Editing Three-valued logic (section)

=== Ternary Post logic ===
: Ternary Post Logic, a particular instance of the more General family of [[Many-valued logic#Post logics Pm|Post Logics]], is a three-valued logic which uses a cyclic negation operation, defined as:
{| style="border-spacing: 10px 0;" align="center"
|- valign="bottom"
|
{| class="wikitable" style="text-align:center;"
|+ NEG(A)
! width="25" | A
! width="25" | ¬A
|-
! scope="row" | 0
| 1
|-
! scope="row" | 1/2
| 0
|-
! scope="row" | 1
| 1/2
|}
|}
:Along with minimum and maximum functions to define conjunction and disjunction connectives, respectively:

{| style="border-spacing: 10px 0;" align="center"
|- valign="bottom"
|
{| class="wikitable" style="text-align:center;"
|+ MIN(A, B)
! colspan="2" rowspan="2" | A ∧ B
! colspan="3" | B
|-
! width="25" | 0
! width="25" | 1/2
! width="25" | 1
|-
! rowspan="3" scope="row" width="25" | A
! scope="row" width="25" | 0
| 0
| 0
| 0
|-
! scope="row" | 1/2
| 0
| 1/2
| 1/2
|-
! scope="row" | 1
| 0
| 1/2
| 1
|}
|
{| class="wikitable" style="text-align:center;"
|+ MAX(A, B)
! colspan="2" rowspan="2" | A ∨ B
! colspan="3" | B
|-
! width="25" | 0
! width="25" | 1/2
! width="25" | 1
|-
! rowspan="3" scope="row" width="25" | A
! scope="row" width="25" | 0
| 0
| 1/2
| 1
|-
! scope="row" | 1/2
| 1/2
| 1/2
| 1
|-
! scope="row" | 1
| 1
| 1
| 1
|}
|}
These functions may also be expressed with arithmetic expressions. Given the set of truth values is <math>\{0, \frac{1}{2}, 1\}</math> they are:
:<math>\begin{align}
               {\neg} x &:= \frac{1}{2}(6x^2 - 7x + 2) := (1,0, \frac{1}{2})\\
  x \wedge y &:= 4y^2x^2 - 4yx^2 - 4y^2x + 5yx := \min(x,y) \\[6pt]
    x \vee y &:= -4y^2x^2 + 4yx^2 + 4y^2x - 5yx + x + y := \max(x,y)
\end{align}</math>
:Unlike the previous system of three-valued logics, Post's 3-valued logic is functionally complete using only the NEG operation and at least one of MIN or MAX operations. This means that any function <math>\{0,\frac{1}{2},1\}^n \rightarrow \{0,\frac{1}{2},1\} : n \in \mathbb{N^+}
</math> can be expressed as a composition of using only the three functions defined above (or any two, as long as one of them is negation).