It depends heavily on how exacting your wording is. It is possible to state "I reject A" in such a way that is equivalent to declaring an acceptance of ~A, because of the Law of the Excluded Middle, which is an accepted axiom of propositional logic (A proposition is always either true or false). However, it is also reasonable to restate the rejection slightly: "I reject this proof of A." This states nothing about the truth or falseness of A, merely that the proof being offered is not sufficient. This is especially important in the handling of axioms. I may believe that Mike is a good dog owner, but not with a sufficient conviction to blindly assume any prepositional logic which may follow from that:

Assume: Mike is a good dog owner

Assume: Good dog owners pick up their dog's poop

Observe: There is dog poop in my back yard

Assume: Mike is the only dog owner with keys to my back yard

Thus: Since Mike is a good dog owner, and good dog owners pick up their dog's poop, Mike would have picked up any poop his dog left in my yard.

Thus: Since no other dog owner has keys to my back yard, my dog must have left the poop

Observe: I did not pick up the dog poop

Thus: I must be a bad dog owner.

You can see why I might like to argue some semantics regarding the validity of these assumptions, but I'd be quick to claim Mike is a good dog owner and good dog owners pick up poop after their dog. Belief is a wiggly thing that way.

1You're asking for a proof that rejecting A is not equivalent to accepting ~A. You've given an example in which it's clearly possible to reject A without accepting ~A. That example

isa proof. What more proof could you need? – WillO – 2015-10-03T22:45:33.413I think an example is a confirmation, not a proof. – Goodies – 2015-10-03T22:50:21.567

3No. A counterexample to X is a

proofof not-X. – WillO – 2015-10-03T22:51:29.833Right. An example of ~X is proof of ~X, but an example of X is not proof of X. That's like a black swan fallacy. – Goodies – 2015-10-03T22:57:02.480

4X is the statement "If you reject A you must accept ~A". You have given a counterexample example to this statement, and hence proved ~X, which is to say you have proved that "It is not true that if you reject A you must accept ~A". I thought this was exactly what you were trying to prove. Were you trying to prove something different? – WillO – 2015-10-03T23:06:37.453

1I see. I think we were discussing two different things, but I see what you mean. Thanks. – Goodies – 2015-10-03T23:09:03.570

2IMO this is more a matter of language than philosophy. – None – 2015-10-04T17:58:06.170