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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 .
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 predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, when used together with a predicate variable, is called an existential quantifier (" ∃x" or "∃(x)" or ...
First-order logic can quantify over individuals, but not over properties. That is, we can take an atomic sentence like Cube(b) and obtain a quantified sentence by replacing the name with a variable and attaching a quantifier: [1] ∃x Cube(x) However, we cannot do the same with the predicate. That is, the following expression: ∃P P(b)
A predicate is a statement or mathematical assertion that contains variables, sometimes referred to as predicate variables, and may be true or false depending on those variables’ value or values. In propositional logic, atomic formulas are sometimes regarded as zero-place predicates. [1] In a sense, these are nullary (i.e. 0-arity) predicates.
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.
Quantifiers differ with respect to whether or not they can be the subject of a collective predicate. For example, quantifiers formed with all the can, while ones formed with every or each cannot. All the students formed a line. — Collective predicate possible with all the. All the students gathered in the hallway.
For example, to express the proposition "this raven is black", one may use the predicate for the property "black" and the singular term referring to the raven to form the expression (). To express that some objects are black, the existential quantifier ∃ {\displaystyle \exists } is combined with the variable x {\displaystyle x} to form the ...