Why should I not believe there are true contradictions?

Question

Kane Baker has a YouTube video in which he introduces the word 'wulture'. 'Wulture' applies to all things that are vultures, and excludes all things which are white. Delia is a white vulture. He asks: What is so objectionable about considering Delia both a wulture and a non-wulture?

This got me thinking: How does one really justify the law of non-contradiction without just appealing to intuition? It could be said that the idea that to reject it would be self-defeating because then you open yourself up to the idea that the law of non-contradiction could also be true. But this issue doesn't really apply when we have limits on what contradictory statements are true. Delia being a wulture and non-wulture doesn't necessitate that 'the law of non-contradiction is true' and 'the law of non-contradiction is false' are both true.

So, can somebody please give me a justification of the law of non-contradiction which doesn't just rely on intuition? Is it possible? I think I have reason to believe that metaphysically speaking there is not contradiction, so I am specifically talking about propositions, even if they're just semantic.

Answer 1

• You may cancel from ordinary logic the prinicple of non-contradiction and admit at least one contradictory statement ’A and non-A’. But then you get for arbitrary statements B the valid statement

  A and non-A = > B

  This makes every statement in the logic a true statement (ex falso quodlibet) and the resulting logic a useless calculus.

• Introducing contradictory statements therefore necessitates a new calculus of logic. Whether such forms of logic are possible is discussed under the heading dialethism. A possible forerunner is Nagarjuna's Tetralemma.

Answer 2

Frame challenge: This is based on a false premise

Your problem is a lot simpler than you think, and your YouTube reference is trivially wrong. You're only confused because they're talking rubbish, and you're under the misapprehension that they're some kind of authority you have to believe.

Consider the basic Boolean logic

"Is a wulture" = "is a vulture" AND (NOT "is white")

This defines the set of all things that are wultures.

Delia is a vulture, so the first condition is met. However Delia is white, so the second condition fails. Delia therefore, by the definition of a wulture, can never be one.

You'll see that the entire proposition of your question fails. Membership of a set is either true or false, and there is no paradox.

Note that Bayesian concepts can allow for a non -binary state, but this still is not a paradox, merely a probability of true/false.

Answer 3

This got me thinking: How does one really justify the law of non-contradiction without just appealing to intuition?

If you accept logic, but specifically deny the law of non contradiction, the problem that arises is called the Principle of Explosion. From a contradiction, anything follows.

So, just imagine the worst most slanderous thing someone could say about you. If logic is generally true, but the law of noncontradiction is not, and there's some statement A that is both true and not true, then someone could use A and not A to prove that slanderous terrible thing about you.

But you know that terrible thing isn't true, so you know that the law of noncontradiction must also be true.

Answer 4

Perhaps we should first check whether the statement, "No contradiction can be true," can be made more precise. We will say, "Propositions (A) and (not A) cannot both be true." But so now what do we mean by "not A"? The explosion argument works in normal logic, over a normal (if academic) definition of "not." However:

And now then elsewhere:

Likewise, there are theories of "not" such that the generic mark of a negation operator is ambiguous enough to sustain multiple operations (though see here for a critique of one example of a disambiguation approach to "not" (it's pretty far into that section, they write "¬_i" and "¬_c" when they get to it)). Or: we could simply define a logical operation called cancellation such that when two propositions are combined and one is the cancellation of the other, the result is the kind of erasure that Routley talked about. We would reformulate the Law of Noncontradiction as the Law of Cancellation, say, then, and speak of it as pertaining to the cancellation operator with conjunction instead of speaking of a negation operator simpliciter with conjunction. And so then see e.g. here for other theories of negation that could be turned into theories of more specified operations subject (or not!) to analogues of the LNC.

Answer 5

Consider the English statement, A = "My eyes are closed." It denotes/symbolizes/represents/names some single proposition, out of the infinity of propositions. Now, some propositions have a truth value that is constant in time, and all others have a truth value that varies in time. "My eyes are closed" denotes a proposition whose truth value varies in time. The key observation is that the proposition it denotes cannot be true and false simultaneously. Symbolically we can express this observation by writing 0 ≠ 1. This leads directly to the Law of Non-Contradiction. A simple truth table reveals this.

A	not A	(A and not A)	not (A and not A)
0	1	0	1
1	0	0	1

Row 1 is the statement row. Rows 2 and 3 are state rows. Each state row of the table corresponds to reality at a single moment in time, i.e at a particular state of the universe (0 and 1 denote the truth values false and true, respectively). So columns 1 and 2 taken together show that there is no moment in time for which the proposition denoted by A is simultaneously true and false. The logical operator 'and' is the simultaneity operator of binary logic. So column 3 all by itself shows there is no moment in time or state, at which the proposition denoted by A is simultaneously true and false.

The statement in column 4 is the Law Of Non-Contradiction. As you can see, there are no moments in time at which the proposition it denotes is false. That proposition is always true. You should accept the Law of Non-Contradiction, because it's always true, and never false.

If a contradiction denotes a true proposition, then you can prove every proposition is true.

Let a be a specific statement, and let B be an arbitrary statement. The following reasoning event proves " if 'a and not a' then B."

Definition. if A then B = not A or B

1. a and not a [open scope of assumption]
2. (a and not a) or B [1; law of addiction]
3. B or (a and not a) [2; commutativity of conjunction]
4. not(not B) or (a and not a) [3; double negation]
5. If not B then (a and not a) [4; def.]
6. not(not B) [5; reductio ad absurdum]
7. B [6; double negation]
8. If a and not a then B [close scope of assumption]

As you can see, this natural deduction demonstrates how to prove ex falso quodlibet. What this says is, if even one contradiction denotes a true proposition, then by modus ponens B denotes a true proposition. Since B is an arbitrary statement, that means any statement you say denotes a true proposition. As Jo said, that makes your logic worthless. As Kaia said, that means 0 = 1.

Now to my point. By understanding temporal logic, and the notion of simultaneity, you know 0 ≠ 1.

I don't consider temporal binary logic intuitive. Nonetheless, it's not complicated. Don't feel obligated to take my word as law, Einstein had misgivings about the concept of simultaneity.

Answer 6

I'll bite on the word belief which you are using in the title. SEP has a lot to say about Belief, but what strikes me as most important is that a belief is something that lives in the brain/mind. Also, if you read the first sentences in that article, and also as a general theme, there is by default no truth value attached to beliefs. A belief can exist as such, in the brain, without the person knowing or caring about whether it is true or not.

In fact, in everyday language, at least in mine, saying someone "believes" something vs. someone "knows" something is separated exactly by whether there is a requirement for some measure of "objective truth" (whatever that may be!) attributed to it or not. In philosophical terms, knowledge is a subset of belief (see SEP's Knowledge Analysis and the concent of "Justified True Belief").

So you can absolutely believe that there are true contradictions. You can literally believe anything. The true questions then become:

• Does this belief help or hinder you? Does it untie some knots you had in your thinking, or does it give you sleepless nights due to the madness looming ahead?
• Does it influence your external actions? Does it have any impact whatsoever on your interactions with the world?

Maybe it helps to take a more Bayesian viewpoint and avoid talking in absolutes so much. Instead of binary true/false values, you would then attach probabilities to your beliefs (and try to avoid the edge cases of exact 0 and 1, as Bayesians usually do). This immediately opens up everything, and always leaves a way out for this kind of problems; you are not constraind by a contradiction anymore.

Answer 7

You won't be able to refute any claims made by trivialists.

Suppose some claim Q is false by direct proof. You mention this to a trivialist. They reply, "Oh, Q is true." They also say, "The proof that Q is false is true." Since you can accept Q as both true and false simultaneously, and you have no principle of explosion, you agree with them. However, they simply cannot agree that Q is false, since all claims are true to trivialists.

Answer 8

All rules of logic are based on "intuition" in the sense you're using the word here. They are based on people agreeing, "This makes sense." There is no way to prove these rules other than by appeal to intuition. It's like the old paradox: "Give me a logical proof that logical proof is valid." It would inevitably be a circular argument, because if logical proofs are not valid, than a logical proof that logical proofs are valid would not be valid.

As another poster said, I don't see how the example you gave proves the proposition. If the definition of "wulture" is "is a vulture and is not white", then Delia, who is a white vulture, is clearly by definition not a wulture. There is no paradox where she is and she isn't. She just isn't. She's white, she doesn't meet the definition.

You could, of course, say, "But this vulture is sort of an off white, so maybe it meets the definition and maybe it doesn't." But that doesn't create a paradox either. It just proves that some definitions are vague, and if you want to be able to clearly determine what is and is not a member of the set, you need a more clear and explicit definition.

You say that allowing contradictions would not create the problem that "contradictions are not allowed" could be both true and false, because you would only allow SOME contradictions. But that begs the question. Once you say that contradictions are allowed, how do you decide which ones? Would you exclude the contradictions that cause logic problems? But that's the whole point of excluding contradictions to begin with: Because they ALL cause logic problems. You'd just end up back where you started.

Answer 9

It is a flaw in logic itself. It's funny I just made a similar post.

Edit: Logic is a type of language that cannot tolerate contradiction. This is why the word "wulture" does not make sense when used in logic.

Logic is a way of describing the world that cannot tolerate contradiction. The idea of a "wulture" is an idea of logic (splitting things that are white and things that are vultures apart) that has no actual influence on the real world, just how people who adhere to it perceive things. As soon as a white vulture appears all of a sudden you can choose what aspect you want to adhere too. Do you want to adhere to the idea that a "wulture" applies to all that is vultures or do you want to appeal to the idea that it excludes all things that are white. It is actually both at the same time, it's just you who thinks that you need to choose.

Logic is based in individualism. The idea that everything is made up of parts. When you use logic to dissect an object you automatically start splitting it into parts. The things is once you've split something into parts you miss what that thing is.

For instance I can list all of the aspects of you but those aspects aren't you. That list of aspects aren't you. The combination of those aspects are you. There are no logical operators for combination. If you have a stick and you get another stick you haven't combined the sticks. You just have two sticks.

If you view the world in holistic terms such as collectivism, everything is part of the whole, you can be okay with contradiction because you don't necessitate that everything needs to be split up into parts.

Something can be true and false, using the language of logic you cannot describe it, but a language based in a more collective perspective may be able to.