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.
