Logical Relationships

A sentence φ is logically equivalent to a sentence ψ if and only if every truth assignment that satisfies φ satisfies ψ and every truth assignment that satisfies ψ satisfies φ. A sentence φ logically entails a sentence ψ (written φ ⊨ ψ) if and only if every truth assignment that satisfies φ also satisfies ψ. A sentence φ is consistent with a sentence ψ if and only if there is a truth assignment that satisfies both φ and ψ.

As with validity and contingency and satisfiability, these definitions are the same for Relational Logic as for Propositional Logic. Moreover, if we treat ground relational sentences as propositions, we get identical results. For example, a ground premise in Relational Logic logically entails a ground conclusion in Relational Logic if and only if the corresponding Propositional Logic premise logically entails the corresponding Propositional Logic conclusion.

For example, we have the following results for logical entailment. The sentence p(a) logically entails (p(a) ∨ p(b)). The sentence p(a) does not logically entail (p(a) ∧ p(b)). However, any set of sentences containing both p(a) and p(b) does logically entail (p(a) ∧ p(b)).

The presence of variables allows for additional relationships. For example, the premise ∃y.∀x.q(x,y) logically entails the conclusion ∀x.∃y.q(x,y). If there is some object y that is paired with every x, then every x has some object that it pairs with, viz. y.

Here is another example. The premise ∀x.∀y.q(x,y) logically entails the conclusion ∀x.∀y.q(y,x). The first sentence says that q is true for all pairs of objects, and the second sentence says the exact same thing. In cases like this, we can interchange variables.

Understanding logical relationships between Relational Logic sentences is complicated by the fact that it is possible to have free variables in those sentences. Consider, for example, the premise q(x,y) and the conclusion q(y,x). Does q(x,y) logically entail q(y,x) or not?

Our definitions and the semantics of Relational Logic give a clear answer to questions like these. Logical entailment holds if and only if every truth assignment that satisfies the premise satisfies the conclusion. A truth assignment satisfies a sentence with free variables if and only if it satisfies every instance. In other words, a sentence with free variables is equivalent to the sentence in which all of the free variables are universally quantified. In other words, q(x,y) is satisfied if and only if ∀x.∀y.q(x,y) is satisfied, and similarly for q(y,x). So, the first sentence here logically entails the second if and only if ∀x.∀y.q(x,y) logically entails ∀x.∀y.q(y,x); and, as we just saw, this is, in fact, the case.