Principle of explosion

From Wikipedia, the free encyclopedia

All or part of this article may be confusing or unclear.
Please help clarify the article. Suggestions may be on the talk page. (December 2007)

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 " $\vdash$ " symbolizes the relation of logical consequence):

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

This can be read as, "If one claims something is both true ( $\phi\,$ ) and not true ( $\lnot \phi$ ), one can logically derive any conclusion ( $ψ$ )."

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) (Latin: "from falsehood/contradiction (follows) anything", literally "... what pleases").

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

[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 $Γ$ only if 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 (2)
$\neg \psi \to \phi$

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

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

from (3) and (7) 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 validity of reductio ad absurdum in such cases, for similar reasons and also 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.

[edit] 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