Conditionals: Kinds of ifthen
connection. In logic there are quite a few kinds of
conditionals. I will later treat some of these, but start with two
remarks on standard conditionals.
The standard bivalent ifthen is defined thus: (if p then q)
is true iff (p) is not true or (q) is true, and (if p then q) is not
true iff (p) is true and (q) is not true. This is adequate for most
mathematical argumentation, and also rather close to most usages of
ifthen in natural language, but for some cases other kinds of ifthen
are useful, while also the standard definition of ifthen comes with
some difficulties. (See: Paradoxes of implication.)
The standard bivalent ifthen corresponds to the ifthen of
inference, indeed provably so by what is known as the Deduction
Theorem: From (if p then q) it follows that if (p) one can infer (q),
and conversely if one can prove that one can infer (q) if (p) is true,
then (if p then q) is true. The difference is that the ifthen of
inference is, besides an ifthen, a rewriting rule, that permits an
action, namely the inference of the
conclusion if the premisses of the
rule are all
true.
