View source for Principle of bivalence

{{Short description|Classical logic of two values, either true or false}}
{{Redirect|Bivalence||Bivalent (disambiguation)}}
{{Use dmy dates|date=June 2020}}
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 [[Truth|true]] or [[false (logic)|false]]. <ref name="Goble2001bis"/><ref name="Tomassi1999">{{cite book|author=Paul Tomassi|title=Logic|url=https://books.google.com/books?id=TUVQr6InyNYC&pg=PA124|year=1999|publisher=Routledge|isbn=978-0-415-16696-6|page=124}}</ref> A logic satisfying this principle is called a '''two-valued logic'''<ref name="Goble2001">{{cite book|author=Lou Goble|title=The Blackwell guide to philosophical logic|url=https://books.google.com/books?id=aaO2f60YAwIC&pg=PA4|year=2001|publisher=Wiley-Blackwell|isbn=978-0-631-20693-4|page=4}}</ref> or '''bivalent logic'''.<ref name="Tomassi1999"/><ref name="Hürlimann2009">{{cite book|author=Mark Hürlimann|title=Dealing with Real-World Complexity: Limits, Enhancements and New Approaches for Policy Makers|url=https://books.google.com/books?id=_BThojvh-XUC&pg=PA42|year=2009|publisher=Gabler Verlag|isbn=978-3-8349-1493-4|page=42}}</ref>

In formal logic, the principle of bivalence becomes a property that a [[formal semantics (logic)|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.<ref name="Tomassi1999"/>

The principle of bivalence is studied in [[philosophical logic]] to address the question of which [[Natural language|natural-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.<ref name="Tomassi1999"/> [[Many-valued logic]]s formalize ideas that a realistic characterization of the [[Entailment|notion of consequence]] requires the admissibility of premises that, owing to vagueness, temporal or [[quantum indeterminacy]], or [[failure to refer|reference-failure]], cannot be considered classically bivalent. Reference failures can also be addressed by [[free logic]]s.<ref name="GabbayWoods2007">{{cite book|author1=Dov M. Gabbay|author2=John Woods|title=The Many Valued and Nonmonotonic Turn in Logic|url=https://books.google.com/books?id=3TNj1ZkP3qEC&pg=PA7|year=2007|publisher=Elsevier|isbn=978-0-444-51623-7|page=vii|series=The handbook of the history of logic|volume=8}}</ref>

== Relationship to 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 of bivalence and the law of excluded middle is important because there are logics that validate the law but not the principle.<ref name="Tomassi1999"/> For example, the [[three-valued logic|three-valued]] [[Logic of Paradox]] (LP) validates the law of excluded middle, and yet also validates the [[law of non-contradiction]], ¬(P ∧ ¬P), and its [[intended interpretation|intended semantics]] is not bivalent.<ref name="Priest2008">{{cite book|author=Graham Priest|title=An introduction to non-classical logic: from if to is|url=https://books.google.com/books?id=rMXVbmAw3YwC&pg=PA124|year=2008|publisher=Cambridge University Press|isbn=978-0-521-85433-7|pages=124–125}} (see also ''[[An Introduction to Non-Classical Logic]]'')</ref>  In [[intuitionistic logic]] the law of excluded middle does not hold. In [[classical logic|classical]] two-valued logic both the law of excluded middle and the [[law of non-contradiction]] hold.<ref name="Goble2001bis">{{cite book|author=Lou Goble|title=The Blackwell guide to philosophical logic|url=https://books.google.com/books?id=aaO2f60YAwIC&pg=PA309|year=2001|publisher=Wiley-Blackwell|isbn=978-0-631-20693-4|page=309}}</ref>

== Classical logic ==
The intended semantics of classical logic is bivalent, but this is not true of every [[formal semantics (logic)|semantics]] 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 Boolean algebra|two-element 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]];<ref name="SørensenUrzyczyn2006">{{cite book|author1=Morten Heine Sørensen|author2=Paweł Urzyczyn|title=Lectures on the Curry-Howard isomorphism|url=https://books.google.com/books?id=_mtnm-9KtbEC&pg=PA206|year=2006|publisher=Elsevier|isbn=978-0-444-52077-7|pages=206–207}}</ref> this is called a [[Boolean-valued 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, Roman Suszko (1977) observes that every structural Tarskian many-valued propositional logic can be provided with a bivalent semantics.<ref>{{cite web |last1=Shramko|first1=Y.|last2=Wansing|first2=H.|title=''Truth Values'', Stanford Encyclopedia of Philosophy|year=2015|url=http://plato.stanford.edu/entries/truth-values/suszko-thesis.html}}</ref>

== Criticisms ==

===Future contingents===
{{main|Problem of future contingents}}
A famous example<ref name="Tomassi1999"/> 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 denies to embrace bivalence for such future contingents;<ref>{{Cite journal|last=Jones|first=Russell E.|date=2010|title=Truth and Contradiction in Aristotle's De Interpretatione 6–9|url=https://www.jstor.org/stable/20720827|journal=Phronesis|volume=55|issue=1|pages=26–67|doi=10.1163/003188610X12589452898804|jstor=20720827|s2cid=53398648 |url-access=subscription}}</ref> [[Chrysippus]], the [[Stoicism|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]].{{citation needed|date=March 2016}}

One of the early motivations for the study of [[many-valued logic]]s 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]];<ref name="Tomassi1999"/> 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.)

===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>
{|class="wikitable"
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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 ".

=== Self-referential statements ===
{{main|Self-reference}}
Some [[Self-reference|self-referential statements]] like the one featured in the [[Liar paradox|liar's paradox]] can not be assigned definite truth values of neither "[[Truth|''True'']]" nor "[[Falsehood|''False'']]" without running into contradictions.<ref name="Andrew Irvine 1992">Andrew Irvine, "Gaps, Gluts, and Paradox", ''Canadian Journal of Philosophy'', supplementary vol. 18 [''Return of the A priori''] (1992), 273–299</ref> The liar paradox can be stated as: {{block indent|A: This statement (A) is false.}}If (A) is true, then "This statement is false" is true. Therefore, (A) must be false. The hypothesis that (A) is true leads to the conclusion that (A) is false, a contradiction.

If (A) is false, then "This statement is false" is false. Therefore, (A) must be true. The hypothesis that (A) is false leads to the conclusion that (A) is true, another contradiction. Either way, (A) is both true and false, which is a paradox.

Some possible resolutions of this paradox include the rejection of [[Boolean logic]] (and thus the '''principle of bivalence'''<ref>{{Citation |last=Beall |first=Jc |title=Liar Paradox |date=2023 |work=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/liar-paradox/ |access-date=2025-05-11 |edition=Winter 2023 |publisher=Metaphysics Research Lab, Stanford University |last2=Glanzberg |first2=Michael and |editor2-last=Nodelman |editor2-first=Uri}}</ref>) and its replacement with any [[Multi-valued logic|many-valued logic]] like [[fuzzy logic]], in which the [[truth value]] of a [[Statement (logic)|statement]] may be any [[real number]] between 0 (denoting "''[[Falsehood]]''") and 1 (denoting "''[[Truth]]''").<ref>{{Citation |last=Cintula |first=Petr |title=Fuzzy Logic |date=2023 |work=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/logic-fuzzy/ |access-date=2025-05-11 |edition=Summer 2023 |publisher=Metaphysics Research Lab, Stanford University |last2=Fermüller |first2=Christian G. |last3=Noguera |first3=Carles |editor2-last=Nodelman |editor2-first=Uri}}</ref><ref>{{cite journal |last1=Hájek |first1=P. |last2=Paris |first2=J. |last3=Shepherdson |first3=J. |date=Mar 2000 |title=The Liar Paradox and Fuzzy Logic |journal=The Journal of Symbolic Logic |volume=61 |pages=339–346 |doi=10.2307/2586541 |jstor=2586541 |s2cid=6865763 |number=1}}</ref>

==See also==
{{Portal|Philosophy|Psychology}}
{{columns-list|colwidth=18em|
* [[Dualism (philosophy of mind)|Dualism]]
* [[Exclusive disjunction]]
* [[Degrees of truth]]
* [[Anekantavada]]
* [[Extensionality]]
* [[False dilemma]]
* [[Fuzzy logic]]
* [[Liar paradox]]
* [[Logical disjunction]]
* [[Logical equality]]
* [[Logical value]]
* [[Multi-valued logic]]
* [[Perspectivism]]
* [[Propositional logic]]
* [[Quantum logic]]
* [[Relativism]]
* [[Self-reference]]
* [[Supervaluationism]]
* [[True and false]]
* [[Truthbearer]]
* [[Truthmaker]]
* [[Truth-value link]]}}

==References==
{{Reflist}}

==Further reading==
* {{Cite journal | last1 = Devidi | first1 = D. | last2 = Solomon | first2 = G. | doi = 10.1017/S0012217300006715 | title = On Confusions About Bivalence and Excluded Middle | journal = Dialogue | volume = 38 | issue = 4 | pages = 785–799| year = 1999| s2cid = 170829533 | language = fr}}.
* Betti Arianna (2002) [https://web.archive.org/web/20110722063257/http://segr-did2.fmag.unict.it/~polphil/PolPhil/Lukas/LukasBival.pdf The Incomplete Story of Łukasiewicz and Bivalence] in T. Childers (ed.) ''The Logica 2002 Yearbook'', Prague: The Czech Academy of Sciences – Filosofia, pp.&nbsp;21–26
* [[Jean-Yves Béziau]] (2003) "[http://www.unine.ch/unilog/jyb/new%20principle%20of%20bivalence.pdf Bivalence, excluded middle and non contradiction]", in ''The Logica Yearbook 2003'', L.Behounek (ed), Academy of Sciences, Prague, pp.&nbsp;73–84.
* {{Cite journal | last1 = Font | first1 = J. M. | title = Taking Degrees of Truth Seriously | doi = 10.1007/s11225-009-9180-7 | journal = Studia Logica | volume = 91 | issue = 3 | pages = 383–406 | year = 2009 | s2cid = 12721181 }}

== External links ==
* {{cite SEP |url-id=truth-values |title=Truth Values |last=Shramko  |first=Yaroslav|last2=Wansing |first2=Heinrich}}

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