In Chapter 6, we introduced the idea that the meaning of an expression can be captured in terms of its extension — the set of entities in a possible world to which we can truthfully apply that expression —, and its intension — the conceptual content associated with the expression. We can truthfully apply an expression to an entity if the entity matches the conceptual content. We can extend this idea to sentences (or, more precisely, clauses), which are, as we have seen in Chapter 7, combinations of smaller expressions.

The basic idea is that the meaning of a sentence is a conceptual structure on whose basis we can determine the conditions under which it would be true in a possible world. The conceptual structure (the intension) of a sentence is called its proposition in linguistics (not to be confused with preposition). The conditions under which the proposition expressed by the sentence is true are called its truth conditions. Look at the following sentence:

The sentence expresses a complex proposition, which consists of the intension of its individual parts: the prepositional verb look after, the noun phrases Zoe and dog, and the possessive phrase Noah’s. Using upper case to represent meaning and the letters x, y and z to stand for potential referents, the proposition is something like (2)

Example (1) is true in any situation where (2a-e) are true.

Now look at the sentence in (3):

It differs from (1) in that it is a passive sentence, systematically related to (1) by a rule like the one we discussed in Section 7.6. But, crucially, it expresses the same propositions as (1), and is therefore true in the same situations.

The relationship between the truth conditions of sentences can be shown using a simple tool called a truth table — a table where each column stands for the propositions associated with a given sentence, and the rows contain the values T (‘true’) and F (‘false’) for each column. It is customary to label the propositions using lowercase letters starting from the letter p (for ‘proposition’). Let example (1) be p and example (2) q, then the truth table for the two sentences would look as follows:

𝗽	𝗾
T	T
F	F

The rows of the tables show relationship between the two sentences in terms of what truth value q has for each truth value of p: if p is true, q is also true, and if p is false, q is also false. Such sentence pairs are called paraphrases of each other. Paraphrases can be purely grammatical, like (1) and (3), or they can be lexical, like (1) and (4), which contains a synonym of look after:

Note that paraphrases do not have to mean exactly the same thing — there is a difference between the active voice in (1) and the passive voice in (2), for example, in that (1) focuses on Zoe, while (2) focuses on Noah’s dog. And while look after and take care are synonyms (dictionaries often use one expression to define the other), they have different connotations — look after has a caring, nurturing ring to it, take care a more organizational one. So propositions (and their truth conditions) do not capture all aspects of a sentence’s meaning, but they capture the most important part — its denotation.

You might think that the table above is a very roundabout way of stating something very simple — that two sentences have the same meaning. But it is also a very transparent and explicit way of stating such relationships, which is useful when it comes to more complex relationships.

Let us look at the most important of these relationships, using sentence (1) above and the sentences in (5a) to (5c):

Let us begin with example (5a). If (5a) is true, (1) is also true (dogs are pets). But if (1) is false, (5a) could be true or false: it would be true if Zoe was looking after Noah’s pet chameleon instead of his dog, and it would be false if Zoe was not looking after any of Noah’s pets. Conversely, if (5a) is false, (1) is also false — if Zoe is not looking after any of Noah’s pets, it follows that she is not looking after his dog. This relationship seems very complex, but let (1) be p and (5a) q, and we can represent it as follows:

𝗽	𝗾
T	T
F	T/F

This relation is called entailment — the sentence in (1) entails the sentence in (5a). In this case, this is due to the relationship between the hyponym dog and the hypernym pet — all dogs are pets, but not all pets are dogs. But entailments can also be due to other aspects of word meaning or grammar. For example, the valency between intransitive and transitive uses of change-of-state verbs which we discussed in Section 7.5.1 is associated with an entailment relation — (6a) entails (6b):

Sentences containing change-of-state verbs also often entail sentences containing an adjective describing an end-state — both (6a) and (6b) entail (7), and (8a) entails (8b):

Let us turn to example (5b). If (1) is p and (5b) is q, the relationship between them is the following:

𝗽	𝗾
T	F
F	T

This relationship is called contradiction: if p is true, q is false, and vice versa. In this case, this is due to the presence of the negative particle not in (5b): simply put, negation flips the truth value of a sentence. Because negation is very important in formal logic, it has its own special symbol, which is also often used in linguistics: ¬. Instead of calling (5b) ‘q’ in the truth table above, we could also call it ‘¬p’. Contradiction can also be due to the use of complementary antonyms — the table above also captures the relationship between sentence pairs like those in (7a, b):

In sentence pairs containing scalar antonyms, we get a different type of relation. Look at sentences (8a, b):

If (8a) is true, (8b) is false and vice versa — just as in the case of complementary antonyms. However, if (8a) is false, then (8b) could be true (Zoe could be sad) or false (Zoe could be in a neutral mood, i.e. neither happy nor sad):

𝗽	𝗾
T	F
F	T/F

This relationship is called contrariety. It does not have to be based on scalar antonymy, it can also be based on world knowledge. For example, (5c) is in a contrary relation to (1) — if Zoe is looking after Noah’s dog, it follows that she is not serving a jail sentence at the same time — inmates are not allowed to bring dogs with them. However, if Zoe is not looking after Noah’s dog, it does not follow that she is in prison.

 

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