ONT Re: Differential Logic -- Series B
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DLOG. Note B2
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Example 1. A Polymorphous Concept
I start with an example that is simple enough that it will allow us to compare
the representations of propositions by venn diagrams, truth tables, and my own
favorite version of the syntax for propositional calculus all in a relatively
short space. To enliven the exercise, I borrow an example from a book with
several independent dimensions of interest, 'Topobiology' by Gerald Edelman.
One finds discussed there the notion of a "polymorphous set". Such a set
is defined in a universe of discourse whose elements can be described in
terms of a fixed number k of logical features. A "polymorphous set" is
one that can be defined in terms of sets whose elements have a fixed
number j of the k features.
As a rule in the following discussion, I will use upper case letters as names
for concepts and sets, lower case letters as names for features and functions.
The example that Edelman gives (1988, Fig. 10.5, p. 194) involves sets of
stimulus patterns that can be described in terms of the three features
"round" 'u', "doubly outlined" 'v', and "centrally dark" 'w'. We may
regard these simple features as logical propositions u, v, w : X -> B.
The target concept Q is one whose extension is a polymorphous set Q,
the subset Q of the universe X where the complex feature q : X -> B
holds true. The Q in question is defined by the requirement:
"Having at least 2 of the 3 features in the set {u, v, w}".
Taking the symbols u = "round", v = "doubly outlined", w = "centrally dark",
and using the corresponding capitals to label the circles of a venn diagram,
we get a picture of the target set Q as the shaded region in Figure 1. Using
these symbols as "sentence letters" in a truth table, let the truth function q
mean the very same thing as the expression "{u and v} or {u and w} or {v and w}".
o-----------------------------------------------------------o
| X |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | U | |
| | | |
| | | |
| | | |
| | | |
| o--o----------o o----------o--o |
| / \%%%%%%%%%%\ /%%%%%%%%%%/ \ |
| / \%%%%%%%%%%o%%%%%%%%%%/ \ |
| / \%%%%%%%%/%\%%%%%%%%/ \ |
| / \%%%%%%/%%%\%%%%%%/ \ |
| / \%%%%/%%%%%\%%%%/ \ |
| o o--o-------o--o o |
| | |%%%%%%%| | |
| | |%%%%%%%| | |
| | |%%%%%%%| | |
| | V |%%%%%%%| W | |
| | |%%%%%%%| | |
| o o%%%%%%%o o |
| \ \%%%%%/ / |
| \ \%%%/ / |
| \ \%/ / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-----------------------------------------------------------o
Figure 1. Polymorphous Set Q
In other words, the proposition q is a truth-function of the 3 logical variables u, v, w,
and it may be evaluated according to the "truth table" scheme that is shown in Table 2.
In this representation the polymorphous set Q appears in the guise of what some people
call the "pre-image" or the "fiber of truth" under the function q. More precisely,
the 3-tuples for which q 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. Polymorphous Function q
o---------------o-----------o-----------o-----------o-------o
| u v w | u & v | u & w | v & w | q |
o---------------o-----------o-----------o-----------o-------o
| | | | | |
| 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 seeing how the word "model" is used in
logic, namely, to distinguish whatever things satisfy a description.
In the venn diagram presentation, to be a model of some conceptual
description !F! is to be a point x in the corresponding region F
of the universe of discourse X.
In the truth table representation, to be a model of a logical
proposition f is to be a data-vector !x! (a row of the table)
on which a function f evaluates to true.
This manner of speaking makes sense to those who consider the ultimate meaning of
a sentence to be not the logical proposition that it denotes but its truth value
instead. From the point of view, one says that any data-vector of this type
(k-tuples of truth values) may be regarded as an "interpretation" of the
proposition with k 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.
Jon Awbrey
| Edelman, Gerald M.,
|'Topobiology: An Introduction to Molecular Embryology',
| Basic Books, New York, NY, 1988.
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