Predicate Logic Practice 1: Q.N. and C.Q.N. Rules (Answers)

Difficulty: What the hell

Here are the answers to our first baby predicate-logic practice. As you probably noticed, Exercises 1 and 2 are basically one-step applications of Q.N. That’s fine. They’re warm-ups. Exercise 3 can also be done as a one-step application of C.Q.N., but I want to make it a little more interesting by breaking it down into multiple steps.

Exercise 1

1. ¬(∀x)Px

∴ (∃x)¬Px

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2. (∃x)¬Px         Q.N. 1

Exercise 2

1. ¬(∃x)Qx

∴ (∀x)¬Qx

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2. (∀x)¬Qx         Q.N. 1

Exercise 3

1. ¬(∀x)(Px ⊃ Qx)

∴ (∃x)(Px • ¬Qx)

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2. (∃x)¬(Px ⊃ Qx)         Q.N. 1

3. (∃x)¬(¬Px ∨ Qx)       conditional exchange 2

4. (∃x)(¬¬Px • ¬Qx)     De Morgan’s law 3

5. (∃x)(Px • ¬Qx)         double negation 4

Of course, Exercise 3 could also have been done in one step by applying C.Q.N. directly. But there’s something satisfying about watching the thing unfold instead of just smashing the correct rule on it like a button.

Next time, we’ll look at the four quantifier rules: universal instantiation, universal generalization, existential instantiation, and existential generalization—along with flagging, which is where predicate logic starts getting a little more annoying. I’m rusty on that stuff myself at the moment, so I’ll have to brush up before pretending to teach it. Until next time, then.