A closed sentential formula is a sentential formula in which none of the variables are free (i.e., all variables are bound). Examples of closed sentential formulas are given by for all x for all y(x + y congruent y + x), which expresses the commutativity of addition, and for all x exists y( for all u for all v(x + y!=(u + 2)(v + 2))), which expresses the infinitude of the primes. A closed sentential formula is called a sentence (Carnap 1958, pp. 24-25 and 85). However, in some language systems, open sentential formulas are also admitted as sentences.
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