# A non-associative three-valued logic

There are three elements: x, y, z and a relation C:

x C y, y C z, z C x, x C x, y C y, z C z.

Let us introduce two binary operations with respect to the C: "the leftmost" (L) and "the rightmost" (R), i.e.

x L x = x L y = y L x = x, y L y = y L z = z L y = y, z L z = z L x = x L z = z

x R x = x R z = z R x = x, y R y = x R y = y R x = y, z R z = z R y = y R z = z.

Similar construction produces a multi-valued logic, if to use a linear order instead of the C, but this non-associative "logic" also has some applications. Yet, I failed to find any notes about that in a book about multi-valued logic. I would be glad to know, if described construction was used somewhere earlier to provide correct references in my works.

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I don't see a question here. –  Cam McLeman Apr 22 '11 at 14:10
Dear qubeat: This post, at present, is "not a real question", meaning that you've rambled a little about an idea you've had (nothing wrong with that! the best questions include some background), but never got to a question. Maybe your question is "where can I read about multi-valued logic?", but if it's only that, then it's only borderline for MathOverflow (I would expect Math.StackExchange to be a better fit). Please read mathoverflow.net/howtoask , and revise this question. If it is closed (and I expect it will be), then once you revise it, you can "flag for moderator attention". –  Theo Johnson-Freyd Apr 22 '11 at 14:18
Does this have anything to do with Trintercal? –  Zsbán Ambrus Apr 22 '11 at 19:40
Trintercal? I do not know. –  Alex 'qubeat' Apr 22 '11 at 20:13
@Theo Johnson-Freyd, thank you for the comments and suggestions, I have seen Math.StackExchange, but afraid it won't help. –  Alex 'qubeat' Apr 24 '11 at 10:45

It sounds like you are describing a situation where $a$ is more true than $b$, $b$ is more true than $c$, but nevertheless $c$ is more true than $a$. I am not sure about the best starting point in looking for relevant references, but maybe Arrow's theorem on the impossibility of a perfect voting scheme, where $a$ represents "candidate $A$ should be elected".

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Yes, relation C itself is known very well due to rock-paper-scissors game, but I may not find anything about L and R operations (analogues of AND, OR) derived from such relation. –  Alex 'qubeat' Apr 22 '11 at 15:33

Just few hours ago I found, that the construction was used in talks of J. B. Nation “How aliens do math”, and “Logic on other planets” (here). Despite of such titles, the works look quite instructive.

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Soon after I posted the answer, initial question was downvoted. Anyway, I think, I must have sent an answer, if I have known about one. –  Alex 'qubeat' Apr 28 '11 at 12:52