What is a "trivial implication"?

Question

This is a philosophical question. 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.

Answer 1

It is trivial to see that P.

Whenever this occurs in a proof, it is essentially making a (meta-logical) claim that the statement "P" is easily implied by the preceding statements. This is usually one of the following types:

• The implication can be proven by a standard or commonly used argument.

• The reader who understands the rest of the proof is likely to be able to fill in the gap.

Example of the first type

It is easy to see that by symmetry we can assume P.

One usually verifies this by observing (meta-logically) that the preceding statements are invariant under a symmetry (usually a permutation of variables) and P is true in one of the cases under the symmetry. For example, if we have reals $x,y$ such that every statement we have proven still holds when we swap $x,y$, then we know that we can choose whether to swap or not such that $x \le y$. So we can say "By symmetry we can assume $x \le y$".

If you think about it, this reasoning about symmetry is not at all a short deduction in the sense that you have to check every single preceding statement. But yet it is such a standard technique that it is not worth writing out all the details to avoid the symmetry argument. An alternative is to show one case and state that the other is similar, which is essentially the same claim of triviality.

Example of the second type

It is trivial to check that P(0) holds.

This is often used in an induction argument, regardless of what P is, since usually it is the case that the intended audience can easily figure out how to prove it on their own and it would be a waste of space and time to write out the details.

Example of both types

A uniformly continuous function on $[0,1]$ is trivially continuous.

The triviality comes down to the basic meta-logical fact that an "$\exists \forall$" statement always implies the corresponding "$\forall \exists$" statement where the quantifiers are swapped. Every reader trained in basic mathematics is expected to know this, and so it is fine to consider it trivial.

Answer 2

"Trivial" is a subjective, not an objective term. It is also a risky term to use, but the rule of thumb would be

You are only ever allowed to use the word "trivially" if you are certain that your reader (and you) is capable of proving the implication with no real cognitive effort.

I wrote cognitive effort because often, the word trivial is used when the proof of a statement is essentially over, but writing the entire thing out would take a lot of time, even though each step would be very simple.

Answer 3

As a supplementary remark, a trivial implication could be an implication of the form $P \Rightarrow Q$ where $P$ is a false statement and $Q$ is a statement; just note that $P \Rightarrow Q$ is equivalent to $\sim P$ or $Q$.