Principle of explosion

In symbolic logic, the principle of explosion can be expressed schematically in the following way:[8][9]

${\displaystyle P,\lnot P\vdash Q}$ For any statements P and Q, if P and not-P are both true, then it logically follows that Q is true.

Below is a formal proof of the principle using symbolic logic.

Step	Proposition	Derivation
1	${\displaystyle P}$	Premise
2	${\displaystyle \neg P}$	Premise
3	${\displaystyle P\lor Q}$	Disjunction introduction (1)
4	${\displaystyle Q}$	Disjunctive syllogism (3,2)

This is just the symbolic version of the informal argument given in the introduction, with ${\displaystyle P}$ standing for "all lemons are yellow" and ${\displaystyle Q}$ standing for "Unicorns exist". We start out by assuming that (1) all lemons are yellow and that (2) not all lemons are yellow. From the proposition that all lemons are yellow, we infer that (3) either all lemons are yellow or unicorns exist. But then from this and the fact that not all lemons are yellow, we infer that (4) unicorns exist by disjunctive syllogism.

Paraconsistent logics have been developed that allow for subcontrary-forming operators. Model-theoretic paraconsistent logicians often deny the assumption that there can be no model of ${\displaystyle \{\phi ,\lnot \phi \}}$ 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.

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, ${\displaystyle \phi \land \lnot \phi }$) 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.

Reduction in proof strength of logics without ex falso are discussed in minimal logic.

• Consequentia mirabilis – Clavius' 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