Cf: Differential Propositional Calculus : 4
At:
http://inquiryintoinquiry.com/2020/02/25/differential-propositional-calculus-%e2%80%a2-4/
We gradually march toward the point where
plain old propositional calculus meets up
with differential propositional calculus.
[Note: As usual, please follow the above link for a better-formatted copy.]
Special Classes of Propositions
===============================
Before moving on, let's unpack some of the assumptions, conventions, and implications involved in the array of concepts
and notations introduced above.
A universe of discourse A+ = [a_1, ..., a_n] based on the logical features a_1, ..., a_n is a set A plus the set of all
possible functions from the space A to the boolean domain B = {0, 1}. There are 2^n elements in A, often pictured as
the cells of a venn diagram or the nodes of a hypercube. There are 2^(2^n) possible functions from A to B, accordingly
pictured as all the ways of painting the cells of a venn diagram or the nodes of a hypercube with a palette of two colors.
A logical proposition about the elements of A is either true or false of each element in A, while a function f:A->B
evaluates to 1 or 0 on each element of A. The analogy between logical propositions and boolean-valued functions is
close enough to adopt the latter as models of the former and simply refer to the functions f:A->B as propositions about
the elements of A.
The full set of propositions f : A -> B contains a number of smaller classes
deserving of special attention.
A "basic proposition" in the universe of discourse [a_1, ..., a_n] is one of the propositions in the set {a_1, ...,
a_n}. There are of course exactly n of these. Depending on the context, basic propositions may also be called
coordinate propositions or simple propositions.
Among the 2^(2^n) propositions in [a_1, ..., a_n] are several families numbering 2^n propositions each which take on
special forms with respect to the basis {a_1, ..., a_n}. Three of these families are especially prominent in the
present context, the "linear", the "positive", and the "singular" propositions. Each family is naturally parameterized
by the coordinate n-tuples in B^n and falls into n+1 ranks, with a binomial coefficient (n choose k) giving the number
of propositions having rank or weight k in their class.
Linear Propositions
https://inquiryintoinquiry.files.wordpress.com/2020/02/linear-propositions-may-be-written-as-sums.png
Positive Propositions
https://inquiryintoinquiry.files.wordpress.com/2020/02/positive-propositions-may-be-written-as-products.png
Singular Propositions
https://inquiryintoinquiry.files.wordpress.com/2020/02/singular-propositions-may-be-written-as-products.png
In each case the rank k ranges from 0 to n and counts the number of positive appearances of the coordinate propositions
a_1, ..., a_n in the resulting expression. For example, when 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), that is, not a_1 and not a_2 and
not a_3.
The basic propositions a_i : B^n -> B are both linear and positive. So these two kinds of propositions, the linear and
the positive, may be viewed as two different ways of generalizing the class of basic propositions.
Finally, it is important to note that all of the above distinctions are relative to the choice of a particular logical
basis {a_1, ..., a_n}. A singular proposition with respect to the basis {a_1, ..., a_n} will not remain singular if the
basis is extended by a number of new and independent features. Even if one keeps to the original set of pairwise
options {a_i} + {not a_i} to pick out a new basis, the sets of linear propositions and positive propositions are both
determined by the choice of basic propositions, and this whole determination is tantamount to the purely conventional
choice of a cell as origin.
Regards,
Jon
inquiry into inquiry: https://inquiryintoinquiry.com/
academia: https://independent.academia.edu/JonAwbrey
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facebook page: https://www.facebook.com/JonnyCache
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