[4.15.] Commutation (Comm). [Chapter 4:8, p.108]

 

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 simply 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 simply 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.

 

As your book points out, this form works only for disjunctions and conjunctions, not for conditionals. “If Lane is three feet tall, then he is the shortest philosophy professor” is not equivalent to “If Lane is the shortest philosophy professor, then he is three feet tall” (since the first is true and the second is false).

 

 

[4.16.] Association (Assoc). [Chapter 4:8, p.108]

 

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, it allows you to change which two are “associated” with each other into a single disjunction or conjunction.

 

 

[4.17.] Distribution (Dist). [Chapter 4:8, p.108]

 

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, or replace a disjunction with a conjunction.

 

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, think about using this rule.

 

 

[4.18.] Contraposition (Contra). [Chapter 4:9, p.109]

 

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 Hillary Clinton 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 Hillary Clinton does not win the election.”

 

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

 

 

[4.19.] Implication (Impl). [Chapter 4:9, p.109]

 

p É q :: ~p Ú q

 

If you’ve 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 (or both). 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.109]

 

(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.109) For example, if the following is true:

 

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

 

then so is this:

 

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

 

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

 

 

[4.21.] Tautology (Taut). [Chapter 4:10, p.110]

 

This equivalence rule 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.110

 

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 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.111-112)

§         [we will do two or three even problems in class]

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

§         no need to write down the entire proof--just fill in the right-hand column (with line numbers and rules)

 

 

 

EXERCISE 4-8 (pp.112-13)

§         [we will do two or three even problems in class]

§         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 Monday February 11. For next time: do exercises 4-7 and 4-8, and read Ch.4:11-12 (pp.113-23).