Changes

Still, one of the aims of formalizing what acts of reasoning that we can is to draw them into an arena where we can examine them more carefully, perhaps to get better at their performance than we can unreflectively, and thus to live, to formalize again another day.  Formalization is not the be-all end-all of human life, not by a long shot, but it has its uses on that behalf.

Still, one of the aims of formalizing what acts of reasoning that we can is to draw them into an arena where we can examine them more carefully, perhaps to get better at their performance than we can unreflectively, and thus to live, to formalize again another day.  Formalization is not the be-all end-all of human life, not by a long shot, but it has its uses on that behalf.

This looks like a good place to pause and take stock.  The question arises:  What's really going on here?  There's all these signs, but what's the object?

This looks like a good place to pause and take stock.  The question arises:  What's really going on here?  There's all these signs, but what's the object? One object worth the candle is simply to study a non-trivial example of a syntactic system, simple in design but not entirely a toy, just to see how similar systems tick. More than that, we would like to understand how sign systems come to exist or come to be placed in relation to object systems, especially those types of object systems that give us compelling cause or independent reason to focus thought on. What is the utility of setting up sets of strings and sets of graphs, and sorting them according to their ''semiotic equivalence class'' (SEC) based on this or that abstract notion of transformational equivalence?

 

One object worth the candle is simply to study a non-trivial example of a syntactic system, simple in design but not entirely a toy, just to see how similar systems tick.

 

More than that, we would like to understand how sign systems come to exist or come to be placed in relation to object systems, especially those types of object systems that give us compelling cause or independent reason to focus thought on.

 

What is the utility of setting up sets of strings and sets of graphs, and sorting them according to their ''semiotic equivalence class'' (SEC) based on this or that abstract notion of transformational equivalence?

Good questions.

Good questions.

I can only begin to tackle these questions in the present frame of work, and I can't hope to answer them in anything like a satisfactory fashion.  Still, it will serve to guide the work if we keep them in mind as we go.

I can only begin to tackle these questions in the present frame of work, and I can't hope to answer them in anything like a satisfactory fashion.  Still, it will serve to guide the work if we keep them in mind as we go.

If you will excuse the bits of autobio-graphical anecdotage, it will help me to reconstruct the steps that I actually took in my thinking as I worked through these problems about logical graphs late in the last millennium.

If you will excuse the bits of autobio-graphical anecdotage, it will help me to reconstruct the steps that I actually took in my thinking as I worked through these problems about logical graphs late in the last millennium. By 1980 my logical graphs were becoming too large and complex to keep within the bounds of 2-dimensional manifolds of paper, and so I started to think once again, with extreme reluctance — given earlier traumatic experiences trying to use Fortran and a CDC 3600 mainframe to do my chem and physics lab work in an era when "turn-around time" was counted in days not microsecs — of representing logical graphs and logical transformations in the computer medium. By a bit of serendipity that still amazes me, it happened that my earlier work on Peirce's use of operator variables, that led in its turn to my discovery of the cactus language, also turned out to provide workable solutions for several problems that arose in the process of trying to find efficient implementations for logical graphs and their logical transformations.

 

By 1980 my logical graphs were becoming too large and complex to keep within the bounds of 2-dimensional manifolds of paper, and so I started to think once again, with extreme reluctance — given earlier traumatic experiences trying to use Fortran and a CDC 3600 mainframe to do my chem and physics lab work in an era when "turn-around time" was counted in days not microsecs — of representing logical graphs and logical transformations in the computer medium.

 

By a bit of serendipity that still amazes me, it happened that my earlier work on Peirce's use of operator variables, that led in its turn to my discovery of the cactus language, also turned out to provide workable solutions for several problems that arose in the process of trying to find efficient implementations for logical graphs and their logical transformations.

For a salient example, consider the existential graph for "p ⇔ q", to wit:

For example, consider the existential graph for <math>p \Leftrightarrow q</math> that is shown below:

{| align="center" style="text-align:center; width:90%"

{| align="center" style="text-align:center; width:90%"

|}

|}

This can be read as "not p without q and not q without p", or what's the same, "[p &rArr; q] and [q &rArr; p]", or "p &hArr; q".

This can be read as "not <math>p\!</math> without <math>q\!</math> and not <math>q\!</math> without <math>p\!</math>", or what's the same thing, <math>(p \Rightarrow q) ~\operatorname{and}~ (q \Rightarrow p).</math>

Graphing the topological dual form, one obtains this rooted tree:

Graphing the topological dual form, one obtains this rooted tree: