ONT Re: Prospects for Inquiry Driven Systems

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

PRO.  Note 53

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

2.3.1.  Propositions and Differences

As a first step, I have taken the problem of propositional calculus
modeling and viewed it from the standpoint of differential geometry.
In this I exploit an analogy between propositional calculus and the
calculus on differential manifolds.

In the qualitative arena "propositions" may be viewed as boolean-valued
functions.  These are associated with the "areas" or arbitrary regions
of a venn diagram, and also with the subsets of an n-dimensional cube.
Logical "interpretations", in the technical sense of boolean-valued
substitutions into propositional expressions, are coded as boolean
vectors.  They correspond to the single "cells" of a Venn diagram,
or to the points of an n-cube.  Put altogether, these linkages
form a three part analogy between the conceptual objects that
are called "propositions" in logic and the two mathematical
domains of functions and sets.  In its pivotal location,
critical function, and isosceles construction, this
analogy suggests itself as the "pons asinorum" of
the subject I think I can see developing here.
But I can't tell till I've crossed it.

Jon Awbrey

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
http://www.cs.bsu.edu/homepages/mighty/history.html
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o