[4.13.] Double Negation (DN). [Chapter 4:7, p.103-104]

 

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

 

And again, 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 component sentence) with another sentence (or component) 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.104]

 

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.105-106)

§  [time permitting, do a couple of even problems in class: 2, 4]

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

§  notice that the final problem, #12, in on p.106

 

 

[4.15.] Commutation (Comm). [Chapter 4:8, p.106]

 

In general, a commutation is a replacement of one thing with another, or an exchange of one thing for another. (The English word “commutation” derives from the Latin for “change.”]

 

The equivalence rule called commutation has two forms:

 

p Ú q :: q Ú p

 

p · q :: q · p

 

The first form indicates that when you have a disjunction, you can reverse the order of the two disjuncts. You can do this because reversing their order will not change the truth-value of the disjunction as a whole.

 

The second form indicates that when you have a conjunction, you can reverse the order of the two conjuncts. You can do this because reversing their order will not change the truth-value of the conjunction as a whole.

 

This rule applies only to disjunctions and conjunctions, not to conditionals. “If Lane is three feet tall, then he is the shortest UWG philosophy professor” is not equivalent to “If Lane is the shortest UWG philosophy professor, then he is three feet tall” (since the first conditional is true and the second conditional is false).

 

 

[4.16.] Association (Assoc). [Chapter 4:8, p.106]

 

This equivalence rule consists of two forms:

 

p Ú (q Ú r) :: (p Ú q) Ú r

 

p · (q · r) :: (p · q) · r

 

In a “string” of three disjuncts or three conjuncts, you are permitted to change which two are “associated” with each other in a single disjunction or conjunction.

 

 

[4.17.] Distribution (Dist). [Chapter 4:8, p.106]

 

This equivalence rule consists of two forms:

 

p · (q Ú r) :: (p · q) Ú (p · r)

 

p Ú (q · r) :: (p Ú q) · (p Ú r)

 

Dist lets you replace a disjunction with a conjunction and vice versa.

 

This rule can be applied only to a sentence containing

·         at least one conjunct that is a disjunction; or

·         at least one disjunct that is a conjunction

 

When you come across a premise that is a conjunction with a disjunction as one of its conjuncts, or that is a disjunction with a conjunction as one of its disjuncts, you should automatically think about using Dist.

 

 

[4.18.] Contraposition (Contra). [Chapter 4:9, p.107]

 

p É q :: ~q É ~p

 

The conditional “~q É~p” is the contrapositive of “p É q”.

 

That the two conditionals in this form are equivalent ought to be intuitively obvious:

 

If this conditional is true:

 

(1) “If Barack Obama wins the election, then our next President is a Democrat.”

 

then so is this one:

 

(2) “If our next President is not a Democrat, then Barack Obama does not win the election.”

 

Conversely, if 2 is true, then so is 1.

 

 

[4.19.] Implication (Impl). [Chapter 4:9, p.107]

 

p É q :: ~p Ú q

 

If you have mastered the truth table for the conditional...

 

 

...then this equivalence rule should make perfect sense. A conditional is true when either its antecedent is false or its consequent is true. So a conditional is equivalent to a disjunction consisting of its consequent and the negation of its antecedent.

 

 

[4.20.] Exportation (Exp). [Chapter 4:9, p.107]

 

(p · q) É r :: p É (q É r)

 

As your textbook says, the rule of Exportation “captures the intuitive idea that if the conjunction of two sentences, (p · q), implies a third, then the first (p) implies that the second (q) implies the third (and vice versa).” (p.107) For example, if the following is true:

 

(1) If Obama runs and Nader runs, Romney will win.

 

then so is this:

 

            (2) If Obama runs, then, if Nader runs, Romney will win.

 

Conversely, if 2 is true, then so is 1.

 

 

[4.21.] Tautology (Taut). [Chapter 4:10, p.107]

 

Recall that a tautology is a sentence that is always true, never false (we used the extended truth-table method to determine whether a given sentence was a tautology, a contradiction, or contingent).

 

The equivalence rule called “tautology” consists of two forms:

 

p :: p · p

p :: p Ú p

 

As your book notes, Taut is used primarily to eliminate redundant letters, either from a conjunction or from a disjunction (although you can also use Simp to eliminate the redundant letter in question from a conjunction).

 

 

[4.22.] Equivalence (Equiv). [Chapter 4:10, p.108]

 

This equivalence rule consists of two forms:

 

p º q :: (p É q) · (q É p)

p º q :: (p · q) Ú (~p · ~q)

 

Notice that this is the only one of the 18 rules that essentially involves a biconditional (represented by the tri-bar). Also not that this rule can be used only to eliminate or introduce a biconditional.

 

This is the last of the first 18 valid forms/rules of the system of propositional logic we’re studying. They are summarized on the inside front cover of your textbook.

 

 

 

 

EXERCISE 4-7 (pp.109-110)

§  here again, you are asked just to fill  in the missing rules and line numbers; there is no need to write down the entire proof--just fill in the right-hand column (with line numbers and rules)

§  complete all of these for next time and check your even-numbered answers against the book; we’ll cover some of the odds in class

 

EXERCISE 4-8 (pp.110-111)

§  complete all of these for next time and check your even-numbered answers against the book; we’ll cover the odds in class

 

 

 

Stopping point for Thursday February 9. For next time: do exercises 4-6, 4-7 and 4-8, and read Ch.4:11-12 (pp.111-21).