G53KRR 2008-2009 answers
1. (a) Give an inductive truth definition for first order logic, that is, define when an
interpretation M = (D, I) and an assignment µ satisfy a formula . Assume that
is defined by the following grammar:
= P(t1, . . . , tn) | ¬ | | x
where terms ti are either variables or constants. (7 marks)
Answer. Let [t]M,µ be the denotation of a term t in M under µ. For a variable x,
[x]M,µ is µ(x) and for a constant a, [a]M,µ is I(a). Then
M, µ |= P(t1, . . . , tn) iff [t1]M,µ, . . . , [tn]M,µ I(P)
M, µ |= ¬ iff M, µ |=
M, µ |= 1 2 iff M, µ |= 1 and M, µ |= 2.
M, µ |= x iff for every assignment µ which differs from µ at most in the value
for x, M, µ |= .
(b) Consider the following set of sentences:
S1 Andrew is the father of Bob.
S2 Bob is the father of Chris.
S3 Every grandfather is someone's father.
S4 Andrew is a grandfather of Chris.
Translate these sentences into first-order logic, using binary predicates Father