Peano only has a universal quantifier.
Most of Aristotle’s logic was concerned with certain kinds of propositions that can be analyzed as consisting of (1) usually a quantifier (“every,” “some,” or the universal negative quantifier “no”), (2) a subject, (3) a copula, (4) perhaps a negation (“not”), (5) a predicate.
On the first reading, the world-quantifier takes wide scope.
similar results holding for other finite-quantifier languages.
The quantifier should express neither “there is” nor “there exists”.
So theories that have quantifier elimination are model-complete, but the converse need not hold.
This section and the next provide an evaluation of quantifier accounts of ontological commitment.
One example is the extension of first order logic by the generalized quantifier known as the Henkin quantifier:
A constant is said to be used by a player if the player chooses it to attack a universal quantifier or to defend an existential quantifier.
The material in the square brackets gives the restriction on the quantifier, and the formula in parentheses after the bracket constitutes the scope of the restricted quantifier.
Since, semantically speaking, the necessity operator is literally a universal quantifier, the definition corresponds exactly to the definition (7) of the existential quantifier.
It can in fact be shown that the presence of this infinite quantifier is essential: the class of well-ordered structures cannot be characterized in any finite-quantifier language.
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.
More generally, the first epsilon theorem establishes that quantifiers and epsilons can always be eliminated from a proof of a quantifier-free formula from other quantifier-free formulae.
We could have used an atomic symbol for this second-order predicate, but the above way of expanding it shows the relation of the generalized quantifier to the ordinary existential quantifier.
The language is very much like the language of classical first-order logic, but rather than the familiar universal and existential quantifier, the language contains a probabilistic quantifier.
A defender of the quantifier account, however, can respond by noting that there is a distinction between ontological commitment to Ks and ontological commitment to the Ks; the quantifier account aims only to analyze the former.
Other quantifier principles, which govern the move of a quantifier from the inside of a formula to governing the entire formula, so called “quantifier containment principles” are also derived as theorems in ∗10.
Montague (Montague 1973) held to a policy of holding fixed the semantic type of lexical items by their category, so that names, falling in the same category as quantifier phrases, were assigned the same type as quantifier phrases.
In (34), with the possessive presupposition trigger “their cars”, there is a global context (outside of the scope of “most”), a local context corresponding to the scope of the quantifier (occupied by the VP “wash their cars on Saturday”), and also an intermediate context in the restrictor of the quantifier (occupied by “Germans”).
quantifier
noun communication
- (logic) a word (such as `some' or `all' or `no') that binds the variables in a logical proposition
noun communication
- (grammar) a word that expresses a quantity (as `fifteen' or `many')
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In 34 with the possessive presupposition trigger their cars there is a global context outside of the scope of most a local context corresponding to the scope of the quantifier occupied by the VP wash their cars on Saturday and also an intermediate context in the restrictor of the quantifier occupied by Germans