# 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):

: $\left\{ phi , lnot phi \right\} vdash psi.$

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 $\left\{phi , lnot phi \right\}$. A fortiori, there is no model of $\left\{phi , lnot phi \right\}$ that is not a model of $psi$. Thus, vacuously, every model of $\left\{phi , lnot phi \right\}$ is a model of $psi$. Thus $psi$ is a semantic consequence of $\left\{phi , lnot phi \right\}$.

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
#$\left(phi wedge eg phi\right) o psi$
#: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
#$\left( eg phi o eg eg psi\right)$
#:from (6) by contraposition
#$eg eg psi$
#:from (3) and (7) by modus ponens
#$psi,$
#:from (8) by double negation elimination
#$\left(phi wedge eg phi\right) o psi$
#: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
#$\left(phi wedge eg phi\right) o psi$
#: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 $\left\{phi , lnot phi \right\}$ 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

* [http://everything2.com/index.pl?node=Ex%20Falso%20Quodlibet Ex Falso Quodlibet] - explanation from Everything2

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Principle of contradiction — In logic, the Principle of contradiction ( principium contradictionis in Latin) is the second of the so called three classic laws of thought. The oldest statement of the law is that contradictory statements cannot both at the same time be true, e …   Wikipedia

• Explosion welding — (EXW) is a solid state process where welding is accomplished by accelerating one of the components at extremely high velocity through the use of chemical explosives. This process is most commonly utilized to clad carbon steel plate with a thin… …   Wikipedia

• Steam explosion — A steam explosion (also called a littoral explosion , or fuel coolant interaction , FCI ) is a violent boiling or flashing of water into steam, occurring when water is either superheated, rapidly heated by fine hot debris produced within it, or… …   Wikipedia

• Dunmurry train explosion — Coordinates: 54°33′07″N 6°00′00″W﻿ / ﻿54.552°N 6.000°W﻿ / 54.552; 6.000 …   Wikipedia

• Paraconsistent logic — A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent (or… …   Wikipedia

• Logic — For other uses, see Logic (disambiguation). Philosophy …   Wikipedia

• Gödel's incompleteness theorems — In mathematical logic, Gödel s incompleteness theorems, proved by Kurt Gödel in 1931, are two theorems stating inherent limitations of all but the most trivial formal systems for arithmetic of mathematical interest. The theorems are of… …   Wikipedia

• Law of noncontradiction — This article uses forms of logical notation. For a concise description of the symbols used in this notation, see List of logic symbols. In classical logic, the law of non contradiction (LNC) (or the principle of non contradiction (PNC), or the… …   Wikipedia

• Dialetheism — is the view that some statements can be both true and false simultaneously. More precisely, it is the belief that there can be a true statement whose negation is also true. Such statements are called true contradictions , or dialetheia.… …   Wikipedia

• Classical logic — identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well.[1][2] They are characterised by a number of properties:[3] Law of the excluded middle and… …   Wikipedia