The Law of Noncontradiction (sometimes called ‘The Principle of Noncontradiction’) is a law of at least three domains. It says,
1) that the same thing or group of things does not both completely have and completely not have some feature at the same time, nor do they partially have and not partially have some feature at the same time
2) that the same thing or group of things cannot be believed or conceived to both completely have and completely not have some feature at the same time, nor can they be believed or conceived to both partially have and not partially have some feature at the same time; and
3) a proposition, which is a sentence that is either true or false and not both, and its negation (typically the version of the proposition with a ‘not’ grammatically placed within it) are not both simultaneously true.
The areas that are covered by these are as follows:
1) is ontological, or about how things are.
2) is epistemological, or having to do with knowledge. In particular, it is about what is possible to believe and, since knowing typically begins with and requires beliefs, has to do with knowledge.
3) is logical, or having to do with propositional truth values.
Aristotle speaks of this principle along similar lines. He mounts an argument in its favor by showing that the Law of Noncontradiction is required in saying anything at all, and thus of making arguments. Thus, claiming that the principle of noncontradiction fails to apply to something is self-defeating. Let me lay out this argument, veering slightly from the terms Aristotle uses.
Let’s take any example of a pair of sentences that are contradictory. What if someone asserts:
The tree is green.
The tree is not green.
Of course, as it stands, the statements are ambiguous. Under the following reading, no contradiction is asserted:
(Part of) The tree is green.
The tree is not (all) green.
These statements are consistent. Therefore they are not examples of claims that something has contradictory properties. A more explicit contradiction would go:
Part of the tree, the green part, is green.
That same part of that same tree is not green.
Aristotle’s challenge is to specify which part of the tree the challenger is speaking about. Maybe the challenger would simply respond, “The green part. I already said that.” Maybe the challenger would give an approximate definition of ‘green’. Maybe the challenger would give a complete, correct, and well-received definition of ‘green’. Maybe the challenger would name the part ‘Bill’. Maybe the challenger would say something completely different.
However, it makes no difference how the challenger responds. What is implied by some specification that does not just amount to everything is that, in specifying something, some things are necessarily excluded from being specified.
So, if the challenger says, “The green part”, or gives some other specification of that part of the tree that does not just amount to anything or everything, what must be excluded by the specification is the not green part(s) of the tree (Note: even if the challenger says “the leaf parts are green” still, we may ask to specify which part of the leaf parts is green; eventually we get to just the green part of the leafs and thus the green part of the tree).
Therefore, specifying that part of the tree that is the green part, however you do that, implies that that green part, being not anything or everything, is itself not not green. Without the double negation: the green part, being not everything, is just the green part and nothing else—it is not what is not green. If it were also what is not green, it would not be the green part of the tree we are talking about. Therefore, the challenger either fails to speak about the green part of a tree, or ends up speaking about something that is not the green part of the tree. For the latter, we need to speak of the green part of the tree, because that is what are talking about in claiming that it has a contradictory property (i.e. is also not green).
The reason we cannot be talking about some non-green part in talking about the green part specifically is that we end up talking about everything anything that is not green, with no demarcation either way, which amounts to failing to specify the green part of the tree, since specifying the green part of the tree would here amount to specifying non-green parts as well.
In order to specify what is green and not green by specifying green, we would have to specify the entire tree. But the tree is not exclusively green (e.g. it’s also made of wood). So the challenger would simply fail to specify anything green at all.
Otherwise, the challenger may succeed in specifying the green part, in a way that does not amount to specifying anything and everything of the tree. But then, in this case of succeeding to specify the green part exclusively, it would therefore not be not green. The claim that the green part is not green would have to be false.
That there is a green part to trees that is not identical to its other properties is one way to validate 1), the ontological position that things do not have contradictory properties. The green part must not be what is not green.
That the green parts to trees may be specified, and that this necessitates claims of its not being green as being false is the way to validate 3), the logical version of the Law of Noncontradiction. For 2), the epistemic version, Aristotle
For when, thinking that it is desirable to drink water and see a man, he goes to look for them, he does not look for and judge all things indifferently; and yet he should, if the same thing were equally man and not-man.
So judging that something is a man is not a matter of indifference to what is selected. Without discriminating, a man cannot be judged to be.
But as we have said, there is no one who does not evidently avoid some things and not others. Hence, as it seems, all men form unqualified judgements, if not about all things, at least about what is better or worse. And if they do this by guesswork and without knowledge, they should be all the more eager for truth; just as a sick man should be more eager for health than a healthy man; for indeed the man who guesses, as contrasted with him who knows, is not in a healthy relation to the truth.
Aristotle also asks along these lines, “why does a man walk to Megara and not stay at home, when he thinks he ought to make the journey? Why does he not walk early one morning into a well or ravine, if he comes to it, instead of clearly guarding against doing so, thus showing that he does not think that it is equally good and not good to fall in?”
Aviccina, the Islamic medieval philosopher, riffs on this passage along similar lines: “As for the obstinate, he must be plunged into fire, since fire and non-fire are identical. Let him be beaten, since suffering and not suffering are the same. Let him be deprived of food and drink, since eating and drinking are identical to abstaining” (Metaphysics I.8, 53.13–15).
It is sometimes thought that the epistemological version is untenable since people can be found to hold contradictory beliefs with a little digging. Socrates himself, for example, often catches his interlocutors in a contradictory bind after they affirm Socrates’ many questions. But how this is a counterexample to version 3) is difficult for me to grasp. After all, the interlocutor ought to report that his own beliefs about how things are have changed. I don’t consider it coherent to think that the interlocutor’s beliefs have remained the same after they explicitly reach different conclusions about the subject under investigation.
Furthermore, our beliefs often carry logical implications which we do not necessarily believe. For example, that the morning star rises implies that the evening star also rises. But maybe I don’t believe that the evening star rises, despite believing that the morning star rises, particularly if I do not believe that the morning star is the evening star (FYI the morning star is the evening star). So it is not surprising that the implications of our beliefs carry contradictions, but this would not show that we believe a contradiction. In this case, coming to believe that the morning star is the evening is inconceivable without also believing that the evening star also rises when the morning star rises.
“As for the obstinate, he must be plunged into fire . . . .”