[4.12.] “Principles of Strategy”, Pt. 1. [Chapter 4:6, pp.99ff.]

 

1.       “look for forms that correspond to valid rules of inference.” (100)

 

2.       “whenever the same sentence occurs on two different lines, look for ways to apply MP, MT, DS, or HS using those two lines.” (100)

 

3.       “small sentences (especially atomic sentences or negated atomic sentences) are your friends. If you have them, use them. If you can get them, do so.” (100)

 

4.       “Once you’ve mastered the rules, you will find completing many proofs much easier by working backward from the conclusion. To do this, begin by examining the logical structure of the conclusion and asking whether there is some line that, together with premises, could be used to derive the conclusion. Ask yourself, ‘Given my premises, what would it take to get my conclusion?” (101)

 

5.       “...usually it is not fruitful to use the same line in a proof over and over again. ‘Fresh information’ (an untapped line in a proof) tends to be most useful.” (101)

 

6.       “...if the conclusion contains an atomic component that does not occur in any premise, you must use Addition to introduce this information into the proof.” (102)

 

7.       “If you get stuck, don’t be afraid to produce a line you end up not using. If there is some rule you can legitimately apply, do it—even though you can’t see how it helps. ... There’s nothing wrong with trial and error.” (p.102)

 

Exercises 4-4 and 4-5, pp.94-95

§         do all for next time; check your even answers, we’ll do the odds in class

§         in ex.4-4, write only what's missing, no need to write entire proof

 

 

[4.13.] Double Negation (DN). [Chapter 4:7, p.105]

 

So far we’ve learned eight implicational (and one-directional) rules / argument forms.

 

There are an additional ten rules / argument forms, all of which are equivalence argument forms, which means that:

(1)    they are two-directional;

(2)    they can be applied to portions of sentences (in addition to entire sentences).

 

The first equivalence rule we will study is:

 

Double Negation (DN):

 

p :: ~~p

 

The double colon indicates that the inference embodied in this argument form can go either way:

 

p   /\~~p

 

and

 

~~p   /\ p

 

 

In a proof, DN works as follows:

 

1.       A É ~~B                p

2.       A                                       p     /\B

3.       ~~B                        1, 2 MP

4.       B                            3 DN

 

In addition to being two-directional, equivalence forms / rules can also be applied to parts of sentences. In this way they are unlike implicational forms / rules, which can be applied only to entire sentences. For example:

 

1. ~~A Ú B                   p   /\A Ú B

2. A Ú B                       1 DN

 

This is a legitimate use of equivalence rules, but not implicational rules, because when you apply an equivalence rule, you are simply replacing one sentence (or piece of a sentence) with another sentence (or piece) with which it is equivalent, i.e., with which it always shares the same truth value.

 

 

[4.14.] DeMorgan’s Theorem (DeM).[1] [Chapter 4:7, p.106]

 

This equivalence rule consists of two forms:

 

~(p · q) :: ~p Ú ~q

 

~(p Ú q) :: ~p · ~q

 

The following illustrates why the first form embodies a valid argument:

 

1. ~(S · C)                    p   /\~S Ú ~C

2. ~S Ú ~C                    1 DeM

 

And because DeM is an equivalence argument form, the following is also valid:

 

1. ~S Ú ~C        p   /\~(S · C)  

2. ~(S · C)        1 DeM

 

And this argument illustrates the second form of DeM:

 

1. ~(D Ú H)                  p   /\ ~D · ~H

2. ~D · ~H                   1 DeM

 

And because DeM is an equivalence argument form, the following is also valid:

 

1. ~D · ~H                   p   /\ ~(D Ú H)

2. ~(D Ú H)                  1 DeM

 

And remember that all equivalence forms / rules can be applied, not just to entire sentences, but to parts of sentences. So the following is a legitimate use of DeM:

 

1.(~D · ~H) É C           p   /\ ~(D Ú H) É C

2. ~(D Ú H) É C           1 DeM

 

 

Exercise 4-6 (pp.107-108)

§         complete all problems and check even-numbered answers; we’ll do the odds next time

§         notice that the final problem, #12, is on p.108

 

 

 

Stopping point for Friday February 8. For next time, exercises 4-4, 4-5 & 4-6; and read ch.4:8-10 (pp.108-111).