Currently stuck on looking at some inferences to identify the basic inference rule of which it is a substitution instance. Examples and questions taken from book "Fuzzy set theory" by Klir/St Clair/Bo Yuan (which frustratingly does not contain the solutions and I cannot find them anywhere on internet).
Example which I understand
Consider the following premises given as true
(P1) p v q (P2) ¬r → ¬p (P3) r → s (P4) ¬s
(c) ∴ q (conclusion to be established)
Proving conclusion c using basic inference forms
(r1) ¬r (P3), (P4), MT
(r2) ¬p (P2), (r1) MP
(r3) q (P1, (r2) DS
where r3 is the proved conclusion, using basic inference rules MT (Modus Tollens), MP (Modus Ponens) and DS (Disjunctive Syllogism).
Stuck getting started on proving the following:
(p1) (p^¬q)→(r v s)
(p2) p^¬q
(c) ∴ r v s
(P1) (¬p→q) ^ (r v ¬s)
(c) ∴ [(¬p→q) ^ (r v ¬s)] v (¬p→s)
(P1) [(p↔¬q) ^ s] ^ (¬p↔q)
(c) ∴(p≡¬q) ^ s
Much obliged for any support, working hard to get my head around this. Thanks!