13.5: Rules of interpretation

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Stating the truth conditions for individual sentences like those in Table 13.1 is a useful first step, but does not yet replicate what speakers can do in their productive use of the language. Ultimately our goal is to provide general rules of interpretation which will predict the correct truth conditions for sentences based on their syntactic structure. As a further step toward this goal, let us return to the sentence in (21a), which we have already discussed several times.

(21) a. King Henry VIII snores.
b. Anne Boleyn snores.

We have already stated an informal rule of interpretation for simple sentences: the proposition expressed by a (declarative) sentence will be true if and only if the referent of the subject NP is a member of the denotation set of the VP. We can now restate this rule in a slightly more formal manner. We will assume that the basic syntactic structure of the clause is [NP VP]. The semantic rule we wish to state operates in parallel with the syntactic rule which licenses this structure, as suggested in (22). (Recall that the semantic value, i.e. the denotation, of a sentence is its truth value.)

(22) syntax: S → NP_subj VP

semantics: The semantic value of a sentence is ‘true’ if the semantic value of the subject is a member of the set which is the semantic value of the VP, and ‘false’ otherwise;

((S)) = ‘true’ iff ((NP_subj)) ∈ ((VP))

Applying this rule to the sentence in (21a), we get the formula in (23). This formula says that the sentence will be true just in case King Henry VIII is a member of the denotation set of snores. Since this is true in our model, the sentence is true relative to this model. The same rule of interpretation allows us to determine that sentence (21b) is false relative to this model.

(23) ((King Henry VIII snores)) = ‘true’ iff ((King Henry VIII))∈((snores))

The statement in (23) can be expressed in logical notation as in (24a). This formula is a specific instance of the general rule for evaluating the truth of propositions involving a one-place predicate. This general rule, shown in (24b), states that the proposition P(α) is true if and only if the entity denoted by α is an element of the denotation set of P.

(24) a. ((SNORE(h))) = ‘true’ iff ((h))∈((SNORE))

b. if α refers to an entity and P is a one-place predicate,
then ((P(α))) = ‘true’ iff ((α))∈((P))

Let us now add a few more vocabulary items to our simple model, calling the new version Model 1ʹ. This revised model presumably reflects the early period of the marriage, ca. 1532–1533 AD, when Henry and Anne were happy and in love. Note also that Thomas More had fallen out of favor with the king around this time.

(25) Model 1ʹ

i. the set of individuals U = {King Henry VIII, Anne Boleyn, Thomas More}

ii. denotations:

((MAN)) = {King Henry VIII, Thomas More}
((WOMAN)) = {Anne Boleyn}
((SNORE)) = {King Henry VIII}
((HAPPY)) = {King Henry VIII, Anne Boleyn}
((LOVE)) = { ⟨King Henry VIII, Anne Boleyn⟩, ⟨Anne Boleyn, King Henry VIII⟩ }
((ANGRY_AT)) = { ⟨King Henry VIII, Thomas More⟩ }
((a)) = Anne Boleyn
((h)) = King Henry VIII
((tK)) = Thomas More

Model 1ʹ includes some two-place (i.e, transitive) predicates, and should allow us to evaluate simple transitive sentences like those in (26). The denotation set of a transitive predicate like LOVE or ANGRY_AT is not a set of individuals, but a set of ordered pairs. Sentence (26a) expresses the proposition stated by the logical formula in (27a). The truth conditions for this proposition are stated in terms of set membership in (27b): the proposition will be true if and only if the ordered pair ⟨King Henry VIII, Anne Boleyn⟩ is a member of the denotation set of LOVE. Since this is true in Model 1ʹ, sentence (26a) is true with respect to this model. The formula in (27b) is an instance of the general pattern stated in (27c).

(26) a. King Henry VIII loves Anne Boleyn.
b. King Henry VIII is angry at Thomas More.

(27) a. LOVE(h,a)

b. ((LOVE(h,a))) = ‘true’ iff ⟨((h)), ((a))⟩∈((LOVE))

c. if α, β refer to entities and P is a two-place predicate,
then ((P(α,β))) = ‘true’ iff ⟨((α)), ((β))⟩∈((P))

So far we have been dealing with the meanings of complete sentences all at once. This is possible only for the very simple kinds of sentences discussed thus far, but more importantly, it misses the point of the exercise. If we hope to account for the compositional nature of sentence meaning, modeling speakers’ and hearers’ ability to interpret novel sentences, we need to pay attention to syntactic structure. The sentences in (26) share the same basic syntactic structure as those in (21), namely [NP VP]. This suggests that the rule of interpretation stated in (22) should apply to the sentences in (26) as well.

The main syntactic difference between the sentences in (26) and those in (21) is the structure of VP: transitive in (26), intransitive in (21). In order to apply rule (22) to the sentences in (26), we need another rule which will provide the semantic value of a transitive VP. Intuitively, rule (22) says that the proposition expressed by a (declarative) sentence will be true if and only if the referent of the subject NP is a member of the denotation set of the VP. So we need to say that sentence (26a) will be true if and only if King Henry VIII belongs to a certain set. What is the relevant set? It would be the set of all individuals that love Anne Boleyn. This set will be the denotation set of the VP loves Anne Boleyn. The standard notation for defining such a set is shown in (28a), which says that the denotation set of this VP will be the set of all individuals x such that the ordered pair ⟨x, Anne Boleyn⟩ is an element of the denotation set of the transitive verb love.

(28) a. ((loves Anne Boleyn)) = {x: ⟨x, Anne Boleyn⟩∈((LOVE)) }

b. syntax: VP → Vtrans NPobj
semantics: The semantic value of a VP containing a transitive verb meaning P together with an object NP meaning α is the set of all individuals x for which P(x,α) is true;

((VP)) = {x: ⟨x, ((NP_obj))⟩∈((V_trans))}

The general rule for deriving denotation sets of transitive VPs is stated in (28b). The denotation sets formed by this rule are sets of individuals, so it makes sense to ask whether the referent of a subject NP is a member of one of these denotation sets. In other words, the denotation sets formed by rule (28b) are the right kind of sets to function as VP denotations in rule (22). So this approach allows us to model the stepwise derivation of sentence denotations. The rule of interpretation stated in (22) applies to both transitive and intransitive sentences. In the case of transitive sentences, rule (28b) “feeds”, or provides the input to, rule (22).

Rule (22) can also be applied to intransitive sentences with non-verbal predicates like those in (29), provided we can determine the denotation set of the VP.

(29) a. King Henry VIII is happy.
b. King Henry VIII is a man.
c. King Henry VIII is a happy man.

We can assume that the semantic contribution of the copular verb is is essentially nil (apart from tense, which we are ignoring for the moment). That means that the denotation set of the VP is happy will be identical to ((HAPPY)) , which is a set of individuals. For now we will also assume that the semantic contribution of the indefinite article in a predicate NP is nil.⁷ So the denotation set of the VP is a man will be identical to ((MAN)), which is also a set of individuals. In general, the denotation sets of common nouns and many adjectives are of the same type as the denotation sets of intransitive verbs; this is observable in the denotations assigned in (25). So no extra work is needed to interpret sentences (29a–b), using rule (22).

Sentence (29c) is more complex, because the predicate NP contains a modifying adjective as well as the head noun. As with transitive verbs, we can determine the denotation set of the VP (in this case, is a happy man) by asking what set the sentence asserts that Henry VIII belongs to? Here the relevant set is the set of happy men, i.e., the set of all individuals who are both happy and men.

The combination of word meanings in happy man follows the same pattern we have already discussed in connection with the phrase yellow submarine. The proposition asserted in (29c) might be represented by the formula in (30a). The truth conditions for this proposition are stated in terms of set membership in (30b). (Recall the definition of intersection given in (19).) The general rule for interpreting modifying adjectives is stated in (30c); we use the category label Nʹ for the constituent formed by A+N. Ignoring once again any possible semantic contribution of the copula and the indefinite article, the denotation set of the VP is a happy man is simply ((HAPPY))∩((MAN)). This is a set of individuals, and so rule (22) will apply correctly to sentence (29c) as well.

(30) a. HAPPY(h) ∧ MAN(h)
b. ((HAPPY(h) ∧ MAN(h))) = ‘true’ iff ((h))∈(((HAPPY))∩((MAN)))
c. syntax: Nʹ → A N
semantics: The semantic value of an Nʹ constituent containing a modifying adjective and a head noun is the intersection of the semantic values of the adjective and noun;
((A N)) = ((A))∩((N))

⁷ This assumption applies only to predicate NPs, and not to indefinite NPs in argument positions.