Here is Wikipedia's list of fallacies. Looking at this list, I can't imagine any non-trivial, non-mathematical (i.e., not a mathematical proof) argument that wouldn't fall afoul of at least one of these fallacies, but I could be wrong.

Can anyone provide any references to non-trivial, non-mathematical arguments that are sound and valid? These can be your own arguments or links to other ones.

  • Is 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
    Commented Jan 7, 2015 at 18:35
  • @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
    Commented Jan 7, 2015 at 19:02
  • 2
    @user74158 what about mathematical proofs? Simple mathematical proofs (like those found in Euclid's Elements) may be a good starting point.
    – R. Barzell
    Commented Jan 7, 2015 at 19:22
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    Thought the same thing. Mathmetical reasoning is valid and non-trivial.
    – Lukas
    Commented Jan 8, 2015 at 11:58
  • 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
    Commented Jan 9, 2015 at 14:59

5 Answers 5


Validity is a strong requirement; to be valid is to roughly ask for guarantee of having no possibble refutation; any refutation would neccessarily be a logical contradiction.

Claiming validity for an invalid argument is a fallacy. But the very invalid argument for which the claim is made is not always a fallacy. (But it is true that some would go as far as to say that all invalid arguments are fallacies. However that opinion is in no way the prevalent one).

Having said that, there are many nontrivial, sound and valid arguments. Unfortunately most of them that I'm aware of have rather long proofs.

Arguments coming from some lightweight math system might be a good example. "Lightweight" because often people don't agree which math axioms are sound (which doesn't really matter for mathematicians, they are exploring systems that needn't have to do with "what's true" or even what's relevant to the real world). So an example should be from a lightweight system, one that doesn't claim much. One such is Heyting arithmetics. This system is about things like "1 + 1 = 2", but with a smaller set of inference rules than what the usual Peano arithemtics offers. For example, you can't even prove "F or not-F" for any given closed formula F.

I don't have experience with Heyting arithmetic, but few things you can prove with it:

  • For all nonnegative integers x and y, x + y = y + x
  • Every nonnegative integer is either zero, or is some other nonnegative number incremented by one

(Proofs would be very uninteresting and would contain long weird formulas, while the inference rules would be modus ponens, generalization and instantiation)


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 have edited my question. Well, I ruled out mathematical arguments because it's not an interesting case in real life. That is, most people are not mathematics savvy, who cares about mathematics? The argument you gave, the gun-ownership argument, is one I would like to see more as an answer. Thanks.
    – hans-t
    Commented Jan 10, 2015 at 4:04

Technically speaking, only formal arguments --arguments posed in a formalized language such as mathematics or formal logic, or arguments in a formal system, such as categorical syllogisms --should be characterized as "sound" or "valid." Informal arguments, which are any arguments not in a formal language or system, are more properly characterized as "strong" or "weak."

Typically, an informal fallacy is a "weak" argument that would be invalid if translated into a formal argument, but that resembles to a deceptive extent a "strong" argument (one that would be valid as a formal argument). Ambiguities remain, however, because no informal argument can be translated with complete fidelity and accuracy into a formal argument --there are irreducible ambiguities in natural language. This is the reason formal languages are necessary in the first place.

Valid arguments that deal purely with logic rather than mathematics are easy enough to come by in any formalized logic, such as FOL. For example, this is a valid logical argument. X -> Y, X : Y (IF X THEN Y, and X, THEREFORE Y). You could make it sound by replacing the variables with statements that make the premises true, for example, X = "Canada is in North America" and Y = "Canada is in the Americas." Please note that although the statements are in natural language, the argument itself is not.


An argument that follows the rules of the categorical syllogism is non-fallacious. The practical effect of the rules is to wash out the fallacies.

Depending on the source, there are five to eight rules of a valid syllogism. Generally, these are: exactly three sentences; exactly three terms; the middle term distributed in at least one premise; a major or minor term is distributed in the conclusion is also distributed in the premises; never two negative premises; when either premise is negative, the conclusion is negative; and no syllogism with a particular conclusion can have two universal premises.

The different configurations of the three terms produce 256 possible syllogisms. But the number that follow all the rules is only 15 exactly or 24 exactly, depending on the assumption about the actual existence of what is being described.

Once an argument confines itself to these rules, it is free from fallacy. Depending on the content of the three terms, the argument might also be non-trivial. The problem arises when human motivation enters the picture, and a person tries to find ways around the rules.

The question refers to Wikipedia's List of fallacies. https://en.wikipedia.org/wiki/List_of_fallacies In this list there is no example of a fallacy that is consistent with the rules of the syllogism and yet yields a false result. Each example is an illustration of human error.

  • Your answer misses something very important. An argument that follows the rules of categorical syllogism will contain no fallacies of inference, but that doesn't mean that this has eliminated all potential for fallacy from the argument. Instead, it merely means that the fallacies which can occur will be restricted to fallacies that occur in translating normal everyday arguments into categorical syllogisms -- and there's a lot informal fallacies that fit that bill.
    – virmaior
    Commented Apr 26, 2017 at 4:44

I think therefore I am

Cogito ergo sum

Descartes asserted that the very act of doubting one's own existence served—at minimum—as proof of the reality of one's own mind; there must be a thinking entity—in this case the self—for there to be a thought


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    This is not at all clear as an answer.
    – virmaior
    Commented Apr 27, 2017 at 3:06

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