Principle of bivalence

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"Bivalence" redirects here. For other uses, see Bivalent (disambiguation).

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 two-valued 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]

The principle of bivalence is studied in

notion of consequence requires the admissibility of premises that, owing to vagueness, temporal or quantum indeterminacy, or reference-failure, cannot be considered classically bivalent. Reference failures can also be addressed by free logics.[5]

Relationship to the law of the excluded middle

The principle of bivalence is related to the

law of non-contradiction hold.[1]

Classical logic

The intended semantics of classical logic is bivalent, but this is not true of every

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.

Assigning Boolean semantics to classical

supremum;[7] 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.[8]

Criticisms

Future contingents

Main article: Problem of future contingents

A famous example

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;

philosophy of time and the philosophy of logic.[citation needed
]

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;[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

multi-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:

This apple is red.[10]

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

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[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 ".

See also

References

  1. ^
    ISBN 978-0-631-20693-4
    .
  2. ^
    ISBN 978-0-415-16696-6
    .
  3. ISBN 978-0-631-20693-4
    .
  4. ISBN 978-3-8349-1493-4
    .
  5. ISBN 978-0-444-51623-7
    .
  6. ISBN 978-0-521-85433-7
    .
  7. ISBN 978-0-444-52077-7
    .
  8. ^ Shramko, Y.; Wansing, H. (2015). "Truth Values, Stanford Encyclopedia of Philosophy".
  9. S2CID 53398648
    – via JSTOR.
  10. ^ 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 edition), p. 91. Russell & Whitehead observe that this " this " indicates "something given in sensation" and as such it shall be considered "elementary".
  11. ISBN 0-7294-2130-9
    .
  12. ^ "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

External links
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