In logic and mathematics, the minimal negation operator is a multigrade operator where each is a kary boolean function defined in such a way that if and only if exactly one of the arguments x_{j} is 0.
In contexts where the initial letter is understood, the minimal negation operators 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.
It may also be noted that is the same function as and , and that the inclusive disjunctions indicated for and for may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint. However, the function is not the same thing as the function .
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.
The next section discusses two ways of visualizing the operation of minimal negation operators. A few bits of terminology will be needed as a language for talking about the pictures, but the formal details are tedious reading, and may already be familiar to many. As a result, the full definitions of the terms marked in bold are relegated to a Glossary at the end of the article.
Contents 
Table 1 is a truth table for the sixteen boolean functions of type f : B^{3} → B, each of which is either a boundary of a point in B^{3} or the complement of such a boundary.
L_{1}  L_{2}  L_{3}  L_{4} 
Decimal  Binary  Sequential  Parenthetical 
p =  1 1 1 1 0 0 0 0  
q =  1 1 0 0 1 1 0 0  
r =  1 0 1 0 1 0 1 0 
f_{104}  f_{01101000}  0 1 1 0 1 0 0 0  ( p , q , r ) 
f_{148}  f_{10010100}  1 0 0 1 0 1 0 0  ( p , q , (r)) 
f_{146}  f_{10010010}  1 0 0 1 0 0 1 0  ( p , (q), r ) 
f_{97}  f_{01100001}  0 1 1 0 0 0 0 1  ( p , (q), (r)) 
f_{134}  f_{10000110}  1 0 0 0 0 1 1 0  ((p), q , r ) 
f_{73}  f_{01001001}  0 1 0 0 1 0 0 1  ((p), q , (r)) 
f_{41}  f_{00101001}  0 0 1 0 1 0 0 1  ((p), (q), r ) 
f_{22}  f_{00010110}  0 0 0 1 0 1 1 0  ((p), (q), (r)) 
f_{233}  f_{11101001}  1 1 1 0 1 0 0 1  (((p), (q), (r))) 
f_{214}  f_{11010110}  1 1 0 1 0 1 1 0  (((p), (q), r )) 
f_{182}  f_{10110110}  1 0 1 1 0 1 1 0  (((p), q , (r))) 
f_{121}  f_{01111001}  0 1 1 1 1 0 0 1  (((p), q , r )) 
f_{158}  f_{10011110}  1 0 0 1 1 1 1 0  (( p , (q), (r))) 
f_{109}  f_{01101101}  0 1 1 0 1 1 0 1  (( p , (q), r )) 
f_{107}  f_{01101011}  0 1 1 0 1 0 1 1  (( p , q , (r))) 
f_{151}  f_{10010111}  1 0 0 1 0 1 1 1  (( p , q , r )) 
Two common ways of visualizing the space B^{k} of 2^{k} points are the hypercube picture and the venn diagram picture. Depending on how literally or figuratively one regards these pictures, each point of B^{k} is either identified with or represented by a point of the kcube and also by a cell of the venn diagram on k "circles".
In addition, each point of B^{k} is the unique point in the fiber of truth [  s  ] of a singular proposition s : B^{k} → B, and thus it is the unique point where a singular conjunction of k literals is 1.
For example, consider two cases at opposite vertices of the cube:
To pass from these limiting examples to the general case, observe that a singular proposition s : B^{k} → B can be given canonical expression as a conjunction of literals, . Then the proposition is 1 on the points adjacent to the point where s is 1, and 0 everywhere else on the cube.
For example, consider the case where k = 3. Then the minimal negation operation , when there is no risk of confusion written more simply as , has the following Venn diagram:
For a contrasting example, the boolean function expressed by the form has the following Venn diagram:
,  
where  
or  , 
for  . 
