Principle of Explosion¶
A logic-relative consequence rule under which a proposition and its negation together entail any formula whatsoever.
Core Idea¶
The principle of explosion is the logic-relative rule that A together with ¬A entails any formula B. Classical and intuitionistic consequence validate it; a paraconsistent consequence relation does not validate it unrestrictedly. The rule must be tested against a specified logic, not inferred from the mere presence of conflicting statements.[ref-5fee83fff750][ref-187f0248c536]
Scope of Application¶
In classical propositional reasoning, P gives P∨Q and P∨Q with ¬P gives unrelated Q by disjunctive syllogism. In a classical description-logic knowledge base, conflicting assertions can leave no model, so every query is formally entailed. These are two formal settings for the same arbitrary-conclusion schema, not proof that any real person believes all its consequences.[ref-168168d30eb7][ref-5fee83fff750]
Clarity¶
Contradiction is the antecedent; explosion is the consequence rule. Deriving one particular claim from inconsistent premises is not enough—the target must be arbitrary. Conversely, a paraconsistent model with p and ¬p but not q shows that a contradictory premise set need not explode under every semantics.[^ref-168168d30eb7]
Manages Complexity¶
Once a classical theory derives both P and ¬P, its deductive closure contains every formula, so derivability alone no longer distinguishes candidate conclusions. An alternative is to repair the premises or change the consequence relation. The latter may retain useful inconsistent information but requires explicit nonclassical rules.[ref-5fee83fff750][ref-168168d30eb7]
Abstract Reasoning¶
To test explosion, state the host logic, choose the same-scope pair A and ¬A, and ask whether every B follows. One unrelated counterexample B defeats the universal schema. A proof route using disjunction introduction and disjunctive syllogism works in suitable systems but is not part of the definition of every explosive logic.[ref-168168d30eb7][ref-187f0248c536]
Knowledge Transfer¶
Propositional proof and classical ontology reasoning preserve the same contradictory-premise/arbitrary-conclusion structure. “One fault contaminates everything” outside logic is only analogy unless a real consequence relation validates the unrestricted step. Contradiction, Paraconsistent Logic and Trivialism are distinct live neighbors, not interchangeable parents; no canonical DAG edge was changed.[ref-5fee83fff750][ref-168168d30eb7]
[^ref-5fee83fff750]: Daniele Sgaravatti, “Explosion and Reasoning”, Episteme (2025), DOI 10.1017/epi.2025.6, especially pp. 1–2. [^ref-187f0248c536]: Pedro Francisco Valencia Vizcaíno, “Relations between ex falso, tertium non datur, and double negation elimination” (2013), §§1–2 and Theorem 1. [^ref-168168d30eb7]: Xiaowang Zhang, Guohui Xiao, Zuoquan Lin and Jan Van den Bussche, “Inconsistency-Tolerant Reasoning with OWL DL”, International Journal of Approximate Reasoning 55, no. 2 (2014), 557–584, author-version PDF §§2.1–2.2.
Neighborhood in Abstraction Space¶
Principle of Explosion sits in a moderately populated region (49th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Formal Models & Logical Foundations (33 abstractions)
Nearest neighbors
- Propositional logic — 0.86
- Modus ponens — 0.86
- Logical Consequence — 0.86
- Formal Theory — 0.86
- Logical or — 0.85
Computed from structural-signature embeddings · 2026-10-08