Symbolic Logic: 10 Replacement Rules

This is part two of our little intro to Symbolic Logic. We’re going to expand our repertoire of rules we can employ in our proofs. These rules are all about putting logic statements into an alternative form. A lot of these rules will be familiar as they’re used in mathematics. One of the differences between these rules and the basic 8 is that these are reversible, hence the :: symbol to denote a two-way operation. The format of this post is going to be similar to the last one; the rules will be listed first, then some simple examples and then a couple of practice problems on the following pages.

1. Double Negation (D.N)

p :: ~~p

2. Commutation (Comm.)

p V q :: q V p

p • q :: q • p

3. Association (Assoc.)

[(p V (q V r)] :: [(p V q) V r)]

*applies to AND operators the same way*

4. Duplication (Dup.)

p :: p V p

5. DeMorgan’s Law (DeM.)

~(p V q) :: ~p • ~q

This law describes how a negation gets distributed into a parenthesised statement. It negates the two variables and switches the operator from an AND to an OR or vice versa. It only works on AND and OR operators though so if you have a (BI)CONDITIONAL operator inside the parenthesis, you’ll need to use one of the later replacement rules to make them into one.

6. Biconditional Exchange (B.E.)

(p ≡ q) :: [(p ⊃ q) • (q ⊃ p)]

7. Contraposition (Contra.)

(p ⊃ q) :: (~q ⊃ ~p)

8. Conditional Exchange (C.E.)

p ⊃ q :: (~p V q)

9. Exportation (Exp.)

[(p • q) ⊃ r] :: [(p ⊃ (q ⊃ r)]

10. Distribution (Dist.)

[p • (q V r)] :: (p • q) V (p • r)

[p V (q • r)] :: (p V q) • (p V r)

Examples

Alright y’all, on to the practice problems.