# Trivalent logic

Trivalent logics (also: ternary logics ) are examples of multi-valued logics , i.e. for non-classical logics that differ from classical logic in that the principle of two- valued is abandoned. This means that instead of two truth values there are three, namely instead of just “true” (or 1) and “false” (or 0) also “unknown”, “indefinite” or “do n't care ” (or . 1/2 or i).

## Various three-valued logics

The first three-valued logic is the system Ł 3 , the Jan Łukasiewicz developed the 1920s. Ł 3 is closely related to intuitionist logic . The system was soon expanded to multi-valued logics by Łukasiewicz and others . A common alternative to Ł 3 is the logic K 3 developed by Stephen Cole Kleene in 1938 .

Dmitrij Analtoljevič Bočvar also presented the three-valued system B 3 in 1938 in order to investigate logical and semantic antinomies that can occur in logic of higher levels. The third truth value was meaningless , paradoxical , meaningless or nonsensical for him .

There are also variants of the three-valued logic, in which, in addition to “true”, “indefinite” is an excellent truth value; H. Consistency in such systems means that conclusions can be derived from true premises that have the truth value “true” or “indefinite”. An alternative to this is the use of the “weak not”, which recognizes the negation of a statement with an indefinite truth value as true.

## Formal similarities between three-valued logics

In addition to the truth values w ( true ) and f ( false ) of classical logic , a third truth value is introduced. For Łukasiewicz, who starts from an epistemological question, the intended meaning of this newly introduced value is something like: “not proven, but also not refuted”; it can be read as m - possible. Interpretations that apply Ł 3 in computer science read the third truth value as u for unknown. Other three-valued logics partly assume that the third truth value is assigned to statements that are “neither true nor false” or “both true and false”. In these cases the truth value i is for “indefinite”.

The following truth tables apply to the joiners and , or and not (unless the "weak not" is used): ${\ displaystyle (\ land)}$ ${\ displaystyle (\ lor)}$ ${\ displaystyle (\ neg)}$

a and b
b

a
f u w
f f f f
u f u u
w f u w
a or B
b

a
f u w
f f u w
u u u w
w w w w
not a
a ${\ displaystyle \ neg}$ a
f w
u u
w f

This can also be summarized as follows:

• The truth value of is the minimum of the truth values ​​of and ;${\ displaystyle A \ land B}$${\ displaystyle A}$${\ displaystyle B}$
• The truth value of is the maximum of the truth values ​​of and ;${\ displaystyle A \ lor B}$${\ displaystyle A}$${\ displaystyle B}$
• The truth value of is the inverse truth value of ;${\ displaystyle \ neg A}$${\ displaystyle A}$

## Ł 3 and K 3

The logics Ł 3 and K 3 differ only in the definition of the subjunction , i.e. H. of the juncture that is to represent the natural language conditional. The corresponding truth tables are: ${\ displaystyle (\ rightarrow)}$

if a then b in Ł 3
b

a
f u w
f w w w
u u w w
w f u w
if a then b in K 3
b

a
f u w
f w w w
u u u w
w f u w

Only the case in which both parts of the subjunction have the truth value 1/2 is disputed. According to Ł 3 the subjunction is true here, according to K 3 it has the truth value 1/2. However, this difference has a significant impact. In particular, there are no tautologies in K 3 , but consistency remains possible. Numerous tautologies of classical logic are preserved in Ł 3 , but there are also paradoxes. These differences can mainly be explained by the fact that Łukasiewicz pursued an epistemological motivation, while Kleene was looking for a way of dealing with statements which, even with objective knowledge of the truth, cannot easily be described as "true" or "false".

## B 3

The logic B 3 differentiates between inner and outer truth value functions. The inner truth value functions correspond to the classic ones if the truth value “u” does not occur and are otherwise always “u”. The inner negation thus corresponds to the negation in Ł 3 and K 3 .

 f u b a Inner conjunction in B 3 f u f u u u f u w
 f u b a Inner alternative in B 3 f u w u u u w u w
 f u b a Inner implication in B 3 w u w u u u f u w

Here the mean truth value is to a certain extent "infectious"; every use of propositions with this truth value in any combination of connectives will lead to the truth value of the overall statement also being u. Therefore there are two more one-digit truth value functions j f and j w in B 3

a Truth value function j w f f w
a Truth value function j f w f f

The truth function j w stands for the utterance of a proposition, j f stands for the negative utterance. 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 use this logic to counter paradoxes such as the liars paradox , which should be assigned the truth value u. So the meaning of u here is “meaningless” or “paradoxical”.

## “U” as an excellent truth value

Another possibility for dealing with the three-valued logic is to allow it as the second excellent truth value in addition to "true". This guarantees consistency when the truth value of the conclusion of an argument is “true” or “indefinite” or “unknown”. It makes sense to change the truth function of the subjunction with respect to Ł 3 in order to limit the number of paradoxes. As examples, here are the truth tables of the subjunction in LP and RM 3 :

 f u b a if a then b in LP w w w u u w f u w
 f u b a if a then b in RM 3 w w w f u w f f w

LP ("Logic of Paradox" by Graham Priest ) uses the same truth function of the subjunction as K 3 , but in contrast to K 3 there are numerous tautologies, but no modus ponens . This is saved in RM 3 , and some paradoxes from LP do not appear here either.

## Strong and weak negation

An alternative to using two distinct truth values ​​is to use two different negations. This is mainly combined with £ 3 . A distinction is made between strong negation and weak negation : ${\ displaystyle \ neg}$ ${\ displaystyle \ thicksim}$

• The truth value of the strong (or inner, presupposing) negation is the inversion of the truth value of , with an indefinite truth value nothing changes here.${\ displaystyle \ neg A}$${\ displaystyle A}$
• The truth value of the weak (or external, non-presuppositioning) negation is "false" if the truth value is "true", and otherwise always "true". This negation corresponds roughly to the formulation "It is not true that P."${\ displaystyle \ thicksim \! \! A}$${\ displaystyle A}$

So the truth tables are:

A. strong negation w u f
A. weak negation w w f

Two sub-functions are defined accordingly:

• The strong subjunction through:${\ displaystyle (\ Rightarrow \! \,)}$${\ displaystyle A \ Rightarrow B: = \ neg A \ lor B}$
• The weak subjunction by:${\ displaystyle (\ rightarrow) \! \,}$${\ displaystyle A \ rightarrow B: = \ \ thicksim \! \! A \ lor B}$

Tautologies are formulas that receive the truth value "w" for each assignment of their elements. In this sense , as well , and tautologies. In general, it can be shown that the tautologies in Ł 3 that do not contain strong joiners correspond exactly to the generally valid formulas of classical two-valued logic. On the other hand, and are not tautologies in Ł 3 , but the inversion and the formula are . Ł 3 thus corresponds to the demands made by the intuitionists . ${\ displaystyle \ thicksim \! \! (A \ land \ thicksim \! \! A)}$${\ displaystyle A \ rightarrow \ \ thicksim \ thicksim \! \! A}$${\ displaystyle A \ lor \ thicksim \! \! A}$${\ displaystyle \ thicksim \ thicksim \! \! A \ rightarrow A}$${\ displaystyle A \ lor \ neg A}$${\ displaystyle \ thicksim \! \ neg A \ rightarrow A}$${\ displaystyle A \ rightarrow \ thicksim \! \ neg A}$${\ displaystyle \ thicksim \! \! (A \ land \ neg A)}$

The “ ex falso quodlibet ” is not only a tautology in the “classical” form , but also in the “intuitionistic” form . In its form , on the other hand, it is not a tautology, as it corresponds to the requirements of the minimal calculus . ${\ displaystyle \ thicksim \! \! A \ rightarrow (A \ rightarrow B)}$${\ displaystyle \ neg A \ rightarrow (A \ rightarrow B)}$${\ displaystyle \ neg A \ Rightarrow (A \ Rightarrow B)}$

## literature

• Ulrich Blau, The Logic of Indeterminacies and Paradoxes, Heidelberg 2008, pp. 191–290.
• Susan Haack , Philosophy of Logics, Cambridge 1978, pp. 204-220.
• Jan Łukasiewicz, Philosophical Remarks on Many-Valued Systems of Propositional Logic, in: Storrs MacCall (ed.), Polish Logic 1920-1939, Oxford 1967.
• Graham Priest, An Introduction to Non-Classical Logic. From If to Is, Cambridge 2008, pp. 120-141.

## Individual evidence

1. Stephen Cole Kleene: On notation for ordinal numbers. In: Journal Symbolic Logic. 3, 1938, pp. 150-155.
2. Дмитрий Анатольевич Бочвар (Dmitry Anatolyevich Bočvar) : Об одном трехзначном исчислении и его применении к анализу парадоксов классического расширенного функционального исчисления . - Whether odnom tréhznačnom isčislénii i égo priménénii k analizu paradoksov klassičéskogo rasširénnogo funkcional'nogo isčisléniá. In: Matématičéskij sbornik. Volume 46-4. 1938, pp. 287-308 (Russian).
3. ^ Siegfried Gottwald: Multi-valued logic. An introduction to theory and application. Akademie-Verlag, Berlin 1989, p. 165 f.
4. ^ Susan Haack: Philosophy of Logics. Cambridge 1978, p. 207.
5. ^ Susan Haack: Philosophy of Logics. Cambridge 1978, p. 211.