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Principle of explosion
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In classical logic, intuitionistic logic, and similar logical systems, the principle of explosion[a][b] is the law according to which any statement can be proven from a contradiction.[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—true or not—can be proven, it trivializes the concepts of truth and falsity.[7] 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–Fraenkel set theory.
As a demonstration of the principle, consider two contradictory statements—"All lemons are yellow" and "Not all lemons are yellow"—and suppose that both are true. If that is the case, anything can be proven, e.g., the assertion that "unicorns exist", by using the following argument:
- We know that "Not all lemons are yellow", as it has been assumed to be true.
- We know that "All lemons are yellow", as it has been assumed to be true.
- Therefore, the two-part statement "All lemons are yellow or unicorns exist" must also be true, since the first part of the statement ("All lemons are yellow") has already been assumed, and the use of "or" means that if even one part of the statement is true, the statement as a whole must be true as well.
- However, since we also know that "Not all lemons are yellow" (as this has been assumed), the first part is false, and hence the second part must be true to ensure the two-part statement to be true, i.e., unicorns exist (this inference is known as the disjunctive syllogism).
- The procedure may be repeated to prove that unicorns do not exist (hence proving an additional contradiction where unicorns do and do not exist), as well as any other well-formed formula. Thus, there is an explosion of provable statements.
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]
Symbolic representation
[edit]In symbolic logic, the principle of explosion can be expressed schematically in the following way:[8][9]
For any statements P and Q, if P and not-P are both true, then it logically follows that Q is true.
Proof
[edit]Below is the Lewis argument,[10] a formal proof of the principle of explosion using symbolic logic.
| Step | Proposition | Derivation |
|---|---|---|
| 1 | Premise[c] | |
| 2 | Conjunction elimination (1) | |
| 3 | Conjunction elimination (1) | |
| 4 | Disjunction introduction (2) | |
| 5 | Disjunctive syllogism (4,3) |
This proof was published by C. I. Lewis and is named after him, though versions of it were known to medieval logicians.[11][12][10]
This is just the symbolic version of the informal argument given in the introduction, with standing for "all lemons are yellow" and 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.
Semantic argument
[edit]An alternate argument for the principle stems from model theory. A sentence is a semantic consequence of a set of sentences only if every model of is a model of . However, there is no model of the contradictory set . A fortiori, there is no model of that is not a model of . Thus, vacuously, every model of is a model of . Thus is a semantic consequence of .
Paraconsistent logic
[edit]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 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.
Usage
[edit]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, ) 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 the principle of explosion is discussed in minimal logic.
See also
[edit]- 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
Notes
[edit]- ^ Latin: ex falso [sequitur] quodlibet, 'from falsehood, anything [follows]'; or ex contradictione [sequitur] quodlibet, 'from contradiction, anything [follows]'.
- ^ Also known as the principle of Pseudo-Scotus (falsely attributed to Duns Scotus).
- ^ Burgess2005 uses 2 and 3 as premises instead of this one
References
[edit]- ^ Carnielli, Walter; Marcos, João (2001). "Ex contradictione non sequitur quodlibet" (PDF). Bulletin of Advanced Reasoning and Knowledge. 1: 89–109.
- ^ Smith, Peter (2020). An Introduction to Formal Logic (PDF) (2nd ed.). Cambridge University Press. Chapter 17.
- ^ MacFarlane, John (2021). Philosophical Logic: A Contemporary Introduction. Routledge. Chapter 7.
- ^ Başkent, Can (2013). "Some topological properties of paraconsistent models". Synthese. 190 (18): 4023. doi:10.1007/s11229-013-0246-8. S2CID 9276566.
- ^ Carnielli, Walter; Coniglio, Marcelo Esteban (2016). Paraconsistent Logic: Consistency, Contradiction and Negation. Logic, Epistemology, and the Unity of Science. Vol. 40. Springer. ix. doi:10.1007/978-3-319-33205-5. ISBN 978-3-319-33203-1.
- ^ Priest, Graham. 2011. "What's so bad about contradictions?" In The Law of Non-Contradiction, edited by Priest, Beal, and Armour-Garb. Oxford: Clarendon Press. p. 25.
- ^ a b McKubre-Jordens, Maarten (August 2011). "This is not a carrot: Paraconsistent mathematics". Plus Magazine. Millennium Mathematics Project. Retrieved January 14, 2017.
- ^ de Swart, Harrie (2018). Philosophical and Mathematical Logic. Springer. p. 47.
- ^ Gamut, L. T. F. (1991). Logic, Language and Meaning, Volume 1. Introduction to Logic. University of Chicago Press. p. 139.
- ^ a b MacFarlane, John (2021). Philosophical Logic: A Contemporary Introduction. Routledge. p. 171. ISBN 978-1-315-18524-8.
- ^ Lewis, C I; Langford, C H (1959). Symbolic Logic (2nd ed.). Dover. p. 250. ISBN 9780486601700.
{{cite book}}: ISBN / Date incompatibility (help) - ^ Burgess, John P (2005). The Oxford Handbook of Philosophy of Mathematics and Logic (ed Stewart Shapiro). Oxford University Press. p. 732. ISBN 9780195325928.
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Principle of explosion
View on Grokipediafrom Grokipedia
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 and its negation are assumed true, then any arbitrary statement 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 , one obtains , then ; combined with , disjunctive syllogism yields .[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]
This sequence establishes for arbitrary , showing that a contradiction entails every proposition.[9][2]
Disjunction introduction allows adding an arbitrary disjunct to a premise without altering its truth conditions in classical semantics, while disjunctive syllogism eliminates one disjunct given its negation, yielding the remaining one. These rules are standard in natural deduction systems for classical propositional logic.[10]
The derivation illustrates that ex falso quodlibet functions as a derived rule rather than a primitive axiom, emerging directly from the inference rules governing disjunction and negation in classical systems. This syntactic justification underscores the principle's validity within the proof-theoretic framework of classical logic, independent of semantic considerations.[9]
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 from premises and —and disjunctive syllogism—which permits inference of from premises and .[2] In symbolic terms, the principle states that if a contradiction is provable, then any proposition is provable: if and , then for arbitrary .[2] The derivation proceeds in two key steps. First, from , apply disjunction introduction to obtain . Second, from and , apply disjunctive syllogism to infer . This shows how the contradiction and "explodes" into any conclusion .[2] An equivalent formulation is the explosion schema, which captures the principle as a tautology in classical logic: .[4] This schema holds because the antecedent is necessarily false under the law of non-contradiction, rendering the implication vacuously true for any consequent .[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 and its negation are provable (i.e., and ), and demonstrates that any arbitrary proposition follows. The proof, often attributed to C. I. Lewis, proceeds in natural deduction style as follows:| Step | Formula | Justification |
|---|---|---|
| 1 | Assumption (given ) | |
| 2 | ∨I from step 1 | |
| 3 | Assumption (given ) | |
| 4 | DS from steps 2 and 3 |
Semantic Interpretation
In classical propositional logic, the semantics is based on the principle of bivalence, according to which every proposition is assigned exactly one of two truth values: true or false.[11] A truth valuation is a function that assigns truth values to atomic propositions and extends recursively to compound formulas using the standard truth-table definitions for connectives such as negation (¬A is true if A is false, and vice versa), conjunction (A ∧ B is true only if both A and B are true), disjunction (A ∨ B is true if at least one of A or B is true), and implication (A → B is false only if A is true and B is false).[12] These valuations provide the models for evaluating the truth of formulas, where a formula is satisfiable if there exists at least one valuation under which it is true, and valid (a tautology) if it is true under every possible valuation.[12] The principle of explosion arises semantically when considering contradictions. Suppose a set of formulas Γ includes both A and ¬A for some proposition A. No truth valuation can satisfy both A and ¬A simultaneously, as this would require A to be both true and false, violating bivalence. Thus, Γ has no models—there is no consistent valuation that makes all formulas in Γ true.[12] In this case, the model collapses in the sense that the assumption of a contradiction renders the entire semantic framework inconsistent for Γ, forcing the entailment of arbitrary statements. This behavior is formalized in Tarski's semantic definition of logical consequence, introduced in 1936, where Γ semantically entails φ (written Γ ⊨ φ) if and only if every model satisfying Γ also satisfies φ.[13] When Γ contains a contradiction and thus has no models, the condition holds vacuously: there are no counterexamples where Γ is satisfied but φ is not, so Γ ⊨ φ for every proposition φ. This vacuous entailment justifies the principle of explosion, as a contradictory premise base implies every possible conclusion in classical semantics.[13] Complementing this semantic view, syntactic proofs demonstrate the same result through inference rules, but the model-theoretic approach underscores the foundational role of bivalence and the absence of models for inconsistencies.[12] The connection to theorems in the logic further ties into this semantics via the empty set. A formula φ is a theorem (⊢ φ) if and only if the empty set of premises semantically entails φ, meaning φ holds in every model (i.e., φ is valid).[13] For the principle of explosion, deriving ⊢ φ from a contradiction aligns with the empty set's entailment, as the contradiction's lack of models propagates to universal validity in the consequence relation.[12]Alternatives and Extensions
Paraconsistent Logic
Paraconsistent logics are non-classical logical systems in which the presence of a contradiction does not entail every possible statement, thereby rejecting the principle of explosion formalized as for arbitrary .[1] This allows for the coherent management of inconsistent information without leading to triviality, where the entire theory collapses into absurdity.[14] The historical development of paraconsistent logic traces back to early 20th-century efforts in Russia by Nikolai Vasiliev (around 1910), who proposed an "imaginary logic" that included contradictory statements like "S is both P and not P," and Ivan Orlov (1929), who provided the first axiomatization of the relevant logic R, a paraconsistent system; though their work was largely overlooked until later.[1] Significant advancements occurred post-World War II, with Stanisław Jaśkowski introducing a discussive logic in 1948 that permitted inconsistent premises without explosion by modeling reasoning as a collective of individual opinions.[15] Independently, Newton C. A. da Costa developed hierarchical paraconsistent systems in the 1960s, starting with his 1963 doctoral dissertation on interpreting classical logic non-explosively, which formalized calculi like to handle inconsistencies in formal systems. These foundations addressed paradoxes such as the Liar paradox, where self-referential statements generate contradictions without necessitating universal entailment.[1] Mechanisms in paraconsistent logics typically involve restricting rules like disjunctive syllogism—from and , infer —to prevent contradictions from propagating arbitrarily, or weakening modus ponens in contexts where premises are inconsistent.[1] For instance, some systems employ non-adjunctive conjunctions or relevance conditions to ensure inferences depend meaningfully on premises.[1] Key examples include Graham Priest's Logic of Paradox (LP), introduced in his 1979 paper, which uses a three-valued semantics (true, false, both) to model dialetheic contradictions where certain statements are both true and false without exploding.[17] Relevance logics, developed by Alan Ross Anderson and Nuel D. Belnap in their 1975 book Entailment: The Logic of Relevance and Necessity, achieve paraconsistency by requiring that premises and conclusions share propositional content, thus blocking irrelevant inferences from contradictions.[3] Applications of paraconsistent logics extend to practical domains involving inconsistency, such as databases where conflicting data entries must be queried without system failure, as explored in works on paraconsistent knowledge bases.[18] They also model vague predicates in natural language, handling sorites paradoxes by tolerating borderline cases without explosive chains of reasoning.[1]Relevant Logic
Relevant logic, also known as relevance logic, constitutes a class of non-classical logics that enforce a relevance constraint on entailments, thereby circumventing the principle of explosion by disallowing derivations where premises bear no informational connection to the conclusion.[3] In these systems, the core requirement is that premises must be relevant to conclusions, prohibiting inferences reliant on irrelevant disjunctions, such as those in classical material implication that validate arbitrary assertions from unrelated assumptions.[3] Relevant logics explicitly reject the unrestricted principle of explosion, permitting a derivation from a contradiction to an arbitrary only if shares a relevant connection to the contradictory pair, thus preserving non-triviality in the presence of inconsistencies.[3] This rejection targets the classical allowance of ex falso quodlibet without qualification, ensuring that logical validity reflects genuine inferential support rather than formal detachment.[3] Key systems within relevant logic include R, which imposes strong relevance conditions on implications, and E, the logic of entailment, which refines these to model strict entailment while avoiding paradoxes of detached implication.[19] These frameworks, pioneered by Alan Ross Anderson and Nuel D. Belnap, prioritize inferences where premises actively contribute to conclusions through shared content.[19] A foundational mechanism in relevant logics is the variable sharing condition, which mandates that for to be valid, the antecedent and consequent must share at least one propositional variable, thereby blocking entailments between semantically disjoint formulas.[3] Many relevant logics qualify as paraconsistent by virtue of rejecting explosion, allowing non-trivial reasoning amid contradictions, although paraconsistency encompasses broader approaches beyond relevance criteria.[1]Applications and Implications
Role in Classical Proofs
In classical logic, the principle of explosion serves as a foundational tool in reductio ad absurdum proofs, where one assumes the negation of a proposition, derives a contradiction from that assumption, and concludes the original proposition by virtue of the fact that a contradiction entails any statement whatsoever.[20] This method ensures that if assuming leads to an inconsistency, then must hold, as the explosion from the contradiction validates the rejection of the assumption.[20] A representative example illustrates this role: to prove , assume ; from this, derive a contradiction such as for some formula ; the principle of explosion then allows derivation of falsehood (or any arbitrary statement), establishing the unsoundness of the assumption and thus confirming .[21] The symbolic derivation of explosion from disjunctive syllogism and related rules underpins such applications in proof construction.[22] In theorem proving, particularly resolution-based automated systems, the principle manifests in refutation procedures: to establish a theorem , the negation is added to the axioms, and resolution steps aim to derive a contradiction (the empty clause), proving unsatisfiability and thereby via the classical inference that inconsistency implies the negation's falsehood.[23] This approach leverages explosion implicitly, as the contradiction entails universal falsehood in the clausal form.[24] In practice, while explosion is theoretically complete, proof designers and automated provers often halt upon detecting a contradiction without invoking the full inferential power of explosion, as generating all consequent propositions would lead to inefficiency and combinatorial explosion in search spaces.[23] The principle connects to broader metatheoretic results, such as Gödel's completeness theorem for classical first-order logic, where the theorem's proof relies on a Hilbert-style system incorporating ex falso quodlibet to ensure that every semantically valid formula is syntactically provable, linking model-theoretic validity to derivability in the explosive framework.[25]Philosophical and Practical Uses
The principle of explosion underscores the philosophical importance of logical consistency, as inconsistencies can lead to the derivation of any proposition, rendering rational discourse trivial. In Aristotle's Metaphysics, the defense of the law of non-contradiction serves to prevent such outcomes, emphasizing that allowing contradictions would undermine meaningful inquiry and argumentation by implying everything follows from falsehood.[26] This view reinforces consistency as a foundational requirement for philosophical reasoning, where contradictions are not merely errors but threats to the coherence of knowledge systems. Dialetheism, a philosophical position advocating for true contradictions, critiques the principle of explosion by arguing that rejecting it allows for the coherent acceptance of certain inconsistencies without triviality. Graham Priest, a leading proponent, contends that explosion is implausible in scenarios involving genuine paradoxes, such as the liar paradox, where both a statement and its negation can hold without implying all propositions.[27] This perspective challenges classical logic's strict avoidance of contradictions, proposing instead that dialetheia—true contradictions—enrich philosophical understanding in areas like metaphysics and semantics. In argumentation theory, the principle is invoked to dismiss inconsistent positions, as a contradictory premise permits the "explosion" of arbitrary conclusions, thereby invalidating the argument's reliability. This application highlights ethical considerations in debate, where exploiting contradictions to derive unintended claims can undermine fair discourse, prompting calls for norms that prioritize non-explosive reasoning to maintain argumentative integrity. Practically, the principle influences AI systems handling inconsistent data, where belief revision techniques aim to avoid explosion by selectively revising beliefs rather than deriving all possibilities from contradictions. For instance, paraconsistent approaches in belief revision ensure that minor inconsistencies, common in real-world data integration, do not collapse the entire knowledge base.[28] In legal reasoning, contradictory evidence often leads to case dismissal or further investigation to prevent the logical triviality of accepting incompatible facts, preserving the system's adjudicative function.[29] Modern applications, including dialetheism and AI, extend beyond classical foundations by exploring non-explosive alternatives for managing real-world inconsistencies.References
- https://www.[researchgate](/page/ResearchGate).net/publication/390318539_Some_remarks_on_two_seminal_approaches_to_paraconsistency_Stanislaw_Jaskowski_and_Newton_da_Costa