Changes

===Commentary Note 11.11===

===Commentary Note 11.11===

The preceding exercises were intended to beef-up our "functional" literacy skills to the point where we can read our functional alphabets backwards and forwards and to ferret out the local functionalites that may be immanent in relative terms no matter where they locate themselves within the domains of relations. I am hopeful that these skills will serve us in good stead as we work to build a catwalk from Peirce's platform to contemporary scenes on the logic of relatives, and back again.

The preceding exercises were intended to beef-up our

functional literacy skills to the point where we can

read our functional alphabets backwards and forwards

and to ferret out the local functionalites that may

be immanent in relative terms no matter where they

locate themselves within the domains of relations.

I am hopeful that these skills will serve us in

good stead as we work to build a catwalk from

Peirce's platform to contemporary scenes on

the logic of relatives, and back again.

By way of extending a few very tentative plancks,

By way of extending a few very tentative plancks, let us experiment with the following definitions:

let us experiment with the following definitions:

1.  A relative term 'p' and the corresponding relation P c X x Y are both

When a relation happens to be a function, it may be excusable to use the same name for it in both applications, writing out explicit type markers like ''P'' : ''X'' × ''Y'', ''P'' : ''X'' → ''Y'', ''P'' : ''X'' ← ''Y'', as the case may be, when and if it serves to clarify matters.

    called "functional on relates" if and only if P is a function at X,

    in symbols, P : X -> Y.

2.  A relative term 'p' and the corresponding relation P c X x Y are both

From this current, perhaps transient, perspective, it appears that our next task is to examine how the known properties of relations are modified when an aspect of functionality is spied in the mix.

    called "functional on correlates" if and only if P is function at Y,

    in symbols, P : X <- Y.

When a relation happens to be a function, it may be excusable

Let us then return to our various ways of looking at relational composition, and see what changes and what stays the same when the relations in question happen to be functions of various different kinds at some of their domains.

to use the same name for it in both applications, writing out

as the case may be, when and if it serves to clarify matters.

Here is one generic picture of relational composition, cast in a style that hews pretty close to the line of potentials inherent in Peirce's syntax of this period.

 

 

Here is one generic picture of relational composition,

cast in a style that hews pretty close to the line of

potentials inherent in Peirce's syntax of this period.

o-----------------------------------------------------------o

o-----------------------------------------------------------o

|                                                          |

|                                                          |

o-----------------------------------------------------------o

o-----------------------------------------------------------o

Figure 16.  Anything that is a 'p' of a 'q' of Anything

Figure 16.  Anything that is a 'p' of a 'q' of Anything

From this we extract the "hypergraph picture" of relational composition:

From this we extract the "hypergraph picture" of relational composition:

o-----------------------------------------------------------o

o-----------------------------------------------------------o

|                                                          |

|                                                          |

o-----------------------------------------------------------o

o-----------------------------------------------------------o

Figure 17.  Relational Composition P o Q

Figure 17.  Relational Composition P o Q

All of the relevant information of these Figures can be compressed

All of the relevant information of these Figures can be compressed into the form of a "spreadsheet", or constraint satisfaction table:

into the form of a "spreadsheet", or constraint satisfaction table:

Table 18.  Relational Composition P o Q

Table 18.  Relational Composition P o Q

o---------o---------o---------o---------o

o---------o---------o---------o---------o

|  P o Q  #    X    |        |    Z    |

|  P o Q  #    X    |        |    Z    |

o---------o---------o---------o---------o

o---------o---------o---------o---------o

So the following presents itself as a reasonable plan of study:

So the following presents itself as a reasonable plan of study: Let's see how much easy mileage we can get in our exploration of functions by adopting the above templates as a paradigm.

Let's see how much easy mileage we can get in our exploration

of functions by adopting the above templates as a paradigm.

===Commentary Note 11.12===

===Commentary Note 11.12===