Sentence examples for free occurrence from inspiring English sources

Any wff of the form (∀a) ⊃ [α ⊃ (∀a)β] is an axiom, provided that α contains no free occurrence of a. Transformation rules: Modus ponens.

(ιx) is analogous to a quantifier in that, when prefixed to a wff α, it binds every free occurrence of x in α.

Any wff of the form (∀a)α ⊃ β is an axiom if β is either identical with α or differs from it only in that, wherever α has a free occurrence of a, β has a free occurrence of some other individual variable b.

The following additions (or some equivalent ones) are made to the axiomatic basis for LPC: the axiom x = x and the axiom schema that, where a and b are any individual variables and α and β are wffs that differ only in that, at one or more places where α has a free occurrence of a, β has a free occurrence of b, (a = b) ⊃ is an axiom.

The main elements of one widely used notation are the following: if α is an expression containing some free occurrence of x, the expression {x : α} is used to stand for the class of objects fulfilling the condition expressed by α.

Every free occurrence of x in ϕ is bound by the operator.

Various predicate calculi of higher order can be formed, however, in which quantifiers may contain other variables as well, hence binding all free occurrences of these that lie within their scope.

The free occurrences of variables in a formula of the form $(B \rightarrow C)$ are the free occurrences of variables in $B$ and $C$.

The free occurrences of variables in the negation of a formula $\lnot B$ are the free occurrences of variables in $B$.

Any free occurrences of v in θ are bound by the initial quantifier.

There is even some nondeterminism in such an approach to substitution: if we need to rewrite free occurrences of y in M before substituting M for free occurrences of x in λy[A], how should we do it?