So let’s talk about sets. That’s sets... with a “t”. Meaning can be a tricky
thing to track down, even for your average noun or adjective. Sure, it’s easy enough
for names, like “Fox Mulder” or “Walter Skinner”, but what about words like “alien”
or “conspiracy”? Or descriptions like “mysterious” or “supernatural”? Not
to mention how we put these meanings together when we make sentences! Luckily, linguists
have picked up the flashlight of set theory and used it to peer around the dark corners
of language. I’m Moti Lieberman, and this is The Ling Space.
If you’ve watched our semantics videos, you already know that linguists have borrowed
a whole lot from philosophy. It’s given us some tools to work out what individual
words contribute to the overall meaning of a sentence. In particular, logic is really
useful when talking about connective words like “and” and “or”. And it also comes
in pretty handy when talking about quantifiers, like “some” and “all”. But
we haven’t really gotten our hands dirty with things like nouns and adjectives yet.
And to do them justice, we’re going to have to get a bit math-y!
And there’s one branch of mathematics that linguists have found pretty darn useful: set theory.
Set theory is the study of sets, or collections, and how they relate to and
interact with each other. It was founded in the late 19th century by German mathematician
Georg Cantor, and it’s gone through a lot of changes since. But to uncover just how
it helps us talk about meaning, we’ll go through some of the most basic ideas, and then
we’ll try applying them to English to see what happens!
The most fundamental concept underlying all of set theory is, well, the set! A set is
any collection of things. And those things can be concrete like people or planets, but they can
also be abstract, like numbers and shapes. To represent sets, we can list all of the
items we want in them, and then close 'em up with curly brackets. So, for example, if we’ve
identified three conspiracy theorists and we want to build a set with them, we could just
list their names and then close it up!
For very big sets, where it gets too cumbersome to list every single thing in it, we can just
be lazy and describe the contents instead. So, if we wanted to collect together everyone
who works for the government, we could write this.
This can be read as “the set of all things ‘x’ such that x works for the government”.
To be even lazier, we can give our sets names for when we want to refer back to them. Like,
we could call the set we just made “W”, for “works”. Or, we could collect together
all of the FBI’s special agents, and call that set “A”, for “agent”. This makes
each group easier to talk about later.
Another fundamental concept is the idea of something being a member of some set. Like,
we know that Dana Scully is a special agent, and so we know that she belongs to the set A.
To represent this relationship in terms of set theory, we write that Dana Scully is
an element of A, like so:
That e-shaped trident symbol means that whatever’s to the left can be found inside of the set on the right.
And speaking of relationships, another important connection in set theory is the subset relation.
We use this one when we find one set forming a part of another one, even if just abstractly.
To put it another way, we say that one set is a subset of another, just so long as everything
in the first one is also in the second. We write this relationship out as a kind of turned
“U”. So, if the set of special agents at the FBI is a subset of all the people who
work for the government, we get this:
Or, to shorten things, we can use the names we’ve given these sets in place of their
descriptions, and write the same relationship out as this:
We can even draw a picture of what’s going on! Here, you can see that the set of all special
agents is a subset of all government employees, because the circle standing in for the special
agents is totally inside the circle standing in for every government worker. That bigger
set includes special agents, but also diplomats and postal workers and everyone else!
And we can do more than just relate sets to each other; we can manipulate them, too. If
we wanted the set of all cigarette-smoking men, we could just build it by taking the
overlap between the set of all smokers and the set of all men.
This operation is known as intersection, and we would write it out in symbols using an
upside-down “u” like this:
We could also combine the whole of these two sets, which is kind of like set theory’s
version of addition. If we take all the smokers and all the men together, we get a group that’s
bigger than either set by itself, since it includes cigarette-smoking men, but also other
cigarette-smokers, and non-smoking men too.
In set-theoretic terms, this can be written out with a plain “U” — for “union”!
Finally, we can use the tools of set theory to count the number of things in a set. To
do this, we just sandwich it in between two vertical bars; like, if we wanted to count how many
conspiracy theorists we had before, we would just write this:
The number of elements inside of a set is called the set’s cardinality, and in case
you were wondering, it can be any number between zero and infinity. It can even BE zero, since
we could just write two curly brackets with nothing in between them, and call it the empty
set. Spooky.
With the basics of set theory out of the way, we can already start seeing some pretty interesting
things going on in language that we might not have noticed before. Like, adjectives
can actually be broken up into a few categories, depending on their set-theoretic characteristics.
For starters, if we keep treating our adjectives and nouns as sets, like we’ve been doing
throughout this episode, some adjectives seem to behave intersectively. That means that
some adjectives, when they combine with nouns, give you the intersection of those two sets.
Words like “extraterrestrial”, “carnivorous”, and “grey” work this way, since an “extraterrestrial
creature” is something that’s both from outer space and a creature! The meaning is
fairly balanced between the two words.
But other adjectives can behave non-intersectively; in fact, this is true of most adjectives.
Words like “brilliant” and “tall” and “famous” are all defined relative
to the set of nouns they’re modifying. A “brilliant pilot” isn’t necessarily
brilliant overall, and it definitely doesn’t mean they’re a brilliant doctor — they
could be a terrible doctor. And a tall agent who plays basketball isn’t necessarily also
tall for a basketball player; they could be really short relative to that set.
We can’t really treat words like “fast” or “tall” as referring to independent
sets on their own; instead, they’re always kind of inside or part of the set associated
with the noun. In other words, nouns with these kinds of adjectives are always
going to be subsets of the noun. For this reason, these kinds of adjectives are sometimes
called “subsective". And the influence on overall meaning here's weighted much more towards the noun.
All the way at the other end of things, we’ve got anti-intersective adjectives. These kinds
of words almost totally ignore the noun that comes after them. We’re talking about words
like “fake” and “imaginary”, and prefixes like “ex-”. After all, an imaginary monster
is much more likely to be found in someone’s head than under a bed. In these cases, we
could say that the intersection of one of these adjectives with a noun is empty, since
the set it creates has nothing in it.
Finally, there are adjectives which don’t bear much of a set-theoretic relationship
with their nouns at all. And they don’t really tell you anything about whether the
noun in question is, well, the noun in question! Adjectives like “alleged” and “potential”
and “likely”, when combined with a noun, kind of leave it open whether that noun is
what it claims to be. An alleged shape-shifter might turn out to be human after all . . .
or they might really be an alien in disguise!
This kind of math is what’s underneath our meanings: subconsciously, it’s how we make
sense of the world. And we can do deeper and more intuitive semantics with sets, as we’ll
come back to talk about in the future. The interaction of adjectives and nouns is just
one of the many realms where set theory can shine a light on how meanings of words come
together to create the meanings of larger expressions, like phrases and sentences. And
linguists will continue exploring these semantic mysteries - after all, the truth is out there!
So, we’ve reached the end of The Ling Space for this week. If you collected all your elements
together into a set, you learned that words can be grouped together according to their
mathematical properties; that some adjectives combine with nouns in a straightforward, intersective
way, where the meaning contribution of each word is about the same; that the meanings
of other adjectives are relative to, or even completely independent of, the noun they modify;
and that the tools of set theory let us explore the semantics of words in novel and powerful ways.
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
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dropping by our store. And if you want to keep expanding your own personal Ling Space,
please subscribe. And we’ll see you next Wednesday. La revedere!