Changes

Because I plan this time around a somewhat leisurely excursion through the primordial wilds of logics that were so intrepidly explored by C.S. Peirce and again in recent times revisited by George Spencer Brown, let me just give a few extra pointers to those who wish to run on ahead of this torturous tortoise pace:

Because I plan this time around a somewhat leisurely excursion through the primordial wilds of logics that were so intrepidly explored by C.S. Peirce and again in recent times revisited by George Spencer Brown, let me just give a few extra pointers to those who wish to run on ahead of this torturous tortoise pace:

:* Jon Awbrey, "Propositional Equation Reasoning Systems (PERS)", [http://forum.wolframscience.com/printthread.php?threadid=297&perpage=35 Online].

:* Jon Awbrey, [http://forum.wolframscience.com/printthread.php?threadid=297&perpage=35 Propositional Equation Reasoning Systems].

:* Lou Kauffman, "Box Algebra, Boundary Mathematics, Logic, and Laws of Form", [http://www.math.uic.edu/~kauffman/Arithmetic.htm Online].

:* Lou Kauffman, [http://www.math.uic.edu/~kauffman/Arithmetic.htm Box Algebra, Boundary Mathematics, Logic, and Laws of Form].

Two paces back I used the word ''category'' in a way that will turn out to be not too remote a cousin of its present day mathematical bearing, but also in way that's not unrelated to Peirce's theory of categories.

Two paces back I used the word ''category'' in a way that will turn out to be not too remote a cousin of its present day mathematical bearing, but also in way that's not unrelated to Peirce's theory of categories.

1. &

1. &

\operatorname{En}_\text{syn} = \operatorname{Ex}_\text{syn} = \operatorname{E}_\text{syn} : Y \to Y_0

\operatorname{En}_\text{syn} = \operatorname{Ex}_\text{syn} = \operatorname{E}_\text{syn} : Y \to Y_0

\\[6pt]

\\[10pt]

2. &

2. &

\operatorname{En}_\text{sem}, \operatorname{Ex}_\text{sem} : Y_0 \to X

\operatorname{En}_\text{sem}, \operatorname{Ex}_\text{sem} : Y_0 \to X

Some will recall the many animadversions that we had on this topic, starting here:

Some will recall the many animadversions that we had on this topic, starting here:

* PS.  Pure Symbols

:* Pure Symbols

** http://stderr.org/pipermail/inquiry/2005-March/thread.html#2465

:: http://stderr.org/pipermail/inquiry/2005-March/thread.html#2465

** http://stderr.org/pipermail/inquiry/2005-April/thread.html#2517

:: http://stderr.org/pipermail/inquiry/2005-April/thread.html#2517

* PS.  Pure Symbols : Discussion

:* Pure Symbols : Discussion

** http://stderr.org/pipermail/inquiry/2005-March/thread.html#2466

:: http://stderr.org/pipermail/inquiry/2005-March/thread.html#2466

** http://stderr.org/pipermail/inquiry/2005-April/thread.html#2514

:: http://stderr.org/pipermail/inquiry/2005-April/thread.html#2514

** http://stderr.org/pipermail/inquiry/2005-May/thread.html#2654

:: http://stderr.org/pipermail/inquiry/2005-May/thread.html#2654

And some will find an ethical principle in this freedom of interpretation.  The act of interpretation bears within it an inalienable degree of freedom.  In consequence of this truth, as far as the activity of interpretation goes, freedom and responsibility are the very same thing.  We cannot blame objects for what we say or what we think.  We cannot blame symbols for what we do.  We cannot escape our response ability.  We cannot escape our freedom.

And some will find an ethical principle in this freedom of interpretation.  The act of interpretation bears within it an inalienable degree of freedom.  In consequence of this truth, as far as the activity of interpretation goes, freedom and responsibility are the very same thing.  We cannot blame objects for what we say or what we think.  We cannot blame symbols for what we do.  We cannot escape our response ability.  We cannot escape our freedom.

Observations like that, made about an arithmetic of any variety, germinated by their summarizations, are the root of all algebra.

Observations like that, made about an arithmetic of any variety, germinated by their summarizations, are the root of all algebra.

Speaking of algebra, and having seen one example of an algebraic law, we might as well introduce the axioms of the ''primary algebra'', once again deriving their substance and their name from Charles S. Peirce and G. Spencer Brown, respectively.

Speaking of algebra, and having seen one example of an algebraic law, we might as well introduce the axioms of the ''primary algebra'', once again deriving their substance and their name from Charles Sanders Peirce and George Spencer Brown, respectively.

{| align="center" cellpadding="10"

{| align="center" cellpadding="10"

A ''sort'' of signs is more formally known as an ''equivalence class'' (EC).  There are in general many sorts of sorts of signs that we might wish to consider in this inquiry, but let's begin with the sort of signs all of whose members denote the same object as their referent, a sort of signs to be henceforth referred to as a ''referential equivalence class'' (REC).

A ''sort'' of signs is more formally known as an ''equivalence class'' (EC).  There are in general many sorts of sorts of signs that we might wish to consider in this inquiry, but let's begin with the sort of signs all of whose members denote the same object as their referent, a sort of signs to be henceforth referred to as a ''referential equivalence class'' (REC).

* Cf: FOLG 5.  http://stderr.org/pipermail/inquiry/2005-October/003113.html

:* [http://stderr.org/pipermail/inquiry/2005-October/thread.html#3104 FOLG]

* In: FOLG.    http://stderr.org/pipermail/inquiry/2005-October/thread.html#3104

:* [http://stderr.org/pipermail/inquiry/2005-October/003113.html FOLG 5]

Toward the outset of this excursion, I mentioned the distinction between a ''pointwise-restricted iconic map'' or a ''pointedly rigid iconic map'' (PRIM) and a ''system-wide iconic map'' (SWIM).  The time has come to make use of that mention.

Toward the outset of this excursion, I mentioned the distinction between a ''pointwise-restricted iconic map'' or a ''pointedly rigid iconic map'' (PRIM) and a ''system-wide iconic map'' (SWIM).  The time has come to make use of that mention.

We are once again concerned with ''categories of structured items'' (COSI's) and the categories of mappings between them, indeed, the two ideas are all but inseparable, there being many good reasons to consider the very notion of structure to be most clearly defined in terms of the brands of "arrows", maps, or morphisms between items that are admitted to the category in view.

We are once again concerned with ''categories of structured items'' (COSIs) and the categories of mappings between them, indeed, the two ideas are all but inseparable, there being many good reasons to consider the very notion of structure to be most clearly defined in terms of the brands of "arrows", maps, or morphisms between items that are admitted to the category in view.

At the level of the ''primary arithmetic'' (PAR), we have a set-up like this:

At the level of the ''primary arithmetic'' (PAR), we have a set-up like this: