Predicates and quantifiers
A predicate is an expression of one or more variables determined on some specific domain. A predicate with variables can be made a proposition by either authorizing a value to the variable or by quantifying the variable.
Definition :
A predicate is statement that contains variables. A predicate may be true or false depending on the values of these variables.
Universal quantifier
means that the predicate
is true for all possible values of
.
A universal quantification is a type of quantifier[1], a logical constant which is interpreted as "given any", "for all", or "for any". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other words, it is the predication of a property or relation to every member of the domain. It asserts that a predicate within the scope of a universal quantifier is true of every value of a predicate variable.
Example :
, i.e., "the square of any number is not negative".
, i.e., "all squares are rectangles."
Existential quantifier
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".
means that there exists an
is true. Sometimes, we will use also
. It means that there exists a unique
where
is true.
Example :
is true.
is true, but
is false.
is true. The order of the quantifiers is very important, this statement
is false.
A quantified propositional function is a statement; thus, like statements, quantified functions can be negated. The negation of a propositional function's existential quantification is a universal quantification of that propositional function's negation.
Example :
Statement:
and negation:
Statement:
and negation:
Statement:
and negation: