Represent the following adjectives as predicates, paying attention to whether they have one or two arguments: happy, proud, guilty, evil, interesting, different. Give an example sentence for each of them.
Section 4.1 Predicates and propositions
In Section 3.1, we introduced the idea that the meaning of a word can be captured in terms of its intension, and the extension that follows from this intension. We limited our discussion to nouns specifically, to nouns referring to entities. The intension of such a noun is a set of properties, and the extension is the set of entities in the world which have these properties (or a relevant subset of the properties).
So, what about other word classes, such as verbs, adjectives and prepositions? Words in these classes have meaning, but they are not used to talk about entities, so their intensions and extensions must be qualitatively different from those of nouns.
Subsection Predicates
Look at the following example sentences:
- (1a)
- Zoe felled the dead tree outside her apartment.
- (1b)
- This entity is a tree.
- (1c)
- This tree is dead.
Example (1a) is the sentence we introduced at the beginning of Section 3.1. We have discussed how we can determine whether an entity is in the extension of the noun tree by checking whether it has the properties “is a plant”, “is woody”, “has a single stem” etc.
How might we transfer this idea to the adjective dead? Note that, unlike a noun like tree, we cannot determine the extension of such an adjective in isolation. We use it to describe a property, and properties must always be properties of something. In other words, its extension is the set of entities that have the property “is dead”, and its intension is that property.
We can represent the meaning of the adjective as follows, where the word DEAD in uppercase letter represents the meaning, and the lowercase letter x in parentheses is a placeholder indicates that the adjective must apply to some entity (such as a person, an animal or a plant):
Words like this, that denote a property (or, as we will see later, a relation) are called predicates, and the entities signaled by the placeholder are called arguments (note that both of these words are used very differently from the way you might use them in everyday language). The phrase dead tree or the sentence The tree is dead in (1c) can be represented as follows:
We will come back to the question how the determine the intension and extension of such a combination of a predicate and an argument in the next section.
The adjective dead and many other adjectives have a single argument — the entity to which the property expressed by the adjective applies. But there are also adjectives that have two arguments:
The adjective fond in (3a) expresses a property of Zoe, but it also requires a second entity that is the target of Zoe’s fondness. It can be represented as FOND(x, y). Likewise, comparative forms like older always need two arguments — the entity with the property expressed by the adjective and the entity to which it is compared. This can be represented as OLDER(x, y) (we will look at comparatives again in Chapter 7).
Question 4.1.1.
Let us now transfer the idea to verbs. As with adjectives, the extension of a verb cannot be determined in isolation — verbs typically describe states, processes or actions, and these always involve entities — entities exist in a certain state, undergo a certain process, perform a certain action (or are affected by it). Take the case of fell: the action of felling something involves two entities: one (typically a person, but also certain animals, such as beavers, or weather phenomena, such as hurricanes) who applies force to another one (typically a tree). We can represent the meaning of the predicate fell as shown in (4a), and that of the sentence Zoe felled a tree as shown in (4b)
The extension of the verb fell is the set of all pairs of entities that are in a relationship that has the properties associated with a felling event, i.e., a relationship where one entity is a tree and one is acting on the tree in a way that seems intended to make the tree fall over (i.e. by using teeth, like a beaver, or a sharp tool, like a lumberjack, or strong air pressure, like a hurricane), etc.
Verbs, too, can differ in their number of arguments. Fell has two, but there are also verbs expressing predicates that have one, three or even four arguments:
- (5a)
- SLEEP(x), e.g. Zoe slept.
- (5b)
- GIVE(x, y, z), e.g. Aylin gave Zoe a bagel.
- (5c)
- BUY(w, x, y, z), e.g. Aylin sold Noah her old phone for 50 Euros.
Question 4.1.2.
Represent the following verbs as predicates, paying attention to whether they have one or two arguments: stumble, watch, melt, hand, drink, buy. Give an example sentence for each of them.
Prepositions, too, typically express relations between two entities and can be thought of as predicates: the preposition outside in (1a) expresses a relation between two entities where one is a concrete entity of any kind, and one is an entity with a boundary enclosing an area or volume, and where the first entity is not located in the enclosed area or volume. It can be represented as shown in (4a), the phrase the tree outside the apartment can be represented as shown in (4b):
Remark 4.1.3.
We have treated verbs, adjectives and prepositions as predicates, and nouns as arguments, and we will mostly continue to do so, but note that we can treat common nouns as predicates too. We said that the intension of the adjective dead is the property “is dead” and that its extension is the set of all entities that have this property. This is very similar to saying that the intension of the noun tree is a set of properties such as “is woody”, “is tall”, etc. and that the extension of the word is the set of all entities that have this property: we could represent the word tree as follows, where the uppercase TREE stands for the intension of the word and the x stands for an entity of which we say that it is a tree:
Strictly speaking, then, proper nouns are the only words that cannot be treated in this way, because they refer directly.
Often, there is nothing gained by this type of representation, but note that there are nouns that must be thought of like this — namely nouns that specify a relation between two entities, like friend in (ii). One cannot simply be a friend, one always has to be a friend of someone, so the word friend has two arguments, as shown in (iii), and the meaning of (ii) can be represented as (iv):
Think about which of the following nouns haveto be represented as predicates (give example sentences): dog, pet, lawyer, boss, daughter, girl, money, capital.
Subsection Propositions
We have made use of the idea that the extension of a noun is the set of entities to which it can be truthfully applied. We then extended this idea to adjectives, verbs and prepositions, arguing that they specify properties or relations, and that their extension is the set of entities that have these properties or that are in related in this way.
Note that we did not apply this idea to isolated nouns, verbs, adjectives etc., but to sentences containing the words we were interested in. For nouns, adjectives and prepositions we used very simple sentences of the form “x (entity) is a N” (for example, This entity is a tree) and “x (entity) is A” (for example, The tree is dead) and “x (entity) is P” (for example, The tree is outside the apartment). For verbs, we used sentences containing the verb in question and words specifying the entities involved in a specific instance of the state, process or event expressed by the verb (for example, Zoe felled the tree).
The reason for this is that words in isolation do not express statements whose truth we can check — we word tree cannot be true or false, but the statement This is a tree can; the word fell certainly cannot be true or false, only a statement that specifies who is performing this action on whom can, etc. The meaning of such a statement is called its proposition (not to be confused with the term preposition). In the simplest case, a proposition consists of a predicate and its arguments (such propositions are sometimes called atomic propositions).
Sentences and propositions are not necessarily in one-to-one relationship: the meaning of a given sentence may (and often does) consist of a number of propositions. Take, again, the sentence cited at the beginning of the preceding section:
The meaning of this sentence encompasses the predicates associated with the verb fell (see 2a), the adjective dead (see 2b), and the preposition outside (see 2c), which we already looked at above, and an additional predicate associated with the possessive pronoun her, which specifies a relation between Zoe and the apartment (see 2d):
In addition, it contains the referring expressions Zoe, tree and apartment, where Zoe is the first argument of (2a) and the second argument of (2d), tree is the second argument of (2a), the argument of (2d) and the first argument of (2c), and apartment is the second argument of (2c) and the first argument of (2b).
Thus, the meaning of the sentence is the set of conditions under which all of the following propositions are true (in the actual or a possible world):
- (3a)
- FELL(ZOE, TREE)
- (3b)
- DEAD(TREE)
- (3c)
- OUTSIDE(TREE, APARTMENT)
- (3d)
- BELONG(APPARTMENT, ZOE)
Or, to paraphrase informally, the set of conditions under which it is true that Zoe felled a tree, where the tree was dead and where the tree was outside an apartment, and where that apartment belonged to Zoe. The set of conditions under which a proposition is true are called its truth conditions.
