For example, both natural deduction and sequent calculus systems facilitate the elimination and introduction of universal and existential quantifiers so that unquantified logical expressions can be manipulated according to the much simpler rules of propositional calculus.
42.
The restriction to formulas in prenex form containing only existential quantifiers does not limit the generality of the theorem, because formulas can be converted to prenex form and their universal quantifiers can be removed by Herbrandization.
43.
To show constructively that a theory has quantifier elimination, it suffices to show that we can eliminate an existential quantifier applied to a conjunction of literals, that is, show that each formula of the form:
44.
It's worth noting that the'scope'of universal quantifiers ( variables ) and existential quantifiers ( BNodes ) is the formula ( or context-to be specific ) in which their statements reside.
45.
For example, it can be shown that there is a polynomial-time many-one reduction that will move all quantifiers to the front of the formula and make them alternate between universal and existential quantifiers.
46.
As a result, after replacing an existential quantifier over variables into an existential quantifiers over functions at the front of the formula, the formula still may be treated as a first-order one by removing these existential quantifiers.
47.
As a result, after replacing an existential quantifier over variables into an existential quantifiers over functions at the front of the formula, the formula still may be treated as a first-order one by removing these existential quantifiers.
48.
As a result, after replacing an existential quantifier over variables into an existential quantifiers over functions at the front of the formula, the formula still may be treated as a first-order one by removing these existential quantifiers.
49.
In " de dicto " claims, any existential quantifiers are within the scope of the modal operator, whereas in " de re " claims the modal operator falls within the scope of the existential quantifier.
50.
In " de dicto " claims, any existential quantifiers are within the scope of the modal operator, whereas in " de re " claims the modal operator falls within the scope of the existential quantifier.
