Sentence examples for formula of the language from inspiring English sources
A sentence is a closed formula of the language.
The definition of a formula of the language of pure quantificational logic proceeds recursively as follows.
Given that specification, the semantics then defines truth conditions for every formula of the language.
where φ(x) can be any formula of the language of PA2 in which X doesn't occur free.
If our way of choosing \(d_{\Gamma }\) can be specified by a formula of the language, we might want to say we have specified an intension.
In first-order formalizations of arithmetic, this is formulated as a scheme: for each first-order arithmetical formula of the language of arithmetic with one free variable, one instance of the induction principle is included in the formalization of arithmetic.
(In these principles we use 'A' and 'B' as metavariables ranging over formulas of the language).
Among the formulas of the language of arithmetic, he isolates what he calls PR- and RE-formulas; the former correspond to the canonical primitive recursive (PR) definitions in arithmetic, and the latter to existential generalizations of the former.
A formula A is logically true iff A is t-entailed by the empty set (iff for every assignment v of algebraic values to the formulas of the language under consideration, v(A) is designated), and A is logically false iff A t-entails the empty set (iff for every assignment v, v(A) is not designated).
It will not do to say: soundness with respect to some V over which the connective in question is truth-functional, since the there may not be enough variety in the truth-values supplied by V to the formulas of the language, in which case its elements may respect the determinants for a truth-function only vacuously because some possibilities do not arise.
Just as the sequents of Set-Fmla can be thought of as those Set-Set sequents Γ ≻Δ with Δ having exactly one element, those of Fmla have the further feature that Γ is empty, making these sequents stand in a one-to-one correspondence with the formulas of the language, with which for practical purposes we may identify them, explaining the following words of the main text.
