Cf: Minimal Negation Operators • 4
https://inquiryintoinquiry.com/2017/09/01/minimal-negation-operators-4/
All,
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.
⁂ ⁂ ⁂
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 ¬.
Formal Definition
=================
To express the general form of νₙ in terms of familiar operations,
it helps to introduce an intermediary concept.
Definition. Let the function ¬ₘ : Bⁿ → B be defined for each
integer m in the interval [1, n] by the following equation.
• ¬ₘ(x₁, …, xₘ, …, xₙ) = x₁ ∧ … ∧ xₘ₋₁ ∧ ¬xₘ ∧ xₘ₊₁ ∧ … ∧ xₙ.
Then νₙ : Bⁿ → B is defined by the following equation.
• νₙ(x₁, …, xₙ) = ¬₁(x₁, …, xₙ) ∨ … ∨ ¬ₘ(x₁, …, xₙ) ∨ … ∨ ¬ₙ(x₁, …, xₙ).
We may take the boolean product x₁ ∙ … ∙ xₙ or the logical conjunction
x₁ ∧ … ∧ xₙ to indicate the point x = (x₁, …, xₙ) in the space Bⁿ, in
which case the minimal negation νₙ(x₁, …, xₙ) indicates the set of points
in
Bⁿ which differ from x in exactly one coordinate. This makes νₙ(x₁, …, xₙ)
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 B = {0, 1}
is interpreted for logic in such a way that 0 = false and 1 = true.
This has the following consequences.
• The operation x + y is a function equivalent to the exclusive
disjunction of
x and y, while its fiber of 1 is the relation of inequality between x
and y.
• The operation ∑ₘ xₘ = x₁ + … + xₙ maps the bit sequence (x₁, …, xₙ)
to its parity.
The following properties of the minimal negation operators
νₙ : Bⁿ → B may be noted.
• The function ν₂(x, y) is the same as that associated with
the operation x + y and the relation x ≠ y.
• In contrast, ν₃(x, y, z) is not identical to x + y + z.
• More generally, the function νₙ(x₁, …, xₙ) for k > 2
is not identical to the boolean sum ∑ₘ xₘ = x₁ + … + xₙ.
• The inclusive disjunctions indicated for the νₙ of more than
one argument may be replaced with exclusive disjunctions without
affecting the meaning since the terms in disjunction are already
disjoint.
Regards,
Jon
