PLIN0009 Semantic Theory Spring 2020 Lecture Notes 4

1 Semantic Types

It is convenient to classify the semantic objects in the model according to what kind of objects they are. To do this, we will use semantic types.

Semantic types are constructed out of two primitive types, e and t. Type e is the type of entities, and type t is the type of truth-values. Everything in the domain of a given model— which is the set of all entities—is of type e (and this set is often denoted by D). We only have two things that are of type t, namely, the two truth-values, 0 and 1.

In our semantic universe, everything else is a function. The types of functions always look like xσ, τy, where σ is the type of possible arguments of this function, and τ is the type of possible output of the function.

For example, any function that takes an entity and returns a truth-value is of type xe, ty, M because its input is of type e and its output is of type t. So vswimsw is a type-xe, ty function (for any model M).

There are more complex semantic types. For example, we can think of a function that maps any entity to a function of type xe, ty. Such a function is of type xe, xe, tyy, because its input is of type e and its output is of type xe, ty. Notice that xe, xe, tyy is an instance of the general format xσ, τy with σ = e and τ = xe, ty.

Keep in mind that things like xe, e, ty or xty are NOT semantic types. Between each pair of angled brackets, you should ind exactly two semantic types separated by ‘,’.

For the sake of explicitness, let us deine all semantic types recursively as follows.

(1) a. e is a semantic type. b. t is a semantic type. c. Whenever σ and τ are semantic types, xσ, τy is also a semantic type. d. Nothing else is a semantic type.

This deinition yields ininitely many semantic types, and at the same time excludes things like xe, e, ty or xty. Besides e, t, xe, ty, and xe, xe, tyy, which are mentioned above, we can also have much more complex semantic types like xxt, ey , xe, xxt, ty , xe, xt, eyyyy. This is probably overkill for natural language semantics, because such complex types seem to be unnecessary for analyzing natural language expressions. But you never know. And having unused semantic types does not do any harm.

It should also be noted that it is common practice to write et for xe, ty. We will not use this abbreviation in this lecture, but it will become handy in the second half of this module. You will be reminded of it then.

Having deined the semantic types, we should also state what they represent. We do so by deining the ‘domain’ of each semantic type. For each type τ, its domain is written as Dτ .

1 (2) a. De is the set of all entities in the given model i.e. De = D b. Dt is the set of all truth-values, i.e. Dt = t 0, 1 u c. For any other type xσ, τy, Dxσ,τy is the set of functions from Dσ to Dτ

The last case (2c) can also be written as Dxσ,τy = t f | f : Dσ Ñ Dτ u, because ‘f : Dσ Ñ Dτ ’ means f is a function whose domain is Dσ and whose range is Dτ . This is just a formal way of stating what we said above. That is, e is the semantic type of entities (in the model), t is the semantic type of truth-values and everything else is a M type of functions. For instance, vswimsw is a function from individuals to truth-values, or equivalently, a function from De to Dt , so its semantic type is xe, ty, or equivalently, the M same thing can be written as vswimsw P Dxe,ty.

2 Transitive Verbs

So far the lexicon of our grammar is limited to proper names and intransitive verbs. Using the semantic types introduced above, we can write them as follows. We will always use the λ-notation for functions from now on.

(3) For any model M M a. vAndyw P De M b. vBeckyw P De M c. vswimsw = [λx P De. 1 iff x swims in M] M d. vcriedw = [λx P De. 1 iff x cried in M]

Proper names denote type-e elements, i.e. entities, and intransitive verbs denote functions of type xe, ty. We often say ‘Proper names are of type e, intransitive verbs are of type xe, ty’, but what we really mean is that their denotations are of type e and type xe, ty.

Because our Lexicon is limited, our semantic theory so far only captures a tiny corner of English, namely, sentences formed with proper names and intransitive verbs. We will add more vocabulary items (and compositional rules, as necessary), so that ultimately we can compute the meanings of all grammatical sentences in English. We of course cannot go that far in this module. Nor have semanticists reached this ultimate goal yet (so they haven’t gotten made redundant!). We will, however, be able to demonstrate how this step-by-step approach works. In this lecture, we will add transitive verbs to our semantic theory.

Transitive verbs are those verbs that take two DPs, (typically) a subject and object, e.g. love, saw, etc. What should be the denotations of the transitive verbs? Let’s start with their semantic types. We can actually infer what their semantic types should be based on what we already know. We assume the following structure (which seems to be reasonable given the syntactic facts we know).

S

Andy VP

loves Becky

2 • We know that the entire sentence denotes a truth-value, i.e. it’s of type t. That is, for any model M, 4 < M S 6 > 6 > 6 > 6 > 6 Andy VP > P Dt 5 = loves Becky • We also know that each proper name denotes an entity, i.e. it’s of type e, as in (3a) and (3b). • Given the Branching Node Rule, the VP loves Becky should denote a function of type xe, ty, because we want to combine the denotation of the VP and the denotation of Andy to get a truth-value. So, for any model M, we want 4

M M • Given the Branching Node Rule, vlovesw should be a function that takes vBeckyw and M returns v[VP loves Becky ]w . That is, for any model M, 4

M Then the argument of vlovesw is of type e, and its output is of type xe, ty. This means M that vlovesw itself should be of type xe, xe, tyy!

Reasoning about semantic types, as demonstrated here, is a very important technique in formal semantics. We will make similar inferences about semantic types many times over in this module.

It is often useful to visualize a process like above by annotating each constituent in the tree diagram with its semantic type. Before the reasoning above, we knew the following:

S t

Andy e VP ?

loves ? Becky e

Given the Branching Node Rule, the VP needs to denote a function of type xe, ty, because its sister constituent (= its input) is of type e (which is not a function) and its mother constituent (= its output) is of type t. So we have:

S t

Andy e VP xe, ty

loves ? Becky e

3 M Similarly, vlovesw needs to be of type xe, xe, tyy, because its sister constituent (i.e. its input) is of type e (and hence not a function) and its mother constituent (i.e. its output) is of type xe, ty. So we get:

t

Andy e VP xe, ty

loves xe, xe, tyy Becky e

2.1 The Denotation of Transitive Verbs

M Now we know the semantic type of vlovesw , namely, xe, xe, tyy. This means that this function will look like [λx P De. [λy P De. 1 iff ???]] M Let’s remind ourselves what x and y are. The irst argument that the function vlovesw combines with is the denotation of the object DP. In the above example, it is Becky. This is going to replace all occurrences of x in the body of the function, when λ-conversion takes place. Then the resulting function of type xe, ty will combine with the denotation of the subject DP, Andy. So Andy is going to replace y in the body of the function.

If this function applies to Becky and then to Andy, we will get a truth-value. What kind of truth-value should we get? We already know the truth-conditions of the entire sentence, so it’s easy to igure this out. That is, if Andy loves Becky in the model, we should get 1, and if not, we should get 0. So what we want is:

(4) [λx P De. [λy P De. 1 iff y loves x in M]]

Importantly, it is ‘y loves x’ rather than ‘x loves y’, because x is going to be the denotation of the object and y is going to be the denotation of the subject, so x should be the person who is loved and y should be the person who loves x. So, (5) is not the right meaning of ‘loves’.

(5) [λx P De. [λy P De. 1 iff x loves y in M]]

This function of also of type xe, xe, tyy, so we could actually use it to compute the meaning of the sentence under question, but the predicted truth-conditions would be wrong. That is, it predicts the sentence Andy loves Becky to denote 1 in M iff Becky loves Andy in M, because it says the irst argument x will be the person who loves the other person, and the second argument y will be the person who is loved.

This demonstrates a very important point: How we construct our semantics depends cru- cially on our syntactic assumptions. If the syntactic structure of the sentence were not as we are assuming, but something like (6) instead, then (5) would be the correct analysis, and (4) would be the wrong one.

4 (6) S

VP Becky

Andy loves

However, by assumption, this is not the right syntax. It is outside the scope of this module to explain why this is not the right structure, but hopefully you can recall the explanations given in the syntax modules you have taken (If you cannot, go to a syntactician and ask them!). The important lession here is that in order to do semantics, you need to know syntax.

Now, we have igured out the denotation of ‘loves’, so let’s add it to our lexicon. It will look like:

(7) For any model M, M vlovesw = [λx P De. [λy P De. 1 iff y loves x in M]]

It is quite easy to generalize this analysis to other transitive verbs.

(8) For any model M, M a. vhatesw = [λx P De. [λy P De. 1 iff y hates x in M]] M b. vsaww = [λx P De. [λy P De. 1 iff y saw x in M]] M c. vkickw = [λx P De. [λy P De. 1 iff y kicks x in M]]

3 Summary

To sum up, we have added transitive verbs to our lexicon:

(9) For any model M, M a. vAndyw P De M b. vBeckyw P De

M c. vswimsw = λx P De. 1 iff x swims in M M d. vleftw = λx P De. 1 iff x left in M

M e. vlovesw = [λx P De. [λy P De. 1 iff y loves x in M]] M f. vsaww = [λx P De. [λy P De. 1 iff y saw x in M]]

We have not changed our compositional rule:

(10) Branching Node Rule 3 ;M 3 ;M A A M M M M For any model M, = vBw (vCw ) or = vCw (vBw ) B C B C

Here is an example computation, demonstrating how meanings are put together. Let us M2 M2 assume that vCatew = c and vDenisw = d, for some particular model M2 (recall that

5 they might denote different entities in different models). 4 < M2 S 4 < 6 > M2 6 > VP ( ) 6 > 5 = M2 6 Cate VP > = vCatew (BNR) 5 = saw Denis saw Denis 4 < M2 VP = 5 = (c) saw Denis M M = vsaww 2 (vDenisw 2 )(c) (BNR) M = vsaww 2 (d)(c) (BNR)

= [λx P De.[λy P De. 1 iff y saw x in M2]](d)(c)

= [λy P De. 1 iff y saw d in M2](c) (λ-conv.)

= 1 iff c saw d in M2 (λ-conv.)

By assumption any speciic model like M2 should be able to tell you whether c saw d or not. So the sentence denotes 1 or 0, depending on the facts in M2.

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