ONT Re: Differential Logic
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DLOG. Note D5
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Special Classes of Propositions
It is important to remember that the coordinate propositions {a_i},
besides being projection maps a_i : B^n -> B, are propositions on
an equal footing with all others, even though employed as a basis
in a particular moment. This set of n propositions may sometimes
be referred to as the "basic" or "simple" propositions that found
the universe of discourse. As typical and collective notations,
we may use the forms {a_i : B^n -> B} = (B^n -i-> B) = (B^n ÷> B)
to indicate the adoption of a set of a_i as a basis for discourse.
Among the 2^2^n propositions or functions in (B^n -> B) are several
fundamental sets of 2^n propositions each that take on special forms
with respect to a given basis !A! = {a_i}. Three of these forms are
especially common, the "linear", the "positive", and the "singular"
propositions. Each set is naturally parameterized by the coordinate
vectors in B^n and falls into n+1 ranks, with a binomial coefficient
C(n, k) giving the number of propositions that have rank or weight k.
The "linear" propositions, {hom : B^n -> B} = (B^n -h-> B) = (B^n ++> B),
may be expressed as sums of the following form:
Sum_i e_i = e_1 + ... + e_n where e_i = a_i or e_i = 0.
The "positive" propositions, {pos : B^n -> B} = (B^n -p-> B) = (B^n >=> B),
may be expressed as products of the following form:
Prod_i e_i = e_1 * ... * e_n where e_i = a_i or e_i = 1.
The "singular" propositions, {x : B^n -> B} = (B^n -s-> B) = (B^n ::> B),
may be expressed as products of the following form:
Prod_i e_i = e_1 * ... * e_n where e_i = a_i or e_i = (a_i).
In each case the rank k ranges from 0 to n and counts the number of
positive appearances of coordinate propositions a_i in the resulting
expression. For example, for n = 3, the linear proposition of rank 0
is "0", the positive proposition of rank 0 is "1", and the singular
proposition of rank 0 is "(a_1)(a_2)(a_3)".
The coordinate projections or simple propositions a_i : B^n -> B are both
linear and positive. So these two kinds of propositions, the linear or the
positive, may be viewed as two different ways of generalizing the class of
simple projections. The linear and the positive propositions are generated
by taking boolean sums and products, respectively, over selected subsets of
the basic propositions in {a_i}. Therefore, each set of functions can be
parameterized by the subsets J of the basic index set I = {1, ..., n}.
Let us define A_J as the subset of A that is given by {a_i : i in J}.
Then we may comprehend the action of the linear and the positive
propositions in the following terms:
1. The linear proposition hom_J : B^n -> B evaluates each cell x of B^n
by looking at x's coefficients with respect to the features that hom_J
"likes", namely those in A_J, and then adds them up in B. Thus, hom_J (x)
computes the parity of the number of features that x has in A_J, yielding
one for odd and zero for even. Expressed in this idiom, hom_J (x) = 1
says that x seems "odd" (or "oddly true") to A_J, whereas hom_J (x) = 0
says that x seems "even" (or "evenly true") to A_J, so long as we recall
that "zero times" is evenly often, too.
2. The positive proposition pos_J : B^n -> B evaluates each cell x of B^n
by looking at x's coefficients with regard to the features that pos_J
"likes", namely those in A_J, and then takes their product in B. Thus,
pos_J (x) assesses the unanimity of the multitude of features that x has
in A_J, yielding one for all and aught for else. In these consensual or
contractual terms, pos_J (x) = 1 means that x is "AOK" or congruent with
all of the conditions of A_J, while pos_J (x) = 0 means that x defaults
or dissents from some condition of A_J.
Jon Awbrey
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