Re: Peirce List • Robert Marty
With a few choice exceptions I have always found Peirce’s earlier writings on categories, relations, and semiotics to be more clear, exact, and fruitful in practice than his last attempts to explain himself without the requisite logical and mathematical supports.
Still, I like Robert Marty’s “podium” picture of the universal categories, comprehend it all or not, and I found myself once using a similar picture to explain the relationships among the big 3 normative sciences of aesthetics, ethics, and logic. I called this “The Pragmatic Cosmos”, using cosmos in the sense of a global order. It looks like most of this stuff has fallen off the live web but here’s a few links I found.
- The Pragmatic Cosmos (Oct 2003)
- Inquiry Oriented Systems (Feb 2004) • (1) … (8)
- The Pragmatic Cosmos (Mar 2012)
I’ll copy and format pieces of this to my blog as I get time.
Resources
cc: Cybernetics • Ontolog Forum • Peirce List • Structural Modeling • Systems Science
and abstract type 
Our inquiry into the differential aspects of logical conjunction will pay dividends as we study the actions of 
and 
on this family of forms.
![\begin{array}{cccl} \text{Let} & X & = & X_1 \times \ldots \times X_k. \\[6pt] \text{Let} & \mathrm{d}X & = & \mathrm{d}X_1 \times \ldots \times \mathrm{d}X_k. \\[6pt] \text{Then} & \mathrm{E}X & = & X \times \mathrm{d}X \\[6pt] & & = & X_1 \times \ldots \times X_k ~\times~ \mathrm{d}X_1 \times \ldots \times \mathrm{d}X_k \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bcccl%7D++%5Ctext%7BLet%7D+%26+X+%26+%3D+%26+X_1+%5Ctimes+%5Cldots+%5Ctimes+X_k.++%5C%5C%5B6pt%5D++%5Ctext%7BLet%7D+%26+%5Cmathrm%7Bd%7DX+%26+%3D+%26+%5Cmathrm%7Bd%7DX_1+%5Ctimes+%5Cldots+%5Ctimes+%5Cmathrm%7Bd%7DX_k.++%5C%5C%5B6pt%5D++%5Ctext%7BThen%7D+%26+%5Cmathrm%7BE%7DX+%26+%3D+%26+X+%5Ctimes+%5Cmathrm%7Bd%7DX++%5C%5C%5B6pt%5D++%26+%26+%3D+%26+X_1+%5Ctimes+%5Cldots+%5Ctimes+X_k+%7E%5Ctimes%7E+%5Cmathrm%7Bd%7DX_1+%5Ctimes+%5Cldots+%5Ctimes+%5Cmathrm%7Bd%7DX_k++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=0&c=20201002)
the (first order) enlargement of 
is the proposition 
defined by the following equation.
are boolean variables of the same type as the ordinary variables 
Although it is conventional to distinguish the (first order) differential variables with the operational prefix 
this way of notating differential variables is entirely optional. It is their existence in particular relations to the initial variables, not their names, which defines them as differential variables.
the enlargement 
is formulated as follows.
To that end, the value of 
at each 
may be computed in graphical fashion as shown below.
is drawn as a loop at the point 
![\begin{array}{rcccccc} f & = & p & \cdot & q \\[4pt] \mathrm{E}f & = & p & \cdot & q & \cdot & \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{(} \mathrm{d}p \texttt{)} \texttt{~} \mathrm{d}q \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \mathrm{d}p \texttt{~} \texttt{(} \mathrm{d}q \texttt{)} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \mathrm{d}p \texttt{~~} \mathrm{d}q \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Brcccccc%7D++f+%26+%3D+%26+p++%26+%5Ccdot+%26+q++%5C%5C%5B4pt%5D++%5Cmathrm%7BE%7Df+%26+%3D+%26+p++%26+%5Ccdot+%26++q++%26+%5Ccdot+%26++%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%29%28%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%29%7D++%5C%5C%5B4pt%5D++%26+%2B+%26++p++%26+%5Ccdot+%26+%5Ctexttt%7B%28%7D+q+%5Ctexttt%7B%29%7D++%26+%5Ccdot+%26++%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%29%7D+%5Ctexttt%7B%7E%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%7E%7D++%5C%5C%5B4pt%5D++%26+%2B+%26++%5Ctexttt%7B%28%7D+p+%5Ctexttt%7B%29%7D+%26+%5Ccdot+%26++q++%26+%5Ccdot+%26++%5Ctexttt%7B%7E%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%7E%7D+%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%29%7D++%5C%5C%5B4pt%5D++%26+%2B+%26++%5Ctexttt%7B%28%7D+p+%5Ctexttt%7B%29%7D+%26+%5Ccdot+%26+%5Ctexttt%7B%28%7D+q+%5Ctexttt%7B%29%7D++%26+%5Ccdot+%26+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%7E%7E%7D+%5Cmathrm%7Bd%7Dq++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=0&c=20201002)
of the proposition 
at the point 
where 
and 
the four propositions 
indicating the “cells”, or the smallest distinguished regions of the universe, are called singular propositions. These serve as an alternative notation for naming the points 
respectively.
so long as we know the frame of reference in force.
indicated by the difference map 
namely, the following six points or singular propositions.
should be clear enough from a survey of these six points — they tell you what you have to do from each point of 
in order to change the value borne by 
that is, the move you have to make in order to reach a point where the value of the proposition 
is different from what it is where you started.
as illustrated by the example 
and 
The resulting difference map 
Abstracting from that picture, the difference map 
and whose arrows are labeled with the elements of 
as shown in the following Figure.
![\begin{array}{rcccccc} f & = & p & \cdot & q \\[4pt] \mathrm{D}f & = & p & \cdot & q & \cdot & \texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Brcccccc%7D++f+%26+%3D+%26+p+%26+%5Ccdot+%26+q++%5C%5C%5B4pt%5D++%5Cmathrm%7BD%7Df+%26+%3D+%26++p+%26+%5Ccdot+%26+q+%26+%5Ccdot+%26++%5Ctexttt%7B%28%28%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%29%28%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%29%29%7D++%5C%5C%5B4pt%5D++%26+%2B+%26++p+%26+%5Ccdot+%26+%5Ctexttt%7B%28%7D+q+%5Ctexttt%7B%29%7D+%26+%5Ccdot+%26++%5Ctexttt%7B%7E%28%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%29%7E%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%7E%7E%7D++%5C%5C%5B4pt%5D++%26+%2B+%26++%5Ctexttt%7B%28%7D+p+%5Ctexttt%7B%29%7D+%26+%5Ccdot+%26+q+%26+%5Ccdot+%26++%5Ctexttt%7B%7E%7E%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%7E%28%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%29%7E%7D++%5C%5C%5B4pt%5D++%26+%2B+%26++%5Ctexttt%7B%28%7D+p+%5Ctexttt%7B%29%7D+%26+%5Ccdot+%26+%5Ctexttt%7B%28%7D+q+%5Ctexttt%7B%29%7D+%26+%5Ccdot+%26++%5Ctexttt%7B%7E%7E%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%7E%7E%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%7E%7E%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=0&c=20201002)
or 
The concrete type takes into account the qualitative dimensions or “units” of the case, which can be explained as follows.
be the set of values 
be the set of values 
as functions of the concrete type 
is a function which takes one function 
written here as 
is constructed according to the following specifications.
![\begin{array}{rcc} X & = & P \times Q \\[4pt] \mathrm{d}X & = & \mathrm{d}P \times \mathrm{d}Q \\[4pt] \mathrm{d}P & = & \{ \texttt{(} \mathrm{d}p \texttt{)}, ~ \mathrm{d}p \} \\[4pt] \mathrm{d}Q & = & \{ \texttt{(} \mathrm{d}q \texttt{)}, ~ \mathrm{d}q \} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Brcc%7D++X+%26+%3D+%26+P+%5Ctimes+Q++%5C%5C%5B4pt%5D++%5Cmathrm%7Bd%7DX+%26+%3D+%26+%5Cmathrm%7Bd%7DP+%5Ctimes+%5Cmathrm%7Bd%7DQ++%5C%5C%5B4pt%5D++%5Cmathrm%7Bd%7DP+%26+%3D+%26+%5C%7B+%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dp+%5Ctexttt%7B%29%7D%2C+%7E+%5Cmathrm%7Bd%7Dp+%5C%7D++%5C%5C%5B4pt%5D++%5Cmathrm%7Bd%7DQ+%26+%3D+%26+%5C%7B+%5Ctexttt%7B%28%7D+%5Cmathrm%7Bd%7Dq+%5Ctexttt%7B%29%7D%2C+%7E+%5Cmathrm%7Bd%7Dq+%5C%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=0&c=20201002)
means “change 
means “change 
”.
requires four logical dimensions, 
but it is possible to project a suggestion of what the differential features 
by drawing arrows crossing the boundaries of the basic circles in the venn diagram for 
reading an arrow as 
in either direction and reading an arrow as 
in either direction, as indicated in the following figure.
says the same thing as 
in other words, there is no change in 
or, written out in full:
in terms of coordinates, at the place where 
or 
and we arrived at the locally applicable law which may be stated and illustrated as follows.
into the following exclusive disjunction.
graphed as two letters attached to a root node, as shown below.
.
may be taken as a boolean function 
where 
is read in such a way that 
means 
and 
means 
and its corresponding graph arise by replacing 
and 
in the boolean product or logical conjunction 
plus and minus are identical operations. This will make the relation between the differential and the integral parts of the appropriate calculus slightly stranger than usual, but we will get into that later.
as shown below.
Inquiry Into Inquiry 