Principle of bivalence

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.

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. In intuitionistic logic the law of excluded middle does not hold. In classical two-valued logic both the law of excluded middle and the law of non-contradiction hold.

Classical logic

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

Criticisms

Future contingents

A famous example 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 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;

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 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":

{|class="wikitable"

|- style="font-size:9pt" align="center" valign="bottom"

|style="background-color:#C0C0C0;font-weight:bold" width="15" Height="9.6" | ~Q

|style="font-weight:bold;font-style:Italic" width="15" |

|style="font-weight:bold" width="15" |

| width="5.4" |

|style="background-color:#C0C0C0;font-weight:bold" width="25.2" | QVR

|style="background-color:#CCFFFF;font-weight:bold;font-style:Italic" width="15" | R

|style="background-color:#CCFFFF;font-weight:bold" width="15" | t

|style="background-color:#CCFFFF;font-weight:bold" width="15" | f

|style="background-color:#CCFFFF;font-weight:bold" width="15" | u

| width="6" |

|style="background-color:#C0C0C0;font-weight:bold" width="28.2" | Q&R

|style="background-color:#CCFFFF;font-weight:bold;font-style:Italic" width="15" | R

|style="background-color:#CCFFFF;font-weight:bold" width="15" | t

|style="background-color:#CCFFFF;font-weight:bold" width="15" | f

|style="background-color:#CCFFFF;font-weight:bold" width="15" | u

| width="5.4" |

|style="background-color:#C0C0C0;font-weight:bold" width="32.4" | Q→R

|style="background-color:#CCFFFF;font-weight:bold;font-style:Italic" width="15" | R

|style="background-color:#CCFFFF;font-weight:bold" width="15" | t

|style="background-color:#CCFFFF;font-weight:bold" width="15" | f

|style="background-color:#CCFFFF;font-weight:bold" width="15" | u

| width="4.2" |

|style="background-color:#C0C0C0;font-weight:bold" width="28.2" | Q=R

|style="background-color:#CCFFFF;font-weight:bold;font-style:Italic" width="15" | R

|style="background-color:#CCFFFF;font-weight:bold" width="15" | t

|style="background-color:#CCFFFF;font-weight:bold" width="15" | f

|style="background-color:#CCFFFF;font-weight:bold" width="15" | u

|- style="font-size:9pt" align="center" valign="bottom"

|style="background-color:#FFFF99;font-weight:bold;font-style:Italic" Height="9.6" | Q