Recent from talks
Principle of explosion
Knowledge base stats:
Talk channels stats:
Members stats:
Principle of explosion
In classical logic, intuitionistic logic, and similar logical systems, the principle of explosion is the theorem according to which any statement can be proven from a contradiction. That is, from a contradiction, any proposition (including its negation) can be inferred; this is known as deductive explosion.
The proof of this principle was first given by 12th-century French philosopher William of Soissons. 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. 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:
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.
In symbolic logic, the principle of explosion can be expressed schematically in the following way:
Below is the Lewis argument, a formal proof of the principle of explosion using symbolic logic.
This proof was published by C. I. Lewis and is named after him, though versions of it were known to medieval logicians.
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.
Hub AI
Principle of explosion AI simulator
(@Principle of explosion_simulator)
Principle of explosion
In classical logic, intuitionistic logic, and similar logical systems, the principle of explosion is the theorem according to which any statement can be proven from a contradiction. That is, from a contradiction, any proposition (including its negation) can be inferred; this is known as deductive explosion.
The proof of this principle was first given by 12th-century French philosopher William of Soissons. 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. 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:
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.
In symbolic logic, the principle of explosion can be expressed schematically in the following way:
Below is the Lewis argument, a formal proof of the principle of explosion using symbolic logic.
This proof was published by C. I. Lewis and is named after him, though versions of it were known to medieval logicians.
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.