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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 .
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.
Thus, for example, quantifiers over sets of individuals may range over only a subset of the powerset of the set of individuals. HOL with these semantics is equivalent to many-sorted first-order logic, rather than being stronger than first-order logic. In particular, HOL with Henkin semantics has all the model-theoretic properties of first-order ...
In mathematical logic, a propositional variable (also called a sentence letter, [1] sentential variable, or sentential letter) is an input variable (that can either be true or false) of a truth function. Propositional variables are the basic building-blocks of propositional formulas, used in propositional logic and higher-order logics.
For example, consider the following expression in which both variables are bound by logical quantifiers: ∀ y ∃ x ( x = y ) . {\displaystyle \forall y\,\exists x\,\left(x={\sqrt {y}}\right).} This expression evaluates to false if the domain of x {\displaystyle x} and y {\displaystyle y} is the real numbers, but true if the domain is the ...
If f is such a k-ary function variable and t 1,...,t k are first-order terms then the expression f(t 1,...,t k) is a first-order term. Each of the variables just defined may be universally and/or existentially quantified over, to build up formulas. Thus there are many kinds of quantifiers, two for each sort of variables.
Sentences without any logical connectives or quantifiers in them are known as atomic sentences; by analogy to atomic formula. Sentences are then built up out of atomic sentences by applying connectives and quantifiers. A set of sentences is called a theory; thus, individual sentences may be called theorems.
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) → ...