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Subsection 3.4.8 Suppose You Want More Rules

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By the way, there is nothing “magic” about the particular list of identities and inference rules that we’ve shown here.

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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

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You may also find more inference rules:

  • Biconditional Introduction

  • Biconditional Elimination

  • Disjunction Elimination

  • Constructive Dilemma

  • Destructive Dilemma

  • Absorption

  • Law of Clavius

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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.

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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:

  1. Write it down. Probably give it a name.

  2. Use a truth table (or the natural deduction technique that we’re about to learn) to prove it.

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Then it’s yours.

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Exercises Exercises

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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.)

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1.

1. Valid identity or not? (( p ∧ q ) → r ) ≡ ( p → ( q → r ))

  1. Yes, this is a valid identity.

  2. No, this isn’t a valid identity.

Answer.
Correct answer: A
Solution.
Explanation: This is a valid identity. It is, in fact, the Exportation identity.
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2.

2. Valid rule or not? p → q

r → s

p ∨ r

∴ q ∨ s

  1. Yes, this is a sound rule.

  2. No, this isn’t a sound rule.

Answer.
Correct answer: A.
Solution.
Explanation: This rule is sound . It is, in fact, the Constructive Dilemma rule.
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3.

3. Valid rule or not? p ∨ q

( p ∧ q ) → r

∴ r

  1. Yes, this is a sound rule.

  2. No, this isn’t a sound rule.

Answer.
Correct answer: B.
Solution.
Explanation: This rule is not sound. It’s easy to see this just by considering a single row of the truth table. Suppose that p is T, q is F, and r is F. Then p  q is T. (p  q)  r is T (because (p  q) is F). So this rule would let us infer r. Sound rules only let us infer things that must be true. So, if the rule were sound, r would have to be T, but it isn’t.