By placing an existential quantifier
[there exists] before x ("for some x") and an universal quantifier [for all] before y ("for all y"), we can bind these variables, as may be seen bellow [Bird, 2009]:
Being, existence, and ontological commitment" presents five theses of Quinian metaontology: (1-3) being is not an activity, is the same as existence, and is univocal; (4) the existential quantifier
adequately captures the sense of being; (5) ontological disputes can be settled by determining the ontological commitments implied by established beliefs (this last is restated and illustrated in chapters four, eight, and ten).
It appears, then, that the property on which K(Q-W) logically depends cannot be adequately formulated by embedding either a universal or existential quantifier
within the abstract with either internal or external scope relative to the modal operator.
Among the joint-carving terms, the most important one in this context is the existential quantifier
Hofweber asserts that we can distinguish between the internal and the external reading of the existential quantifier
checking in the process if he is in possession of or has developed a rudimentary but accurate feeling for syntax to concentrate afterwards in semantics, once we have informed him that the traditional symbol for the universal quantifier "for any" is "[for all]", an inverted letter "A", and for the existential quantifier
"there exists" is "[there exist]", a rotated letter "E".
Let the existential quantifier
'[there exists]x' range over the domain of all propositions [alpha] (that is, statements which have a definite truth-value).
It is pertinent to observe that the relative order of the existential quantifier
and negation in (3) is the opposite of the order of the indefinite some and negation in (1): adult English only licenses the "nonisomorphic" interpretation of sentences like (1).
The for all and every English phrases can be satisfied in SQL through negating its existential quantifier
Now consider a quantifier that behaves exactly like the universal quantifier (over individuals) in models with domains of cardinality [greater than or equal to] n, but like the existential quantifier
in models with domains of cardinality < n.
Following this, Kolmogorov does not give the interpretation of the existential quantifier
, as we would expect, since intuitionistically the existential quantifier
cannot be defined using the universal and negation; but elsewhere in the paper he gives ample explanations of the meaning of existential claims in intuitionistic mathematics, and in particular, of the central point concerning them: that the person who makes the claim must be able to indicate a particular instance of it.
To eliminate an existential quantifier
[exists]z [element of] U, we use the lemma above, by translating its statement into first-order logic.