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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 in the first order formula () expresses that everything in the domain satisfies the property denoted by .
In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it is possible to say that an object is a whole number using a formula of L ω 1 , ω {\displaystyle L_{\omega _{1},\omega }} such as
The Skolem term () contains , but not , because the quantifier to be removed is in the scope of , but not in that of ; since this formula is in prenex normal form, this is equivalent to saying that, in the list of quantifiers, precedes while does not. The formula obtained by this transformation is satisfiable if and only if the original formula is.
In classical first-order logic, quantifiers are only applied to individuals. The formula " (() ()) " (some apples are sweet) is an example of the existential quantifier " " applied to the individual variable " ". In higher-order logics, quantification is also allowed over predicates.
Example requires a quantifier over predicates, which cannot be implemented in single-sorted first-order logic: Zj → ∃X(Xj∧Xp). Quantification over properties Santa Claus has all the attributes of a sadist. Example requires quantifiers over predicates, which cannot be implemented in single-sorted first-order logic: ∀X(∀x(Sx → Xx) → ...
In standard semantics, also called full semantics, the quantifiers range over all sets or functions of the appropriate sort. A model with this condition is called a full model, and these are the same as models in which the range of the second-order quantifiers is the powerset of the model's first-order part. [3]
In semantics and mathematical logic, a quantifier is a way that an argument claims that an object with a certain property exists or that no object with a certain property exists. Not to be confused with Category:Quantification (science) .
Conditional quantifiers are meant to capture certain properties concerning conditional reasoning at an abstract level. Generally, it is intended to clarify the role of conditionals in a first-order language as they relate to other connectives , such as conjunction or disjunction.