So let’s talk about what if.
It can be a lot of fun thinking about all the ways things could be different.
Like, imagine if you’d grown up in some other place, half a world away; would you
even recognize your alternate self, or would this counterpart be a complete stranger?
Happily, language provides the tools we need to make this sort of journey.
But how exactly does it go about charting our path?
I’m Moti Lieberman, and this is The Ling Space.
We’ve talked in the past about some of the defining traits that make human language so
special, like how a small set of rules put together in the right way can generate an
infinite number of expressions.
And one of the most distinguishing features we’ve uncovered so far is displacement — the
power language has to leave the here and now, and talk about events hundreds, or even millions
of years in the future or past, or just as many light-years away.
And maybe the ultimate example of displacement is the conditional sentence — so, basically
any sentence that takes the form “if A, then B.” Conditional sentences let us navigate
all the possibilities our brains can cook up, and range from the plausible, like “if
no one had smartphones, then the world would be a very different place,” to the preposterous,
like “if you hadn’t become a violinist, then you would have been an inter-dimensional hitman."
Figuring out how that little “if” lets us peek into different realities, though,
can be harder than it seems.
If you were to ask anyone who’s studied a bit of logic, they might tell you to look
no further than the idea of material implication.
Material implication is the little arrow symbol that logicians typically use to represent
these kinds of sentences, so that “if A, then B” comes out looking like this.
That little arrow acts as a connective, binding two simple sentences A and B together to make
a bigger, more complex one.
And its meaning, at least in logic, is very well understood.
In particular, whether a conditional sentence comes out true or false is just a product
of the value of its parts, and whether they’re true or false.
The relationship can even be summed up with a handy chart!
The first row, for instance, tell us that a conditional sentence will come out true
when both of its parts are true, while the last row tells us we get the same result if
both parts are false.
And this truth-value-based approach to the meanings of conditional sentences actually
seems to work pretty well, at least some of the time.
Take “if you study enough history, then you’ll pass the test.”
It feels like this chart is a good description of when that sentence comes out true, and
when it doesn’t.
If you think of the sentence as a kind of promise, then the promise is kept if the person
studies hard and passes, and it’s broken if they study hard but end up failing.
The last two rows cover those cases where the person doesn’t study much at all, and
either passes or fails based on luck.
But this theory starts to make weird predictions when you exploit the fact that putting any
false sentence in the “if” half renders the whole thing true, no matter what the second "then" half says.
Take the statement “if there’s a secret underground portal in Berlin, then Emily absolutely loves poetry."
Because we can be reasonably sure no such portal exists, according to our table, the
whole thing is guaranteed true.
But it doesn’t really feel true — at least, not without a whole lot of context.
Like, what do clandestine German gateways have to do with a person’s reading habits?
There’s no obvious causal connection there.
What’s worse is that even if you could imagine a way for the two ideas to connect up in some
way, because the first sentence is still false, the opposite also comes out true, so that
“if there’s a secret underground portal in Berlin, then Emily absolutely hates poetry.”
So how can we fix this mess?
Well, for one, we know that modern theories about how meaning works tend to rely on a
more sophisticated understanding of what sentences really are, beyond them simply being true or false.
This is something we’ve touched on before, when we talked about the difference between
a sentence’s extension, which is its truth-value, and its intension, which gets closer to its core meaning.
For instance, a verb like “believe,” which takes a whole other sentence as its object,
needs to be able to see more than that sentence’s truth-value; it needs to know something about
its content.
In the sentence “Howard believes his wife knows nothing about his work,” it doesn’t
matter whether or not she actually does; what matters is the relationship between his beliefs
and the idea that the embedded sentence expresses.
To put it in a way that’ll be useful for understanding conditionals, we can represent
Howard’s beliefs as a set — the set of all the different ways the world might be,
that correspond to what Howard thinks.
If Howard believes Andrei is better than him at board games, the set will only have worlds
where that’s true; if he doesn’t have an opinion either way, the set will have worlds
where it’s true and worlds where it’s false, since both possibilities are in play.
Then, the sentence “Howard believes his wife knows nothing about his work” simply
means that Howard’s beliefs fit neatly into the set of worlds where his wife doesn’t
know anything about his job.
In other words, all the worlds that makes up his beliefs are worlds where his wife is
in the dark about what he does.
If we apply this template to conditionals, a sentence like “if there’s a secret underground
portal in Berlin, then Emily absolutely loves poetry” basically claims that all secret-German-portal
scenarios are also Emily-loving-poetry scenarios.
And that statement seems plainly false, because we can easily imagine cases where she loves
poetry and cases where she hates it, independent of what’s going on beneath the ground.
There shouldn’t be a connection, and in our more sophisticated theory, there isn’t one.
So we get what we need: a theory that still covers the simpler cases, like how to pass
a history test, but excludes the problematic ones that our first hypothesis couldn’t rule out.
If sentences are just collections of possibilities -- like all the possible scenarios involving
secret German portals, or those involving someone liking poetry -- conditional sentences
compare those possibilities, and check to see if they’re connected in the right way;
specifically, checking to see if the first set fits inside the second.
This theory of conditional sentences, known as strict implication, fares a lot better
than our original, truth-value-based one.
But unfortunately, it still has problems of its own, especially when it comes to words like
“can” and “might.”
Imagine a couple driving through a dark forest, searching out a training school for deadly assassins.
They know the school is just inside the borders of the city of Potsdam, but they don’t know
quite where they are right now, or if they’re there yet.
Then, they come upon a big building and one of them says “if we’re in Potsdam, then
that might be the school we’re looking for.”
The good news is that, in our little scenario, the sentence is true under our new theory.
The bad news is that it isn’t true in quite the right way.
The strict implication account says this sentence essentially means that in all cases where
they’re in Potsdam, it’s possible — as far as they know — that they’ve found
the school.
And it’s true, but only because their being in Potsdam, or anywhere similar, has no direct
effect on their knowledge in this story.
Remember that we already said they don’t have any clues about where they are.
So whatever sentence we put at the start makes no difference according to the theory, as
long as its being true doesn’t give them some new information that tells them something
more; it just comes out to mean “that might be the school.”
The sentence “if we unknowingly took a wrong turn and ended up in the town of Templin,
then that might be the school we’re looking for” would also come out true, and that’s
bad news, because it’s actually nowhere close to Templin.
It looks like we’re back where we started!
We still haven’t managed to make the right connection between the “if” and the “then.”
But it’s not really as bad as it seems, because we’ve actually got one more trick
left up our sleeve, to help narrow down what’s been going on.
Consider this particular use of a conditional clause: “usually, if he visits the hospital, he brings flowers."
What that sentence seems to be saying is that, at most times — let’s say more than half —
when he visits the hospital, he brings flowers.
The adverb “usually” works to quantify time, while the if-clause restricts the occasions
that we need to consider by focusing in on only those involving hospital visits.
It’s just like how a relative clause restricts the meaning of the word “everyone”
in the phrase “everyone who was replaced,” which now obviously doesn’t mean literally
every possible person, but only a certain subset.
Okay, so maybe if-clauses are restrictors of some kind.
But what exactly do they have to restrict?
There doesn’t seem to be any quantifying words like “usually” or “every” in our examples.
Except, if we look carefully, we notice that every sentence we’ve seen so far has some
kind of modal word in it.
That is, each example is saying something about the possibility or certainty of some idea.
And if you remember back to our episode on modal verbs, they can be thought of as another
kind of quantifying word, like usually or every.
If we zoom in on the second half of our problem sentence, “that might be the school we’re
looking for,” we can think of its meaning like this.
Inside the set of possible worlds that represent our knowledge, there’s at least one where
that building we just found is our school.
The word “might” acts like an existential quantifier, and says that at least one world
in the first set can also be found in the second.
If the clause “if we’re in Potsdam” really is a restrictor, like in that sentence
with “usually,” then it comes in and temporarily shrinks down the first set, so that now the
sentence is only dealing with worlds that are in line with the speaker’s knowledge
and where they’re in Potsdam.
In other words, the sentence means something like “if we learned that we’re in Potsdam,
we'd conclude that that might be our school.”
And that seems to be what the sentence actually means.
And, importantly, other sentences with different cities in them come out false, just like we
want, since they shrink the first set down so much that it squeezes out any possibility
of being anywhere near the destination.
Of course, for this restrictor theory to work across the board, even sentences without modal
words, like “if there’s a mole, then it’s Peter,” must have some kind of modal meaning
hiding beneath the surface, so that the sentence really means “If there’s a mole, it must be Peter."
But that doesn’t seem like too much of stretch, given that those two sentences seem pretty
interchangeable.
As the linguist Angelika Kratzer famously wrote, “the history of the conditional is
the story of a syntactic mistake.
There is no two-place ‘if…then’ connective in the logical forms of natural languages.”
And we’ve seen here just how long and winding that history has been!
But with the map of linguistics pointing us down the right path, our journey has taken
us right where we need to be.
So, we’ve reached the end of The Ling Space for this week.
If you met all our conditions, you learned that one of the defining features of language
is displacement, which lets us talk not only about what is, but what could be; that logic
doesn’t do the best job of explaining how displacement works, especially when it comes
to conditional sentences; and that it makes more sense to think of if-clauses as tools
that restrict the possibilities we’re considering.
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See you next time!
Salamatta kal!