Principle of explosion


Principle of explosion

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" - 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 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 (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) by conjunction elimination
# eg phi,
#:from (1) by conjunction elimination
#phi vee psi,
#:from (2) by disjunction introduction
#psi,
#:from (3) and (4) by disjunctive syllogism
#(phi wedge eg phi) o psi
#:from (5) by conditional proof (discharging assumption 1)

Or:

#phi wedge eg phi,
#:hypothesis
#phi,
#:from (1) by conjunction elimination
# eg phi,
#:from (1) by conjunction elimination
# eg psi,
#:hypothesis
#phi,
#:reiteration of (2)
# eg psi o phi
#:from (4) to (5) by deduction theorem
#( eg phi o eg eg psi)
#:from (6) by contraposition
# eg eg psi
#:from (3) and (7) by modus ponens
#psi,
#:from (8) by double negation elimination
#(phi wedge eg phi) o psi
#:from (1) to (9) by deduction theorem

Or:

#phi wedge eg phi,
#:assumption
# eg psi,
#:assumption
#phi,
#:from (1) by conjunction elimination
# eg phi,
#:from (1) by conjunction elimination
# eg eg psi,
#:from (3) and (4) by reductio ad absurdum (discharging assumption 2)
#psi,
#:from (5) by double negation elimination
#(phi wedge eg phi) o psi
#:from (6) by conditional 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, and reductio ad absurdum). See the article on paraconsistent 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 from Everything2


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