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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.

user56834
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  • Applying the axiom that $(\forall x) \phi \to \phi[t / x]$ pretty much always produces a theorem which is a "trivial" consequence of $(\forall x) \phi$. – Patrick Stevens Sep 12 '16 at 06:46
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    That depends on the context and assumed known knowledge of the reader. – Asaf Karagila Sep 12 '16 at 06:47
  • Is this an example of a way to produce a trivial consequence, or a theory about what "trivial consequences" are? – user56834 Sep 12 '16 at 06:48
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    "Is there a way to objectively determine...": of course not, what is obvious to Paul Erdös or Srinivasa Ramanujan may not be obvious to us mortals. –  Sep 12 '16 at 06:49
  • Depends on the context. Either of the following could qualify as "trivial" in different circles: (a) $3,4,5$ is a pythagorean triangle, (b) if integers $a^n + b^n = c^n$ then $n \le 2$. – dxiv Sep 12 '16 at 06:51
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    @dxiv, in no circle is Fermat's last theorem considered trivial. – Mariano Suárez-Álvarez Sep 12 '16 at 07:40
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    @MarianoSuárez-Álvarez it's trivial, but the proof just doesn't fit into this comment. – Glorfindel Sep 12 '16 at 14:06
  • @MarianoSuárez-Álvarez I didn't mean to imply that FLT was trivial, but rather that knowledge of its proof has become common enough. If one had a problem that reduced to FLT, they could conclude it nowadays with "known to have no integer solutions Q.E.D.". – dxiv Sep 12 '16 at 15:30

3 Answers3

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"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.

Asaf Karagila
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5xum
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  • Perhaps you are using the word trivial different from what I meant by it. Of course often mathematicians will say that something is trivial if it is provable by an elementary theorem. If something is very easily provable by the pythagorean theorem, it would be a waste of time to actually prove it. However, I am talking about a trivial implication in the sense that there is no way to prove it at all, but that it simply "directly" follows. – user56834 Sep 12 '16 at 07:17
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    @Programmer2134 Oh. Well, then a trivial implication is anything that directly follows from the rules of logic and your axioms, I guess... – 5xum Sep 12 '16 at 07:18
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    @Programmer2134, there is no such thing as something for which "there is no way to prove it at all", that simply makes no sense. – Mariano Suárez-Álvarez Sep 12 '16 at 07:41
  • @MarianoSuárez-Álvarez, yes there is. In at least two ways. Firstly, Godel's incompleteness theorem proves that there are unprovable statements that are true. secondly, some statements must be taken as true in order to derive anything at all. – user56834 Sep 12 '16 at 08:19
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    Godel's theorem has absolutely nothing to do with this (and no one sanely refers to unprovable statements as trivial). Axioms, on the other hand, «do not simply "directly" follow» from anything. Trust me. What you wrote does not make any sense.ç – Mariano Suárez-Álvarez Sep 12 '16 at 08:23
  • It's true that Godel's theorem has nothing to do with this, I shouldn't have brought it up, but it does literally contradict what you said, so perhaps you should have formulated it more clearly. However, I did not say anything about axioms. What I am talking about is this: Whenever you write down a proof that A follows from B, you are using "steps" in the proof. Each step in turn may be a nontrivial statement that needs proof as well. however, because of the problem of infinite regress, there comes a point at which you must simply accept that one step follows from the one before. – user56834 Sep 12 '16 at 08:38
  • At that point, it seems to me that you can really do nothing except appeal to the intuition of the reader that one step or proposition, follows from the one before. – user56834 Sep 12 '16 at 08:42
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    @Programmer2134: Usually you appeal not to the intuition of the reader but to the basic rules of logic. – Javier Sep 12 '16 at 10:14
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    There is an old joke, of a lecturer who writes on a blackboard "it is obvious that ". The lecturer then stops, thinks for a moment, walks out of the room, comes back 30 minutes later announces, "yes, I'm satisfied that it is obvious", and continues with the proof on the blackboard. – Steve Jessop Sep 12 '16 at 10:43
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    @SteveJessop You forgot the part where a student asks the lecturer "are you sure this is obvious?" :) – 5xum Sep 12 '16 at 10:44
  • @Programmer2134 Trivial means that it's easy to prove, or that it's assumed to be easy to prove, with "easy" being undefined. An unprovable statement is not trivial. That "trivial implication in the sense that there is no way to prove it at all" does not exist, according to my understanding, unless you can provide a concrete example. – Peter Sep 12 '16 at 10:48
  • @Programmer2134 Unless of course you refer to the cases where "trivial" is used deceptively to draw the wool over the reader's eyes, and the writer is just making things up. – Peter Sep 12 '16 at 10:51
  • @5xum: It is obviously obvious that it was not obviously obvious. – user21820 Sep 12 '16 at 11:06
  • here is a concrete example of an inference that is not "provable", and is trivial: (A) and (B), therefore (A and B). This could be a step in a proof that you cannot further break down. I know that this is a logical axiom, so perhaps I shouldn't have called it a "trivial" proof, but an "undividable" proof or something. – user56834 Sep 16 '16 at 06:54
  • @Programmer2134: Just to clarify a little. Probably your notion of "proof" involves more than one step, which is why you are excluding this kind of trivial inference as provable. However, in the field of logic a proof is simply defined as a sequence of deductions (possibly none!) that lead to a proven conclusion. So even an axiom would be classified as provable (using no inference rules). – user21820 Sep 16 '16 at 15:43
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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.

user21820
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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 $Q$.

Yes
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