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The principle of non-contradiction is that contradictory propositions cannot both be true, e. g. the two propositions "A is B" and "A is not B" are mutually exclusive.

However, whenever something is used as a criterion for truth (like the principle of non-contradiction), it suffers from the problem that it cannot justify itself. For example, if we believe that scientifically verified things are true, then there's the problem that the statement that "scientifically verified things are true" is itself not scientifically verified and therefore not true by its own standards, which, in turn, defeats the belief that scientifically verified things are true. This is, as far as I know, called the problem of the criterion.

But I was pondering on whether the following statements make the principle of non-contradiction self-evident or not:

  1. The principle of non-contradiction is valid.
  2. The principle of non-contradiction is invalid.

Are both of the above statements mutually exclusive? If so, then would that not mean that the principle of non-contradiction is self-evidently true?

I'm so confused. I feel like there's an error somewhere but I cannot pinpoint it if there is one.

user3776022
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    Most people feel that (1) and (2) are mutually exclusive. But that does not imply that the principle is valid. You could consistently believe that (1) and (2) are mutually exclusive and (2) is true. – Jishin Noben Mar 09 '19 at 08:56
  • Can I consistently believe that (1) & (2) are mutually exclusive and simultaneously believe that (2) is nevertheless true? The belief that (1) & (2) are mutually exclusive hinges on (1) alone being true, so it rules out the possibility of still believing that (2) is also true. Right? – user3776022 Mar 09 '19 at 09:00
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    There are plenty of invalid "principles" that validate themselves, for example, "everything I say is true", so self-validation does not make something self-evident. Conversely, many reputable normative principles do not validate themselves simply because they do not apply to themselves, e.g. "empirical conjectures should be tested". Moreover, self-evidence is not a guarantee of validity. LNC has been challenged, some believe that Liar sentence "I am not true" is both true and not true. There are [systems of logic](https://en.wikipedia.org/wiki/Paraconsistent_logic) that explicitly reject LNC. – Conifold Mar 09 '19 at 09:17
  • I'd say it is self-evident. This does not mean we can't set it to one side for other systems of logic when we feel like it, but I doubt anybody could understand a theory for which it is abandoned, as in the case of Dialethism say, since it is the way we think. . –  Mar 09 '19 at 12:19
  • @user3776022 To your comment: No, if (1) and (2) are mutually exclusive, that does not mean that the principle of non-contradiction is valid. That's because the principle of non-contradiction says that **every** pair of contradictory propositions are mutually exclusive. Even if (1) and (2) are mutually exclusive, that's just one such pair, and it doesn't say anything about the rest of all pairs of contradictory propositions. – E... Mar 10 '19 at 00:58
  • I thought that the LNC was one of the Laws of Thought. As such, it is valid because without it, thinking itself is not possible. I have not looked at systems that deny LNC. – Mark Andrews Mar 10 '19 at 04:45
  • You can see [Aristotle on Non-contradiction](https://plato.stanford.edu/entries/aristotle-noncontradiction/) and the study of Jan Lukasiewicz, [On the Principle of Contradiction in Aristotle](https://www.jstor.org/stable/20125812?seq=1#page_scan_tab_contents). – Mauro ALLEGRANZA Mar 10 '19 at 14:18
  • There is a basic problem here: "Self-evident" doesn't necessarily mean 'true'. That things naturally stop moving unless something causes them to continue seems self-evident, until you do a lot of observations and decide it makes the math too bizarre. And then you discover vacuums and outer space, and you decide you are clearly wrong. Does that mean that the *evidence* evaporates? No, because children continue to have to be disabused of this notion, which no one has necessarily taught them. Classical logic is self-evident -- and wrong. –  Mar 14 '19 at 14:04
  • @jobermark - I feel your example is not quite right. That moving objects stop is not self-evident however plausible it may seem. The LNC is different since it emerges from our mental process and is connected to 'me' and my thoughts, thus has the possibility of being self-evident. It's seems incorrect to equate 'self-evident' with ';seemingly obvious'. However, I would say self-evident does mean true so it seems we come at this from different directions, and maybe it's all about definitions as usual. . –  Apr 13 '19 at 09:34

1 Answers1

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"A is B" and "A is not B"…. Are both of the above statements mutually exclusive? If so, then would that not mean that the principle of non-contradiction is self-evidently true?

Assume the truth of statements "A is B" and "A is not B". Now assume A is true. The combination of premises means that the following is true: B and not-B.

Such a conclusion is absurd, of course, but there is a problem. This reduction to absurdity assumes what it sets out to prove; it assumes that the law of non-contradiction is valid.

Now continue using the conclusion. Because B and not-B can be true together, it becomes impossible to distinguish anything in the world from anything else. A chair can be blue, or not-blue, or it can be the Planet Saturn. But this collapse of everything into everything is still noncontroversial, on a hypothetical assumption that everything is one, and that differences, however great they might appear, are truly superficial.

The only solution I can offer is this: the law of non-contradiction is so basic that if it were untrue, language would become impossible to use and the untrue statement itself could never be thought. Thus the law of non-contradiction is a law of human thought.

How about that?

Mark Andrews
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