Relations between Propositions: Consistency, Entailment, and Equivalence

On the preceding pages, we saw how to use truth tables and the truth assignment method to determine whether arguments are valid or invalid, and to determine whether an individual proposition is a tautology, contradiction, or contingency. We can use similar methods to study the logical relations between propositions or sets of propositions. On this page, we’ll consider three important logical relations: consistency, entailment, and equivalence.

Logical Consistency

Two or more propositions are logically consistent if it is possible for them to all be true at the same time. If there is no way for them all to be true at once, they are inconsistent. Inconsistent propositions are said to contradict one another.

To determine whether propositions are consistent or inconsistent, we can use either a truth table or the truth assignment method:

Logical Entailment

One proposition entails another if (and only if) there is no way for the former to be true while the latter is false.

Similarly, a set of propositions entails another set if (and only if) it is impossible for all the propositions in the first set to be true while any one of the propositions in the second set is false.

The set of propositions (A ∨ B), (A ⊃ B), and (B ⊃ A) together entail the set (A • B) and (A ≡ B). As shown in the following truth table, the fourth row is the only row were the former three propositions are all true, and the latter two propositions are both true in that row as well:
AB(A ∨ B)(A ⊃ B)(B ⊃ A)(A • B)(A ≡ B)
0001101
0111000
1010100
1111111
There is no way for all three propositions in the first set to be true while any one of the last two propositions is false, so the first set entails the latter set.
The premises of a valid argument together entail its conclusion, because there is no way for the conclusion to be false while all the premises are true. This provides another way to define validity: a valid argument is one in which the premises entail the conclusion.

Logical Equivalence

Two propositions are logically equivalent if and only if they entail each other. In other words, their truth values match in all possible circumstances: whenever one is true, the other must be true as well; and whenever one is false, the other must be false as well.