Principle of explosion

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The principle of explosion (Latin: ex falso (sequitur) quodlibet (EFQ), "from falsehood, anything (follows)", or ex contradictione (sequitur) quodlibet (ECQ), "from 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 (including their negations) can be inferred from it. This is known as deductive explosion. The proof of this principle was first given by 12th century French philosopher William of Soissons.

As a demonstration of the principle, consider two contradictory 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 is the case, anything can be proven, e.g. "unicorns exist", by using the following argument:

1. We know that "All lemons are yellow" as it is defined to be true.
2. Therefore, the statement that ("All lemons are yellow" OR "unicorns exist”) must also be true, since the first part is true.
3. However, if "Not all lemons are yellow" (and this is also defined to be true), unicorns must exist – otherwise statement 2 would be false (in rigor, given that at least one lemon exists). It has thus been "proven" that unicorns exist. The same could be applied to any assertion, including the statement "unicorns do not exist".

Due to the principle of explosion, the existence of a contradiction (inconsistency) in a formal axiomatic system is disastrous; since any statement can be proved true it trivializes the concepts of truth and falsity. Around the turn of the 20th century, the discovery of contradictions such as Russell's paradox at the foundations of mathematics thus threatened the entire structure of mathematics. Mathematicians such as Gottlob Frege, Ernst Zermelo, Abraham Fraenkel, and Thoralf Skolem put much effort into revising set theory to eliminate these contradictions, resulting in the modern Zermelo–Fraenkel set theory.

In a different solution to these problems, a few mathematicians have devised alternate theories of logic called paraconsistent logics, which eliminate the principle of explosion. These allow some contradictory statements to be proved without affecting other proofs.

Symbolic representation

In symbolic logic, the principle of explosion can be expressed in the following way

$\forall P\forall Q:(P\land \lnot P)\vdash Q$

(For any statements P and Q, if P and not-P are both true, then Q is true)

Proof

Below is a formal proof of the principle using symbolic logic

1. $P\wedge \neg P$

   assumption

2. $P$

   from (1) by conjunction elimination

3. $\neg P$

   from (1) by conjunction elimination

4. $P\vee Q$

   from (2) by disjunction introduction

5. $Q$

   from (3) and (4) by disjunctive syllogism

6. $(P\wedge \neg P)\to Q$

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

This is just the symbolic version of the informal argument given in the introduction, with $P$ standing for "all lemons are yellow" and $Q$ standing for "Unicorns exist". 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 unicorns exist" (4); and from "not all lemons are yellow" (3) and "all lemons are yellow or unicorns exist" (4), we infer "unicorns exist" (5). Hence, if all lemons are yellow and not all lemons are yellow, then unicorns exist.

Semantic argument

An alternate argument for the principle stems from model theory. A sentence $P$ is a semantic consequence of a set of sentences $\Gamma$ only if every model of $\Gamma$ is a model of $P$. But there is no model of the contradictory set $(P\wedge \lnot P)$. A fortiori, there is no model of $(P\wedge \lnot P)$ that is not a model of $Q$. Thus, vacuously, every model of $(P\wedge \lnot P)$ is a model of $Q$. Thus $Q$ is a semantic consequence of $(P\wedge \lnot P)$.

Paraconsistent logic

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.

Use

The metamathematical value of the principle of explosion is that for any logical system where this principle holds, any derived theory which proves ⊥ (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.

See also

• Consequentia mirabilis – Clavius's Law
• Dialetheism – belief in the existence of true contradictions
• Law of excluded middle – every proposition is true or false
• Law of noncontradiction – no proposition can be both true and not true
• Paraconsistent logic – a family of logics 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