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I saw the other day something similar to the following:

  1. one of the following is true.
  2. the above is false.
  3. $ 1 + 1 = 5 $

You can probably see the problem with this. I can clearly state that $3$ is false, but what would I call $1$ and $2$?

To clarify, I really meant if there were some state between true and false that could make these consistent.

hjpotter92
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Simply Beautiful Art
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  • Presumably, you mean "exactly one of the following is true" – Ben Grossmann Dec 07 '16 at 16:10
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    A similar "paradox" is attained with 1) statement 2 is false, 2) statement 1) is true. – Ben Grossmann Dec 07 '16 at 16:11
  • @Omnomnomnom you could have at least one of the following is true, but I don't see the affect either would have. – Simply Beautiful Art Dec 07 '16 at 16:12
  • I'd say that it lacks a definite truth value. – Wojowu Dec 07 '16 at 16:16
  • @SimpleArt are you saying that you don't understand the problem with my setup? If 1 is true, then 2 is false. But since 2 is false, 1 can't be true. – Ben Grossmann Dec 07 '16 at 16:29
  • Another interesting case is Yablo's paradox – Ben Grossmann Dec 07 '16 at 16:29
  • @Omnomnomnom Yes, I understood that much, so I've tried to clear up that this is not what I want. – Simply Beautiful Art Dec 07 '16 at 17:11
  • Not a technical response so I won't post an answer, but I have heard this "true/false blend" being referred to as "tralse". Related image: Frue – machine yearning Dec 07 '16 at 18:18
  • Even more simply: "This statement is false." – hBy2Py Dec 07 '16 at 18:42
  • @SimpleArt Are many-valued logics something you would find relevant? That is, are you asking for truth values between true and false that could be used to analyze such statements? – Noah Schweber Dec 07 '16 at 19:08
  • The answer is clearly "file not found" - http://thedailywtf.com/articles/What_Is_Truth_0x3f_ – Joe Dec 07 '16 at 19:13
  • I think a logician- , and I am not one, would say that there are other ways that something can fail to be true or false. Imagine you have a superdooper computer that you can go on as far as you like hunting for special numbers (you never run out of memory, but like the rest of us you will run out of time.) Just using this computer you will never be able to show that 65537 is the largest Fermat prime, but you might be able to show it's not. The statement can be refuted, but not affirmed.. Its negation can be affirmed but not refuted. Another neither true nor false situation arises (cont.) – Airymouse Dec 07 '16 at 20:19
  • from statements that depend on axioms. The sentence, "every countably compact perfectly normal space is compact," sounds reasonable enough. If you assume the axiom of choice, there is a counterexample. If you assume the negation of this axiom and another axiom, the statement is true. There is yet a worse problem. If the axioms everyone has been assuming turn out to be inconsistent, there may be some theorem that is also false. – Airymouse Dec 07 '16 at 20:28
  • @NoahSchweber That sounds like what I'm trying to understand, so probably yes. – Simply Beautiful Art Dec 07 '16 at 21:14
  • @amWhy I am younger than a freshman, so no, I have not encountered such content before. And this area of mathematics I haven't given much attention to on this site... its just not my area... yet. :D – Simply Beautiful Art Dec 08 '16 at 00:56
  • @SimpleArt When you say, "[...] some state between true and false that could make these consistent.", are you thinking about a kind of complex-like consideration which would internalize the inconsistency in a imaginary-unit ? – keepAlive Dec 08 '16 at 03:25
  • @SimpleArt My Apology I'll delete my comment. – amWhy Dec 08 '16 at 11:52
  • How about "meaningless?" No meaningful statement can be true if and only if it is false. A simpler example, the sentence: This sentence is false. – Dan Christensen Dec 08 '16 at 16:09
  • @DanChristensen but what if such a similar statement describes something? Like the continuum hypothesis? – Simply Beautiful Art Dec 08 '16 at 17:54

5 Answers5

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These statements taken together are called inconsistent. That means that they cannot all be simultaneously true. But the first and second are neither true nor false without broader context. Using the language of first order logic, they might be said to be formulas with "free variables." Here's an example of a formula with free variables $$ 4x+3y=9$$ This is neither true nor false because I haven't told you what $x$ or $y$ are. If I use quantifiers to get rid of all the free variables, then I have a sentence which may be true or false: $$\forall x\forall y (4x+3y=9)$$ $$\forall x\exists y (4x+3y=9)$$ $$\exists x\exists y (4x+3y=9)$$ The first statement is false, while the second two are true.

D Wiggles
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    I think that technically your statements can't be made into first order formulas or sentences because in first order logic, "you can only quantify over variables, but not sets." The idea of inconsistency is still what you are looking for. Two relevant (and totally awesome) results related to this are Godel's completeness and incompleteness theorems. – D Wiggles Dec 07 '16 at 16:29
  • Thank you for this, I've tried to clear up my question. – Simply Beautiful Art Dec 07 '16 at 17:08
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In the 3 statements you've shown, statement #3 is definitely false, as you have already mentioned.

To clarify, I really meant if there were some state between true and false that could make these consistent.

No, statements #1 and #2 are inconsistent statements, they contradict each other.

Actually, since there's no discrepancy that statement #3 is false, it is a bit of a red herring, irrelevant. Alternately, statement #3 can be omitted and statement #1 and #2 can be re-written as:

1) Statement 2 is true.
2) Statement 1 is false.

or

The next statement is true.
The previous statement is false.

If you are looking for a word or phrase to describe the relationship of the statements, I would say the statements are "paradoxical".

You can also use the term: non sequitur.

noun: non sequitur; plural noun: non sequiturs; noun: nonsequitur; plural noun: nonsequiturs

a conclusion or statement that does not logically follow from the previous argument or statement.

Kevin Fegan
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5

Too long to repeat the whole Wikipedia article.

What you are looking for are the Liar paradox and its variants. Here are the possible resolution.

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    This doesn't answer the question, namely: what truth value do you assign to such a statement, or is there a word for the inability to assign such a truth value. – Ben Grossmann Dec 07 '16 at 16:19
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This may not satisfy anyone reading this... but here goes...

In the software world, something that is neither true or false is likely not yet to have been determined. In which case it is null (or unset).

joespr
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    That's just called three-valued logic and has no context to this question (which is using law of excluded middle) – q.Then Dec 07 '16 at 19:03
  • @Ephemeral I'm not sure about that - the question is asking what sort of truth value ("state") 1 and 2 may have, and to me that sounds like many-valued logic. – Noah Schweber Dec 07 '16 at 19:06
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    Haha, I know what null is! Though I'm not sure how it fits in mathematically. – Simply Beautiful Art Dec 07 '16 at 21:12
  • I am not sure this is true; I am playing with reinventing if/else in terms of ifTrue (Coproduct, fThen), and ifFalse(Coproduct, fThen), so that (Coproduct ifTrue fThen1) ifFalse fThen2. I want to make a random Coproduct generator(Left ()) or (Right ()) depending on if it rolls 0 or 1. I want to make a variable name to describe a value which is always either Left () or Right () but I never know what it is at time of call; So it is not null or unset, it is an actual function, but it is undecidable at time of getting it. – Dmytro Dec 19 '16 at 20:23
  • I am currently curious if there is a word for "something that is undecidable". – Dmytro Dec 19 '16 at 20:25
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The answer you are looking for is "undecidable". What is neither true nor false is called an undecidable proposition. This is about the liar paradox and Gödel's incompleteness theorems.

Every attempt to establish the truth of the first proposition leads to a contradiction in the second. And you would never want inconsistency, ever. Not in mathematics and logic.

So, you decide it as "undecidable" to avoid contradiction.

allwind
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