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A statement can be called valid, i.e. logical truth, in some systems of logic like in Modal logic if the statement is true in all interpretations. In Aristotelian logic statements are not valid per se. Validity refers to entire arguments. The same is true in propositional logic (statements can be true or false but not called valid or invalid).
The following example demonstrates why this line of reasoning is a logical fallacy: I've seen a person shoot someone dead. Therefore, all people are murderers. In the common discourse, a proof by example can also be used to describe an attempt to establish a claim using statistically insignificant examples. In which case, the merit of each ...
The following famous example of a nonconstructive proof shows that there exist two irrational numbers a and b such that is a rational number. This proof uses that 2 {\displaystyle {\sqrt {2}}} is irrational (an easy proof is known since Euclid ), but not that 2 2 {\displaystyle {\sqrt {2}}^{\sqrt {2}}} is irrational (this is true, but the proof ...
A sentence is said to be a logical consequence of a set of sentences, for a given language, if and only if, using only logic (i.e., without regard to any personal interpretations of the sentences) the sentence must be true if every sentence in the set is true.
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 ...
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements.For example, in the conditional statement: "If P then Q", Q is necessary for P, because the truth of Q is guaranteed by the truth of P.
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.
A fully interpreted language L which does not have a truth predicate can be extended to a fully interpreted language Ľ that contains a truth predicate T, i.e., the sentence A ↔ T(⌈A⌉) is true for every sentence A of Ľ, where T(⌈A⌉) stands for "the sentence (denoted by) A is true". The main tools to prove this result are ordinary and ...