🚀 go-pugleaf

In article <1992Nov16.162733.1831@falcon.aamrl.wpafb.af.mil> cmitchell@falcon.
aamrl.wpafb.af.mil writes:

   >My expectation was that:
   >if   it is not sunny,
   >then I would have no way of determining the truth of the implication.

  You're asking why "if A then B" is a consequence of "not A", as we
understand "if-then" in (classical or constructive) mathematics.

  Here's one answer: although we don't often explicitly make this
inference, it is implicit in ordinary mathematical reasoning of a kind
that does not usually strike people as puzzling. Or more properly, if
we try to make the rules of ordinary mathematical reasoning precise
and explicit, we will be faced with such consequences as the one you
are asking about.

   Consider the following three rules of logic:

  1) From A follows "A or B"

  2) If B follows from some premisses together with the assumption A,
     then "if A then B" follows from those premisses alone.

  3) From "A or B" together with "not A" follows B.

  We reason in mathematics in accordance with these rules all the
time.  It's instructive to verify that "if A then B" can be derived
from "not A" using these rules. What this tells us is that if we want
to reason in mathematics in a way that does not justify the step from
"not A" to "if A then B", then we must think hard about which rules of
reasoning we should accept. As soon as one does this one finds that it
is no simple matter to formulate a logic that does not contain the
rules 1)-3) above in full generality. Such logical systems do exist -
e.g.  relevant logic - but they have little to do with ordinary mathematical
reasoning.