So let’s talk about exploring. If you were all alone on your spaceship with no one to
talk to and nothing to do, you might go looking around for some friendly crewmates and a place
where you could have some fun in the sun. Unfortunately for words like “some” and
“all”, there’s no guidance system to help them find meaning in the universe; when
it comes to the vast expanse of natural language, basic logic just won’t cut it. So what’s
a poor quantifier to do? I’m Moti Lieberman, and this is The Ling Space.
When you want to explain the way the universe works, coming up with mathematical models
can be really useful, and this applies to language, too. The most basic bits of a system
like predicate logic can capture the meanings of adjectives and verbs, and it does a good
job with connectives like “and” and “or”. On top of that, predicate logic uses quantifiers —
symbols that can represent the meanings of words like “all” and “some”.
But we have a bunch of quantifiers, not just those two. How about words like “few”
and “many” and “most”? Can we use logic to represent these, too? Well, like
a bathroom break on a backwards planet, that’s easier said than done.
In fact, in the 1980s, a couple of linguists mathematically proved that no matter how hard
you tried, you’d never make it work. Even for a really simple sentence like “most
toasters just make toast”, no combination of quantifiers and connectives could ever
come close to mathematically meaning what we want these expressions to mean.
Well, what about if you invent a new quantifier — say, one that means “more than half”?
Even with that, no matter what curry of symbols we try to cook up, none manage to exactly
match speakers’ intuitions about what “most toasters just make toast” really means.
The problem is that we have no way of limiting where quantifiers have to look to calculate
whether something is true or false. That upside down “∀” and backwards “∃” in logic
work by scanning through the whole universe and checking every single thing in it, to
see whether they’ve found what they’re looking for. That’s how they’re defined,
and that’s how our made-up quantifier “M” works, too.
Say there’s just one talking toaster in the whole universe, while the rest stay silent
and just make toast. Our intuition is that “Most toasters just make toast” feels
true. But since most things in the universe *aren’t* toasters, the logical version of
this sentence comes out false - it counts everything that’s not a toast-devoted toaster
against the sentence, whether it’s an appliance or not. And since that doesn’t match our
own intuitions about what the sentence means, that’s not the best result. It’s also
not just this particular logical setup using “and” - any other construction, like with
“or” or the conditional, will fail in the same kind of way. So we want a way
to restrict the space a quantifier needs to look through.
Thankfully, we have set theory to rescue us, which we talked about a while back. Instead
of thinking of the quantifiers in logic as bounty hunters scouring the whole universe
to get the job done, we can think of them as hardworking technicians, relating and comparing
sets. For instance, in the sentence “all of the crew escaped the squid”, we can ditch
our logical version and replace it with something simpler!
We can just think of the set of all crew members as being a subset of all the people who escaped
the squid. So it turns out the meaning of the word “all” is just like the subset
relation. And in the same way, in the sentence “some members of the crew are holograms”,
the meaning of “some” can be expressed using the operation of set intersection.
The meaning of “some” just says that the intersection of the two sets it’s comparing
isn’t empty. So there’s at least one thing that’s both a member of the crew and a hologram!
And we can use these tools to talk about other quantifiers too, like “no”. In “no cat
wants to be poorly dressed”, we can represent the meaning of this sentence by saying that
the intersection of these two sets is empty. Obviously, there’s nothing that’s both
a cat, and that also wants to be poorly dressed.
Most importantly, we can finally get a grip on the meaning of “most”. For a sentence
like “most crew members are afraid of Mr. Flibble”, we can write out its meaning by
saying that the number of crew members who are afraid of Mr. Flibble makes up more than
half of all crew members.
By mixing a little bit of set theory into the recipe, we’ve now redefined quantifiers
to be fundamentally binary; they take two sets and, depending on how they’re related
to each other, spit out either “true” or “false”. We’ve taken our original
logical quantifiers and generalized the underlying idea to make it easier to talk about real
human language. We made a Generalized Quantifier Theory.
But the coolest thing to come out of this is that once we look at word meanings in this
way, we find patterns that seem to be universal across all human languages. For one thing,
all quantifiers seem to have the property of conservativity. Which isn’t a politics
thing - it’s a mathematical property we can spell out using set theory, like this:
That means quantifiers only care about the elements in the first set they combine with,
and so they ignore anything in the second set that’s not already in the first. Think
back to our sentence “all crew members escaped the squid”. We know that “all” is conservative
because that sentence doesn’t care what else may have escaped the squid, and the meaning
doesn’t change when we reword the sentence as “all crew members are crew members that
escaped the squid”. It passes our test with flying colours! And we can do the same for
“some”, “no”, “most”, and the rest.
Now, nothing’s stopping us from inventing quantifiers that aren’t conservative. Like,
we could say that “gelf” means the two sets it compares have the same size, so “gelf
androids have hair” means there are as many androids in the universe as things with hair --
and that’s a lot. Since our made-up word’s all about numbers, though, it has to count through
both sets, in their entirety. If we try to shrink that second set down to only those
things with hair that are androids -- and there aren’t a lot of those -- we get “gelf
androids are hairy androids”, which seems to mean something different. In ignoring everything
else that has hair, it fails the same test all the real quantifiers pass. Words like
this turn out to be easy enough to imagine, but since they aren’t conservative, we don’t
see anything like them out in the wild; it really looks like this is hardwired into language!
Linguists have looked into this by doing learnability experiments on young children, between the
ages of 4 and 6. Like, take this one: first, the researchers split the kids into two groups,
and trained them on fake words, like “gleeb”. For one group, the kids got a version of the word that
had a conservative definition, but in the other, they were exposed to an “impossible”, non-conservative word.
Then afterwards, the children were asked to apply “gleeb” in different scenarios,
to see if they had properly acquired what it means. Turns out, the group who’d been
exposed to the conservative version managed to learn it, but the unfortunate kids that
trained with the impossible version didn’t perform any better than chance.
So, even though we might not be conscious of the math that underlies our sentences,
it still manages to constrain what’s actually possible, guiding and shaping language in
ways that most of us aren’t even aware of! So it’s definitely worth it to keep exploring.
So, we’ve reached the end of The Ling Space for this week. If you quantified everything
conservatively, you learned that even something as sophisticated as predicate logic can’t
really do justice to natural language; that set theory gives us an invaluable way to talk
about the meanings of quantifiers like “all” and “most”; and that when we apply the
tools of math, we uncover deep connections that link together all human languages.
The Ling Space is produced by me, Moti Lieberman, and directed by Adèle-Elise Prévost. This
week’s episode was written by Stephan Hurtubise. Our editor is Georges Coulombe, our music
is by Shane Turner, and our graphics team is atelierMUSE. We’re down in the comments
below, or you can bring the discussion back over to our website, where we’ll have some
extra material on this topic.
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want to keep expanding your own personal Ling Space, please subscribe. And we’ll see you
next Wednesday. Shala bimpa!