Three-valued logic

Trivalent logics (also: ternary logics ) are examples of multi-valued logic, ie for non-classical logics, which differ from classical logic by the fact that the principle of bivalence is abandoned. This means that there are three instead of two truth values ​​, namely, instead of just " true" ( or 1 ) and "false" (or 0 ) is also still "unknown", "undefined " or "Do not Care " ( or. half or i).

Various trivalent logics

The first three-valued logic, the system is L3, which developed Łukasiewicz January 1920. A completely different approach pursued eg Bolesław Sobociński. L3 is closely related to intuitionischen logic. The system was soon extended by Łukasiewicz and others to multi-valued logics. A common alternative to L3 is the 1938 Stephen Cole Kleene logic developed by K3.

Dmitry Analtoljevič Bočvar has also in 1938, the trivalent system B3 presented to investigate logical and semantic paradoxes that may occur in logic at higher levels. The third truth value stood by him senseless, paradoxical, meaningless or nonsensical.

There are also variants of the three-valued logic, in addition to "true" and "undetermined" an excellent truth value, ie Consistency means in such systems that conclusions may be derived from true premises that have the truth value "true" or "undefined ." An alternative to this is the use of " not weak", which recognizes the negation of a statement with undetermined truth value as true.

Formal similarities trivalent logics

In addition to the truth values ​​of w ( true) and f ( false) of classical logic, a third truth value is introduced. In Łukasiewicz emanating from an epistemological question, is the intended meaning of this newly introduced value in about: "not proven, but not refuted "; he may as m - read possible. Interpretations which apply in the computer science L3, read the third truth value as u for unknown. Other trivalent logics have partially from the fact that the third truth value is given for statements that were " neither true nor false " or " both true and false." In such cases, the truth value is for i " indefined ".

( Provided there is no the " weak non " used ) for the logical connectives and, or and not are the following truth tables:

This can be summarized as follows:

• The truth value of the minimum of the truth values ​​of and;
• The truth value of the maximum of the truth values ​​of and;
• The truth value of is the reverse of truth value;

L3 and K3

The logics L3 and K3 differ only in the definition of subjunction, ie of connective intended to reflect the natural language conditional. The corresponding truth tables are:

So controversial is the only case in which both parts of the subjunction have the truth value 1/2. After the L3 subjunction here is true, according to K3 it bears the truth value 1/2. However, this difference has a significant impact. In particular, there are no tautologies in K3, consistency remains possible. In L3 numerous tautologies of classical logic are preserved, but there is the possibility to paradoxes. These differences are mainly explained with the fact that Łukasiewicz an epistemological motivation pursued, while Kleene rather a deal with statements sought, which can also be referred to an objective knowledge of the truth is not readily as "true" or "false".

B3

The logic B3 distinction between inner and outer truth value functions. The inner truth-value functions correspond to the classical, if the truth value of u does not exist, otherwise they are always u The inner negation thus corresponds to the negation in the L3 and K3.

Here the average truth value is quasi " infectious", any use of propositions with this truth value will result in some combination of Connectives to the fact that the truth value of the overall presentation is also u. Therefore, in B3 two -digit boolean functions jf and jw

The truth function jw stands for the utterance of a proposition, jf is the negative statement. Thus, the assertion of a proposition P with the truth value u can be evaluated as false, the rejection of P is evaluated as true. Bočvar wanted to meet with this logic paradoxes such as the liar paradox that should be u assigned the truth value. The importance of u is here " meaningless " or " paradoxical ".

"u" as an excellent truth value

Another option for dealing with the three-valued logic, "true" to allow him next as the second excellent truth value. This consistency is guaranteed if the truth value of the conclusion of an argument is "true" or " indeterminate" or " unknown " is. It is natural to change the truth function of the subjunction against L3 to limit the number of paradoxes. As examples, here the truth tables of the subjunction in LP and RM3

LP ( " Logic of Paradox" by Graham Priest ) used the same truth function of subjunction as K3, but there is in contrast to K3 numerous tautologies, but not for modus ponens. This is backed up to RM3, some paradoxes of LP do not appear here.

Strong and weak negation

An alternative to the use of two excellent truth values ​​is the use of two different negations. This is especially in combination with L3. Here, a strong negation and weak negation can be distinguished:

• The truth value of the strong (or inner, präsupponierenden ) Negation is the reverse of the truth value of, at undetermined truth value here, nothing changes.
• The truth value of the weak (or outer, non - präsupponierenden ) negation is "false " if the truth value of " true", and otherwise always " true." This negation corresponds approximately to the phrase " It is not true that P. "

The truth tables are the following:

Accordingly, two Subjunktionen are defined:

• The strong subjunction by:
• The weak subjunction by:

In the As of tautologies are formulas denotes the assignment obtained at each of its elements the truth value of "w". In this sense, but also, and tautologies. In general it can be shown that the tautologies in L3 which do not contain strong connectives, exactly correspond to the general formulas of the classical two-valued logic. In contrast, and no tautologies in L3, but probably the reversal and the formula. L3 meeting the market demand, which have established the intuitionists.

The " ex falso quodlibet " is not only in the " classic" form a tautology, but also in the " intuitionistic " Form. 's Form however, it is not a tautology, as this corresponds approximately to the demands of the minimal calculus.