Quantifiers in Formal Logic
Familiarity with classical quantification theory is presupposed here. Some proposed amendments are considered, as are several additions.
Alternatives to Classical Quantification Theory
First-order logic can be reformulated so as to avoid quantifiers and variables. This is only partially done in modal logic, which avoids explicit quantification over possible states of the world in favor of operators □ and ♢. However, in principle all quantification is avoidable, if one is willing to admit enough operators and does not worry about their having ordinary-language readings. In practice, however, few have preferred this predicate-functor approach (see Quine 1960, Benthem 1977). Thus, even such dissidents as the intuitionists adopt the classical quantificational language, though the properties they ascribe to the quantifiers are nonclassical. (Thus, while classically ∀ and ¬¬∀ and ∀¬¬ are equivalent, intuitionistically the first is stronger than the second and the second stronger than the third.)
Classical logic allows terms formed from constants and function symbols, subject to the restriction that each term must denote some element of the domain over which the quantifiers range; but terms are eliminable using Bertrand Russell's theory of descriptions. On the classical Tarskian definition of truth in a model, truth of ∀xϕ(x) (respectively, ∃xϕ(x)) is equivalent to the truth of ϕ(t) for all (respectively, some) terms t only in special cases, as when each element of the domain is the denotation of some term of the language (which is never so if the domain is uncountable and the language countable).
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