[5.3.] “Strategy Hints for Using CP and IP.” (Ch.5:3)

 

This short section of chapter 5 (5:3, p.139) provides the following hints:

 

Conditional Proof

§         If your conclusion is a conditional, try using CP.

§         Try using CP if your conclusion is not a conditional but if it equivalent to a conditional; e.g., if your conclusion has the form ~p Ú q, you can use CP to prove p É q, then use Impl to convert that conditional to the conclusion you want.

§         Sometimes it makes for a shorter proof to assume the negation of the consequent of the conditional you want to prove (i.e., assume ~q), derive the negation of the antecedent (~p) thereby proving ~q É ~p, and then use Contra to convert that conditional to p É q.

 

Indirect Proof

§         “If all else fails, try using IP” -- this is because all proofs can be solved using this rule (although not all proofs can be solved in the fewest steps using it).

§         IP is the preferred method of solving a proof when the conclusion is an atomic sentence or the negation of an atomic sentence.

 

Exercise 5-5

§         complete all of these; we’ll check the odds next time

 

 

[5.4.] “Theorems.”

 

In this section, you will learn a new use for the proof method:  to prove that a sentence is a theorem of sentential logic.

 

theorem: “a tautology derived by a zero-premise deduction” (149)

 

Up to now, we have not encountered a zero-premise deduction. All of the proofs you’ve worked up to this point have had at least one undischarged premise.

 

Recall that a premise is discharged when it is set aside as no longer being eligible for use in the proof. Anytime you use CP or IP, you begin by assuming a premise. That premise and all premises that follow from it must be discharged before the proof is complete. In all the proofs we’ve examined up to now, there has always remained behind at least one premise with which you started and which remained undischarged.

 

If you can prove a conclusion using CP or IP, based on nothing but premises that you yourself assume (i.e., without any starting premises), then you will have shown that sentence to be a theorem of sentential logic.

 

In exercise 5-6 (p.141) you are asked to prove that a given sentence is a theorem. You do this by constructing a proof that has no starting premises. To begin the proof, just use CP or IP as you normally would.

 

For example:

 

Problem #2 asks you to show that (A  B) É A is a theorem. You do this by beginning a proof with an assumed premise, just as in past uses of CP:

 

(2)        1. A · B                        AP

2. A                              1 Simp

3. (A · B) É A              1-2 CP

Problem #2 asks you to show that [A É (B É C)            ] É [(A É B) É (A É C)] is a theorem. You do this by beginning a proof with an assumed premise:

 

(4)        1. A É (B É C)                                                 AP

2. A É B                                                          AP

3. A                                                                  AP

4. B É C                                                           1, 3 MP

5. A É C                                                          2, 4 HS

6. C                                                                  3, 5 MP

7. A É C                                                          3-6 CP

8. (A É B) É (A É C)                                       2-7 CP

9. [A É (B É C)           ] É [(A É B) É (A É C)]          1-8 CP

 

Exercise 5-6 (p.141):

·         do all of these; we’ll check the odds next time

 

 

 

[5.5.] “Proving Premises Inconsistent.”

 

Any given set of sentences is either consistent (it is possible for all the sentences to be true at the same time) or inconsistent (it is impossible for all the sentences to be true at the same time).

 

Now we will use the proof method to demonstrate that a set of premises is inconsistent. We will do this by deriving a contradiction from those premises.

 

Remember that a contradiction is a sentence that cannot possibly be true. If a set of sentences implies a contradiction, then at least one of those sentences must be false. So to demonstrate that a set of sentences implies a contradiction is to show that it is impossible for all of those sentences to be true at the same time.

 

For example, let’s do #2 in exercise 5-7 (p.142). You are given an argument:

 

1.       ~A Ú B                   p

2.       ~B Ú ~A                 p

3.       A                           p          /\B

 

and asked to show that the three premises are inconsistent. To do this, use the rules that you’ve learned to derive a contradiction (a sentence of the form p · ~p) from the premises (important: the conclusion of the argument is irrelevant; to prove the premises to be inconsistent, you need only derive a contradiction from the premises themselves, so in these problems you should simply ignore the conclusion of the argument you are given):

 

4. ~~A                          3 DN

5. ~B                            2, 4 DS

6. ~A                            1, 5 DS

7. A · ~A                     3, 6 Conj

 

 

Exercise 5-7, p.142

§         do all of these for next time; we’ll check the odd problems in class

 

 

Stopping point for Wednesday February 20. For next time, complete ex. 5-5, 5-6 and 5-7, and read ch.3:6-8 (pp.75-81)