Compositionality

Wilhelm von Humbolt famously described language as a system that “makes infinite use of finite means.”

Infinite Noun Phrases

There are infinitely many noun phrases: you can always make another one by adding another adjective:• Dog• Old dog• Smelly, old dog• Brown, smelly, old dog• Big, brown, smelly, old dog

Infinite Adjective Phrases

• Old• Extremely old• Probably extremely old• Invariably probably extremely old• Predictably invariably probably extremely oldUsed in a sentence: “Residents of nursing homes are predictably invariably probably extremely old.”

Infinitely Many VPs and DPs

And of course there are infinitely many verb phrases, and even “determiner phrases”:

• John’s mother.• John’s wife’s mother.• John’s wife’s lawyer’s mother.• John’s wife’s lawyer’s dog’s mother.

Infinitely Many Sentences

It follows that there are infinitely many sentences, because (for example) each sentence NP + VP can be lengthened to AP + NP + VP, then to AP + AP + NP + VP, and so on.

Infinitely Many Sentences

In addition, there are infinitely many sentences because you can take any sentence S and add “so-and-so believes that” to the front:• S• John believes that S• Mary hopes that John believes that S• Sam doubts that Mary hopes that John believes

that S

Recursion

This is in general possible because language is recursive. Suppose I’m throwing a party. I start writing the invite list:• My friends are invited.• Friends of my friends are invited.• Friends of friends of my friends are invited.• Friends of friends of friends…It seems like I’ll never finish!

Recursive Loop

But suppose instead I said:INVITE LIST:(i) My friends are invited.(ii) If x is a friend of someone who is invited,

then x is invited.

This captures all the cases, by going in a loop. (ii) defines who is invited in terms of who is invited.

Recursion and Language

Here’s how language might do it:NOUN PHRASE:(i) “man” is a noun phrase(ii) If NP is a noun phrase then “old” + NP is a

noun phrase.

From this recursive definition, it follows there are infinitely many noun phrases.

Infinite Use of Finite Means

This is one sense in which language “makes infinite use of finite means.” There are finitely many words, and the rules of grammar are presumably finite. But recursion generates infinite complex expressions from a finite “base.”

Understanding

But this raises another question. Each of the infinite distinct sentences in English has a different meaning. We cannot learn the meaning of each one separately. But we can understand any English sentence, even one we’ve never heard before.

Novel Utterance

“Yesterday, on my way to the plastic cow hat factory, I witnessed on two separate occasions police selling cupcakes out of empty space shuttles that had been painted in red and blue stripes.”

Compositionality

How is it possible for us to understand a potential infinitude of novel utterances? The most common solution in philosophy and linguistics is to maintain that the meanings of complex expressions depend on– and depend only on– the meanings of their simple parts and the ways that those parts are organized (put together by the grammar). This is called compositionality.

Compositionality

How does this solution work? Since there are only finitely many simple expressions (words/ morphemes) in English (or any other language), each language user only has to learn finitely many meaning facts: what all the simple expressions mean.

Compositionality

Then when that user encounters a novel utterance she just uses the already-learnt meaning facts about words and the grammar of the utterance to work out its meaning. Compositionality says that’s all she needs! “The meaning of the whole depends on (and only on) the meanings of the parts and the way they are combined.”

Non-Compositionality

What does this claim rule out? Here’s an example of a non-compositional arithmetical function #:(n # m) := n + m, if (n # m) appears alone.(n # m) := n + m + x, in the context (x # (n # m)).In this example the value of (n # m) depends not just on the values of n and m, but sometimes on other values (e.g. x).

Locality

In this sense, compositionality is local. In the expression [old [brown dog]] what “brown dog” means cannot depend on what “old” means, even though that’s also part of the expression containing “brown dog.”

Non-Compositionality

The second thing compositionality rules out is that the meaning of a complex depends on more than just the meanings of the parts. For example consider the count function C, it counts the number of symbols after it, ignoring their values:C(8) = 1C(5 + 3) = 3C(2 + 2 + 2 + 2) = 7

Semantic Closure

So if these sentences have different meanings: 1. “Lois Lane loves Clark Kent.”2. “Lois Lane loves Superman.”We cannot say that the reason they have different meanings is that “Clark Kent” and “Superman” are different words. Compositionality says that different meanings are possible for complex expressions only if their parts have different meanings (not diff words).

The Substitutability Criterion

Another way of saying “the meaning of the whole depends only on the meanings of the parts and the way they’re combined” is this:

“For any sentence S(E) containing some expression E as part, if E and E* have the same meaning, then S(E) and S(E*) have the same meaning.”

ARGUMENTS FROM COMPOSITIONALITY

What’s at Stake?

Before we consider arguments for or against compositionality, let’s look at what’s at stake. At various points, compositionality has been used to argue against all of the theories of meaning we have considered in class.

Vs. the Idea Theory

According to the idea theory, the meaning of a word is an idea, where ideas are construed as something like “little colored pictures in the mind.” Let’s consider an example: what’s your idea of a pet?

Idea of a Pet

OK, now what’s your idea of a fish?

Idea of Fish

Now try to combine those ideas into the idea of “pet fish.”

Vs. the Idea Theory

That clearly doesn’t work. Notice that we cannot say that in the context of “____ fish” “pet” means something other than . This would make the meaning of “pet” non-local (depend on surrounding context) and that’s not allowed on any compositional theory. Conclusion: the idea theory violates the principle of compositionality.

Vs. Verificationism

Let’s get a little more specific than we have before in discussing verificationism (the point I’ll make is general, but the details help us see it). Let’s suppose that the meaning of a sentence is the set of experiences that it probably causes you to have. So a cow will probably cause you to hear cow-sounds, so cow-sounds are part of the meaning of “cow.” In other words the probability of cow-sounds is increased by the presence of cows.

Color Animal Threat LevelBrown Dog SafeBrown Ant SafeBrown Pig SafeBrown Goat SafeBrown Cow DANGER!

Red Cow SafeWhite Cow SafeBlack Cow Safe

Orange Cow Safe

Cows are Safe

Let’s suppose that 99% of cows are safe, so the probability of experiencing bodily injury given a cow is 1%: P(injury/ cow) = 1%. The unconditional probability of injury is much higher, say P(injury) = 5%. So the meaning of “cow” does not include the experience of bodily harm, because cows lower, rather than raise, the chances that you’ll experience bodily harm.

Brown Things are Safe

Let’s also suppose that brown things are in general safe, only 1 in 100 of them are dangerous: P(injury/ brown) = 1% < P(danger). So again, “brown” doesn’t have the experience of bodily harm as part of its meaning either. You’re less likely to experience this around brown things than around other-colored things.

Brown Cows are Dangerous!

However, suppose that the 1% of dangerous cows and 1% of dangerous brown things are 100% brown cows. All brown cows are dangerous, and the probability of being mauled by one is 100%, if one is around. So P(injury/brown cow) = 100%. Thus the meaning of “brown cow” contains the experience of bodily harm. That experience confirms the presence of brown cows.

Vs. Verificationism

But how is this possible? Neither the set of experiences that is the meaning of “brown” nor the set of experiences that is the meaning of “cow” contains the experience of bodily harm. The meaning of “brown cow” thus seems to depend on something other than the meanings of its parts, “brown” and “cow”: verificationism violates the principle of compositionality.

Vs. Causal Theories

Recall how Kripke’s causal-historical theory worked. A baby is born. The parents point to it and say “that baby over there shall henceforth be named ‘Richard Feynman.’” As new people learn the name, they inherit the referent of the name from whoever they acquired it from. Thus the name succeeds in referring to Richard Feynman when I use it, even if I know nothing, or only false things, about Feynman.

No Connotations

Where do connotations fit in to this picture? Well, lots of philosophers have thought that they don’t, and they’re not needed. Connotations (ideas, confirming experiences, definitions) were supposed to explain how expressions got their referents. But Kripke’s story does the explaining without anything mental (a connotation) being involved.

Direct Reference Theory

The resulting view is known as direct reference theory. This is just the claim that names and natural kind terms “directly” refer to their denotations, and that connotations aren’t involved in mediating the process. The meaning of a name, for example, is the person named. There is nothing more to meaning than reference/ denotation.

Vs. Direct Reference

But now we have a problem with compositionality. Here are two sentences that intuitively have two distinct truth values:TRUE: Lois Lane believes Superman can fly.FALSE: Lois Lane believe Clark Kent can fly.But notice that the two sentences differ only in that parts with the same meaning (reference) have been swapped: ‘Superman’ and ‘Kent.’

Vs. Direct Reference

This is a violation of the substitutability criterion, which is equivalent to the principle of compositionality (Pagin & Westerstahl prove this in the reading). So it seems like if compositionality is true, then the direct reference theory is false.

Vs. the Use Theory

Does knowing how word W1 is used and how W2 is used suffice for knowing how [W1 W2] is used? This seems unlikely. You might teach a Martian how the word “black” is used by showing it color samples or whatever, and you might teach it how “person” is used, but I doubt the Martian would be able to work out itself that Alicia Keys counts as a “black person.”

Fundamental Acceptance Property

Recall that for Horwich, the fundamental acceptance property underlying all uses of ‘black’ is to apply ‘black’ to surfaces that are clearly black. Suppose we taught a Martian this. And suppose we taught a Martian how to apply ‘human’ or ‘person’ (distinguishing us from other apes). Could the Martian work out how to use ‘black person’? I think not.

Interests in Exaggeration

We (humans) have a vested interest in exaggerating differences in skin tone in order to effect a certain constructed social order. Unless you know about this proclivity to exaggerate, (which doesn’t affect normal color ascriptions), then you can’t predict ascriptions of the form ‘COLOR + person.’ Using ‘black’ (or ‘red’ or ‘yellow’) for a color and using ‘person’ for a certain sort of animal doesn’t determine how to use the ‘COLOR + person’ form.

Vs. The Use Theory

The point is that complex expressions can acquire uses that aren’t determined by the uses of their parts. Consider the English insult “Mama’s boy.” If you’re a child, and a male, it’s not insulting to be called a ‘boy.’ Nor is ‘Mama’ an insult. But put them together, and that’s insulting, even if you are a Mama’s boy. Just learning the use of the parts won’t tell a second-language learner of English that the whole is insulting.

IS THE PRINCIPLE OF COMPOSITIONALITY TRUE?

Argument from Understanding

As I said before, the principal argument in favor of compositionality is simply the argument from our ability to understand a potential infinitude of novel utterances to the purported best explanation of this fact, compositionality. Sometimes things are stated in terms of our ability to learn or master a language with a potential infinitude of novel utterances, but that’s just the same argument from a different angle.

A Different Perspective

I personally (and I’m an exception in philosophy and linguistics on this issue) think that (i) compositionality is not the best explanation for our ability to understand a potential infinitude of novel utterances, (ii) that it’s not even an explanation, and (iii) that it’s probably not even true of English. Still, I think that the arguments we’ve reviewed so far are pretty good ones. Let me explain…

Against Locality

As we saw before, compositionality is local. In the expression [old [brown dog]] what “brown dog” means cannot depend on what “old” means, even though that’s also part of the expression containing “brown dog.”

But there are reasons for thinking English might not be local in this sense and that it’s no big deal that it isn’t.

Donkey Sentences

Normally, sentences S(‘a donkey’) are made true by the existence of a donkey who satisfies S(x). For example:

• A donkey pooped on the train.• John punched a donkey.

Geach Sentence

However, consider the following sentence (due to Peter Geach):

• Every farmer who owns a donkey beats it.

This sentence is (emphatically!) not made true by a donkey who satisfies “Every farmer who owns x beats x.”

Universal Interpretation

It means something more like “For every farmer F and every donkey D, if F owns D, then F beats D.”

Notice, in particular, that this sentence talks about every donkey, not some donkey. But that’s not normally how ‘a donkey’ works, as we saw. (REMINDER: ‘a donkey ate my shoes.’)

Non-Local Donkeys

Without going into deep details, you can see how the non-compositionalist crowd might take this data. In normal contexts ‘…a donkey…’ means AT LEAST ONE DONKEY, whereas in contexts like ‘if… a donkey…., then ….’ things are different. Here, ‘…a donkey…’ means EVERY DONKEY. You can calculate what ‘a donkey’ means in each context, but its meaning is non-local and hence non-compositional.

Against Semantic Closure

There might also be reason to deny that English satisfies the principle of semantic closure (that the meanings of expressions depend only on the meanings of their parts and how they’re combined.) Furthermore, this might not be obviously a bad thing.

Quotation

In English we use quote marks in lots of different ways:

Pure: “Cat” has 3 letters.Direct: John said “it’s raining.”Scare: Mommy and Daddy are having “special time.”Greengrocer: “Fresh” fruit!

Pure Quotation

Pure quotation is an interesting phenomena. Consider that “bachelor” and “unmarried man” are synonymous. The substitutability criterion (compositionality) says:

“For any sentence S(E) containing some expression E as part, if E and E* have the same meaning, then S(E) and S(E*) have the same meaning.”

A Counterexample

So letE = “bachelor”E* = “unmarried man”S(E) = “‘bachelor’used to mean squire.”S(E*) = “‘unmarried man’ used to mean squire.”

The substitutability criterion fails!

No Semantic Closure

Intuitively, the failure of the substitutability of synonyms here arises from the fact that the meanings of “‘bachelor’” and of “‘unmarried man’” do not depend on the meanings of “bachelor” and “unmarried man,” respectively. Instead they depend on the words themselves. This is unproblematic: we can still calculate the meaning of any pure quotation, even though the s.c. fails and pure quotation isn’t compositional.

The Lesson

I don’t want to endorse the specific contents of either of these objections– there are compositional treatments of both donkey sentences and pure quotation in the literature. But the important lesson I think is that if non-locality or semantic closure failed in these particular ways, it wouldn’t matter as far as the understandability argument goes. We could still calculate the meanings of donkey sentences and pure quotation.

Computability

What I’d suggest then is that the real constraint is not that the meaning of any expression is determined by and only by the meanings of its parts and the way they’re combined. Rather, it’s that the meaning of sentences (and not necessarily their parts) is computable from the sentences themselves and the meanings of the simple parts.

Computability

This suggestion rejects semantic closure in favor of “semantic and syntactic closure”: both the words themselves and their meanings can determine the meaning of the whole. It also encapsulates a limited form of locality: a sentences meanings can’t depend on the meanings of things outside the sentence, but parts of a sentence can depend for their meanings on the meanings of other parts of the same sentence (e.g. in donkey sentences).

Computability

In addition to what we’ve seen, I think there’s a very powerful reason to replace compositionality with computability. Compositionality does not explain how we can understand a potential infinitude of novel utterances. There are compositional functions that are not computable. Thus there are (possible) languages with compositional meaning functions where we cannot in principle work out the meanings of expressions in those languages!

Computability

What computability requires is that we actually be able to work out the meanings of sentences from the meanings of their parts and how they’re combined, not that their meanings merely depend on or co-vary with the meanings of their parts and how they’re combined.

Does It Matter to Our Arguments?

While I think this is right, in some ways it doesn’t even matter. Why? Because locality and semantic closure didn’t really enter into our compositionality-flavored objections earlier.

For instance, you can’t calculate the pet fish idea from the pet idea and the fish idea. And you can’t calculate the use of “mama’s boy” from the use of “mama” and “’s” and “boy.”

Except!

Well, there’s one exception. Our argument against the Direct Reference theorist crucially relied on semantic closure:1. Lois Lane believes that Superman can fly.2. Lois Lane believes that Clark Kent can fly.

So the Direct Reference theorists has the resources to respond by rejecting compositionality as a constraint and instead endorsing computability.