Principle of explosion

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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"; that is, once a contradiction has been asserted, any proposition (or its converse) can be inferred from it. In symbolic terms, the principle of explosion can be expressed in the following way (where "" symbolizes the relation of logical consequence):

This can be read as, "If one claims something is both true () and not true (), 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").

Arguments for explosion

An informal statement of the argument for explosion is this. Consider two inconsistent statements, “Lemons are yellow” and "It is not the case that lemons are yellow", and suppose for the sake of argument that both are true. We can then prove anything, for instance that Santa Claus exists: Since the statement that "Lemons are yellow and it is not the case that they are yellow" is true, we can infer that lemons are yellow. And from this we can infer that the statement “Either lemons are yellow or Santa Claus exists” is true (one or the other has to be true for this statement to be true, and we just showed that it is true that lemons are yellow, so this expanded statement is true). And since either lemons are yellow or Santa Claus exists, and since it is not the case that lemons are yellow (this was our first premise), it must be true that Santa Claus exists.

In more formal terms, there are two basic kinds of argument for the principle of explosion, semantic and proof-theoretic.

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 . A fortiori, there is no model of that is not a model of ψ. Thus, vacuously, every model of is a model of ψ. Thus ψ is a semantic consequence of .

The proof-theoretic argument

The second type of argument is proof-theoretic in nature. Consider the following derivations:

1. 

   assumption

2. 

   from (1) by conjunction elimination

3. 

   from (1) by conjunction elimination

4. 

   from (2) by disjunction introduction

5. 

   from (3) and (4) by disjunctive syllogism

6. 

   from (5) by conditional proof (discharging assumption 1)

Or:

1. 

   hypothesis

2. 

   from (1) by conjunction elimination

3. 

   from (1) by conjunction elimination

4. 

   hypothesis

5. 

   reiteration of (2)

6. 

   from (4) to (5) by deduction theorem

7. 

   from (6) by contraposition

8. 

   from (3) and (7) by modus ponens

9. 

   from (8) by double negation elimination

10. 

    from (1) to (9) by deduction theorem

Or:

1. 

   assumption

2. 

   assumption

3. 

   from (1) by conjunction elimination

4. 

   from (1) by conjunction elimination

5. 

   from (3) and (4) by reductio ad absurdum (discharging assumption 2)

6. 

   from (5) by double negation elimination

7. 

   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 the three arguments above.

As for the semantic argument, paraconsistent logicians often deny the assumption that there can be no model of 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 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