The use of universal quantifiers to comprehend and express the for all and every phrases is more natural and intuitive than negating existential quantifiers. Consider the following statement: Every company, which is destroying at least one forest, is savage, and every person who lives in Canada is concerned (1) about such companies.
In addition, [for all], the universal quantifier, and [there exists], the existential quantifier, can occur in formulas.
(We need at most m extra variable for existential quantifiers, where m is the maximum arity of a schema relation.)
This continues to hold for any fragment of second-order that allows existential quantifiers over unary predicates (e.g., existential second-order monadic [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]).
The same argument shows that even if "natural number", "zero" and "successor" are logical terms, they are not definable using negation, identity, and the first-order existential quantifier alone.
If negation, identity, and the first-order existential quantifier are indeed logical terms, and if the analytic (or logical) truths are closed under consequence, then no analytic (or logical) truth can entail (Two).
Hintikka adds a proviso that 'moves connected with
existential quantifiers are always independent of earlier moves with
existential quantifiers' (p.
The theme of this book is that many of the fundamental notions of logic, including the connectives of propositional logic, the universal and
existential quantifiers and various modalities, are definable in terms of implication.
For the other direction, replace the variables ranging over level n open sentences with variables ranging over type n objects (or sets of rank n in the non-cumulative hierarchy), replace the symbol for satisfaction with that of predication (or membership), and, of course, replace constructibility quantifiers with
existential quantifiers. The systems seem to be definitionally equivalent.
