Principle of explosion

In classical, intuitionistic, and related logical systems, the principle of explosion states that a contradiction allows any statement whatsoever to be proven. This means that if you have both a proposition and its negation, you can logically deduce any other proposition—including the opposite of what you just proved. This effect is called deductive explosion.

The first known proof of this principle came from the 12th-century French philosopher William of Soissons. Because of this principle, any inconsistency within a formal axiomatic system is catastrophic: if a single contradiction exists, every statement—whether true or false—becomes provable, which destroys any meaningful distinction between truth and falsehood. At the turn of the 20th century, when contradictions like Russell's paradox were discovered in the foundations of mathematics, this threatened the entire field. To fix this, mathematicians such as Gottlob Frege, Ernst Zermelo, Abraham Fraenkel, and Thoralf Skolem worked to revise set theory and remove these contradictions, leading to the modern Zermelo–Fraenkel set theory.

As an example, take two contradictory claims: "All lemons are yellow" and "Not all lemons are yellow." If we assume both are true, we can prove anything—say, that unicorns exist. The argument goes like this: Since "Not all lemons are yellow" is assumed true, and "All lemons are yellow" is also assumed true, the compound statement "All lemons are yellow or unicorns exist" must be true (because the first part is true, and an "or" statement is true if at least one part is). But we also know the first part is false (since "Not all lemons are yellow" is true), so the second part—"unicorns exist"—must be true to keep the compound statement true.

This step is called disjunctive syllogism. The same method can prove that unicorns do not exist, creating yet another contradiction, and can prove any other well-formed formula. Hence, an explosion of provable statements occurs.

Some mathematicians have responded by developing alternative logics called paraconsistent logics, which allow certain contradictions without making every other statement provable.

In symbolic logic, the principle of explosion is written as: ⊥ → P

Proof

A formal proof, known as the Lewis argument (published by C. I. Lewis, though medieval logicians knew similar versions), goes as follows: Assume (1) P and (2) ¬P. From (1), we infer (3) P ∨ Q (by disjunction introduction). From (3) and (2), we infer (4) Q (by disjunctive syllogism). Here, P stands for "all lemons are yellow" and Q for "unicorns exist."

Semantic argument

Another argument comes from model theory. A sentence ψ is a semantic consequence of a set of sentences Σ only if every model of Σ is also a model of ψ. But a contradictory set Σ has no models. So, vacuously, every model of Σ is a model of ψ, meaning ψ follows semantically from Σ.

Paraconsistent logic

Paraconsistent logics handle this differently. Model-theoretic versions often deny that a contradictory set has no models, creating semantic systems with such models. Others reject the idea that propositions are simply true or false. Proof-theoretic paraconsistent logics typically reject one of the steps needed for explosion, such as disjunctive syllogism, disjunction introduction, or reductio ad absurdum.

Usage

The metamathematical significance of the principle of explosion is that in any logical system where it holds, a theory that proves a contradiction (⊥ or its equivalent) becomes worthless: all statements become theorems, making truth and falsehood indistinguishable. This principle thus supports the law of non-contradiction in classical logic, because without it, all truth claims lose meaning. Logics that lack the principle of explosion are discussed in minimal logic.