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.
Contents
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:
- We know that "All lemons are yellow" as it is defined to be true.
- Therefore, the statement that (“All lemons are yellow" OR "unicorns exist”) must also be true, since the first part is true.
- However, if "Not all lemons are yellow" (and this is also defined to be true), unicorns must exist – otherwise statement 2 would be false. 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-Frankel set theory.
In a different solution to these problems, some 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. In artificial intelligence and models of human reasoning it is common for such logics to be used. Truth maintenance systems are AI models which try to capture this process.
Symbolic representation
In symbolic logic, the principle of explosion can be expressed in the following way
(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
-
P ∧ ¬ P assumption -
P from (1) by conjunction elimination -
¬ P from (1) by conjunction elimination -
P ∨ Q from (2) by disjunction introduction -
Q from (3) and (4) by disjunctive syllogism -
( P ∧ ¬ P ) → Q from (5) by conditional proof (discharging assumption 1)
This is just the symbolic version of the informal argument given in the introduction, with
Semantic argument
An alternate argument for the principle stems from model theory. A sentence
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
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,