[1][2][3] That is, from a contradiction, any proposition (including its negation) can be inferred; this is known as deductive explosion.
[4][5] The proof of this principle was first given by 12th-century French philosopher William of Soissons.
[6] Due to the principle of explosion, the existence of a contradiction (inconsistency) in a formal axiomatic system is disastrous; since any statement can be proven, it trivializes the concepts of truth and falsity.
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.
If that is the case, anything can be proven, e.g., the assertion that "unicorns exist", by using the following argument: In a different solution to the problems posed by the principle of explosion, some mathematicians have devised alternative theories of logic called paraconsistent logics, which allow some contradictory statements to be proven without affecting the truth value of (all) other statements.
[7] In symbolic logic, the principle of explosion can be expressed schematically in the following way:[8][9] Below is the Lewis argument,[10] a formal proof of the principle of explosion using symbolic logic.
This proof was published by C. I. Lewis and is named after him, though versions of it were known to medieval logicians.
But then from this and the fact that not all lemons are yellow, we infer that (4) unicorns exist by disjunctive syllogism.
An alternate argument for the principle stems from model theory.
Model-theoretic paraconsistent logicians often deny the assumption that there can be no model of
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.
) 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.