Subsection 3.4.8 Suppose You Want More Rules
By the way, there is nothing “magic” about the particular list of identities and inference rules that we’ve shown here.
If you decide you need to look at some other “Learn Logic” resources (but why would you?), you may find more identities:
Exportation
Negation of Conditional
You may also find more inference rules:
Biconditional Introduction
Biconditional Elimination
Disjunction Elimination
Constructive Dilemma
Destructive Dilemma
Absorption
Law of Clavius
We chose the ones we did because they’re the most useful. And at some point, it just isn’t worth memorizing a longer list.
But if you want more, you can have them. You can create them even in the middle of a proof. Whenever you’re working with Boolean expressions and you’d like to apply a rule that you’re pretty sure is sound but that does not show up on our list, all you have to do is:
Write it down. Probably give it a name.
Use a truth table (or the natural deduction technique that we’re about to learn) to prove it.
Then it’s yours.
Exercises Exercises
Exercise Group.
Indicate, for each of these proposed “identities” and “inference rules”, whether it’s one we could have added to our list. (Hint: Use a truth table.)
1.
1. Valid identity or not? (( p ∧ q ) → r ) ≡ ( p → ( q → r ))
Yes, this is a valid identity.
No, this isn’t a valid identity.
2.
2. Valid rule or not? p → q
r → s
p ∨ r
∴ q ∨ s
Yes, this is a sound rule.
No, this isn’t a sound rule.
3.
3. Valid rule or not? p ∨ q
( p ∧ q ) → r
∴ r
Yes, this is a sound rule.
No, this isn’t a sound rule.