Search results
Results from the WOW.Com Content Network
As an example, the only difference in the definition of uniform continuity and (ordinary) continuity is the order of quantifications. First order quantifiers approximate the meanings of some natural language quantifiers such as "some" and "all".
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) → ...
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.
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 ...
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. To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpretation of the theory.
However, we cannot do the same with the predicate. That is, the following expression: ∃P P(b) is not a sentence of first-order logic, but this is a legitimate sentence of second-order logic. Here, P is a predicate variable and is semantically a set of individuals. [1] As a result, second-order logic has greater expressive power than first ...
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.
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 ...