Editing Many-valued logic (section)

=== Gödel logics ''G<sub>k</sub>'' and ''G''<sub>∞</sub> ===

In 1932 [[Kurt Gödel|Gödel]] defined<ref>{{cite journal
 | last = Gödel | first = Kurt
 | title = Zum intuitionistischen Aussagenkalkül
 | journal = Anzeiger der Akademie der Wissenschaften in Wien
 | date = 1932 | issue = 69 | pages = 65f
}}</ref> a family <math>G_k</math> of many-valued logics, with finitely many truth values <math>0, \tfrac{1}{k - 1}, \tfrac{2}{k - 1}, \ldots, \tfrac{k - 2}{k - 1}, 1</math>, for example <math>G_3</math> has the truth values <math>0, \tfrac{1}{2}, 1</math> and <math>G_4</math> has <math>0, \tfrac{1}{3}, \tfrac{2}{3}, 1</math>. In a similar manner he defined a logic with infinitely many truth values, <math>G_\infty</math>, in which the truth values are all the [[real number]]s in the interval <math>[0, 1]</math>. The designated truth value in these logics is 1.

The conjunction <math>\wedge</math> and the disjunction <math>\vee</math> are defined respectively as the [[minimum]] and [[maximum]] of the operands:

: <math>\begin{align}
  u \wedge v &:= \min\{u, v\} \\
    u \vee v &:= \max\{u, v\}
\end{align}</math>

Negation <math>\neg_G</math> and implication <math>\xrightarrow[G]{}</math> are defined as follows:

: <math>\begin{align}
  \neg_G u &= \begin{cases}
                1, & \text{if }u = 0 \\
                0, & \text{if }u > 0
              \end{cases} \\[3pt]
  u \mathrel{\xrightarrow[G]{}} v &= \begin{cases}
                             1, & \text{if }u \leq v \\
                             v, & \text{if }u > v
                           \end{cases}
\end{align}</math>

Gödel logics are completely axiomatisable, that is to say it is possible to define a logical calculus in which all tautologies are provable. The implication above is the unique [[Heyting implication]] defined by the fact that the suprema and minima operations form a complete lattice with an infinite distributive law, which defines a unique [[complete Heyting algebra]] structure on the lattice.