Editing Principle of bivalence (section)

===Vagueness===
Such puzzles as the [[Sorites paradox]] and the related continuum fallacy have raised doubt as to the applicability of classical logic and the principle of bivalence to concepts that may be vague in their application.  [[Fuzzy logic]] and some other [[multi-valued logic]]s have been proposed as alternatives that handle vague concepts better. Truth (and falsity) in fuzzy logic, for example, comes in varying degrees. Consider the following statement in the circumstance of sorting apples on a moving belt:

: This apple is red.<ref>Note the use of the (extremely) definite article: "This" as opposed to a more-vague "The". If "The" is used, it would have to be accompanied with a pointing-gesture to make it definitive. Ff ''Principia Mathematica'' (2nd ed.), p.  91. Russell & Whitehead observe that this " this " indicates "something given in sensation" and as such it shall be considered "elementary".</ref>

Upon observation, the apple is an undetermined color between yellow and red, or it is mottled both colors. Thus the color falls into neither category " red " nor " yellow ", but these are the only categories available to us as we sort the apples. We might say it is "50% red". This could be rephrased: it is 50% true that the apple is red. Therefore, P is 50% true, and 50% false. Now consider:

: This apple is red and it is not-red.

In other words, P and not-P. This violates the law of noncontradiction and, by extension, bivalence. However, this is only a partial rejection of these laws because P is only partially true. If P were 100% true, not-P would be 100% false, and there is no contradiction because P and not-P no longer holds.

However, the law of the excluded middle is retained, because P [[And (logic)|and]] not-P implies P [[Inclusive or|or]] not-P, since "or" is inclusive. The only two cases where P and not-P is false (when P is 100% true or false) are the same cases considered by two-valued logic, and the same rules apply.

'''Example of a 3-valued logic applied to vague (undetermined) cases''': Kleene 1952<ref>Stephen C. Kleene 1952 ''Introduction to Metamathematics'', 6th Reprint 1971, North-Holland Publishing Company, Amsterdam, NY, {{isbn|0-7294-2130-9}}.</ref> (§64, pp.&nbsp;332–340) offers a 3-valued logic for the cases when algorithms involving partial recursive functions may not return values, but rather end up with circumstances "u" = undecided. He lets "t" = "true", "f" = "false", "u" = "undecided" and redesigns all the propositional connectives. He observes that:

{{Blockquote|We were justified intuitionistically in using the classical 2-valued logic, when we were using the connectives in building primitive and general recursive predicates, since there is a decision procedure for each general recursive predicate; i.e. the law of the excluded middle is proved intuitionistically to apply to general recursive predicates.

Now if Q(x) is a partial recursive predicate, there is a decision procedure for Q(x) on its range of definition, so the law of the excluded middle or excluded "third" (saying that, Q(x) is either t or f) applies intuitionistically on the range of definition. But there may be no algorithm for deciding, given x, whether Q(x) is defined or not. [...] Hence it is only classically and not intuitionistically that we have a law of the excluded fourth (saying that, for each x, Q(x) is either t, f, or u).

The third "truth value"  u is thus not on par with the other two t and f in our theory. Consideration of its status will show that we are limited to a special kind of truth table".}}

The following are his "strong tables":<ref>"Strong tables" is Kleene's choice of words. Note that even though " u " may appear for the value of Q or R, " t " or " f " may, in those occasions, appear as a value in " Q V R ", " Q & R " and " Q → R ". "Weak tables" on the other hand, are "regular", meaning they have " u " appear in all cases when the value " u " is applied to either Q or R or both. Kleene notes that these tables are ''not'' the same as the original values of the tables of Łukasiewicz 1920. (Kleene gives these differences on page 335). He also concludes that " u " can mean any or all of the following: "undefined", "unknown (or value immaterial)", "value disregarded for the moment", i.e. it is a third category that does not (ultimately) exclude " t " and " f " (page 335).</ref>
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|style="font-weight:bold;font-style:Italic" width="15" |
|style="font-weight:bold" width="15" |
| width="5.4" |
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| width="4.2" |
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For example, if a determination cannot be made as to whether an apple is red or not-red, then the truth value of the assertion Q: " This apple is red " is " u ". Likewise, the truth value of the assertion R  " This apple is not-red " is " u ". Thus the AND of these into the assertion Q AND R, i.e. " This apple is red AND this apple is not-red " will, per the tables, yield " u ". And, the assertion Q OR R, i.e. " This apple is red OR this apple is not-red " will likewise yield " u ".