Introduction to Logic

Relational Logic: Satisfaction and Logical Entailment

Many students struggle with Exercise 8.3. Therefore, we've provided the following exercise that presents four different variants of the Exercise 8.3 along with hints regarding the satisfiability and entailment of the supplied relational logic sentences. Our hope is that after trying out and / or seeing the answers of this exercise, the students can better understand the relationship and the differences between consistency (also called satisfiable) and logical entailment of sentences.

You can access the four variants of Exercise 8.3 by choosing the appropriate option from the following dropdown menu. By default, Variant 1 is loaded.

Let's suppose that our vocabulary consists of the object constants: a, b and c and the binary relation constants p and r. The p relation is axiomatized below.

 ¬p(a,a) p(a,b) ¬p(a,c) ¬p(b,a) ¬p(b,b) p(b,c) ¬p(c,a) ¬p(c,b) ¬p(c,c)

Consider the following sentence about r.

Say whether each of the following sentences is (a) consistent with the above sentence about r, and (b) logically entailed by the above sentence about r.

Sentence \ Property: Consistent? Logically Entailed?
r(a, b)    r(b, c)    r(a, c)    r(a, a)    r(b, b)    r(c, c)    r(b, a)    r(c, a)    r(c, b)    Reference: What is consistency? What is logical entailment?

A sentence φ is consistent with a set of sentences Δ if and only if there exists a truth assignment that satisfies all the sentences in Δ ∪ {φ}. To show that a sentence is consistent with a supplied set of sentences, it suffices to find one truth assignment that satisfies all the supplied sentences.

A sentence φ is logically entailed by a set of sentences Δ if and only if all the truth assignments that satisfy the sentences in Δ, also satisfy φ.