So let’s talk about pieces. Linguists and philosophers often think of sentences as being
either true or false; it’s just part of their meaning! But what about the bits and
pieces that make the sentences up? Words like “all” or “some”, or phrases like “lives
in LA” or “help the helpless”? Our success in finding out how true or false they can
be is predicated on donning our logic hats and doing a little investigating. And maybe
checking out our previous episode on the topic to cover the basics, cause it’s gonna get
technical in here. I’m Moti Lieberman, and this is The Ling Space.
If you think back to when we introduced logic, we talked about how it was first developed
by the Greek thinker Aristotle to get a better handle on why some kinds of arguments were
convincing, while others weren’t. We also learned that philosophers developed sentential
logic as a way of expanding on what that original system could do. Aristotle’s approach couldn’t
really deal with complex sentences, and it didn’t really have a good way of talking
about individuals -- only groups.
So, the artificial language of sentential logic was designed to use the letters of the
alphabet to stand in for any of the sentences you can find in a natural language. Take a
pair of sentences like, “The Powers That Be sent Cordelia a vision. She received it”.
We could symbolize the first sentence with, say, an S for “sent”, and the second one
with an R, for “received”. To symbolize words like “and” and “or”, people
invented logical connectives, so we could represent the way that bigger sentences can
be built up out of smaller ones. All we need to put our two sentences together
in a natural way is to use this symbol for “and”.
With this system in place, linguists can then use the well-understood properties of logic
to think and talk about the less well-understood properties of language. Like, take our now-completed
stuck together sentence. In logic, the whole sentence is true only when both halves are
true. And since English seems to mostly work the same way, and the word “and” acts just
like its logical cousin, we can suppose their meanings -- their semantics -- are the same.
And we can represent the semantics of a logical connective with a truth table that describes
it, so we can do the same for English!
With this method, “and” in English becomes just like our little logical truth monster,
munching on truths and falsehoods, and spitting out the results according to the table.
And we’ve managed to chip away a little bit at the semantics of a natural language
using the chisel of logic. Since this works for words like “or” and “if” and “not” as well,
it could turn out that just by speaking a language, you know a lot more logic than you ever realized!
Now, if we want to get the most out of using logic as a model for understanding natural
human language, we need to tweak it a bit. After all, language is more than just some
letters strung together with a handful of symbols: we’ve got verbs and prepositions,
and tense and number and person; language is jam-packed! We need to get inside sentences,
too. And sentential logic just isn’t up to the task.
Thankfully, philosophers noticed this too. They saw its limitations, and realized that
just like Aristotle’s logic, there are some things that sentential logic can’t do.
Like, take the fact that the following argument is definitely a valid one:
None of Angel’s friends hang out with vampire hunters.
Connor hangs out with Holtz, and Holtz is a vampire hunter.
So, Connor is no friend of Angel.
If we symbolize this using only the tools of sentential logic, we get this:
But that argument doesn’t hold up! None of the symbols even have anything to do with
each other! Like, how do we get from an “N”, a “C”, and an “H”, to an “F”?
Sentential logic treats sentences as indivisible atoms, and so it can’t see what’s going
on inside the sentences. It has no way of relating the contents of these sentences to
each other, so we need to crank up the power. To deal with a problem like this, we need predicate logic.
Predicate logic has all the tools of sentential logic, but it replaces the simple sentences
we’ve been using so far with something a bit more fine-grained: predicates. Intuitively,
predicates are like properties of things, like “green” or “immortal”, but predicates
can also be actions, like “bites”, or even states of being, like “is a vampire”.
Generally, predicates are the sorts of things that can be true of something, like how Lorne
is green or how Angel is a vampire. So you can cut up the meaning within a sentence into
idea chunks that fit together.
To symbolize predicates, we use the same alphabet we were using before, but with some extra
variables. We add the variables after the predicate so it’s easier to keep track of
how many things are missing before the predicate can become a full-fledged sentence.
So, “green” could be “Gx”, where the “x” lets us know that it needs one thing
to be green. And “bites” could be “Bxy”, where the “x” says this predicate needs
a biter, and the “y” tells us that something needs to get bitten!
Unlike sentences in sentential logic, though, predicates on their own can’t be true or
false. If we want them to have some value, we have to apply them to something, to fill in those variables.
So, we introduce a few individuals into the mix, using lowercase letters, like writing
“Gl” to mean “Lorne is green”, or “Bda” for “Darla bit Angel”.
Now, both sentences are complete at last, and they’re both true! And since predicate
logic otherwise works just like sentential logic, we can stick the two sentences together.
And so we can say “Lorne is green, and Darla bit Angel” is also true!
Predicate logic is so powerful that it can fully capture almost every sentence we can
think of. But we still need to cover the area that Aristotle had mastered with his logic:
describing groups of individuals. That’s because even though predicate logic is much
more useful and much less ambiguous than Aristotle’s, it still has no way to explicitly talk about
collections of things. For that, we have to add one more piece to our puzzle: quantifiers.
Quantifiers are the symbols that stand in for the words “all” and “some”, and
historically we write them out with an upside down “A” and a backwards “E”.
Quantifiers let us create all sorts of new sentences. Like, “All of Angel’s friends
are good guys” would look something like this:
Since that little arrow stands in for the phrase “if . . . then”, this new quantified
sentence literally means “for all the things under discussion, if those things are friends
of Angel, then those things are also good guys”. Clunky, but clear and unambiguous.
And we can use quantifiers to talk about smaller groups of things, too. Like, for the sentence
“some vampire has a soul”, we would get this:
Literally translated back to plain English, this sentence means “for at least one thing
under discussion, that thing is a vampire and that thing has a soul”. Again, it’s
hardly Shakespeare, or even Joss Whedon, but it delivers the goods.
And with quantifiers in the system, predicate logic can even explain why the argument we
brought up earlier actually works, even when sentential logic made it look like it didn’t.
If it’s true that “none of Angel’s friends hang out with vampire hunters, that Connor
hangs out with Holtz, and that Holtz is a vampire hunter”, predicate logic guarantees
that “Connor is no friend of Angel". That’s exactly what we want!
It might look a little scary at first, but we can understand how predicate logic works
for us here by thinking of it as being a kind of subtraction. The predicates “O”
and “H” in the first and second lines cancel each other out. And we drop the quantifiers,
because by the second line we’ve replaced all of our variables with individuals, so
we don’t need them anymore. All that’s left over is negation and the sentence “Connor
is a friend of Angel”; put it together, and we get “Connor is no friend of Angel”.
And even if you haven’t mastered predicate logic yet, the connections between the sentences
are a lot more obvious here than they were with sentential logic, so it’s much easier
to see where the conclusion comes from; it doesn’t just pop magically into existence
out of nowhere! And it’s all because predicate logic lets us peek inside of our sentences to
see what they’re made of.
Finally, just like how sentential logic gave us the tools to understand connectives, predicate
logic gives us what we need to start thinking about the meanings of other words. Verbs and
adjectives and all these other predicates all have these logical meanings.
And that means, if we wanted to, we could build truth tables for our predicates, too,
just like with the connectives. So, the meaning of a predicate like “is a vampire,” or
“Vx” would just be a list of all the individuals that it could possibly apply to, followed
by “true” in all the cases where it does!
So, Angel is a vampire, Buffy is not, neither is Cordelia, but Darla is, etc. And if a phrase
like “is a vampire” acts enough like its symbolic version, we can suppose that the
meaning of the phrase is the same sort of thing -- a kind of truth table that picks
out all the real-world individuals the phrase applies to. This way, Logic gets us closer
and closer to understanding what speakers must know when they know the meanings of words.
Of course, there’s still more work to be done, since we haven’t even come close to
covering all the categories and concepts that you find across the world’s languages. But
now that we’re able to crack open our sentences and take a look at the pieces inside, our
investigation has become a lot more meaningful!
So we’ve reached the end of the Ling Space for this week. If you subtracted out the right
variables, you learned that sentential logic isn’t powerful enough to describe a natural
language like English; that predicate logic improves our understanding by breaking sentences
up into pieces — adding variables and quantifiers to the mix; and that with careful investigation,
it could turn out that the semantics of logic offers greater insight into the semantics 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. Mirupafshim!