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Encyclopedia > Minimal negation operator

In logic and mathematics, the minimal negation operator nu! is a multigrade operator (nu_{k})_{k in mathbb{N}} where each nu_{k}! is a k-ary boolean function defined in such a way that nu_{k}(x_1, ldots , x_k) = 1 if and only if exactly one of the arguments xj is 0. Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of criteria for the evaluation of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... Euclid, detail from The School of Athens by Raphael. ... In mathematics, a boolean function is usually a function F(b1, b2, ..., bn) of a number n of boolean variables bi from the two-element boolean algebra B = {0, 1}, such that F also takes values in B. A function on an arbitrary set X taking values in B is...


In contexts where the initial letter nu! is understood, the mno's can be indicated by argument lists in parentheses. The first four members of this family of operators are shown below, with paraphrases in a couple of other notations, where tildes and primes, respectively, indicate logical negation.

begin{matrix} ( ) & = & 0 & = & mbox{false} (x) & = & tilde{x} & = & x' (x, y) & = & tilde{x}y lor xtilde{y} & = & x'y lor xy' (x, y, z) & = & tilde{x}yz lor xtilde{y}z lor xytilde{z} & = & x'yz lor xy'z lor xyz' end{matrix}

It may also be noted that (x, y)! is the same function as x + y! and x ne y, and that the inclusive disjunctions indicated for (x, y)! and for (x, y, z)! may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint. However, the function (x, y, z)! is not the same thing as the function x + y + z!.


The minimal negation operator (mno) has a legion of aliases: logical boundary operator, limen operator, threshold operator, or least action operator, to name but a few. The rationale for these names is visible in the Venn diagrams of the corresponding operations on sets. Venn diagrams are illustrations used in the branch of mathematics known as set theory. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...


See also



 

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