Sentence (mathematical Logic)

Sentence (mathematical Logic)

In mathematical logic, a sentence of a predicate logic is a boolean-valued well formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that may 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.

Sentences without any logical connectives or quantifiers in them are known as atomic sentences; by analogy to atomic formula. Sentences are then built up out of atomic sentences by applying connectives and quantifiers.

A set of sentences is called a theory; thus, individual sentences may be called theorems. To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpretation of the theory. For first-order theories, interpretations are commonly called structures. Given a structure or interpretation, a sentence will have a fixed truth value. A theory is satisfiable when all of its sentences are true. The study of algorithms to automatically discover interpretations of theories that render all sentences as being true is known as the satisfiability modulo theories problem.

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Famous quotes containing the word sentence:

    In how few words, for instance, the Greeks would have told the story of Abelard and Heloise, making but a sentence of our classical dictionary.... We moderns, on the other hand, collect only the raw materials of biography and history, “memoirs to serve for a history,” which is but materials to serve for a mythology.
    Henry David Thoreau (1817–1862)