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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 old
Used 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) = 1
C(5 + 3) = 3
C(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
Brown
Brown
Brown
Brown
Brown
Red
White
Black
Orange
Animal
Dog
Ant
Pig
Goat
Cow
Cow
Cow
Cow
Cow
Threat Level
Safe
Safe
Safe
Safe
DANGER!
Safe
Safe
Safe
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 nonlocal 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 let
E = “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 nonlocality 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.