By the article content, isn't this the exact same thing as affirming the consequent (which is a logical fallacy)? If so, it should redirect there. Otherwise, what does "trivially true" actually mean?

However, if one reads the article for affirming the consequent, it seems to only apply to arguments with hypothetical propositions--not valid propositions. In the case of valid propositions, there is no logical fallacy committed. The logical fallacy occurs only when the ${\displaystyle p\implies q}$ proposition is hypothetical. Any ideas?

Slartoff 16:44, 18 Mar 2005 (UTC)

What the present revision means is that p→q is trivially true under the assumption that q is true. This is not a logical fallacy. However, the article seems to suggest that only formulas of this form and under these circumstances qualify as being "trivially true". This is wrong, IMO. Here are some other arguments that are trivially true as I understand the term:

• ${\displaystyle \vdash \top }$
• ${\displaystyle p\vdash p\lor q}$ and in particular ${\displaystyle \vdash \top \lor q}$
• ${\displaystyle p\vdash \lnot \lnot p}$ (but not the converse!)
• ${\displaystyle p\land q\vdash p}$
• ${\displaystyle \vdash \bot \rightarrow p}$
• ${\displaystyle \vdash p\rightarrow \top }$
• ${\displaystyle (\forall x)\phi }$ if the domain of quantification is empty

This all raises the fundamental question of what is to be considered "trivial". I'm not aware of a precise definition of this term, and I think it boils down to what kinds of inferences philosophers, logicians, mathematicians etc. would consider trivial. Strictly speaking, every single thing listed above requires the application of one or more rules of inference in, say, a natural deduction system (or a model-theoretic argument regarding ${\displaystyle \forall }$ quantification over empty domains).

So overall I agree that the factual accuracy of this article is in dispute, but I think the reason for the dispute is different: as I see it, the article gives just one example of a trivial inference, strongly suggesting that nothing else could be considered to be triviall true. This seems wrong. --MarkSweep 17:14, 18 Mar 2005 (UTC)

The article's individual statements are not factually false, but in its present 'stub' state, it is too short to a fault. It [correctly] does not describe affirming the consequent as suggested. It does, however, omit other forms of trivial truths. By only listing a single logic formula as the general form of a trivial truth, it incorrectly implies that other logical forms to not qualify. As noted earlier in MarkSweep's talk entry, there are many other examples within propositional logic.

Arguably outside of 'logic proper', operations involving trivial truths are often searched for and eliminated during optimization and streamlining of computer programs. For example, any instance of a conditional that tests the truth value a complex statement obviously wastes time if the statement being tested is found to be a tautology. As in that example, the unnecessary conditionals are typically logically-distinct from ' Q, therefore if P then Q ', as ' P then Q ' alone is not a tautology. The term is also often used for statements that are 'right for the wrong reasons' in mathematics proper: whereas ' 2+x=4, therefore x=2 ' is true, ' 2+2=x, therefore 4=4 ' is trivially true.

Aside: I personally feel this term does not deserve a Wikipedia entry, as I have never seen it used as anything other than the 'common sense' combination of two words with their own meanings, trivial and truth. A Mountain Trail is certainly a thing, but it does not have its own entry. How idiomatic do two adjacent words have to get for the combination to be considered a single term? —Joel D. Reid 18:48, 2005 Jun 2 (UTC)