+++
title = "Łukasiewicz logic"
author = "Yann Herklotz"
tags = []
categories = []
backlinks = ["4e2a"]
forwardlinks = ["4e2a"]
zettelid = "4e2b"
+++

This logic has a different definition of implication. This has many
benefits, especially when one wants to differentiate between undefined
values and truth/false values. The main benefit is that one can actually
have tautologies in this logic, even with undefined values, whereas in
the Kleene logic ([\#4e2a]) there can be no tautologies, because
assigning all variable to `U` will end up with `U` every time.

| A-\>B | F   | U   | T   |
|-------|-----|-----|-----|
| F     | T   | T   | T   |
| U     | U   | T   | T   |
| T     | F   | U   | T   |

This logic has the same exact definition of AND and OR than Kleene logic
([\#4e2a]), and these connectives can be expressed in terms of Ł3
implication.

```{=latex}
\begin{align}
  A \lor B &= (A \rightarrow B) \rightarrow B \\
  A \land B &= \neg (\neg A \lor \neg B) \\
  A \Leftrightarrow B &= (A \rightarrow B) \land (B \rightarrow A)
\end{align}
```
We can then define additional unary operators using the following:

```{=latex}
\begin{align}
  \mathcal{M} A &= \neg A \rightarrow A \\
  \mathcal{L} A &= \neg \mathcal{M} \neg A \\
  \mathcal{I} A &= \mathcal{M} A \land \neg \mathcal{L} A
\end{align}
```
Especially the last, $\mathcal{I} A$, has interesting properties,
because it will be 1 iff A is 0, and will be -1 otherwise.

  [\#4e2a]: /zettel/4e2a