epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

8 The Nuts and Bolts of Proof, Third Edition
conclusion is true). But, on the other hand, Joe might inherit some money
(or he might be already wealthy) and he will be able to afford a new car
(the conclusion is false, whereas the hypotheses are still true). Thus, the
conclusion does not follow logically from the hypotheses, because the
hypotheses do not state what Joe will do if he does not win the lottery. So,
any conclusion we reach in this case is just speculation as it is not the only
possible logical conclusion. This is why the reasoning process is not vahd.
When we are working on the proof of a statement, we strive for a sound
proof; a proof that uses vahd arguments, under true hypotheses. It is not
unusual to be able to construct more than one sound proof for a true
statement, especially when the chain of steps required is rather lengthy.
Very often the construction of a sound proof takes considerable time and
effort, and usually the first attempts produce Httle more than "scratch work."
Thus, we must be ready to work on several drafts. While doing this, the
elegance of the construction is not the most important issue. After the
soundness of a proof is established, it is easier to keep working on it to make
it flow well and to remove useless details.