## 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:

• Truth table test for consistency: Two or more propositions are consistent if and only if there is at least one row in which they are all true. Otherwise, they are inconsistent.

The three propositions (A ⊃ B), (A ∨ B), and ~A are consistent, because there is a row in which all three are true:
AB(A ⊃ B)(A ∨ B)~A
00101
01111
10010
11110
The two propositions (A • B) and (~A ∨ ~B) are inconsistent, because there is no row in which both are true:
AB(A • B)(~A ∨ ~B)
0001
0101
1001
1110

• Truth assignment test for consistency: Assign “1” to the main connective of each proposition, then calculate the truth values of any other connectives and sentence letters that can be determined based on that assumption. If some letters cannot be calculated, try all possible combinations of values for those letters. If there is a way to assign truth values to the letters while keeping all of the propositions true, the propositions are consistent. Otherwise, they are inconsistent.

### Logical Entailment

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

• Truth table test for entailment: One proposition entails another if (and only if) there is no row where the former is true but the latter isn’t.

The proposition (A ≡ B) doesn’t entail (A • B), because there is a row where (A ≡ B) is true but (A • B) isn’t:
AB(A ≡ B)(A • B)
0010
0100
1000
1111
However, (A • B) does entail (A ≡ B), because there is no way for (A • B) to be true without (A ≡ B) being true as well. The proposition (A • B) is true only in the last row, and (A ≡ B) is true in that row too.

Thus, (A • B) entails (A ≡ B), but not vice versa.

• Truth assignment test for entailment: To determine whether one proposition entails another, assign “1 ” to the first proposition and assign “0” to the second proposition. Then calculate the truth values of any other connectives and sentence letters that can be determined based on that assumption. If some letters cannot be calculated, try all possible combinations of values for those letters. If there is a way to assign truth values to all the letters while keeping the first proposition true and the second false, then the first proposition doesn’t entail the second. On the other hand, if you can’t assign truth values to everything while keeping the first proposition true and the second false, then the first does entail the second.

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.
• Truth table test for equivalence: Two propositions are equivalent if and only if their truth values match in every row. In other words, equivalent propositions have identical columns in the truth table.

The proposition ~(A • B) is equivalent to (~A ∨ ~B). Both propositions are true in the first three rows and false in the last row:
AB~(A • B)(~A ∨ ~B)
0011
0111
1011
1100

• Truth assignment test for equivalence: First use the truth assignment test described above to determine whether the first proposition entails the second; then test whether the second entails the first. The two propositions are equivalent if (and only if) they entail each other.