ONT Re: Differential Logic
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
DLOG. Note D3
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Qualitative Logic and Quantitative Analogy
| Logical, however, is used in a third sense, which is at once more
| vital and more practical; to denote, namely, the systematic care,
| negative and positive, taken to safeguard reflection so that it
| may yield the best results under the given conditions.
|
| John Dewey, 'How We Think', [Dew, 56]
These concepts and notations can now be explained in greater detail.
In order to begin as simply as possible, I distinguish two levels of
analysis and set out initially on the easier path. On the first level
of analysis, I take spaces like B, B^n, and (B^n -> B) at face value and
treat them as the primary objects of interest. On the second level of
analysis, I use these spaces as coordinate charts for talking about
points and functions in more fundamental spaces.
A pair of spaces, of types B^n and (B^n -> B), give typical expression
to everything that we commonly associate with the ordinary picture of
a venn diagram. The dimension, n, counts the number of "circles" or
simple closed curves that are inscribed in the universe of discourse,
corresponding to its relevant logical features or basic propositions.
Elements of type B^n correspond to what are often called propositional
"interpretations" in logic, that is, the different assignments of truth
values to sentence letters. Relative to a given universe of discourse,
these interpretations are visualized as its "cells", in other words,
the smallest enclosed areas or undivided regions of the venn diagram.
The functions f : B^n -> B correspond to the different ways of shading
the venn diagram to indicate arbitrary propositions, regions, or sets.
Regions included under a shading indicate the "models", and regions
excluded represent the "non-models" of a proposition. To recognize
and formalize the natural cohesion of these two layers of concepts
into a single universe of discourse, I introduce the type notations
[B^n] = B^n +-> B to stand for the pair of types (B^n, (B^n -> B)).
The resulting "stereotype" serves to frame the universe of discourse
as a unified categorical object, and makes it subject to prescribed
sets of evaluations and transformations (categorical morphisms or
"arrows") that affect the universe of discourse as an integrated
whole.
Most of the time we can serve the algebraic, geometric, and logical interests
of our study without worrying about their occasional conflicts and incidental
divergences. The conventions and definitions already set down will continue
to cover most of the algebraic and functional aspects of our discussion, but
to handle the logical and qualitative aspects we will need to add a few more.
In general, abstract sets may be denoted by gothic, greek, or script capital
variants of A, B, C, and so on, with elements denoted by a corresponding set
of subscripted letters in plain lower case, for example, !A! = {a_i}. Most
of the time, a set such as !A! = {a_i} will be employed as the "alphabet" of
a formal language. These alphabet letters serve to name the logical features
(properties or propositions) that generate a particular universe of discourse.
When we want to discuss the particular features of a universe of discourse,
beyond the abstract designation of a type like (B^n +-> B), then we may use
the following notations. If !A! = {a_1, ..., a_n} is an alphabet of logical
features, then A = <|!A!|> = <|a_1, ..., a_n|> is the set of interpretations,
A^ = (A -> B) is the set of propositions, and A% = [!A!] = [a_1, ..., a_n] is
the combination of these interpretations and propositions into the universe of
discourse that is based on the features {a_1, ..., a_n}.
As always, especially in concrete examples, these rules may be dropped whenever
necessary, reverting to a free assortment of feature labels. However, when we
need to talk about the logical aspects of a space that is already named as a
vector space, it will be necessary to make special provisions. At any rate,
these elaborations can be deferred until actually needed.
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o