Your wording makes the answers tricky. Most valid arguments have some form of math in them because First Order Logic (FOL) is so common and so intuitive for us. Arguments with 'if' or 'and' in them usually are held to the FOL definitions of those words.

**One method I have seen for what I would consider a "non-trivial, non mathematical" argument without fallacies is one which does not come to a final actionable conclusion, but rather boils down the "logical" portions of the argument and leaves the fuzzier parts as axioms for the other side to work through.**

*This is about to be fun! I get to make a contentious argument, and use it to answer a question! Please, if you are a highly-anti gun person, consider this an example. If you find a fallacy that I missed, try to think of how to word around it rather than pinning it to me. This is the sort of debate which should be had in person, with lots of back and forth, rather than as written. As written it makes it look like my stance is set in stone.*

Consider a gun-ownership argument. If one is considering owning a gun (assume it is legal in your state for now), one should consider the question "how can I become a responsible gun owner?" This is not an easy question to answer. However, there is a nontrivial argument that arises from this which generates a question that is easier to think through.

It is well accepted by gun owners that you should not shoot for an arm or a leg unless you are *extremely* confident in your shooting -- the adrenaline makes it likely that all you do is miss, and missing is very bad when your life is on the line. Most gun owners will argue that you should usually shoot for center of mass. So the first part of this argument is a very complicated "trust those with experience" argument... meeting your non-trivial condition. It would be reasonable for you to debate with my "most gun owners" wording, but it would not be a fallacy yet. I just might have to tweak my wording one way or another.

The next part of the argument is that, in a home defense situation, you rarely have time to think through the implications of your actions completely. You need to have pre-wired yourself to get reasonably close to a solution *before* the break in occurs. It doesn't have to be completely thought out, but you have to recognize how little room for thought there will be. This is another appeal to trust the experts, but it is also an appeal to your history. Most people have been in surprise situations where they are forced to admit that they could not think through the situation as well as they can in normal low-stress situations. This also is clearly non-trivial, and very un-mathematical unless you dig way deep in to the physics of how our brain works.

The conclusion that is drawn is that every potential gun owner should enumerate a list of crimes, and whether they are willing to kill over them. Most people considering a gun will be very quick to claim they will use a gun to defend against murder of themselves, but what about murder of a 3rd party? There's a known psychological test which shows they aren't the same. What about a thief, just stealing property? What about rape? Will your list change when you have kids?

I would say that argument avoids fallacies, is non-trivial, and non-mathematical. The way it gets away with it is that it answers a question with a question: "How can I be a responsible gun owner?" was answered "One first step along that path is to ask 'what crimes would I kill over?'" I have not completely answered their question, but yet this answer is still highly useful. I have debated with individuals for a long period about what different crimes "deserve." And I have had people who decide they are too uncomfortable making the list, and that is sufficient for *them* to arrive at the conclusion, "gun ownership is not for me," even though they have no problem with gun ownership in general.

The argument wraps up the parts which are most susceptible to fallacies and hands them back in one big gift-wrapped package, but hopefully teases apart a very difficult question into smaller questions.

I am afraid you are right, if we approach non-mathematical arguments with the mathematical standard of rigor they are all, technically, non-sequiturs and/or unsound. Philosophical premises can neither be stated with mathematical precision nor ascertained, even plain facts can be disputed, however implausibly. As Peirce put it in reference to Hegel, "

reductions of the different opinions to absurdity are in almost all cases of the flimsiest texture, and permit a mind of any subtlety to escape by every mesh". Real-world arguments must rely on plausibility and good faith in addition to logic. – Conifold – 2017-04-26T04:18:28.430Is there any chance you could share a little bit more regarding the context and motivations of this question? What is the specific problem are you encountering in your study (that you are you hoping for someone here to provide an explanation about to you)? – Joseph Weissman – 2015-01-07T18:35:47.383

@JosephWeissman looks like another topic that (if read a certain way) can have philosophical import and fascinating implications. Time for me to get to work :) – R. Barzell – 2015-01-07T19:02:59.707

2@user74158 what about mathematical proofs? Simple mathematical proofs (like those found in Euclid's Elements) may be a good starting point. – R. Barzell – 2015-01-07T19:22:06.790

1Thought the same thing. Mathmetical reasoning is valid and non-trivial. – Lukas – 2015-01-08T11:58:14.080

Just a simple motivation. There are too many fallacies, I wonder if someone could make a perfect argument. I would like a non-mathematical arguments. I'm sorry I didn't mention that. – hans-t – 2015-01-09T14:59:18.087

I think some of the challenge of a non-mathematical "perfect" argument is that English, like all spoken languages, is

designedwith ambiguity in mind. That makes it really hard to even define what a perfect argument is, much less make one. Mathematical arguments are popular because you can at least write an argument and not have it mistranslated; a "perfect argument" means something. – Cort Ammon – 2015-01-10T22:52:10.750A perfect argument is an argument that has no fallacy. – hans-t – 2015-01-12T02:34:45.033