Consider the following predicates defined over the domain of people: S(x): x is a student W(x): x works hard Now express the following expressions in English: (a) S(Natalie) (b) Vx. (S(x) ⇒ W(x)) (c) ExW(x) ⇒¬S(David) (d) 3x (S(x) ^ W(x))
Formal Specification Predicate Logic
By quantification
A quantifier is a
are three quantifiers that we can use, each with its own symbol:
–The universal quantifier ∀
This quantifier enables a predicate to make a statement about all the elements in
a particular set. For example, if M(x) is the predicate x chases mice, we could write:
∀x Cats ● M(x)
This reads For all the x which are members of the set Cats, x chases mice, or, more
simply, All cats chase mice.
– The existential quantifier ∃
In this case, a statement is made about whether or not at least one element of a set
meets a particular criterion. For example, if, as above, P(n) is the predicate n is a
prime number, then we could write:
∃n ● P(n)
This reads There exists an n in the set of natural numbers such that n is a prime
number, or, put another way, There exists at least one prime number in the set of
natural numbers.
–The unique existential quantifier !
This quantifier modifies a predicate to make a statement about whether or not
precisely one element of a set meets a particular criterion. For example, if G(x) is the
predicate x is green, we could write
∃!x Cats ● G(x)
This would mean There is one and only one cat that is green.
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