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:

1. $phi\; wedge\; neg\; phi,$
2. :assumption
3. $phi,$
4. :from (1) by conjunction elimination
5. $neg\; phi,$
6. :from (1) by conjunction elimination
7. $phi\; vee\; psi,$
8. :from (2) by disjunction introduction
9. $psi,$
10. :from (3) and (4) by disjunctive syllogism
11. $(phi\; wedge\; neg\; phi)\; to\; psi$
12. :from (5) by conditional proof (discharging assumption 1)

Or:

1. $phi\; wedge\; neg\; phi,$
2. :hypothesis
3. $phi,$
4. :from (1) by conjunction elimination
5. $neg\; phi,$
6. :from (1) by conjunction elimination
7. $neg\; psi,$
8. :hypothesis
9. $phi,$
10. :reiteration of (2)
11. $neg\; psi\; to\; phi$
12. :from (4) to (5) by deduction theorem
13. $(neg\; phi\; to\; neg\; neg\; psi)$
14. :from (6) by contraposition
15. $neg\; neg\; psi$
16. :from (3) and (7) by modus ponens
17. $psi,$
18. :from (8) by double negation elimination
19. $(phi\; wedge\; neg\; phi)\; to\; psi$
20. :from (1) to (9) by deduction theorem

Or:

1. $phi\; wedge\; neg\; phi,$
2. :assumption
3. $neg\; psi,$
4. :assumption
5. $phi,$
6. :from (1) by conjunction elimination
7. $neg\; phi,$
8. :from (1) by conjunction elimination
9. $neg\; neg\; psi,$
10. :from (3) and (4) by reductio ad absurdum (discharging assumption 2)
11. $psi,$
12. :from (5) by double negation elimination
13. $(phi\; wedge\; neg\; phi)\; to\; psi$
14. :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.

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

External links

• Ex Falso Quodlibet - explanation from Everything2