Shantila's Inside logic #8

Conditional Proof

You are getting impatient. Your plane was supposed to have left twenty minutes ago. You are getting bored watching the people around you. How long can you listen to these people? One couple is still arguing and the other couple is still smooching. Who cares? You don't know these people. You just want to get going.

You walk over to the check-in counter to find where the airplane is. The attendant says, "Thank you for the help with that guy."

"Well, no problem," you say. "I just showed him to the bathroom."

"He seems to have some serious problems," the attendant says.

"Yes," you say. "I believe he may have a logic block."

"Oh, that's terrible!" the attendant says. "I wonder how he gets dressed."

"That's a good question," you say. "But, frankly, I don't know. And I don't really care either."

"Can I help you with something?" she asks.

"Do you know when we're going to leave?" you ask.

You are in the Detroit airport trying to get to Atlanta because it is your sister's birthday. There is a dinner planned for tonight. The dinner will be followed by a party.

"I'm sorry. I don't really know," the attendant says. "If it snowed this morning in Denver, the plane will be at least two hours late. I should be getting the information soon, and I will let everybody know."

"Oh. Ok. Thanks," you say. Snow in Denver?! This is disappointing news. Assume it did snow in Denver this morning. Then, as she just said, the plane will be at least two hours late; and if so, I won't get to Atlanta until after 8 p.m.-- which is too late!

--So if it snowed in Denver this morning, you think, I will be too late for the dinner!

*

Your conclusion here is a conditional. So far we have used conditionals as premises (for use with the rules MP and MT). As our next rule, we now introduce a general method for obtaining conditional conclusions.

During those moments at the airport, you do not know whether or not it snowed in Denver; but still, for the sake of reasoning about your situation, you can assume it temporarily. As we naturally say,

-- what follows from that? Well, it follows that you will be late for the dinner, given these two additional premises:

The reasoning so far is familiar and can be symbolized using the obvious letters:

S

S>T

T>L

From these three assumptions we use MP twice in order to derive L, you will be late.

But remember -- we don't really know whether or not S is true. We only assumed it for the sake of the argument. So the simple conclusion L is too strong. All that we want to conclude is the conditional, if it snowed in Denver, then you will be late. (For all you know, it did not snow in Denver and the plane will get to Detroit in time, after all, so that you will not be late to the dinner.) That is, you want to conclude S>L based on the other two premises.

Our new rule of derivation makes precise this form of reasoning. If Q depends on P as one of its assumptions, the new rule permits us to derive P>Q based on the remaining assumptions.

The example can be proved as follows. Notice that our two initial assumptions of the proof are the two conditional premises. But on line 3 we also make the additional assumption S -- we do this precisely because we plan later to use the rule CP.

S>T, T>L } S>L

The numbers listed on the right on line 6 refer, first, to the line where we assumed the antecedent (line 3), and second, to the line where we derived the consequent (line 5). Notice that the number of assumptions for line 6 is one fewer than that for line 5. That is, the number of assumptions goes down by one number in the transition from line 5 to line 6. The number of assumptions always falls by one in this way in our use of this new rule CP. In this case, we "discharged" the assumption S that we made on line 3, and we did so by "absorbing" it into the conditional itself on line 6 as its antecedent.

Here is an example in which CP is used with MT.

37 P>Q } ~Q>~P

The assumption ~Q made on line 2 is discharged on line 4 as ~Q is absorbed into the conditional statement itself. Notice how this proof is related to the proof we did earlier for sequent 10.

10 P>Q, ~Q } ~P

The first three lines of these two proofs are exactly the same! And notice, then, what happens on line 4 of the proof of 37 -- that is what the CP rule permits us to do.

Proof of the following sequent relies on our new rule CP as well as the rules MP and &E.

38 P>(Q>R) } (P&Q)>R

Notice that P&Q was assumed on line 2 because it is the antecedent of the conditional we want to derive.

If you are trying to prove a sequent whose conclusion is a conditional, the CP strategy usually is a good one to try. Assume the antecedent of the conditional and try to derive the consequent. Whenever you see that you are trying to derive a conditional sentence, it is a good idea to consider using the CP strategy.

Here is an example in which CP is used twice, as is MP.

47 P>(Q>R) } Q>(P>R)

We assumed Q on line 2 because we saw it was the antecedent of the conditional Q>(P>R) we want to prove. Having assumed Q we now need to derive P>R. And to do that we assume P and try to derive R. Using MP twice we derive R on line 5. Notice that the assumptions on the left decrease by one in each of the two uses of CP. If the consequent itself is a conditional (as in sequent 47 that was just proved) then assume its antecedent and try to prove its consequent. We can repeat this process until what we are trying to prove is a non-conditional statement.