What would the consequences be, if we suspended some of the usual laws of logic so that they were no longer available as tools in proofs?
(a) The Law of Self-Implication
A ⇒ A for any proposition A
This is a consequence of the rule of inference of conditional introduction: because A ⊢ A for any proposition A, the deduction theorem implies that ⊢ A ⇒ A. What forms of logic where conditional introduction of this sort is not allowed, and what features do they have?
(b) The Law of Excluded Middle
A∨¬A for any proposition A
This is mostly used in proofs of things by dilemma, to show that if A ⇒ B and ¬A ⇒ B, then B is true because A∨¬A. Using the rule of inference of disjunctive syllogism (i.e. material implication), this is equivalent to A ⇒ A, because A ⇒ B is equivalent to ¬A∨B. What forms of logic prevent us from having A∨¬A as a theorem for free, for every A?
(c) The Law of Non-contradiction
¬(A&¬A) for any proposition A
This proposition is essentially built into how we approach reductio ad absurdum, and comes close to being the defining property of logical negation. Using de Morgan's Laws, it's equivalent to the Law of the Excluded Middle. What forms of logic prevent us from having ¬(A&¬A) as a theorem for free, for every A?