Introduction to Logic

Exercise 8.3 - Consistency

Consider a version of the Blocks World with just three blocks - a, b, and c. The on relation is axiomatized below.

¬on(a,a)   on(a,b)   ¬on(a,c)
¬on(b,a)   ¬on(b,b)   on(b,c)
¬on(c,a)   ¬on(c,b)   ¬on(c,c)

Let's suppose that the above relation is defined as follows. This is almost the same as in the notes except that we have replaced an occurrence of on with above and we have dropped the axiom ∀xabove(x,x).

x.∀z.(above(x,z) ⇔ on(x,z) ∨ ∃y.(above(x,y) ∧ above(y,z)))

A sentence φ is consistent with a set Δ of sentences if and only if there is a truth assignment that satisfies all of the sentences in Δ ∪ {φ}. Say whether each of the following sentences is consistent with the sentences about on and above shown above. Be careful. It's tricky.

a. above(a,c)
b. above(a,a)
c. above(c,a)