For quantifiers of this kind, a world-relative domains are appropriate.
If so, this would presumably require a different semantics for these quantifiers.
., ranging over the set of declarative sentences – and propositional quantifiers.
The connection between model-completeness and elimination of quantifiers is as follows.
as primitive, the free inner quantifiers can be defined in terms of the classical outer ones as follows:
This is unlike most modern theories, which introduce quantifiers only when names and predicates are brought in.
This also has the advantage that the same framework is used for count quantifiers and mass quantifiers.
And spelling out those world quantifiers in turn using Plantinga’s definition will re-introduce those same modal operators yet again.
The acknowledgement of different meanings for the quantifiers is not enough by itself to explain away the intuitiveness of the Metaphysics 101 argument.
If the (apparent) quantifiers in (13) and (14) can bind variables in sentences after those in which they occur, why can’t the quantifiers in (15) and (16)?
We did not include this requirement in the definition of generalized quantifiers however, since there are natural language quantifiers that do not satisfy it; see below.
However, unlike “ordinary” quantifiers, these anaphoric pronouns qua quantifiers have their forces, restrictions and relative scopes determined by features of their linguistic environments.
If a quantifier account of ontological commitment is to be applied directly to ordinary language, we need to know which ordinary language quantifiers or uses of quantifiers (if any) are ontologically committing.
(The ‘method of elimination of quantifiers’ discussed in section 2.2 of Tarski’s truth definitions) was a syntactic and pre-model-theoretic method for proving elimination of quantifiers down to a particular set of formulas.)
Thus the unscoped LF above would be knows(⟨∃poem⟩, ⟨∀person⟩), and scoping of quantifiers, along with their restrictors, now involves “raising” quantifiers to take scope over a sentential formula, with simultaneous introduction of variables.
After all, the metalanguage in which the semantics is developed already has universal and existential quantifiers, and these quantifiers need not be interpreted as providing ontological commitment any more than the object language quantifiers do.
Suppose we use the actualist semantics, so each state has an associated domain of actually existing things, but suppose we allow quantifiers to range over the members of any domain, without distinction, which means quantifiers are ranging over the same set, at every state.
The logical expressions in these languages are standardly taken to be the symbols for the truth-functions, the quantifiers, identity and other symbols definable in terms of those (but there are dissenting views on the status of the higher-order quantifiers; see 2.4.3 below).
To make a logic that fits these games, we use the same first-order language as in the previous section, except that a notation is added to some quantifiers (and possibly also some connectives), to show that the Skolem functions for these quantifiers (or connectives) are independent of certain variables.
Actually, a simple special case of ω-consistency suffices here; namely, the assumption is only needed with respect to what logicians call Σ01-formulas; these are, roughly, the purely existential formulas; more exactly, formulas of the form ∃x1∃x2…∃xnA, where A does not contain any unbounded quantifiers (A may contain bounded universal quantifiers ∀x < t and bounded existential quantifiers ∃x < t).
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Actually a simple special case of ω-consistency suffices here namely the assumption is only needed with respect to what logicians call Σ01-formulas these are roughly the purely existential formulas more exactly formulas of the form ∃x1∃x2…∃xnA where A does not contain any unbounded quantifiers A may contain bounded universal quantifiers ∀x quantifiers ∃x