(intro music)
Hello, my name is Justin Khoo
and I'm an assistant professor[br]of philosophy at MIT.
This is the third part of[br]our series on conditionals.
Last time, we saw a classic problem
with the material conditional theory.
It predicts that it's just too easy
for conditionals to be true.
Recall Mitt Romney's conditional,
(1): "If the safety net needs repair,
"I will fix it."
The material conditional theory predicts
that Romney says something[br]true in uttering this sentence
if in fact the safety[br]net does not need repair.
But this seems wrong.
For it to be true,
the conditional requires[br]that there be some connection
between its antecedent (that[br]the safety net needs repair)
and its consequent (that[br]Romney will fix it).
But how should we[br]understand this connection
between the antecedent and[br]consequent of a conditional?
One promising theory, due[br]to Charles Sanders Peirce,
is that conditionals tell us about more
than how things actually are.
They tell us about how[br]they must or have to be.
Peirce's theory is known as
the "strict conditional[br]theory of conditionals."
According to this theory,[br]a conditional "If A, then B"
is true just if it is[br]impossible for "A" to be true
and "B" to be false.
Thus, on this theory, (1) means
that it is impossible for[br]the safety net to need repair
and Romney not fix it.
So when Romney utters one,
he tells us something both[br]about how things actually are,
as well as about how they must be.
The words that I used to characterize
Pierce's strict conditional[br]theory are known as "modals."
"Must," "have to,"
"impossible," "possible," and so on.
These words are used to express claims
about how things could have gone,
even if they didn't actually go that way.
Here's an example.
As a matter of fact,
you are watching a video on conditionals.
But you could have been[br]doing something else,
maybe watching a video about cats
or having a chat with friends.
Since you could've done something else,
it's merely a contingent fact
that you are now watching[br]a video on conditionals.
It's true, but didn't have to be true.
It could've easily been false,
say if you had decided to click the button
for the cat video instead.
Contrast the fact
that you're now watching[br]a video on conditionals
with the fact that two[br]plus two equals four.
This is also true,
but there is a strong sense[br]in which it has to be true.
There's just no way that two[br]plus two could equal five.
That's impossible.
Now, remember that what's[br]intuitively missing
from the material conditional theory
is a special connection between[br]antecedent and consequent.
According to the strict[br]conditional theory,
the special connection is the[br]necessity of the consequent
given the assumption of its antecedent.
Consider Romney's conditional, (1), again.
According to the strict[br]conditional theory,
(1) says that it is necessary
that Romney will fix the safety net
on the assumption that it needs repair.
Or, as we put it earlier,
in no way can it be that[br]the safety net needs repair
and Romney not fix it.
Thus, according to the[br]strict conditional theory,
(1) tells us that there is[br]a very strong connection
between the safety net needing repair
and Romney fixing it.
Unlike the material conditional theory,
it being false that the[br]safety net needs repair
is not enough to ensure that[br]the conditional is true.
With that in mind,
let's return to one of the sentences
which we found problematic
for the material conditional theory,
(2): "If the Earth is flat, I[br]will win the lottery tomorrow."
According to the material[br]conditional theory,
this sentence is true
simply because the Earth isn't flat.
But that seems wrong.
For (2) to be true,
there must be some connection
between the Earth being flat[br]and my winning the lottery.
Since there is no such connection,
that's why it's false, so we think.
According to the strict[br]conditional theory,
(2) says that it's impossible
for the Earth to be flat[br]and I not win the lottery.
But intuitively it is possible
for the Earth to be flat and I[br]not win the lottery tomorrow.
That's because there's[br]no connection at all
between flat Earths[br]and my future winnings,
so the strict conditional[br]theory predicts, correctly,
that (2) is false.
It seems then that the[br]strict conditional theory
marks a serious advance
over the material conditional theory.
That's some good news.
However, now for the bad news.
It turns out that the[br]strict conditional theory
is subject to a very similar problem
as the material conditional theory.
The problem is that it seems like
there could be false conditionals
that have impossible antecedents.
The reason this is a problem
for the strict conditional[br]theory is that it says that
a conditional "If A, then B" is[br]true just if it is impossible
for "A" to be true and "B" to be false.
Therefore, if "A" is itself impossible,
then of course it's[br]impossible for "A" to be true
and "B" to be false, in[br]which case the theory says
that "If A, then B" must be true.
So, what sort of conditional[br]raises this kind of problem
for the strict conditional theory?
Consider Goldbach's Conjecture.
This is the claim that every[br]even integer greater than two
is equal to the sum of two prime numbers.
This is a mathematical claim,
and hence if it is true,[br]it is necessarily true,
and if false, necessarily false.
Now, we don't know whether[br]Goldbach's conjecture is true.
That's what makes it a conjecture.
But suppose for a moment that it is false,
and consider the following conditional:
"If Goldbach's conjecture is true,
"I will win the lottery tomorrow."
This conditional seems false,
since my winning the lottery tomorrow
has nothing whatsoever to do
with Goldbach's conjecture[br]being true or false.
However, given our assumption
that Goldbach's conjecture is false,
then it is impossible that[br]Goldbach's conjecture be true,
and therefore it is impossible
for Goldbach's conjecture be true
and I not win the lottery tomorrow.
Therefore, according to the[br]strict conditional theory,
(3) is true.
But this prediction is wrong.
(3) is intuitively false.
The reason is, again, because[br]there's just no connection
between Goldbach's conjecture being true
and my winning the lottery tomorrow.
Furthermore, we arrive at this conclusion
regardless of whether we assume
that Goldbach's conjecture[br]is true or false.
I leave it as an exercise to the listener
to run the argument.
Notice the parallel between this problem
for the strict conditional theory
and the problem facing the[br]material conditional theory.
Both theories ran into[br]trouble with conditionals
which have false or[br]impossible antecedents.
This common core to their problems
suggests that perhaps we need
a radical new way of[br]thinking about conditionals,
on which they are not categorical claims
that have well-defined truth values
in situations where their[br]antecedents are false.
This insight will form the basis
of the last theory of conditionals
that we'll consider in the[br]final video in this series.
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