Recall our discussion above: necessarily, B in (16) is true, if A is true. This is an instance of entailment:
(16) A. Donald Duck is a duck.
B. Donald Duck is a bird.
The relation of logical entailment is defined by one crucial condition: it is impossible that B is false if A is true. (This is the case with A and B in (16): it is impossible that Donald is a duck (A true) and not a bird (B false). Hence, if he is a duck, he necessarily is a bird.
DEFINITION A B
A logically entails B/B logically follows from A 1 1
A
c
B 1 0 impossibleif and only if6: 0 1
necessarily, if A is true, B is true. 0 0
Two arbitrary sentences A and B may be independently true or false. This yields four possible combinations of truth values. Logical entailment rules out one of these four combinations, A-true-B-false. If A entails B, the truth values of A and B depend on each other in a particular way: B cannot be false if A is true, and A cannot be true if B is false. Thus, logical entailment results in a certain link between the truth conditions of the two sentences.
The definition of entailment does not tell us anything about the remaining three combinations of truth values. When we say that in (16) a entails b, we say so because of the general condition that ducks are birds.
Remember that we found it difficult to decide whether Donald Duck is a duck or not. For the question whether A entails B this does not matter because logical entailment means that if A is true, then B must be true. If Donald is a duck, he must be a bird. But if he is not a duck, he still may be a bird or not. Donald might be a raven; then a is false and b is true. This is
04-UnderSemantics-Ch4-cp 04/04/2002 8:36 am Page 64
M E A N I N G A N D L O G I C 65
admissible. He might be a cow; then both a and b are false. This too is admissible. What is not admissible is his being a duck but not a bird. Let me give you three more examples of entailments.
(17) A It’s raining heavily.
c
B It’s raining.(18) A Ann is a sister of my mother.
c
B Ann is an aunt of mine.(19) A Today is Monday.
c
B Today isn’t Wednesday.If it is raining, but not heavily so, one can say (17B), but not (17A). Likewise, (18B) can be true without (18A) being true: Ann could as well be a sister of my father or the wife of an uncle of mine. That (19B) may be true and (19A) at the same time false, is immediately clear. In principle, the relation of entailment is asymmetric: A may entail B without B entailing A. (In general, a relation is symmetric if and only if x is in the relation to y entails y is in the relation to x, it is asymmetric if this does not hold.) Applying the definition of entailment to the case of B entailing A yields the picture in Table 4.1 (it rules out the combination B-true-A-false). Accordingly, B does not entail A iff B-true-A-false is possible.6If we add this condition to the condition for A entailing B, we obtain a table for A unilaterally entailing B (Table 4.2).
There is one way of reversing an entailment: if A entails B, then A is nec-essarily false, if B is false. Table 4.3 shows how the truth values of not-A and not-B co-vary with those of A and B. Ruling out the combination A-true-B-false is obviously the same as ruling out (not-B)-true-(not-A)-A-true-B-false.
B entails A
A B
1 1
1 0
0 1 impossible
0 0
Table 4.1
A unilaterally entails B
A B
1 1
1 0 impossible
0 1 possible
0 0
Table 4.2
A entails B = not-B entails not-A
A B not-B not-A
1 1 0 0
1 0 1 0
0 1 impossible 0 1
0 0 1 1
Table 4.3
04-UnderSemantics-Ch4-cp 04/04/2002 8:36 am Page 65
66 U N D E R S T A N D I N G S E M A N T I C S
Hence, if Donald Duck is not a bird, he cannot be a duck; if it is not rain-ing, it cannot be raining heavily, and so on. A
c
B is equivalent to not-Bc
not-A. For example, (16) yields:
(20) not-B Donald Duck is not a bird.
c
not-ADonald Duck is not a duck.Let us now take a look at a few examples which are not cases of logical entailment, although in each case sentence B would under normal circumstances be inferred from A. What matters, however, is whether the consequence is really necessary or whether it is based on some additional assumptions.
(21) A Mary is John’s mother.
c
/ B Mary is the wife of John’s father.(22) A John said he is tired.
c
/ B John is tired.(23) A The beer is in the fridge.
c
/ B The beer is cool.There are no logical reasons for drawing these conclusions. It is logically possible that parents are not married, that John was lying, or that the fridge does not work or the beer has not been in it long enough. In most cases we draw our conclusions on the basis of our world knowledge, i.e. of what we consider normal, plausible or probable. The notion of logical entailment does not capture all these regularities and connections. It just captures the really ‘hard’ cases of an if-then relation, those based on the Principle of Polarity and the semantic facts alone.
What does logical entailment mean for the meanings of A and B? If A and B are contingent and A unilaterally entails B, both sentences contain infor-mation about the same issue, but the inforinfor-mation given by A is more specific than the information given by B. The (truth) conditions that B imposes on the situation are such that they are always fulfilled if A is true. Therefore, the truth conditions of B must be part of the truth conditions of A. In general, if no further logical relation holds between A and B, A will impose additional conditions on the situation referred to. In this sense, A contains more information, i.e. is more informative and more specific, than B. The situation expressed by A is a special case of the situation expressed by B. As we shall see in 4.3.5, this does not hold if A and/or B are not contingent.
One further property should be noted here: logical entailment is what is called a transitive relation. The general property of transitivity7is defined as follows: a relation R is transitive if and only if ‘x is in relation R to y’ and
‘y is in relation R to z’ entails ‘x is in relation R to z’. Applied to entailment, this means that if A entails B and B entails C then A entails C. For example, Donald is a duck
c
Donald is a bird; Donald is a birdc
Donald is an animal;hence Donald is a duck
c
Donald is an animal. The property of transitivity immediately follows from the way entailment is defined. Suppose Ac
Band B
c
C; then if A is true, necessarily B is true; if B is true, necessarily C is true, hence: if A is true, necessarily C is true, i.e. Ac
C.04-UnderSemantics-Ch4-cp 04/04/2002 8:36 am Page 66
M E A N I N G A N D L O G I C 67