The ontological commitments we require to show the necessity of ultimately founded propositions are, on the one hand, the possibilist interpretation of the existential quantifier
or the assumption of a constant domain of the possible worlds W*; and, on the other hand, the modal logic S5.
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).
In addition, [for all], the universal quantifier, and [there exists], the existential quantifier
, can occur in formulas.
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.
60) 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.
Although it is a statement that there are at least two objects, (Two) is composed only of standard logical terminology: negation, identity, and first-order existential quantifiers
. Since (Two) is a logical consequence of 0 [not equal to] 0, then according to our logicist (Two) is itself analytic and logically true, provided that analyticity and logical truth are closed under logical consequence, or at least the introduction rule for the first-order existential quantifier
The rules for the independent existential quantifier
elimination require the introduction of function letters (to express the independence).
"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).
Fitch's Paradox is presented as a problem for realism, and is 'solved', I think, by denying that existential quantifier
elimination is legitimate.
However, the relation between these two representations, as well as the relation between the universal and the existential quantifier
representation, is not made clear (it seems to me that the two exercises devoted to these issues are not very helpful).
For (ii), we slightly modify the previous formula [Phi] by turning the existential quantifier
[exists]z into a universal quantifier [inverted] Az, and by replacing the last two conjuncts with