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In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier ∀ {\displaystyle \forall } in the first order formula ∀ x P ( x ) {\displaystyle \forall xP(x)} expresses that everything in the domain satisfies the property denoted by P ...
An open formula can be transformed into a closed formula by applying a quantifier for each free variable. This transformation is called capture of the free variables to make them bound variables. For example, when reasoning about natural numbers, the formula "x+2 > y" is open, since it contains the free variables x and y.
Example. In a given propositional logic, a formula can be defined as follows: Every propositional variable is a formula. Given a formula X, the negation ¬X is a formula. Given two formulas X and Y, and a binary connective b (such as the logical conjunction ∧), the expression (X b Y) is a formula. (Note the parentheses.)
[1] [2] Examples in English include articles (the and a), demonstratives (this, that), possessive determiners (my, their), and quantifiers (many, both). Not all languages have determiners, and not all systems of grammatical description recognize them as a distinct category.
The T-schema interprets the logical connectives using truth tables, as discussed above. Thus, for example, φ ∧ ψ is satisfied if and only if both φ and ψ are satisfied. This leaves the issue of how to interpret formulas of the form ∀ x φ(x) and ∃ x φ(x). The domain of discourse forms the range for these quantifiers.
In formal semantics, a generalized quantifier (GQ) is an expression that denotes a set of sets. This is the standard semantics assigned to quantified noun phrases . For example, the generalized quantifier every boy denotes the set of sets of which every boy is a member: { X ∣ ∀ x ( x is a boy → x ∈ X ) } {\displaystyle \{X\mid \forall x ...
For example, if the domain is the set of all real numbers, one can assert in first-order logic the existence of an additive inverse of each real number by writing ∀x ∃y (x + y = 0) but one needs second-order logic to assert the least-upper-bound property for sets of real numbers, which states that every bounded, nonempty set of real numbers ...
A second-order propositional logic is a propositional logic extended with quantification over propositions. A special case are the logics that allow second-order Boolean propositions, where quantifiers may range either just over the Boolean truth values, or over the Boolean-valued truth functions.