Editing Many-valued logic (section)

=== Łukasiewicz logics {{mvar|L<sub>v</sub>}} and {{math|''L''<sub>∞</sub>}}===

Implication <math>\xrightarrow[L]{}</math> and negation <math>\underset{L}{\neg}</math> were defined by [[Jan Łukasiewicz]] through the following functions:

: <math>\begin{align}
             \underset{L}{\neg} u &:= 1 - u \\
  u \mathrel{\xrightarrow[L]{}} v &:= \min\{1, 1 - u + v\}
\end{align}</math>

At first Łukasiewicz used these definitions in 1920 for his three-valued logic <math>L_3</math>, with truth values <math>0, \frac{1}{2}, 1</math>. In 1922 he developed a logic with infinitely many values <math>L_\infty</math>, in which the truth values spanned the real numbers in the interval <math>[0, 1]</math>. In both cases the designated truth value was 1.<ref>{{cite book
|last1= Kreiser |first1= Lothar
|last2 = Gottwald |first2 = Siegfried
|last3 = Stelzner |first3 = Werner
|date= 1990
|title= Nichtklassische Logik. Eine Einführung
|location= Berlin
|publisher= Akademie-Verlag
|pages= 41ff – 45ff
|isbn= 978-3-05-000274-3
}}</ref>

By adopting truth values defined in the same way as for Gödel logics <math>0, \tfrac{1}{v-1}, \tfrac{2}{v-1}, \ldots, \tfrac {v-2} {v-1}, 1</math>, it is possible to create a finitely-valued family of logics <math>L_v</math>, the abovementioned <math>L_\infty</math> and the logic <math>L_{\aleph_0}</math>, in which the truth values are given by the [[rational number]]s in the interval <math>[0,1]</math>. The set of tautologies in <math>L_\infty</math> and <math>L_{\aleph_0}</math> is identical.