This is from the Daniel Bonevac Deduction text, page 122 #12.
Given premise (p -> q) -> p show p using deduction.
I can do this using made up rules on steps 4+5, not given in the book. The other steps are proper steps given in the book.
1. (p -> q) -> p. A 2.
Showp 3. | ¬p. AIP 4. | ¬p -> ¬(p -> q) made up rule. take contrapositive of conditional 1 5. | ¬(p -> q). ->E, 4, 3 6. | ¬(¬p ∨ q). made up rule. convert conditional 5 to disjunction form. 7. | (¬p ∨ q). ∨I, 3
We've only covered conjunction exploitation/introduction, negation exploitation/introduction, indirect proof, reiteration, conditional exploitation, conditional proof, biconditional exploitation/introduction.
The fundamental rules covered:
&E: Conjunction Exploitation &I: Conjunction Introduction ¬¬: Negation introduction/exploitation AIP: Indirect Proof (Show p, then assume ¬p, derive contradiction, conclude and cancel Show) R: Reiteration ->E: Conditional Exploitation ACP: Conditional Proof (Show p->q, assume p, derive q, conclude and cancel Show) I: Biconditional introduction E: Biconditional exploitation
How could the the above proof be completed using only the given rules?