Today's lecture is about conditionals, to introduce the topic of conditionals.
Let me start by telling you a story. Imagine that my next door neighbor Travis
has been breaking into my house while I'm gone, and stealing my stuff.
I get suspicious about this, so I hire a private investigator to follow Travis
everywhere he goes. He follows Travis to his house at night.
He follows Travis out of his house in the morning.
And he follows Travis everywhere he goes during the day.
Now, you might ask me is the private investigator right now having lunch at
that Cuban Restaurant, New Havana?. I'll say, well if Travis is having lunch
there, then the private investigator is having lunch there.
Notice that we just used the words if and then to connect two propositions.
We connected the proposition, Travis is eating lunch at New Havana, with the
proposition, the private investigator is eating lunch at New Havana.
By saying, if Travis is eating lunch at New Havana,
then the private investigator is eating lunch at New Havana.
Thus, we built a new proposition, out of those first two propositions connecting
them with the words if and then. So, the words if and then together work
as a propositional connective. Let's consider how well you understand
the propositional connective if then. Suppose you have four cards,
this card has the number three on it, this card has the number eight on it,
this card has the letter A on it, and this card has the letter X on it.
Now consider this claim, if a card has a vowel on one side, then
it has an even number on the other side. Now I ask you, which card or cards, would
you need to flip over in order to verify that this claim is true?
Well, the right answer is this card with the letter A on it and this card with the
number three on it. Now, statistics tell me that 70% of you
answered that you would need to flip over the card with the number eight on it.
And for that reason, you need to learn something, about
conditionals. But do they work as a truth functional
connective? Well, remember they do workers of
truth-functional connective. Only if the truth value of the
proposition that you use them to build depends solely on the truth values of the
propositions that you use them to connect.
So, does the truth value of if Travis is eating lunch at new Havana, then the
private investigator is eating lunch at New Havana, depends solely on the truth
value of Travis's is eating lunch at new Havana?
And on the truth value of the private investigator is eating lunch at new
Havana? Well, let's see.
We can try to construct a truth table and see if the truth value of the whole
proposition depends solely on the truth values of its two ingredient
propositions. So, to see whether the propositional
connective if then is a truth functional connective or not, let's see if we can do
a truth table for it. So, we consider four possible situations.
In one situation, it's true that Travis is at New Havana, and true that the
private eye is at New Havana. In the second possible situation, it's
true that Travis is at New Havana, but false that the private eye is at New
Havana. And the third, it's false that Travis is
at New Havana, but true that the private eye is at New
Havana. And in the fourth possible situation,
it's false that Travis is at New Havana, and also false that the private eye is at
New Havana. Now, what can we say about the situations
in which it's true that if Travis is at New Havana, then the private eye is at
New Havana. Well, one thing that's clear is that if
it's true that Travis is at New Havana, but it's false that the private eye is at
New Havana. Then this proposition right here is going
to be false. So in the situation represented on the
second line, the proposition if Travis is at New Havana, then the private eye is at
New Havana is going to be false. But what about the other situations?
Suppose, Travis is at New Havana and the private eye is at New Havana.
Well, if they're both at New Havana, then it seems pretty clear that the
statement if Travis is at New Havana, then the private eye is at New Havana is
going to turn out to be true. But you might worry.
Wait a second, just because the first proposition is
true and the second proposition is true, does that mean that combining them with
the propositional connective if then results in a true proposition?
You might think not necessarily, because consider the following example.
Consider this example, if two plus two equals four, then Pierre
is the capital of South Dakota. Now what I've done there is used the
propositional connective if then to connect two propositions, both of which
are true. It's true that two plus two equals four,
and it's also true that Pierre is that capital of South Dakota.
But just because I've used the proportional connective if then to
connect two propositions that are true, does that mean that the resulting
proposition, if two plus two equals four then Pierre
is the capital of South Dakota. That the resulting proposition is true.
Well, it's not clear. If I tell you, if two plus two equals
four, then Pierre is the capital of South Dakota.
Do you want to say that's true? Or that's false?
Well, it's not obviously true, but it's also not obviously false.
It's just kind of baffling. But just because this proposition is
baffling doesn't mean it's not true, look it's baffling that Pierre is that capital
of South Dakota but that doesn't mean it's not true, as it happens it's true
that Pierre is the capital of South Dakota, baffling as that might be.
So, just because this proposition of two2=4, plus two euqals four, then Pierre
is the capital of South Dakota. Just because that proposition is
baffling, doesn't mean that it's not true.
So is it true? Let's find out.
So suppose you have two propositions, P and Q.
P could be the proposition, Travis is at New Havana and Q could be the
proposition, the private eye is in New Havana.
Well, let's suppose, that the negation of P, and the negation of Q is true.
Suppose that, for a moment. What follows?
What, what follows is that P and the negation of Q, that proposition right
there is false. Remember, because every proposition has
the truth value opposite the truth value of its negation.
So, if the negation of that proposition is true then that proposition itself must
be false. Now if that proposition is false, then
what does that tell us? Well it tells us that P and not Q cannot
both be true. So if P is true, than not Q has got to be
false. Right, if P and not Q were both true,
then this conjunction would be true and not false.
So if P is true, then not Q has got to be false.
But if not Q is false, then Q is true. because Q is got to have the opposite
truth value of not Q. So what can we conclude?
We can conclude that if the negation of P and not Q is true, then if P is true then
Q is true. If this proposition right here is true,
then this proposition right here is true. But notice, we could conclude something
else. Suppose that if P is true, then Q is
true. What follows from that?
Well, one thing that follows from it IS that, you can't have it be the case.
That P, ends up being true but Q is false.
Let's write that down. It can't be the case,
it's not the case that P is true and Q is false, in other words not Q is true.
It can't be the case that P is true and that not Q is true.
But that's just what we started with. So here's what we've shown.
We've shown that if the negation, of P and not Q is true.
Then it follows, that if P is true then Q is true.
And we've also shown, that if the proposition.
If P is true then Q is true, if that proposition is true?
Then it follows, that the negation.
of P and not Q is going to be true. So, this proposition implies this one and
this one implies this one. Which means, that whenever this one is
true this one is got to be true and whenever this one is true this one is got
to be true. So in other words, they are true in all
the same situations. So if there is a truth table for this its
going to be the very same as the truth table for this.
But is there a truth table for this? Lets figure out whether there is a truth
table for this. So remember if you're going to do the
truth table for this we have to start by considering the four possible situations
we have with the propositions P and Q. It could be true that P and true that Q,
true that P and false that Q, false that P and true the Q or false that P and
false that Q. Okay, so those are our four, four
possible situations. Well consider the truth table for the
negation of Q, for not Q. Well that's going to be false whenever Q
is true, and it's going to be true whenever Q is false.
So that's the, the truth table for not Q. Now how about the conjunction of P and
not Q. That's only going to be true when P is
true and not Q is true. In other words, in this situation.
This is the only situation where P and Q will be true.
In every other situation it'll be false. It'll be false wherever P is false and
then will be false wherever not Q is false.
Now how about the negation of P and not Q.
Well the negation of P and not Q, is going to true, whenever P and not Q is
false and its going to be false whenever P and not Q is true.
So, its true here, false here, true here and true here.
Now does this truth table look familiar to you?
Yeah, you remember, it's the same as the truth table for not P or Q.
The truth table that we constructed in last lecture.
So remember, the negation of P and not Q is true.
In just those cases, when if P then Q is true.
These two propositions are going to be true or false in all the same situations.
So if there is a truth table for the negation of P and not Q, then there is
going to be a truth table for if P then Q and its going to be the same truth table.
So lets get back to our Travis example and see how to fill it out.
So applying that discussion to this example, we can now say that whenever its
false that Travis is at New Havana. That no matter where the private eye is,
its going to be, true that if Travis is at New Havana, then the private eye is at
New Havana. So, this is the truth table for if then.
And so if then is a truth-functional connective.
I've just argued that in English the phrase if then can work as a
truth-functional connective. But there are other phrases in English that work in
exactly the same way, to express exactly the same truth-functional connective.
One such phrase is only if. If I say,
if Travis is in New Havana, then the private eye is in New Havana.
What I'm saying means the same thing as Travis is in New Havana only if the
private eye is in New Havana. Those two propositions are going to be
true, under just the same circumstances. Another word that we can use in English
to express the very same truth functional connective is just the word if.
We could say the private eye is in New Havana, if Travis is in New Havana.
And that means the very same thing. And it expresses the very same
proposition, as if Travis is in New Havana, then the private eye is in New
Havana. We're going to use a horseshoe to
symbolize the truth functional connective, if then.
And we can use the truth table for if then to show that a certain kind of
inference, a kind of inference that we're going to call modus ponens is always
valid. Now a modus ponens inference is an
inference that has two premises. The first premise has the form if P then
Q. The second premise has the form P,
in other words the second premise just says that the first proposition in if P
then Q is true. And the conclusion is here.
So the first premise tells us that we're in a situation in which P horseshoe Q is
true. So the first premise rules out this
situation right here. It tells us that this situation right
here isn't the actual situation. The second premise tells us that we're in
a situation in which P is true. So, the second premise rules out this
situation where P is false, and it rules out this situation where P is false.
So, the two premises together rule out all three of these possible situations.
In other words, according to the premise P horseshoe Q and the premise P together,
the only possible situation is this one. But notice, if this is the only possible
situation, then Q has gotta be true, because Q is true in this situation.
So whenever we have an inference, the first premise of which is P horseshoe Q,
the second premise of which is P. We know we can draw a conclusion to the
effect that Q and that inference is going to be valid.
There's no possible way for the two premises of that inference to be true
while the conclusion is false. And that's why modus ponens is a valid
inference. There's another kind of inference called
the modus tollens. A modus tollens inference is one in which
the first premise says if P then Q. The second premise says not Q,
and the conclusion is not P. Now we can use the truth table for if
then to show that a modus tollens inference is always valid.
Here's how, in order to see that modus tollens is a
valid form of inference, we need a slightly more extensive truth table than
we used to show that modus ponens is a valid form of inference.
So again, remember we have four possible situations.
P and Q are both true, P is true and Q is false, P is false and Q is true or P and
Q are both false. Now, when P and Q are both true, then P
horseshoe Q is true. When P is true and Q is false, then P
horseshoe Q is false and whenever P is false then P horseshoe Q is true.
Now the negation of Q is going to be false whenever Q is true and true
whenever Q is false. And the negation of P is going to be
false whenever P is true and true whenever P is false.
Now, using this truth table, we can prove that modus tollens is a valid form of
inference. Remember, the first premise of a modus
tollens inference is, if P then Q, P horseshoe Q.
So, that first premise rules out the possible situation where P horseshoe Q is
false. So, we can rule out that situation right
here. The second premise of a modus tollens
inference, says, not Q and that rules out the situation in which not Q is false.
There're two subsituations. There's this one,
that's one where not Q is false and there's this one where not Q is false.
So, the two premises of a modus tollens inference rule out these three possible
situations, and they leave the only possible situation as this one.
This situation where P is false, Q is false, P horseshoe Q is true, not Q is
true and not P is true. But since that's the only possible
situation left open by the premises of a modus tollens inference, and since that's
the situation in which not P is true. The conclusion of a modus tollens
inference, namely not P is going to have to be true whenever the premises of that
inference are true. So every modus tollens inference is
valid. The truth functional connective if then,
is what we call a conditional. Each conditional has two propositions
that it connects, a proposition that comes before the horseshoe, right after
the if part. That's called the antecedent.
And then there's a proposition that comes after the horseshoe.
Right after the then part, that's called the consequent.
For every conditional has an antecedent and a consequent.
And the conditional is true whenever the antecedent is false, or when the
antecedent is true and the consequent is also true.
The only time a conditional is false is when the antecedent is true and the
consequence is false. Finally,
I want to wrap up by introducing another kind of truth functional connective that
we call the biconditional. The biconditional is really a conjunction
of two conditionals. It's the conjunction of if P then Q and
if Q then P. What the by conditional says, is, two
propositions P and Q. Have exactly the same truth value.
P is true if and only if Q is true. We were going to use these three
horizontal lines to symbolize the biconditional.
Now, here's the truth table for the biconditional.
Remember we have four possible situations.
P and Q are both true, P is true and Q is false, P is false and Q is true, or P and
Q are both false. Well P biconditional Q is true whenever P
and Q have the same truth value. So, P biconditional Q is going to be true
in this situation, when P and Q are both true.
And it's going to be true in this situation, where P and Q are both false.
But in these two situations, the propositions, P and Q, have different
truth values. So P biconditional Q is going to be false
in both of those two situations. In English,
we express the biconditional using the phrase if and only if.
That concludes our lectures on truth functional connectives and propositional
logic. Next week we'll talk about categorical
logic.