Editing Fuzzy logic (section)

===Gödel G<sub>∞</sub> logic===
{{further|Many-valued logic#Gödel logics Gk and G∞}}

Another logical system where truth values are real numbers between 0 and 1 and where AND & OR operators are replaced with MIN and MAX is Gödel's G<sub>∞</sub> logic. This logic has many similarities with fuzzy logic but defines negation differently and has an internal implication. 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>

which turns the resulting logical system into a model for [[intuitionistic logic]], making it particularly well-behaved among all possible choices of logical systems with real numbers between 0 and 1 as truth values. In this case, implication may be interpreted as "x is less true than y" and negation as "x is less true than 0" or "x is strictly false", and for any <math>x</math> and <math>y</math>, we have that <math> AND(x, x \mathrel{\xrightarrow[G]{}} y) = AND(x,y) </math>. In particular, in Gödel logic negation is no longer an involution and double negation maps any nonzero value to 1.