The following rules (also to be memorized!) exhibit various equivalencies. In each case one expression is interchangeable with another. Since they are truly equivalent, then either one of the equivalencies could be listed first. Thus, to take Rule 9, Double Negation, p is interchangeable with ~~p and vice versa.

9. Double Negation (DN):    p is interchangeable with ~~p

You will be expected to make explicit any interchanges involving a proposition and its double negation. It is not enough to think it; you should show it on paper.

10. Transposition (Trans):    p É q is interchangeable with ~q É ~p

You can swap the antecedent and the consequent, provided you negate each.

11. Commutation (Comm):    p Ú q is interchangeable with q Ú p and
                                            p & q is interchangeable with q & p

Another rule that students are tempted to ignore, but should be shown explicitly.

12. Association (Assoc):      p Ú (q Ú r) is interchangeable with (p Ú q) Ú r and
                                           p & (q & r) is interchangeable with (p & q) & r

13. Distribution (Distr):       p & (q Ú r) is interchangeable with (p & q) Ú (p & r)
                                           p Ú (q & r) is interchangeable with (p Ú q) & (p Ú r)

14. De Morgan's Law (DeM):    ~(p & q) is interchangeable with ~p v ~q and
                                                  ~(p Ú q) is interchangeable with ~p & ~q

15. Conditional (Cond):       p É q is interchangeable with ~p Ú q and
                                           ~p É q is interchangeable with p Ú q

16. Biconditional (Bicond):   p º q is interchangeable with (p É q) & (q É p)

17. Exportation (Exp):          (p & q) É r is interchangeable with p É (q É r)

18. Absorption (Abs):           p É q is interchangeable with p É (p & q)

19. Tautology (Taut):            p is interchangeable with p Ú p and
                                            p is interchangeable with p & p

A Derivation

To illustrate briefly the use of these rules, we will derive a conclusion from its premises, using some of the rules. Here is an argument:

If anything is possible [P], then elephants can fly [F].
Elephants can fly, if, and only if, they have wings [W].
Elephants do not have wings.
Therefore, it is not the case that anything is possible.

The task is to get from the premises to the conclusion, showing the logical steps involved. First we display the symbolized argument, numbering the premises and putting the conclusion to the left of the last numbered line:

1. P É F

2. F º W

3. ~W   /\ ~P

Then we derive the conclusion (~P) from the premises, showing which lines and rules we are using:

4. (F É W) & (W É F)            2     Bicond

5. F É W                                4     Simpl

6. ~F                                      5,3   MT

7. ~P                                      1,6   MT

Here we used three rules and seven lines to prove that ~P could be derived from the premises given. In the next lesson we learn some strategies for derivations, but for now we try to understand and learnthe rules. Accordingly, Exercise 12 will ask you to construct truth tables to test the validity or equivalency of some of the rules. Also you will be asked to supply lines and rules for some derivations.

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Copyright © 1999, Michael Eldridge