Consider the pair of sentences in (5). The meanings of these two sentences are related in an important way. Specifically, in any situation for which (5a) is true, (5b) must be true as well; and in any situation for which (5b) is false, (5a) must also be false. Moreover, this relationship follows directly from the meanings of the two sentences, and does not depend on the situation or context in which they are used.
(5) a. Edward VIII has abdicated the throne in order to marry Wallis Simpson.
b. Edward VIII is no longer the King.
This kind of relationship is known as entailment; sentence (5a) entails sentence (5b), or more precisely, the proposition expressed by (5a) entails the proposition expressed by (5b). The defining properties of entailment are those mentioned in the previous paragraph. We can say that proposition p entails proposition q just in case the following three things are true:4
(a) whenever p is true, it is logically necessary that q must also be true;
(b) whenever q is false, it is logically necessary that p must also be false;
(c) these relations follow directly from the meanings of p and q, and do not depend on the context of the utterance.
This definition gives us some ways to test for entailments. Intuitively it seems clear that the proposition expressed by (6a) entails the proposition expressed by (6b). We can confirm this intuition by observing that asserting (6a) while denying (6b) leads to a contradiction (6c). Similarly, it would be highly unnatural to assert (6a) while expressing doubt about (6b), as illustrated in (6d). It would be unnaturally redundant to assert (6a) and then state (6b) as a separate assertion; this is illustrated in (6e).
(6) a. I broke your Ming dynasty jar.
b. Your Ming dynasty jar broke.
c. #I broke your Ming dynasty jar, but the jar didn’t break.
d. #I broke your Ming dynasty jar, but I’m not sure whether the jar broke.
e. #I broke your Ming dynasty jar, and the jar broke
Now consider the pair of sentences in (7). Intuitively it seems that (7a) entails (7b); whenever (7a) is true, (7b) must also be true, and whenever (7b) is false, (7a) must also be false. But notice that (7b) also entails (7a). The propositions expressed by these two sentences mutually entail each other, as demonstrated in (7c–d). Two sentences which mutually entail each other are said to be synonymous, or paraphrases of each other. This means that the propositions expressed by the two sentences have the same truth conditions, and therefore must have the same truth value (either both true or both false) in any imaginable situation.
(7) a. Hong Kong is warmer than Beijing (in December).
b. Beijing is cooler than Hong Kong (in December).
c. #Hong Kong is warmer than Beijing, but Beijing is not cooler than Hong Kong.
d. #Beijing is cooler than Hong Kong, but Hong Kong is not warmer than Beijing.
A pair of propositions which cannot both be true are said to be inconsistent or incompatible. Two distinct types of incompatibility have traditionally been recognized. Propositions which must have opposite truth values in every circumstance are said to be contradictory. For example, any proposition p must have the opposite truth value from its negation (not p) in all circumstances. Thus the pair of sentences in (8) are contradictory; whenever the first is true, the second must be false, and vice versa.
(8) a. Ringo Starr is my grandfather.
b. Ringo Starr is not my grandfather.
On the other hand, it is possible for two propositions to be inconsistent without being contradictory. This would mean that they cannot both be true, but they could both be false in a particular context. We refer to such pairs as contrary propositions. An example is provided in (9a–b). These two sentences cannot both be true, so (9c) is a contradiction. However, they could both be false in a given situation, so (9d) is not a contradiction.”
(9) a. Al is taller than Bill.
b. Bill is taller than Al.
c. #Al is taller than Bill and Bill is taller than Al.
d. Al is no taller than Bill and Bill is no taller than Al.
Finally, two sentences are said to be independent when they are neither incompatible nor synonymous, and when neither of them entails the other. If two sentences are independent, there is no truth value dependency between the two propositions; knowing the truth value of one will not provide enough information to know the truth value of the other.
These meaning relations (incompatibility, synonymy, and entailment) provide additional benchmarks for evaluating a possible semantic analysis: how successful is it in predicting or explaining which pairs of sentences will be synonymous, which pairs will be incompatible, etc.?
4 Cruse (2000: 29)
