[4.23.] Strategy.

 

Chapter 4:11 of your textbook gives the following principles of strategy for completing proofs in sentential logic:

 

1.       Use equivalence rules, especially DeMorgan's (DeM), to break down complex sentences into simpler sentences, which are easier to exploit.

 

For example (p.113):

 

1. (B É C) É A                   p

2. ~(D Ú A)                                    p / \ B

3. ~D · ~A                         2 DeM

4. ~A                                  3 Simp

5. ~(B É C)                                    1, 4 MT

6. ~(~B Ú C)                       5 Impl

7. ~~B · ~C                        6 DeM

8. ~~B                                7 Simp

9. B                                    8 DN

 

2.       "A letter that occurs only once in the premises of an argument and not at all in the conclusion usually is excess baggage, to be gotten rid of or (if possible) ignored." (pp.113-14)

 

1.        ~B · C             p

2.       (A É D) É B    p  /\ A

3.       ~B                    1 Simp

4.       ~(A É D)         2, 3 MT

5.       ~(~A Ú D)        4 Impl

6.       ~~A · ~D         5 DeM

7.       ~~A                 6 Simp

8.       A                     7 DN

 

“C” plays no role in the logic of the argument and can be gotten rid of immediately.

 

 

3. “Break apart any nonrepeated conjunctions, using Simplification. Break down nonrepeated biconditionals, using Equivalence.” (p.114)

 

 

4. “If you have some premises that are disjunctions and some that are conditionals, it will usually help either to change the conditionals into disjunctions, or the disjunctions into conditionals, using Implication.”(p.114)

 

For example: Exercise 4-12, #4 (p.121):

 

1. S Ú (~R · T)                 p
2. R É ~S                         p          /\ ~R
3. ~~S Ú (~R · T)             1 DN
4. ~S É (~R · T)               3 Impl
5. R É (~R · T)                2, 4 HS
6. ~R Ú (~R · T)               5 Impl
7. (~R Ú ~R) · (~R Ú T)   6 Dist
8. ~R Ú ~R                       7 Simp
9. ~R                                8 Taut

 

[Notice that this is a different solution than the one given in the back of your textbook.]

 

5. [This principle was given in an earlier section]: “work backward from the conclusion.” (p.115)

 

 

Exercise 4-9 (p.116)

§         Do all of these and check your even-numbered answers. We’ll do the odds next time.

 

 

[4.24.] “Common Errors in Problem Solving.”

 

Your textbook lists a number of common mistakes in ch.4:12. Make sure you understand why all these are mistakes, so that you can avoid them:

 

1.      “Using Implicational Forms on Parts of Lines.”  Implications forms/rules (the first eight forms/rules we learned) can only be applied to entire lines, not to parts of lines. Only the ten equivalence forms can be applied to whole lines AND to parts of lines.

 

2.      “Reluctance to Use Addition.” Sometimes the conclusion of a proof contains constants that do not appear in any of the proof’s premises. In such cases, the only way to introduce that constant is to use this rule.

 

3.      “Reluctance to Use Distribution.” It can be difficult to identify lines in a proof where Dist can be applied profitably. Study this rule so that you can more easily recognize premises having the forms which allow you to apply this rule:

·         conjunctions, at least one conjunct of which is a disjunction

·         disjunctions, at least one disjunct of which is a conjunction

 

4.      “Trying to Prove What Cannot be Proved.” Do not waste your time attempting to “prove” substitution instances of invalid forms. The best way to guard against this mistake is to become extremely familiar with the 18 valid argument forms we’ve learned so far and take care to use only those forms in your proofs.

 

5.      “Failure to Notice the Scope of a Negation Sign.” To avoid this, simply take care in reading sentences. Also be sure you understand that a tilde immediately preceding parentheses (or brackets, or braces), negates the entire sentence within the parentheses (or brackets, or braces).

 

 

Exercise 4-10 (pp.118-19)

§         [we will do #6 and #10 in class]

§         do all of these for next time; we’ll cover the odds next time

 

Exercise 4-11 (p.121)

§         do all of these for next time; we’ll cover the odds next time

 

Exercise 4-12 (pp.121-22)

§         do all of these for next time; we’ll cover the odds next time

 

[I have not assigned 4-13, although you can still attempt these. They are very difficult and could take you a lot of time.]

 

 

 

Stopping point for Wednesday February 13. For next time, do exercises 4-9, 4-10, 4-11, 4-12; no new reading.