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In the natural logic system we're using, every logical statement is either true($T$) or false($F$), so there are only two possible states. Can there be other possibilities? I mean, for example, can we form a sufficiently interesting logical system with $T, F, X$ (where $X$ is another possible state)?

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    $\begingroup$ I think you should search for fuzzy logic, if that helps. $\endgroup$ – thanasissdr Sep 25 '14 at 10:32
  • $\begingroup$ $\text{"I am a Liar"}$ $\endgroup$ – Nick Sep 25 '14 at 10:39
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There are plenty of schemes for three-valued logics as well as many-valued logics. Beyond those there are things such as intuitionistic logic whose proof system can't be explained by any finite set of generalized truth values.

These systems are generally not used to model ordinary mathematical reasoning, but they have niche applications where much of the machinery that was invented for metamathematics are used to model something else than the mathematical search for deductive truth.

(Intuitionistic logic is kind of an exception to that -- it was originally developed as an attempt to model the kind of non-formal reasoning that Brouwer and his followers found acceptable for general mathematical purposes. However, very few working mathematicians nowadays insist on the intuitionistic strictures, and the continuing interest in intuitionistic logic is for its niche technical applications).

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Among the most widely known alternative is "fuzzy logic", where a statement can be given any real-number truth value on a continuum from 0 to 1 (where 0 = false and 1 = true). Probabilistic logic does basically the same, and largely expresses our relative certainty of a statement in the face of partial information (probability that the statement is true, or how confident we are that it is true).

https://en.wikipedia.org/wiki/Fuzzy_logic

https://en.wikipedia.org/wiki/Probabilistic_logic

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