Sentence examples for universal closure from inspiring English sources

Let \(\forall \overline{P}\phi \) denote the universal closure of an \(L_{2}\ -formula \(\phi\) over aL_{2}\ -formulariables occurring in it.

This is exactly the same as to saying that the universal closure of φ ≈ ψ is a sentence true in A according to the usual semantics for first-order logic with equality.

An existence and uniqueness claim must hold: the universal closure of the formula must be provable.[9] In a logic that allows for vacuous names, the specific condition on the definiens of (7) would be weaker: the existence condition would be dropped.

An L-quasiequation is valid in an L-algebra A, or the algebra is a model of it, if the universal closure of the quasiequation is true in A. A quasivariety of L-algebras is a class of algebras which is the class of the models of some set of L-quasiequations.

Note that the inferential force of adding definition (10) to the language is the same as that of adding as an axiom, the universal closure of However, this similarity in the logical behavior of (10) and (11) should not obscure the great differences between the biconditional ('\(\leftrightarrow\)') and definitional equivalence ('\(\eqdf\)').

The formula ψ X,Y) is a deductive generalization of ϕ(X), if it holds in Σ that the less general ϕ implies the more general ψ where for the free variables X (the ones that occur in ϕ and possibly in ψ) the universal closure and for free variables Y (the ones that occur in ϕ only) the existential closure is taken: Σ ⊨ ∀ X ∃ Y ( ϕ ( X ) → ψ ( X, Y ) ).

Note that we can regard a recursive definition such as (15) as an implicit definition by a theory that consists of the universal closures of the equations.

In the first epsilon theorem, "quantifier-free predicate logic" is intended to include the substitution rule above, so quantifier-free axioms behave like their universal closures.

Annular algebras admit a universal C⁎-algebra closure analogous to the universal C⁎-algebra for groups.

In order to obtain reproducibility of experiments with natural and man-made CES (with respect to degree of closure) some universal estimate needs to be developed.

They craved closure, a universal human need after any tragedy.