Compositionality

1. Compositionality

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

3. 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

4. Infinite Adjective Phrases • Old • Extremely old • Probably extremely old • Invariably probably extremely old • Predictably invariably probably extremely old Used in a sentence: “Residents of nursing homes are predictably invariably probably extremely old.”

5. 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.

6. 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.

8. 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

9. 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!

10. Recursive Loop But suppose instead I said: INVITE LIST: • My friends are invited. • 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.

11. Recursion and Language Here’s how language might do it: NOUN PHRASE: • “man” is a noun phrase • 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.

12. 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.”

13. 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.

14. 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.”

15. 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.

16. 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.

17. 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.”

18. 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).

19. 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.”

20. 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) = 1 C(5 + 3) = 3 C(2 + 2 + 2 + 2) = 7

21. Semantic Closure So if these sentences have different meanings: • “Lois Lane loves Clark Kent.” • “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).

22. 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.”

23. Arguments from compositionality

24. 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.

25. 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?

26. Idea of a Pet

27. OK, now what’s your idea of a fish?

28. Idea of Fish

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

31. 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.

32. 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.

34. 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.

35. 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.

36. 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.

37. 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.

38. 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.

39. 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.

40. 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.

41. 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.’

42. 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.

43. 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.”

45. 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.

46. 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.

47. 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.

48. Is the principle of compositionality true?

49. 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.

50. 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…