A sentence is a logic formula in which every variable is quantified. The concept of a sentence is important because formulas with variables that are not quantified are ambiguous.

The concept of the sentence can be illustrated as follows (Enderton 1977). The formula  exists (x, forall (y,y in x)), in which each variable is quantified, can be translated into English as the complete sentence "There exists a set which has every set as an element." However, the formula  forall (y,(y in x)), in which x is not quantified, can only be translated as the sentence fragment "Every set is an element of ___," where "___" is unspecified because x is not quantified.

Because a "quantified variable" (or "quantifier") is just a more descriptive name for a bound variable, a sentence can also be defined as a logic formula with no free variables (Enderton 1977). A sentence can also be defined as a closed sentential formula (Carnap 1958, pp. 24 and 85), although in some language systems, open sentential formulas are also admitted as sentences (Carnap 1958, p. 25).

See also

Bound Variable, Closed Sentential Formula, Free Variable, Open Sentential Formula, Quantifier, Sentential Formula, Sentential Variable, Theory

This entry contributed by Matthew Szudzik

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Carnap, R. Introduction to Symbolic Logic and Its Applications. New York: Dover, 1958.Enderton, H. B. Elements of Set Theory. New York: Academic Press, 1977.

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Cite this as:

Szudzik, Matthew. "Sentence." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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