Note. I’m including a more detailed definition of minimal negation operators in terms of conventional logical operations largely because readers of particular tastes have asked for it in the past. But it can easily be skipped until one has a felt need for it. Skimmed lightly, though, it can serve to illustrate a major theme in logic and mathematics, namely, the Relativity of Complexity or the Relativity of Primitivity to the basis we have chosen for constructing our conceptual superstructures.
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Defining minimal negation operators over a more conventional basis is next in order of exposition, if not necessarily in order of every reader’s reading. For what it’s worth and against the day when it may be needed, here is a definition of minimal negations in terms of and
To express the general form of in terms of familiar operations, it helps to introduce an intermediary concept.
Definition. Let the function be defined for each integer in the interval by the following equation.
Then is defined by the following equation.
We may take the boolean product or the logical conjunction to indicate the point in the space in which case the minimal negation indicates the set of points in which differ from in exactly one coordinate. This makes a discrete functional analogue of a point-omitted neighborhood in ordinary real analysis, more precisely, a point-omitted distance-one neighborhood. Viewed in that light the minimal negation operator can be recognized as a differential construction, an observation opening a very wide field.
The remainder of this discussion proceeds on the algebraic convention making the plus sign and the summation symbol both refer to addition mod 2. Unless otherwise noted, the boolean domain is interpreted for logic in such a way that and This has the following consequences.
- The operation is a function equivalent to the exclusive disjunction of and while its fiber of is the relation of inequality between and
- The operation maps the bit sequence to its parity.
The following properties of the minimal negation operators may be noted.
- The function is the same as that associated with the operation and the relation
- In contrast, is not identical to
- More generally, the function for is not identical to the boolean sum
- The inclusive disjunctions indicated for the of more than one argument may be replaced with exclusive disjunctions without affecting the meaning since the terms in disjunction are already disjoint.
- Logic Syllabus
- Boolean Domain
- Boolean Function
- Multigrade Operator
- Minimal Negation Operator
- Survey of Animated Logical Graphs