Changes

===Note 2===

===Note 2===

'''Example 1.  A Polymorphous Concept'''

Example 1.  A Polymorphous Concept

I start with an example that is simple enough that it will allow us to compare

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 <math>k\!</math> of logical features.  A "polymorphous set" is one that can be defined in terms of sets whose elements have a fixed number <math>j\!</math> of the <math>k\!</math> features.

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

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.

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

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" <math>u\!</math>, "doubly outlined" <math>v\!</math>, and "centrally dark" <math>w\!</math>.  We may regard these simple features as logical propositions <math>u, v, w : X \to \mathbb{B}.</math>  The target concept <math>\mathcal{Q}</math> is one whose extension is a polymorphous set <math>Q\!</math>, the subset <math>Q\!</math> of the universe <math>X\!</math> where the complex feature <math>q : X \to \mathbb{B}</math> holds true.  The <math>Q\!</math> in question is defined by the requirement: "Having at least 2 of the 3 features in the set <math>\{ u, v, w \}\!</math>".

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",

Taking the symbols u = "round", v = "doubly outlined", w = "centrally dark",

and using the corresponding capitals to label the circles of a venn diagram,

and using the corresponding capitals to label the circles of a venn diagram,