ONT Re: Differential Logic & Dynamic Systems
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Dif Log Dyn Sys Group:
| Notes On Approach And On Notation
|
| To make the present discussion more self-contained, I will need to insert
| my discussion of an Example from the psych research literature of the times,
| what was known in the Cognitive Science, Immune System Models, and Neural Net
| circles as a "polymorphous set".
|
| In some parts of this old discussion, I find myself occasionally guilty
| of engaging in the rather sloppy practice of using the same symbol for
| any of the correlated notions of a concept, a set, or a truth function.
| Please be forgiving of this less sad and less wise person who did this.
| I have tried to go back through the text, marking the concept symbols
| in bold letters, as #A#, the proposition names or truth-function names
| in plain letters, as A, and the set names in primed symbols, as A', but
| I may have missed a few, or sometimes have left off the extra markings
| in Figures and in Tables.
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Example 1. A Polymorphous Set
We start with an example simple enough to compare
the representations by venn diagrams, truth tables,
and our version of the syntax for propositional calculus
all in a relatively short space. To enliven the exercise,
we borrow an example from a book with several independent
dimensions of interest, 'Topobiology' by Gerald M. Edelman.
We find discussed there the notion of a polymorphous set.
This is defined relative to a universe of discourse whose
elements can be characterized by a number N of features.
In such a universe, a polymorphous set is one that can
be characterized in terms of the sets whose elements
have exactly M of the N features.
The example Edelman gives (Ref, Fig. 10.5, p. 194) concerns
sets of stimulus patterns described in terms of the features
"round" (A), "doubly outlined" (B), and "centrally dark" (C).
The target concept #T# is one whose extension is a particular
polymorphous set, namely, the extension of the description:
"Having at least 2 of the 3 features in the set {A, B, C}".
Taking the symbols A = "round", B = "doubly outlined", C = "centrally dark",
and using them to label the circles of a venn diagram, we get a picture of
the target set T' as the shaded region in Figure 1. Using these symbols as
sentence letters of a truth table, let the truth function T mean the very
same thing as the expression "[A and B] or [B and C] or [C and A]".
o-----------------------------------o
| U |
| o-----------------------o |
| | A | |
| | o o | |
| | /%\ /%\ | |
| | /%%%\ /%%%\ | |
| | /%%%%%.%%%%%\ | |
| | /%%%%%/%\%%%%%\ | |
| | /%%%%%/%T%\%%%%%\ | |
| | /%%%%%/%%%%%\%%%%%\ | |
| o-----------------------o |
| / /%%%%%%%%%\ \ |
| o o%%%%%%%%%%%o o |
| \ \%%%%%%%%%/ / |
| \ B \%%%%%%%/ C / |
| \ \%%%%%/ / |
| \ \%%%/ / |
| \ \%/ / |
| \ . / |
| \ / \ / |
| \ / \ / |
| o o |
| |
o-----------------------------------o
Figure 1. A Polymorphous Set
In other words, the proposition T is a truth-function of the
three logical variables A, B, C, and may be evaluated according
to the "truth table" scheme that is shown in Table 2. In this
representation the polymorphous set T' appears in the guise of
what some people call the "pre-image" or the "fiber of truth"
under the function T. More precisely, the three-tuples for
which T evaluates to true are in an obvious correspondence
with the shaded cells of the venn diagram. No matter how
we get down to the level of actual information, it's all
pretty much the same stuff.
Table 2. A Polymorphous Function
o-------------o-----------o-----------o-----------o-------o
| A B C | A & B | A & C | B & C | T |
|-------------|-----------|-----------|-----------|-------|
| | | | | |
| 0 0 0 | 0 | 0 | 0 | 0 |
| | | | | |
| 0 0 1 | 0 | 0 | 0 | 0 |
| | | | | |
| 0 1 0 | 0 | 0 | 0 | 0 |
| | | | | |
| 0 1 1 | 0 | 0 | 1 | 1 |
| | | | | |
| 1 0 0 | 0 | 0 | 0 | 0 |
| | | | | |
| 1 0 1 | 0 | 1 | 0 | 1 |
| | | | | |
| 1 1 0 | 1 | 0 | 0 | 1 |
| | | | | |
| 1 1 1 | 1 | 1 | 1 | 1 |
| | | | | |
o-------------o-----------o-----------o-----------o-------o
With the pictures of the venn diagram and the truth table
before us, we have come to the verge of sensing how the
word "model" is used in logic, namely, to distinguish
whatever thing it may be that satisfies a description.
In the venn diagram, to be a thing of some description is to
be a point #x# of some region A' in the universe of discourse.
In the truth table, to be a model of a proposition is to
be a data-vector #x# (an n-tuple row of the table) on which
a function A evaluates to true.
This makes sense to those who consider the meaning of a sentence letter
to be not the sentence but its truth value instead. In this view, we may
say that any data-vector of this type (an n-tuple of truth values) is an
"interpretation" of the proposition with n variables. An interpretation
that yields a value of true is then called a "model".
For the most threadbare kind of logical system that we find residing
in Propositional Calculus, this notion of model is almost too simple
to deserve the name, yet it can be of service to fashion some form
of continuity between the simple and the complex.
Reference:
| Edelman, Gerald M.,
|'Topobiology: An Introduction to Molecular Embryology',
| Basic Books, New York, NY, 1988.
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