Re: Laws Of Form Discussion • Minimal Negation Operators
The brief description of minimal negation operators given in the previous post is enough to convey the rule of their construction. For future reference, a slightly more formal definition is given below.
Initial Definition
The minimal negation operator 
In contexts where the initial letter
The first four members of this family of operators are shown below. The third and fourth columns give paraphrases in two other notations, where tildes and primes, respectively, indicate logical negation.
![\begin{matrix} \texttt{()} & = & \nu_0 & = & 0 & = & \mathrm{false} \\[6pt] \texttt{(x)} & = & \nu_1 (x) & = & \tilde{x} & = & x^\prime \\[6pt] \texttt{(x, y)} & = & \nu_2 (x, y) & = & \tilde{x}y \lor x\tilde{y} & = & x^\prime y \lor x y^\prime \\[6pt] \texttt{(x, y, z)} & = & \nu_3 (x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x^\prime y z \lor x y^\prime z \lor x y z^\prime \end{matrix}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Bmatrix%7D++%5Ctexttt%7B%28%29%7D+%26+%3D+%26+%5Cnu_0+%26+%3D+%26+0+%26+%3D+%26+%5Cmathrm%7Bfalse%7D++%5C%5C%5B6pt%5D++%5Ctexttt%7B%28x%29%7D+%26+%3D+%26+%5Cnu_1+%28x%29+%26+%3D+%26+%5Ctilde%7Bx%7D+%26+%3D+%26+x%5E%5Cprime++%5C%5C%5B6pt%5D++%5Ctexttt%7B%28x%2C+y%29%7D+%26+%3D+%26+%5Cnu_2+%28x%2C+y%29+%26+%3D+%26+%5Ctilde%7Bx%7Dy+%5Clor+x%5Ctilde%7By%7D+%26+%3D+%26+x%5E%5Cprime+y+%5Clor+x+y%5E%5Cprime++%5C%5C%5B6pt%5D+%5Ctexttt%7B%28x%2C+y%2C+z%29%7D+%26+%3D+%26+%5Cnu_3+%28x%2C+y%2C+z%29+%26+%3D+%26+%5Ctilde%7Bx%7Dyz+%5Clor+x%5Ctilde%7By%7Dz+%5Clor+xy%5Ctilde%7Bz%7D+%26+%3D+%26+x%5E%5Cprime+y+z+%5Clor+x+y%5E%5Cprime+z+%5Clor+x+y+z%5E%5Cprime++%5Cend%7Bmatrix%7D&bg=ffffff&fg=000000&s=1&c=20201002 "\begin{matrix} \texttt{()} & = & \nu_0 & = & 0 & = & \mathrm{false} \\[6pt] \texttt{(x)} & = & \nu_1 (x) & = & \tilde{x} & = & x^\prime \\[6pt] \texttt{(x, y)} & = & \nu_2 (x, y) & = & \tilde{x}y \lor x\tilde{y} & = & x^\prime y \lor x y^\prime \\[6pt] \texttt{(x, y, z)} & = & \nu_3 (x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x^\prime y z \lor x y^\prime z \lor x y z^\prime \end{matrix}")
Resources
- Minimal Negation Operator • InterSciWiki • MyWikiBiz • Wikiversity
