# What is a “trivial implication” in mathematics?

Most often when a certain axiom or proposition implies another proposition, this implication is not trivial, or "immediate". That is, you need a proof consisting of some number of steps to reach the second proposition.

So in that case we can say, proposition A implies but does not trivially imply proposition B.

My (philosophical) question is: what does it mean for a proposition to trivially imply another, or in other words, that it follows "immediately"? Is there a way to objectively determine whether an implication is trivial? Does a trivial implication only rely on an appeal to an intuition about "primitive notions"?

Perhaps we should exclude from this analysis cases where a theorem is implicitly used but simply not explicitly stated for the sake of brevity.

• I do not think you can find a precise definition. An example can be an implication P→Q where the consequent Q is true (e.g. a theorem); in this case, the said implication is true whatever P is. – Mauro ALLEGRANZA Sep 12 '16 at 8:38
• @Programmer2134: Do you have a concrete example for such a trivial implication? – Moritz Sep 12 '16 at 14:01
• the q is ambiguous. "Implication" (->) is a logical constant that does not involve inference. with truth tables A->B is either true or false, full stop, which is arguably trivial. you can use A->B in an inference (A->B; but A, therefore B), but that is not implication - modus poems is the standard term for the rule of inference that licenses the reasoning. it sounds like you're really interested in trivial consequence? – user20153 Sep 12 '16 at 18:19