Summary: Express the following knowledge in first-order logic and add enough common
sense statements (e.g. everyone has at most one spouse, nobody can be married
to himself or herself, Tom, Sue and Mary are different people) to make it entail
"Mary is not married" in first order logic.
Knowledge: There are exactly three people in the club, Tom, Sue and Mary.
Tom and Sue are married. If a member of the club is married, their spouse is
also in the club.
Answer We are going to use constants tom, sue, mary, unary predicate C
for member of the club, and binary predicate Married.
Translation of the sentences:
S1 There are exactly three people in the club, Tom, Sue and Mary:
C(tom)C(sue)C(mary)x(C(x) (x = tomx = suex = mary))
S2 Tom and Sue are married:
S3 If a member of the club is married, their spouse is also in the club:
xy(C(x) Married(x, y) C(y))
S4 We need to show that this entails "Mary is not married":
In first-order logic, S1-S3 do not entail S4. Consider for example an inter-
pretation where Mary is married to Tom (nothing in S1-S3 forbids this). Then