Ramesh: The story I am sure is apocryphal, but it is indeed the case that
the proposition “If P, then Q” is invariably true if P is false. In other
words, from any given false proposition, ALL OTHER propositions follow. So,
of course, do their negations. If 2 + 2 = 5, then I am NOT the Pope. (If 2
+ 2 = 5, then 0 = 1, so the Pope, being one person, is actually zero persons
and does not exist. I however, do exist…) For much, much more on this,
see Russell’s fascinating book INQUIRY INTO MEANING AND TRUTH.
Russell, by the way, defined mathematics to be: “The class of all
propositions of the form ‘If P, then Q’.” He meant this to be understood as
the class of all enquiries into the structure of, and connection between,
such propositions, of all chains of deduction, regardless of whether P is
true in the sense of everyday veridicality, and regardless of what actual
things P might refer to. Whence his other definition of mathematics: “That
subject in which we do not know what we are talking about, nor whether what
we say is true.” Modern philosophers of mathematics are not quite so
reductionist…