A quantified statement consists of three parts: the quantifier, the quantification, which describes the variables and the type of variable with which the statement is concerned and the predicate which is normally some statement about the quantified variables.
Every predicate logic statement can be considered as follows
As universal quantification is used to assert that a certain property holds of every element of a set, existential quantification is used to assert that such a property holds of some (or at least) elements of a set
“ Some natural numbers are divisible by 3” may be written as
A predicate p is satisfiable if and only if it is true for some values of the appropriate type. That is, if a predicate p is associated with a variable x of type X, then p is satisfiable if, and only if ,
When negation is applied to a quantified expression it flips quantifiers as it moves inwards(i.e negation turns all universal quantifiers to existential quantifiers and vice versa, and negates all predicates)
Exercise, Determine the scope of the universal and existential quantifiers of the following predicate,
The distinction between predicates and propositions can be stated as predicates are logical statements which can contain free variables while propositions are logical statements which contain no such holes
we may replace the name n with any term t, provided that t is of the same type as n. This process is called substitution: the expression p[t/n] denotes the fact that the term t is being substituted for the variable name n in the predicate p.
Example, we may denote the substitution of 3 for n in the predicate n>5 by