
Introduction to Logic

Tools for Thought

A set of Relational Logic sentences Δ logically entails a sentence φ (written Δ ⊨ φ) if and only if every truth assignment that satisfies Δ also satisfies φ.

As with validity and contingency and satisfiability, this definition is the same for Relational Logic as for Propositional Logic. As before, if we treat ground relational sentences as propositions, we get similar results. In particular, a set of ground premises in Relational Logic logically entails a ground conclusion in Relational Logic if and only if the corresponding set of Propositional Logic premises logically entails the corresponding Propositional Logic conclusion.
For sentences without variables, we have the following results. 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)).
p(a) ⊨ (p(a) ∨ p(b))
p(a) ⊭ (p(a) ∧ p(b))
{p(a), p(b)} ⊨ (p(a) ∧ p(b))

The presence of variables allows for additional logical entailments. 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.
∃y.∀x.q(x,y) ⊨ ∀x.∃y.q(x,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.
∀x.∀y.q(x,y) ⊨ ∀x.∀y.q(y,x)

Understanding logical entailment for Relational Logic is complicated by the fact that it is possible to have free variables in Relational Logic 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 definition for logical entailment and the semantics of Relational Logic give a clear answer to this question. 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.

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