Principle of explosion

This article is about the ex falso quodlibet logical principle. For the file tagging software, see Ex Falso (software).

The principle of explosion, (Latin: ex falso quodlibet or ex contradictione sequitur quodlibet, "from a contradiction, anything follows") or the principle of Pseudo-Scotus, is the law of classical logic, intuitionistic logic and similar logical systems, according to which any statement can be proven from a contradiction. That is, once a contradiction has been asserted, any proposition (or its negation) can be inferred from it. In symbolic terms, the principle of explosion can be expressed in the following way (where "$\vdash$" symbolizes the relation of logical consequence):

$\{\phi ,\lnot \phi \}\vdash \psi$

or

$\bot \to P$.

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 metamathematical value of the principle of explosion is that for any logical system where this principle holds, any derived theory which proves $\bot$ (or an equivalent form, $\phi \land \lnot \phi$) is worthless because all its statements would become theorems, making it impossible to distinguish truth from falsehood. That is to say, the principle of explosion is an argument for the law of non-contradiction in classical logic, because without it all truth statements become meaningless.

Arguments for explosion

An informal arguement

Consider two inconsistent statements - “All lemons are yellow” and "Not all lemons are yellow" - and suppose for the sake of argument that both are simultaneously true. If that's the case we can prove anything, for instance that "Santa Claus exists", by using the following argument: 1) We know that "All lemons are yellow". 2) From this we can infer that (“All lemons are yellow" OR "Santa Claus exists”) is also true. 3) If "Not all lemons are yellow", however, this proves that "Santa Claus exists" (or the statement ("All lemons are yellow" OR "Santa Claus exists") would be false).

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

The semantic arguement

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 arguement

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

1. $\phi \wedge \neg \phi \,$

   assumption

2. $\phi \,$

   from (1) by conjunction elimination

3. $\neg \phi \,$

   from (1) by conjunction elimination

4. $\phi \vee \psi \,$

   from (2) by disjunction introduction

5. $\psi \,$

   from (3) and (4) by disjunctive syllogism

6. $(\phi \wedge \neg \phi )\to \psi$

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

This is just the symbolic version of the informal argument given above, with $\phi$ standing for "all lemons are yellow" and $\psi$ standing for "Santa Claus exists". From "all lemons are yellow and not all lemons are yellow" (1), we infer "all lemons are yellow" (2) and "not all lemons are yellow" (3); from "all lemons are yellow" (2), we infer "all lemons are yellow or Santa Claus exists" (4); and from "not all lemons are yellow" (3) and "all lemons are yellow or Santa Claus exists" (4), we infer "Santa Claus exists" (5). Hence, if all lemons are yellow and not all lemons are yellow, then Santa Claus exists.

Or:

1. $\phi \wedge \neg \phi \,$

   hypothesis

2. $\phi \,$

   from (1) by conjunction elimination

3. $\neg \phi \,$

   from (1) by conjunction elimination

4. $\neg \psi \,$

   hypothesis

5. $\phi \,$

   reiteration of (2)

6. $\neg \psi \to \phi$

   from (4) to (5) by deduction theorem

7. $(\neg \phi \to \neg \neg \psi )$

   from (6) by contraposition

8. $\neg \neg \psi$

   from (3) and (7) by modus ponens

9. $\psi \,$

   from (8) by double negation elimination

10. $(\phi \wedge \neg \phi )\to \psi$

    from (1) to (9) by deduction theorem

Or:

1. $\phi \wedge \neg \phi \,$

   assumption

2. $\neg \psi \,$

   assumption

3. $\phi \,$

   from (1) by conjunction elimination

4. $\neg \phi \,$

   from (1) by conjunction elimination

5. $\neg \neg \psi \,$

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

6. $\psi \,$

   from (5) by double negation elimination

7. $(\phi \wedge \neg \phi )\to \psi$

   from (6) by conditional proof (discharging assumption 1)

Addressing the principal

Paraconsistent logics have been developed that allow for sub-contrary forming operators. Model-theoretic 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. Proof-theoretic paraconsistent logics usually deny the validity of one of the steps necessary for deriving an explosion, typically including disjunctive syllogism, disjunction introduction, and reductio ad absurdum.

See also

• Consequentia mirabilis - Clavius's Law
• 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 – a modal logic used to address contradictions
• 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