedge \neg \phi) \to \psi$

from (1) to (9) by deduction theorem

[edit] 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 $\{\phi , \lnot \phi \}$ 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.