Principle of explosion
The principle of explosion, also known as ex falso quodlibet ("from falsehood, anything follows") or ex contradictione quodlibet ("from contradiction, anything follows"), is a fundamental theorem in classical logic asserting that a contradiction logically entails every possible proposition.[1] In formal terms, if both a statement A and its negation \neg A are assumed true, then any arbitrary statement B follows from them.[2] This principle underscores the importance of consistency in classical deductive systems, as accepting a contradiction renders the entire theory trivial by implying all statements, thereby collapsing meaningful inference.[2] Its validity in classical logic derives from basic rules such as disjunction introduction and disjunctive syllogism: from A \land \neg A, one obtains A, then A \lor B; combined with \neg A, disjunctive syllogism yields B.[3] Historically, while ancient logicians like Aristotle implicitly opposed unrestricted explosion through connexive principles linking antecedents and consequents, the explicit derivation emerged in medieval Europe, with the 12th-century Parisian logician William of Soissons providing the first known proof.[1] It solidified as a cornerstone of modern classical logic during the 19th and early 20th centuries, amid formalizations by George Boole, Gottlob Frege, and Bertrand Russell, who treated consistency as essential for rigorous mathematics and philosophy.[1] The principle's acceptance in classical systems has faced challenges in alternative logics developed to handle inconsistencies without triviality.[1] Relevance logics, pioneered in the mid-20th century by C.I. Lewis, Alan Anderson, and Nuel Belnap, reject explosion by requiring premises to share propositional content with conclusions, thus avoiding irrelevant inferences from contradictions.[3] Similarly, paraconsistent logics, formalized from the 1940s onward by Stanisław Jaśkowski and Newton da Costa, explicitly block explosion to permit non-trivial reasoning amid contradictory data, such as in databases or dialetheic philosophies that tolerate true contradictions like the liar paradox.[1] These developments highlight explosion's role in defining classical logic's boundaries while enabling more flexible systems for inconsistent information.[1]Definition and Formulation
Symbolic Representation
The principle of explosion, known in Latin as ex falso quodlibet ("from falsehood, anything follows"), encodes the idea that a contradiction in classical logic entails any arbitrary proposition.[4][2] This principle relies on fundamental inference rules of classical propositional logic, including modus ponens—which allows inference of B from premises A and A \to B—and disjunctive syllogism—which permits inference of B from premises A \lor B and \neg A.[2] In symbolic terms, the principle states that if a contradiction is provable, then any proposition B is provable: if \vdash A and \vdash \neg A, then \vdash B for arbitrary B.[2] The derivation proceeds in two key steps. First, from A, apply disjunction introduction to obtain A \lor B. Second, from A \lor B and \neg A, apply disjunctive syllogism to infer B. \begin{align*} & \vdash A \\ & \therefore \vdash A \lor B \quad (\text{disjunction introduction}) \\ & \vdash \neg A \\ & \therefore \vdash B \quad (\text{disjunctive syllogism}) \end{align*} This shows how the contradiction A and \neg A "explodes" into any conclusion B.[2] An equivalent formulation is the explosion schema, which captures the principle as a tautology in classical logic: (A \land \neg A) \to B.[4] This schema holds because the antecedent A \land \neg A is necessarily false under the law of non-contradiction, rendering the implication vacuously true for any consequent B.[2]Informal Explanation
The principle of explosion holds that if a logical system or set of premises includes even a single contradiction—such as both a statement and its negation being true—then any possible statement can be logically derived from it, rendering the system entirely trivial and useless for distinguishing truth from falsehood.[5] This means that once inconsistency arises, the logic "blows up," allowing proofs of contradictory or irrelevant claims alike, which underscores why maintaining consistency is fundamental in reasoning.[6] Known historically by the Latin phrase ex falso quodlibet, meaning "from a falsehood, anything follows," the principle emerged in medieval scholastic logic as a way to handle contradictory premises.[7] The term reflects the idea that falsehood, once admitted, unleashes boundless inferences, a concept debated by logicians like John Buridan in the 14th century.[7] Intuitively, this can be likened to a logical short circuit: just as a single fault in an electrical system can cause widespread failure and erratic behavior, a contradiction propagates falsehood throughout the entire framework, shorting out any reliable conclusions. The principle goes beyond merely detecting an inconsistency; it emphasizes the dramatic fallout, where the system's explosive derivation of everything eliminates its capacity for coherent inference, turning a minor error into total collapse.[5] This intuitive basis is later rigorized through symbolic forms in formal logic.[8]Justification
Disjunctive Syllogism Derivation
In classical logic, the principle of explosion, also known as ex falso quodlibet or ex contradictione quodlibet, can be derived syntactically from a contradiction using the basic rules of disjunction introduction (∨I) and disjunctive syllogism (DS). This derivation assumes that both a proposition A and its negation \neg A are provable (i.e., \vdash A and \vdash \neg A), and demonstrates that any arbitrary proposition B follows. The proof, often attributed to C. I. Lewis, proceeds in natural deduction style as follows:| Step | Formula | Justification |
|---|---|---|
| 1 | A | Assumption (given \vdash A) |
| 2 | A \lor B | ∨I from step 1 |
| 3 | \neg A | Assumption (given \vdash \neg A) |
| 4 | B | DS from steps 2 and 3 |