The 9th edition of Patrick Hurley’s Concise Introduction to Logic (Wadsworth, 2006) was a relatively friendly logic course but it had the potential to baffle a discerning student too – I discovered then and reassured myself recently having reread parts of it.
One such example is 7.2.III.15 where we are asked to derive the conclusion using the eight rules of implication.
1. (S v B) ⊃ (S v K) 2. K v ~ D) ⊃ (H ⊃ S) 3. ~ S • W ∴ ~ H | |
I have copied the task as it stands in the book, with the opening bracket in the antecedent of the conditional missing on line 2.
Bracket or no bracket, the argument resists the application of the rules of inference Hurely is talking about. But the argument seems ripe for indirect proof, so if we could derive the conclusion using indirect proof, which often unlocks perplexing problems, we could safely say that we have not tried hard enough to reach the conclusion using the basic rules. Suppose the opening bracket is there on line 2, and we use indirect proof:
1. (S v B) ⊃ (S v K) 2. (K v ~ D) ⊃ (H ⊃ S) 3. ~ S • W ∴ ~ H 4. ~ ~ H 5. H 6. ~ S 7. H • ~ S 8. ~ ~ H • ~ S 9. ~ (~ H v S) 10. ~ (H ⊃ S) 11. ~ (K v ~ D) 12. ~ K • D 13. ~ K 14. ~ S • ~ K 15. ~ (S v K) 16. ~ (S v B) 17. ~ S • ~ B |
| AIP 4 DN 3 Simp 5,6 Conj 7 DN 8 DM 9 Impl 2,10 MT 11 DM 12 Simp 6,13 Conj 14 DM 1,15 MT 16 DM |
At this point, we have isolated all propositions (H, ~ S, ~ K, D, ~ B, (W is irrelevant)) and not reached a contradiction. If we start reapplying these propositions, we’ll be going round in circles.
Another example is the translation Hurley suggests in 8.6.I.19 for:
Every person admires some people he or she meets.
This is rendered as:
(x){Px ⊃ (∃y)[Py • (Mxy ⊃ Axy)]}
If the horseshoe is turned into a disjunction by the rule of implication and subsequently rewritten using distribution, the translation will read:
For every person there is a person whom he doesn’t meet or a person he admires.
(x){Px ⊃ (∃y)[(Py • ~ Mxy) v (Px • Axy)]}
This doesn’t look to me like the English sentence in 8.6.I.19. Here is how I would translate it:
(x)[Px ⊃ (∃y)(Py • Mxy • Axy)]