# Principle of explosion

The

**principle of explosion**is the law ofclassical logic and a few other systems (e.g.,intuitionistic logic ) according to which "anything follows from a contradiction" - i.e., once you have asserted a contradiction, you can infer any proposition, or its converse. In symbolic terms, the principle of explosion can be expressed in the following way (where "$vdash$" symbolizes the relation oflogical 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 ($psi$)."

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

**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 $psi$ is a "

semantic consequence " of a set of sentences $Gamma$ only if every model of $Gamma$ is a model of $psi$. 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 $psi$. Thus, vacuously, every model of $\{phi\; ,\; lnot\; phi\; \}$ is a model of $psi$. Thus $psi$ is a semantic consequence of $\{phi\; ,\; lnot\; phi\; \}$.**The proof-theoretic argument**The second type of argument is "proof-theoretic" in nature. Consider the following derivations:

#$phi\; wedge\; eg\; phi,$

#:assumption

#$phi,$

#:from (1) byconjunction elimination

#$eg\; phi,$

#:from (1) byconjunction elimination

#$phi\; vee\; psi,$

#:from (2) bydisjunction introduction

#$psi,$

#:from (3) and (4) bydisjunctive syllogism

#$(phi\; wedge\; eg\; phi)\; o\; psi$

#:from (5) byconditional proof (discharging assumption 1)Or:

#$phi\; wedge\; eg\; phi,$

#:hypothesis

#$phi,$

#:from (1) byconjunction elimination

#$eg\; phi,$

#:from (1) byconjunction elimination

#$eg\; psi,$

#:hypothesis

#$phi,$

#:reiteration of (2)

#$eg\; psi\; o\; phi$

#:from (4) to (5) bydeduction theorem

#$(\; eg\; phi\; o\; eg\; eg\; psi)$

#:from (6) bycontraposition

#$eg\; eg\; psi$

#:from (3) and (7) bymodus ponens

#$psi,$

#:from (8) bydouble negation elimination

#$(phi\; wedge\; eg\; phi)\; o\; psi$

#:from (1) to (9) bydeduction theorem Or:

#$phi\; wedge\; eg\; phi,$

#:assumption

#$eg\; psi,$

#:assumption

#$phi,$

#:from (1) byconjunction elimination

#$eg\; phi,$

#:from (1) byconjunction elimination

#$eg\; eg\; psi,$

#:from (3) and (4) byreductio ad absurdum (discharging assumption 2)

#$psi,$

#:from (5) bydouble negation elimination

#$(phi\; wedge\; eg\; phi)\; o\; psi$

#:from (6) byconditional proof (discharging assumption 1)**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. Alternatively, they reject the idea that propositions can be classified as true or false.

As for the proof-theoretic arguments, they reject some of the assumptions typically including the following:

disjunctive syllogism ,disjunction introduction , andreductio ad absurdum ). See the article onparaconsistent logic .**ee 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*** [

*http://everything2.com/index.pl?node=Ex%20Falso%20Quodlibet Ex Falso Quodlibet*] - explanation fromEverything2

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