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Formally the law of non-contradiction is written as ¬(P ∧ ¬P) and read as "it is not the case that a proposition is both true and false". The law of non-contradiction neither follows nor is implied by the principle of Proof by contradiction. The laws of excluded middle and non-contradiction together mean that exactly one of P and ¬P is true.
The use of this fact forms the basis of a proof technique called proof by contradiction, which mathematicians use extensively to establish the validity of a wide range of theorems. This applies only in a logic where the law of excluded middle A ∨ ¬ A {\displaystyle A\vee \neg A} is accepted as an axiom.
This resolution technique uses proof by contradiction and is based on the fact that any sentence in propositional logic can be transformed into an equivalent sentence in conjunctive normal form. [4] The steps are as follows. All sentences in the knowledge base and the negation of the sentence to be proved (the conjecture) are conjunctively ...
Consistent sentence: A sentence of is consistent if it is true under at least one interpretation. It is inconsistent if it is not consistent. [ 66 ] [ 68 ] An inconsistent formula is also called self-contradictory , [ 1 ] and said to be a self-contradiction , [ 1 ] or simply a contradiction , [ 81 ] [ 82 ] [ 83 ] although this latter name is ...
A Gödel sentence G for a system F makes a similar assertion to the liar sentence, but with truth replaced by provability: G says "G is not provable in the system F." The analysis of the truth and provability of G is a formalized version of the analysis of the truth of the liar sentence.
A term used to describe non-standard or alternative logical systems that deviate from classical logic. diagonalization lemma A lemma used in the proof of Gödel's incompleteness theorems, stating that for any formula with one free variable, there exists a sentence that asserts its own unprovability. dialetheism
A sentence can be viewed as expressing a proposition, something that must be true or false. The restriction of having no free variables is needed to make sure that sentences can have concrete, fixed truth values : as the free variables of a (general) formula can range over several values, the truth value of such a formula may vary.
The same type of relationship is shown in (2), where the first sentence can be interpreted as implying that by giving a party for the new students, the hosts will serve drinks. This is, of course, a defeasible inference based on world knowledge, that is then contradicted in the following sentence.