Inlogic, the semanticprinciple (orlaw)of bivalence states that every declarative sentence expressing aproposition (of a theory under inspection) has exactly onetruth value, eithertrue orfalse.[1][2] A logic satisfying this principle is called atwo-valued logic[3] orbivalent logic.[2][4]
In formal logic, the principle of bivalence becomes a property that asemantics may or may not possess. It is not the same as thelaw of excluded middle, however, and a semantics may satisfy that law without being bivalent.[2]
The principle of bivalence is studied inphilosophical logic to address the question of whichnatural-language statements have a well-defined truth value. Sentences that predict events in the future, and sentences that seem open to interpretation, are particularly difficult for philosophers who hold that the principle of bivalence applies to all declarative natural-language statements.[2]Many-valued logics formalize ideas that a realistic characterization of thenotion of consequence requires the admissibility of premises that, owing to vagueness, temporal orquantum indeterminacy, orreference-failure, cannot be considered classically bivalent. Reference failures can also be addressed byfree logics.[5]
The principle of bivalence is related to thelaw of excluded middle though the latter is asyntactic expression of the language of a logic of the form "P ∨ ¬P". The difference between the principle of bivalence and the law of excluded middle is important because there are logics that validate the law but not the principle.[2] For example, thethree-valuedLogic of Paradox (LP) validates the law of excluded middle, and yet also validates thelaw of non-contradiction, ¬(P ∧ ¬P), and itsintended semantics is not bivalent.[6] InIntuitionistic logic the law of excluded middle does not hold. Inclassical two-valued logic both the law of excluded middle and thelaw of non-contradiction hold.[1]
The intended semantics of classical logic is bivalent, but this is not true of everysemantics for classical logic. InBoolean-valued semantics (for classicalpropositional logic), the truth values are the elements of an arbitraryBoolean 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 thetwo-element algebra, which has no intermediate elements.
Assigning Boolean semantics to classicalpredicate calculus requires that the model be acomplete Boolean algebra because theuniversal quantifier maps to theinfimum operation, and theexistential quantifier maps to thesupremum;[7] this is called aBoolean-valued model. All finite Boolean algebras are complete.
In order to justify his claim that true and false are the only logical values, Roman Suszko (1977) observes that every structural Tarskian many-valued propositional logic can be provided with a bivalent semantics.[8]
A famous example[2] is thecontingent sea battle case found inAristotle's work,De Interpretatione, chapter 9:
The principle of bivalence here asserts:
Aristotle denies to embrace bivalence for such future contingents;[9]Chrysippus, theStoic logician, did embrace bivalence for this and all other propositions. The controversy continues to be of central importance in both thephilosophy of time and thephilosophy of logic.[citation needed]
One of the early motivations for the study ofmany-valued logics has been precisely this issue. In the early 20th century, the Polish formal logicianJan Łukasiewicz proposed three truth-values: the true, the false and theas-yet-undetermined. This approach was later developed byArend Heyting andL. E. J. Brouwer;[2] seeŁukasiewicz logic.
Issues such as this have also been addressed in varioustemporal 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.)
Such puzzles as theSorites 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 othermulti-valued 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:
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:
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 Pand not-P implies Por 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[11] (§64, pp. 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:
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":[12]
| ~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 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 ".
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