Principle of explosion
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In
The proof of this principle was first given by 12th-century French philosopher
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 true statements.
In a different solution to the problems posed by the principle of explosion, some mathematicians have devised alternative theories of
Symbolic representation
In
Proof
Below is a formal proof of the principle using
Step | Proposition | Derivation |
---|---|---|
1 | Premise | |
2 | Premise | |
3 | Disjunction introduction (1) | |
4 | Disjunctive syllogism (3,2) |
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
An alternate argument for the principle stems from model theory. A sentence is a
Paraconsistent logic
Usage
The
Reduction in proof strength of logics without ex falso are discussed in minimal logic.
See also
- 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
References
- ^ Carnielli, Walter; Marcos, João (2001). "Ex contradictione non sequitur quodlibet" (PDF). Bulletin of Advanced Reasoning and Knowledge. 1: 89–109.[permanent dead link]
- ^ Smith, Peter (2020). An Introduction to Formal Logic (2nd ed.). Cambridge University Press. Chapter 17.
- ^ MacFarlane, John (2021). Philosophical Logic: A Contemporary Introduction. Routledge. Chapter 7.
- S2CID 9276566.
- ISBN 978-3-319-33203-1.
- ^ Priest, Graham. 2011. "What's so bad about contradictions?" In The Law of Non-Contradicton, 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.