Principle of bivalence
Encyclopedia
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
. A logic satisfying this principle is called a two-valued logic or bivalent logic.
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.
The principle of bivalence is studied in philosophical logic to address the question of which natural-language statements have a well-defined 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 natural-language statements. Many-valued 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 reference-failure, cannot be considered classically bivalent. Reference failures can also be addressed by free logic
s.
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. For example, the three-valued Logic of Paradox (LP) validates the law of excluded middle, but not the law of non-contradiction, ¬(P ∧ ¬P), and its intended semantics
is not bivalent. In classical two-valued logic both the law of excluded middle and the law of non-contradiction hold.
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.
for classical logic. In Boolean-valued 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 two-element algebra, which has no intermediate elements.
Assign 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
; this is called a Boolean-valued model
. All finite Boolean algebras are complete.
's work, De Interpretatione, chapter 9:
The principle of bivalence here asserts:
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 metaphysics and the philosophy of logic.
One of the early motivations for the study of many-valued logics has been precisely this issue. In the early 20th century, the Polish formal logician Jan Łukasiewicz proposed three truth-values: the true, the false and the as-yet-undetermined. This approach was later developed by Arend Heyting
and L. E. J. Brouwer; see Łukasiewicz logic.
Issues such as this have also been addressed in various temporal logic
s, 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.)
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:
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:
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 not-P implies P 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 (§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:
The following are his "strong tables":
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 ".
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
, 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
True
True may refer to:* Truth, the state of being in accord with fact or reality-Music:* True , 1996* True , 2002* True , 1983** "True"...
or false
False
False or falsehood may refer to:*False *Lie or falsehood, a type of deception in the form of an untruthful statement*Falsity or falsehood, in law, deceitfulness by one party that results in damage to another...
. A logic satisfying this principle is called a two-valued logic or bivalent logic.
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
Law of excluded middle
In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....
, however, and a semantics may satisfy that law without being bivalent.
The principle of bivalence is studied in philosophical logic to address the question of which natural-language statements have a well-defined 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 natural-language statements. Many-valued logics formalize ideas that a realistic characterization of the notion of consequence
Entailment
In logic, entailment is a relation between a set of sentences and a sentence. Let Γ be a set of one or more sentences; let S1 be the conjunction of the elements of Γ, and let S2 be a sentence: then, Γ entails S2 if and only if S1 and not-S2 are logically inconsistent...
requires the admissibility of premises which, owing to vagueness, temporal or quantum indeterminacy, or reference-failure, cannot be considered classically bivalent. Reference failures can also be addressed by free logic
Free logic
A free logic is a logic with fewer existential presuppositions than classical logic. Free logics may allow for terms that do not denote any object. Free logics may also allow models that have an empty domain...
s.
Relationship with the law of the excluded middle
The principle of bivalence is related to the law of excluded middleLaw of excluded middle
In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....
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. For example, the three-valued Logic of Paradox (LP) validates the law of excluded middle, but not the law of non-contradiction, ¬(P ∧ ¬P), and its intended semantics
Intended interpretation
One who constructs a syntactical system usually has in mind from the outset some interpretation of this system. While this intended interpretation can have no explicit indication in the syntactical rules - since these rules must be strictly formal - the author's intention respecting...
is not bivalent. In classical two-valued logic both the law of excluded middle and the law of non-contradiction hold.
Many modern logic programming
Logic programming
Logic programming is, in its broadest sense, the use of mathematical logic for computer programming. In this view of logic programming, which can be traced at least as far back as John McCarthy's [1958] advice-taker proposal, logic is used as a purely declarative representation language, and a...
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 semanticsSemantics
Semantics is the study of meaning. It focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for, their denotata....
for classical logic. In Boolean-valued 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 two-element algebra, which has no intermediate elements.
Assign Boolean semantics to classical predicate calculus requires that the model be a complete Boolean algebra
Complete Boolean algebra
In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum . Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing...
because the universal quantifier maps to the infimum
Infimum
In mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...
operation, and the existential quantifier maps to the supremum
Supremum
In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...
; this is called a Boolean-valued model
Boolean-valued model
In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory. In a Boolean-valued model, the truth values of propositions are not limited to "true" and "false", but instead take values in some fixed complete Boolean...
. All finite Boolean algebras are complete.
Future contingents
A famous example is the contingent sea battle case found in AristotleAristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
'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 not true that there will be a sea battle tomorrow.
Aristotle hesitated to embrace bivalence for such future contingents; Chrysippus
Chrysippus
Chrysippus of Soli was a Greek Stoic philosopher. He was a native of Soli, Cilicia, but moved to Athens as a young man, where he became a pupil of Cleanthes in the Stoic school. When Cleanthes died, around 230 BC, Chrysippus became the third head of the school...
, the Stoic
STOIC
STOIC was a variant of Forth.It started out at the MIT and Harvard Biomedical Engineering Centre in Boston, and was written in the mid 1970s by Jonathan Sachs...
logician, did embrace bivalence for this and all other propositions. The controversy continues to be of central importance in both metaphysics and the philosophy of logic.
One of the early motivations for the study of many-valued logics has been precisely this issue. In the early 20th century, the Polish formal logician Jan Łukasiewicz proposed three truth-values: the true, the false and the as-yet-undetermined. This approach was later developed by Arend Heyting
Arend Heyting
Arend Heyting was a Dutch mathematician and logician. He was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic logic on a footing where it could become part of mathematical logic...
and L. E. J. Brouwer; see Łukasiewicz logic.
Issues such as this have also been addressed in various temporal logic
Temporal logic
In logic, the term temporal logic is used to describe any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time. In a temporal logic we can then express statements like "I am always hungry", "I will eventually be hungry", or "I will be hungry...
s, 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 paradoxSorites paradox
The sorites paradox is a paradox that arises from vague predicates. The paradox of the heap is an example of this paradox which arises when one considers a heap of sand, from which grains are individually removed...
and the related continuum fallacy
Continuum fallacy
The continuum fallacy is an informal logical fallacy closely related to the sorites paradox, or paradox of the heap...
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
Fuzzy logic
Fuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1...
and some other multi-valued logic
Multi-valued logic
In logic, a many-valued logic is a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values for any proposition...
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.
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 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 not-P implies P 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 (§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":
| ~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 ".
See also
Further reading
.- 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
- Jean-Yves BéziauJean-Yves BéziauJean-Yves Béziau is a professor and researcher of the Brazilian Research Council - CNPq - at the Federal University of Ceara, Brazil. Béziau is a dual citizen of France and Switzerland...
(2003) "Bivalence, excluded middle and non contradiction", in The Logica Yearbook 2003, L.Behounek (ed), Academy of Sciences, Prague, pp. 73-84.