Linguistics An Essential Introduction (Version 1.5)

Look at the sentences in (1):

They differ in that (1a) is an active sentence, while (1b) is a passive sentence. We will discuss their formal relationship in detail in Section 8.6, but for now, let us focus on their meaning. If we call the proposition of (1a) “p” and the proposition of (1b) “q”, we can construct the following truth table:

In a truth table, each column stands for one proposition (or set of propositions) associated with a given sentence, while 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’). In this case, the table shows that if the proposition of (1a) is true, then the proposition of (1b) is also true, and if the proposition of (1a) is false, then the proposition of (1b) is also false.

In other words, the two sentences are true under the same conditions in the same actual and possible worlds — their propositions are identical. They are paraphrases, or, to put it in terms that we already know, synonyms. In this case, this is due to the grammatical synonymy between the active and the passive voice, but truth tables can also represent the kind of lexical synonymy that we have already seen in Section 3.5. The sentence in (1c) also has the same truth conditions as (1a) and (1b):

Here, this is due to the fact that cut down is a synonym of fell (at least in the word sense relevant to the context). In other words, we can define lexical synonyms as words that can replace each other in a sentence without changing its truth conditions!

Note that paraphrases do not have to mean exactly the same thing — there is a difference between the active voice in (1a) and the passive voice in (1b), for example, in that (1a) focuses on Zoe, while (1b) focuses on the tree. And while fell and cut down do not change the truth conditions, they have different connotations — fell evokes a context where a tree is cut down in order to use its wood, typically by a lumberjack, while cut down is a more generic way of describing the same action. 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.

Look at the following pair of sentences:

A dog that is owned by someone is a pet, so if Zoe is looking after Noah’s dog, both (2a) and (2b) are true. However, not all pets are dogs, so (2b) can also be true in situations where (2a) is false — for example, if Zoe is looking after Noah’s pet chameleon. And if (2b) is false, then (2a) must also be false — if Zoe is not looking after any of Noah’s pets, it follows that she is not looking after his dog.

This sounds very complicated, but if we use a truth table where (2a) is p and (1b) is q, we can represent it very simply:

This type of relation is called entailment — the sentence in (2a) entails the sentence in (2b). In this case, this entailment is caused by the systematic relationship between a hyponym (dog) and its hypernym (pet), but entailments can also be caused by less systematic relations between word meanings:

Here, the entailment is due to the fact that adopt means “legally take the role of parent to a person who is not one’s biological child” — if it is true that Barbara went through this process with Noah (as 3a states), then he is now her child. Conversely, adoption is only one way of establishing a parent-child relationship, others are giving birth to a child, being married to a person giving birth to a child, declaring oneself to be the biological father of a child, etc., and (3b) could be true due to any of these reasons.

Entailment can also be due to systematic grammatical relations between sentences. Look at the following examples (Laika is Noah’s dog):

The verb dry can be used transitively (with a subject and an object), as in (4a), or intransitively (with just a subject), as in (4b). We will look at formal aspects of such verbs in more detail in Chapter 8, but here, we are interested in the semantic relation: if the transitive version in (1a) is true, the intransitive version in (1b) is also true, but (1b) can also be true if (1a) is false — Noah could have used a towel to dry Laika’s fur, Laika could have crawled under a radiator, etc.

Sentences containing change-of-state verbs also often entail sentences containing an adjective describing an end-state — both (4a) and (4b) entail (4c).

Next, look at the following pair of sentences:

The relationship is quite straightforward: if (5a) is true, then (5b) is false, and if (5a) is false, then (5b) is true. The corresponding truth table looks like this:

This relationship is called contradiction. In this case, the contradiction is due to the use of the complementary antonyms alive and dead, but, as in the case of entailment, it can also be due to less systematic relations between words. Look at the following:

To quit means “to stop doing”, and for good means “never changing back to the previous situation”. Thus, if Noah quit vaping for good, it means he cannot be vaping, and if he is vaping, he has not quit.

Finally, a contradictory relationship can be produced through grammatical means, specifically, negation:

Simply put, the negative particle not flips the truth value of a sentence from true to false. Because negation is very important in formal logic, it has its own special symbol, which is also often used in linguistics: ¬. Instead of calling (7b) ‘q’ in the truth table above, we could also call it ‘¬p’.

Next, look at the following pair of sentences:

Like (5a, b), these sentences contain antonyms, but in this case, they are scalar antonyms rather than complementary ones. This has an effect on the relationship between the corresponding propositions. 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). This relationship is represented in the following truth table:

This relationship is called contrariety. It does not have to be based on scalar antonymy, it can also be based on world knowledge, as in the following case:

To serve a jail sentence means to be locked up in a jail for a certain period of time, and we known that inmates are not allowed to keep pets. Thus, if (9a) is true, (9b) must be false, and if (9b) is true, (9a) must be false. However, if (9b) is false, this does not tell us anything about (9a) - it could be true or false, as there are indefinitely many reasons why Zoe might not be looking after Noah’s dog.

CC-BY-NC-SA 4.0. Written by Anatol Stefanowitsch