So let’s talk about logic. Maybe you think it’s only Logic-Bots who care
about this, but you might be surprised. In one form or another, logic has been around
for thousands of years, and it’s extra interesting to linguists as a tool for studying language.
But, just what is logic anyway? I’m Moti Lieberman, and this is The Ling Space.
So logic and language might not jump out at you as being related, but actually, the two
are super connected. When we try to understand people’s sentences,
we crunch together their words and other bits into a form that we can take our meanings from.
And if you want to do semantics, where people study exactly what language means, there’s
a whole lot of logic in there.
At the least, most of us want to be thinking rationally, or at least sound like we're
making sense when we talk. The conclusions we reach and the things we
believe should probably have a solid foundation, or we might end up sounding like chumps.
Great Eastern and Western thinkers alike have spent thousands of years worrying about how
to think and argue clearly. But it wasn’t until the Greek philosopher, Aristotle,
that we took the first real steps towards understanding why some reasoning works better than others.
It turns out that the base of the system for how we understand sentences, before
considering real world knowledge or pragmatic concerns, can be boiled down to a system of logic.
Let’s try this: If I tell you that all carnivores eat meat,
and that some dinosaurs are carnivores, then that must mean that some dinosaurs eat meat, too!
These three sentences, taken together, make up what’s called a logical argument.
Each of the first two is called a premise, and they both support the third, which is the argument's conclusion.
And while this particular argument isn’t exactly groundbreaking, there’s still something
interesting there: any argument with that shape is a good one.
You could plug in whatever you want, and your conclusion will always follow from your premises.
You could even use nonsense words, and the argument still works! If you know that
all bleeps are bloops, and that some blops are bleeps, then you can be pretty sure that some blops are bloops!
So, Aristotle discovered that some kinds of arguments were automatically valid -- meaning
that if you had true premises, you'd always get a good conclusion.
But his idea of logic was very limited; it only applied to very short arguments, and
it only works for groups of things, like “all raccoons” or "some cephalopods."
Aristotle’s logic couldn’t handle sentences about individuals like “Utahraptor” or “Dromiceiomimus”.
And it wasn’t accommodating enough for complex sentences
built out of simpler ones, like “T-Rex stomped on the house, and he stomped on the car."
And you may have noticed, in most conversations, people like building big multipart sentences,
and they love talking about specific people. So for dealing with real-life conversations
with our pals, we’re going to need something a little more well-rounded.
That’s why in the following centuries, philosophers developed exactly the kind of system we’re looking for:
sentential logic. Sentential logic gets its name from the fact
that it takes the simple sentence as its fundamental building block.
It’s not restricted to talking about groups, and it can take smaller atom-like sentences,
and combine them into larger molecular ones, using sticky words like “and” and “or”.
Sentential logic has its own little language, but it’s not a natural language like what
linguists study. Instead, you can think of it like an artificial language, where we have
complete control over its inner workings. And that means we have a very good understanding
of how it works! Let’s zoom in on this setup.
First off, sentential logic is completely symbolic, which means it doesn't have any of the
messiness of English or any other natural language.
It just uses a few basic symbols, along with some rules that tell us how to combine simple
sentences into more complex ones. So, to start, sentential logic uses the letters of the alphabet
to represent sentences. For example, we could say “H” stands
for “T-Rex stomped on the house”, while “C” stands for “T-Rex stomped on the car."
These letters are just handy space-saving tools.
They aren’t tied to any particular proposition, and so we can link them up to pretty much whatever
sentence we want. Once we have our sentences all good n’ symbolized,
the structure-building rules of sentential logic use another group of five symbols to
join them together. These are the logical connectives, and if
we want to be technical, they’re called: the “conditional”, the “biconditional”,
“conjunction”, “disjunction”, and “negation”.
And you can see that each of these logical connectives roughly corresponds to some bit
of English, but they actually have strict, logical meanings.
So now, we have a way of building complex ideas in sentential logic.
Our earlier examples of “T-Rex stomped on the house, and he stomped on the car” would
look like this, with that upside-down v to mean “and”.
On the other hand, “Either Utahraptor confronted T-Rex, or he didn’t” would look something
like this. This uses the regular v to mean “or”, and that little flipped L to mean "not",
because in the second sentence, Utahraptor did not confront T-Rex.
All of these expressions, and an infinite number of longer ones, are part of sentential
logic. They’re allowed by its syntactic rules. And all of these sentences, simple
or complex, have a semantic meaning, too.
So for simple, atomic sentences, it’s easy: they’re either true or false. Depending
on the circumstances, it could be true that T-Rex stomped on the house, but false that
he stomped on the car.
But for bigger sentences, we need semantic rules that tell us how to combine the values
of smaller sentences. And, that depends on what connectives we choose.
To do this, we treat the logical connectives a bit like little truth-monsters, feasting
on truth and falsehood, and then expelling out the digested remains.
And we represent what their diets look like using truth tables, with the food that they eat
on the left-hand side, and the expelled parts on the right. Here’s the one for “and”.
So, for our earlier sentence “T-Rex stomped on the house, and he stomped on the car”,
we can use look at the right row on the truth table to find out the result. Since we decided that “H” was
true but “C” was false, we can look on the truth table and see that the whole sentence was false!
If we’d had “T-Rex stomped on the house, or he stomped on the car”, we could
look at the truth table for “or” and figure out that this sentence is true!
These truth tables show up a lot in semantics work, because they can show us at first
glance exactly when a sentence is true. And now that we have a grasp on sentential
logic, we not only understand why Aristotle’s arguments worked, but also a whole slew of
other ones. Like, take an argument that goes something
like this: “If Dromiceiomimus talks to T-Rex, then T-Rex will stomp on someone;
Dromiceiomimus did talk to T-Rex; so, he will definitely stomp on someone”.
Written out in sentential logic, it would look like this:
So let’s examine the truth table for the conditional connective, symbolized by this little arrow.
We see that the only row where the first
input and the result are true is the first one. And there we can see that the second
input must also be true. The argument works because of the way sentential logic is set up.
So, by creating symbols for our sentences and the connectives that join them together,
we’ve increased the precision and clarity of our thought. But, so what?
It’s fair to wonder why any of this matters much at all.
But! Logic has a very special use in linguistics that goes above and beyond its
use in other sciences. That’s because logic is able to model language.
Like the mathematical models that physicists use to study the universe, it’s something
that we understand well, so it’s easy to work with. Plus, it’s close enough to
natural language to be helpful.
And it lets us ask some interesting questions that otherwise, we wouldn't be able to!
Like, we know that the logical connective “and” can swap the order of the things
without changing the meaning; so like, “A and B” means the same thing as “B and A."
But we can wonder whether “and” in English
works the same way. And if you think about it, at first, the answer seems to be “no”.
Like, I can say “T-Rex has friends, and all of his friends are dinosaurs”, but it’s
weird and redundant to say “All of T-Rex’s friends are dinosaurs, and he has friends."
And “T-Rex stomped on the woman, and she died” means something very different
from “The woman died, and T-Rex stomped on her”. So maybe English “and” isn’t
like logical “and”, after all. Or, maybe it is! If we take into account
pragmatic knowledge, like Grice’s Cooperative Principle, we know that we usually want to talk in an orderly way,
one thing coming after another, because of the Maxim of Manner. And we know that we give
out just the right amount of information because of the Maxim of Quantity.
So maybe the fact that English’s version of “and” doesn’t seem entirely logical
is because Grice’s Maxims are imposed on top of the logic of English.
If Grice’s Maxims can explain why redundant sentences seem weird, or why things have to go in
a particular order to make sense, maybe language is more like logic than it first seems!
Truth is, questions about natural language connectives, and topics like it, are still open
ones. And we’ll be exploring more about that when we come back to talk about language & logic in the future.
But even with just these basics, logic has already proven itself to be awesomely handy. Way to go, Logic-Bot!
So, we’ve reached the end of The Ling Space for this week.
If you followed this episode to its logical conclusion, you learned that logic was first
developed as a way of understanding how sentences relate to each other and why some arguments
are better than others; that philosophers systematized logic into
sentential logic, by symbolizing sentences and their logical connections;
and that what we learn about the properties of logic can help us to investigate the properties
of natural language!
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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to keep expanding your own personal Ling Space, please subscribe.
And, we’ll see you next Wednesday. Do zustrichi!