In logic, the semantic principle (or law) of bivalence states that every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false.^{[1]}^{[2]} A logic satisfying this principle is called a twovalued logic^{[3]} or bivalent logic.^{[2]}^{[4]}
In formal logic, the principle of bivalence becomes a property that a semantics may or may not possess. It is not the same as the law of excluded middle, however, and a semantics may satisfy that law without being bivalent.^{[2]} It may be written in the secondorder sentence as: \forall P\,\forall x (x \in P \lor x \notin P), demonstrating similarity yet differing mainly by quantified set elements.
The principle of bivalence is studied in philosophical logic to address the question of which naturallanguage statements have a welldefined truth value. Sentences which predict events in the future, and sentences which seem open to interpretation, are particularly difficult for philosophers who hold that the principle of bivalence applies to all declarative naturallanguage statements.^{[2]} Manyvalued logics formalize ideas that a realistic characterization of the notion of consequence requires the admissibility of premises which, owing to vagueness, temporal or quantum indeterminacy, or referencefailure, cannot be considered classically bivalent. Reference failures can also be addressed by free logics.^{[5]}
Relationship with the law of the excluded middle
The principle of bivalence is related to the law of excluded middle though the latter is a syntactic expression of the language of a logic of the form "P ∨ ¬P". The difference between the principle and the law is important because there are logics which validate the law but which do not validate the principle.^{[2]} For example, the threevalued Logic of Paradox (LP) validates the law of excluded middle, but not the law of noncontradiction, ¬(P ∧ ¬P), and its intended semantics is not bivalent.^{[6]} In classical twovalued logic both the law of excluded middle and the law of noncontradiction hold.^{[1]}
Many modern logic programming systems replace the law of the excluded middle with the concept of negation as failure. The programmer may wish to add the law of the excluded middle by explicitly asserting it as true; however, it is not assumed a priori.
Classical logic
The intended semantics of classical logic is bivalent, but this is not true of every semantics for classical logic. In Booleanvalued semantics (for classical propositional logic), the truth values are the elements of an arbitrary Boolean algebra, "true" corresponds to the maximal element of the algebra, and "false" corresponds to the minimal element. Intermediate elements of the algebra correspond to truth values other than "true" and "false". The principle of bivalence holds only when the Boolean algebra is taken to be the twoelement algebra, which has no intermediate elements.
Assigning Boolean semantics to classical predicate calculus requires that the model be a complete Boolean algebra because the universal quantifier maps to the infimum operation, and the existential quantifier maps to the supremum;^{[7]} this is called a Booleanvalued model. All finite Boolean algebras are complete.
Suszko's thesis
In order to justify his claim that true and false are the only logical values, Suszko (1977) observes that every structural Tarskian manyvalued propositional logic can be provided with a bivalent semantics.^{[8]}
Criticisms
Future contingents
A famous example^{[2]} is the contingent sea battle case found in Aristotle's work, De Interpretatione, chapter 9:

Imagine P refers to the statement "There will be a sea battle tomorrow."
The principle of bivalence here asserts:

Either it is true that there will be a sea battle tomorrow, or it is false that there will be a sea battle tomorrow.
Aristotle hesitated to embrace bivalence for such future contingents; Chrysippus, the Stoic logician, did embrace bivalence for this and all other propositions. The controversy continues to be of central importance in both the philosophy of time and the philosophy of logic.
One of the early motivations for the study of manyvalued logics has been precisely this issue. In the early 20th century, the Polish formal logician Jan Łukasiewicz proposed three truthvalues: the true, the false and the asyetundetermined. This approach was later developed by Arend Heyting and L. E. J. Brouwer;^{[2]} see Łukasiewicz logic.
Issues such as this have also been addressed in various temporal logics, where one can assert that "Eventually, either there will be a sea battle tomorrow, or there won't be." (Which is true if "tomorrow" eventually occurs.)
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 multivalued logics 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.^{[9]}
Upon observation, the apple is an undetermined color between yellow and red, or it is motled 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 notred.
In other words, P and notP. 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, notP would be 100% false, and there is no contradiction because P and notP no longer holds.
However, the law of the excluded middle is retained, because P and notP implies P or notP, since "or" is inclusive. The only two cases where P and notP is false (when P is 100% true or false) are the same cases considered by twovalued logic, and the same rules apply.
Example of a 3valued logic applied to vague (undetermined) cases: Kleene 1952^{[10]} (§64, pp. 332–340) offers a 3valued 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:

"We were justified intuitionistically in using the classical 2valued 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":
^{[11]}
~Q




QVR

R

t

f

u


Q&R

R

t

f

u


Q→R

R

t

f

u


Q=R

R

t

f

u

Q

t

f


Q

t

t

t

t


Q

t

t

f

u


Q

t

t

f

u


Q

t

t

f

u


f

t



f

t

f

u



f

f

f

f



f

t

t

t



f

f

t

u


u

u



u

t

u

u



u

u

f

u



u

t

u

u



u

u

u

u

For example, if a determination cannot be made as to whether an apple is red or notred, 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 notred " is " u ". Thus the AND of these into the assertion Q AND R, i.e. " This apple is red AND this apple is notred " will, per the tables, yield " u ". And, the assertion Q OR R, i.e. " This apple is red OR this apple is notred " will likewise yield " u ".
See also
References

^ ^{a} ^{b} Lou Goble (2001). The Blackwell guide to philosophical logic. WileyBlackwell. p. 309.

^ ^{}a ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} Paul Tomassi (1999). Logic. Routledge. p. 124.

^ Lou Goble (2001). The Blackwell guide to philosophical logic. WileyBlackwell. p. 4.

^ Mark Hürlimann (2009). Dealing with RealWorld Complexity: Limits, Enhancements and New Approaches for Policy Makers. Gabler Verlag. p. 42.

^ Dov M. Gabbay; John Woods (2007). The Many Valued and Nonmonotonic Turn in Logic. The handbook of the history of logic 8. Elsevier. p. vii.

^ Graham Priest (2008). An introduction to nonclassical logic: from if to is. Cambridge University Press. pp. 124–125.

^ Morten Heine Sørensen; Paweł Urzyczyn (2006). Lectures on the CurryHoward isomorphism. Elsevier. pp. 206–207.

^ "Stanford Encyclopedia of Philosophy".

^ Note the use of the (extremely) definite article: " This " as opposed to a morevague " The ". " The " would have to be accompanied with a pointinggesture to make it definitive. Ff Principia Mathematica (2nd edition), p. 91. Russell & Whitehead observe that this " this " indicates "something given in sensation" and as such it shall be considered "elementary".

^ Stephen C. Kleene 1952 Introduction to Metamathematics, 6th Reprint 1971, NorthHolland Publishing Company, Amsterdam NY, ISBN 0729421309.

^ "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).
Further reading

Devidi, D.; Solomon, G. (1999). "On Confusions About Bivalence and Excluded Middle". Dialogue (in French) 38 (4): 785–799. .

Betti Arianna (2002) The Incomplete Story of Łukasiewicz and Bivalence in T. Childers (ed.) The Logica 2002 Yearbook, Prague: The Czech Academy of Sciences—Filosofia, pp. 21–26

JeanYves Béziau (2003) "Bivalence, excluded middle and non contradiction", in The Logica Yearbook 2003, L.Behounek (ed), Academy of Sciences, Prague, pp. 73–84.

Font, J. M. (2009). "Taking Degrees of Truth Seriously". Studia Logica 91 (3): 383–406.
External links
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