Directory:Jon Awbrey/Papers/Futures Of Logical Graphs - Revision history

Jon Awbrey: /* Categories of structured individuals */ waybak links

2017-08-06T12:41:01Z

Categories of structured individuals: waybak links

Jon Awbrey: /* Categories of structured individuals */ waybak links

2017-08-06T12:16:15Z

Categories of structured individuals: waybak links

Jon Awbrey: sub "File" for "Image" + use "Rooted Node Big" & "Rooted Edge Big"

2017-08-04T18:18:44Z

sub "File" for "Image" + use "Rooted Node Big" & "Rooted Edge Big"

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Jon Awbrey: /* Categories of structured individuals */ use "Rooted Node Big" & "Rooted Edge Big"

2017-08-04T18:08:52Z

Categories of structured individuals: use "Rooted Node Big" & "Rooted Edge Big"

Jon Awbrey: sub ~\! for \!

2017-08-04T18:00:19Z

sub ~\! for \!

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Jon Awbrey: sub \mathrm for \operatorname

2017-08-04T17:45:20Z

sub \mathrm for \operatorname

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Jon Awbrey: /* Categories of structured individuals */ formats

2017-08-04T17:20:40Z

Categories of structured individuals: formats

Jon Awbrey: /* Introduction */ workaround for MathJax's improper handing of “~” in \texttt{...} environments

2017-08-04T14:32:47Z

Introduction: workaround for MathJax's improper handing of “~” in \texttt{...} environments

Jon Awbrey: /* Introduction */ spacing ~\!

2017-08-04T13:02:47Z

Introduction: spacing ~\!

Jon Awbrey: /* Introduction */ workaround for MathJax's bad spacing of subscripts after \mathit{...} -- use \mathit{...}_{\,k}

2017-08-04T12:46:02Z

Introduction: workaround for MathJax's bad spacing of subscripts after \mathit{...} -- use \mathit{...}_{\,k}

@@ Line 429: / Line 429: @@
 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).
-:* [http://stderr.org/pipermail/inquiry/2005-October/thread.html#3104 FOLG]
+:* [http://web.archive.org/web/20150224133200/http://stderr.org/pipermail/inquiry/2005-October/thread.html#3104 Inquiry List &bull; Futures Of Logical Graphs]
-:* [http://stderr.org/pipermail/inquiry/2005-October/003113.html FOLG 5]
+:* [http://web.archive.org/web/20120206123011/http://stderr.org/pipermail/inquiry/2005-October/003113.html Inquiry List &bull; Futures Of Logical Graphs &bull; Note 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.

@@ Line 333: / Line 333: @@
 :* Pure Symbols
-:: http://stderr.org/pipermail/inquiry/2005-March/thread.html#2465
+:: [http://web.archive.org/web/20141124153003/http://stderr.org/pipermail/inquiry/2005-March/thread.html#2465 2005 &bull; March]
-:: http://stderr.org/pipermail/inquiry/2005-April/thread.html#2517
+:: [http://web.archive.org/web/20120601160642/http://stderr.org/pipermail/inquiry/2005-April/thread.html#2517 2005 &bull; April]
 :* Pure Symbols : Discussion
-:: http://stderr.org/pipermail/inquiry/2005-March/thread.html#2466
+:: [http://web.archive.org/web/20141124153003/http://stderr.org/pipermail/inquiry/2005-March/thread.html#2466 2005 &bull; March]
-:: http://stderr.org/pipermail/inquiry/2005-April/thread.html#2514
+:: [http://web.archive.org/web/20120601160642/http://stderr.org/pipermail/inquiry/2005-April/thread.html#2514 2005 &bull; April]
-:: http://stderr.org/pipermail/inquiry/2005-May/thread.html#2654
+:: [http://web.archive.org/web/20120421003708/http://stderr.org/pipermail/inquiry/2005-May/thread.html#2654 2005 &bull; May]
 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.

@@ Line 206: / Line 206: @@
 Returning to <math>\mathrm{En}</math> and <math>\mathrm{Ex},</math> the two most popular interpretations of logical graphs, ones that happen to be dual to each other in a certain sense, let's see how they fly as ''hermeneutic arrows'' from the syntactic domain <math>S~\!</math> to the object domain <math>O,~\!</math> at any rate, as their trajectories can be spied in the radar of what George Spencer Brown called the ''primary arithmetic''.
-Taking <math>\mathrm{En}~\!</math> and <math>\mathrm{Ex}~\!</math> as arrows of the form <math>\mathrm{En}, \mathrm{Ex} : S \to O,</math> at the level of arithmetic taking <math>S = \{ \text{rooted trees} \}~\!</math> and <math>O = \{ \mathrm{falsity}, \mathrm{truth} \},~\!</math> it is possible to factor each arrow across the domain <math>S_0~\!</math> that consists of a single rooted node plus a single rooted edge, in other words, the domain of formal constants <math>S_0 = \{ \ominus, \vert \} = \{</math>[[Image:Rooted Node.jpg|16px]],&nbsp;[[Image:Rooted Edge.jpg|12px]]<math>\}.~\!</math>  This allows each arrow to be broken into a purely syntactic part <math>\mathrm{En}_\text{syn}, \mathrm{Ex}_\text{syn} : S \to S_0</math> and a purely semantic part <math>\mathrm{En}_\text{sem}, \mathrm{Ex}_\text{sem} : S_0 \to O.</math>
+Taking <math>\mathrm{En}~\!</math> and <math>\mathrm{Ex}~\!</math> as arrows of the form <math>\mathrm{En}, \mathrm{Ex} : S \to O,</math> at the level of arithmetic taking <math>S = \{ \text{rooted trees} \}~\!</math> and <math>O = \{ \mathrm{falsity}, \mathrm{truth} \},~\!</math> it is possible to factor each arrow across the domain <math>S_0~\!</math> that consists of a single rooted node plus a single rooted edge, in other words, the domain of formal constants <math>S_0 = \{ \ominus, \vert \} = \{</math>[[File:Rooted Node Big.jpg|16px]],&nbsp;[[File:Rooted Edge Big.jpg|12px]]<math>\}.~\!</math>  This allows each arrow to be broken into a purely syntactic part <math>\mathrm{En}_\text{syn}, \mathrm{Ex}_\text{syn} : S \to S_0</math> and a purely semantic part <math>\mathrm{En}_\text{sem}, \mathrm{Ex}_\text{sem} : S_0 \to O.</math>
 As things work out, the syntactic factors are formally the same, leaving our dualing interpretations to differ in their semantic components alone.  Specifically, we have the following mappings:
@@ Line 213: / Line 213: @@
 | width="5%" | &nbsp;
 | width="5%" | <math>\mathrm{En}_\text{sem} :</math>
-| width="5%" align="center" | [[Image:Rooted Node.jpg|16px]]
+| width="5%" align="center" | [[Image:Rooted Node Big.jpg|16px]]
 | width="5%" | <math>\mapsto</math>
 | <math>\mathrm{false},</math>
@@ Line 219: / Line 219: @@
 | &nbsp;
 | &nbsp;
-| align="center" | [[Image:Rooted Edge.jpg|12px]]
+| align="center" | [[Image:Rooted Edge Big.jpg|12px]]
 | <math>\mapsto</math>
 | <math>\mathrm{true}.</math>
@@ Line 225: / Line 225: @@
 | &nbsp;
 | <math>\mathrm{Ex}_\text{sem} :</math>
-| align="center" | [[Image:Rooted Node.jpg|16px]]
+| align="center" | [[Image:Rooted Node Big.jpg|16px]]
 | <math>\mapsto</math>
 | <math>\mathrm{true},</math>
@@ Line 231: / Line 231: @@
 | &nbsp;
 | &nbsp;
-| align="center" | [[Image:Rooted Edge.jpg|12px]]
+| align="center" | [[Image:Rooted Edge Big.jpg|12px]]
 | <math>\mapsto</math>
 | <math>\mathrm{false}.</math>

@@ Line 178: / Line 178: @@
 |}
-The practical use of Peirce's categories is simply to organize our thoughts about what sorts of formal models are demanded by a material situation, for instance, a domain of phenomena from atoms to biology to culture.  To say that "k-ness" is involved in a phenomenon is simply to say that we need k-adic relations to model it adequately, and that the phenomenon itself appears to demand nothing less.  Aside from this, Peirce's realization that k-ness for k = 1, 2, 3 affords us with a sufficient basis for all that we need to model is a formal fact that depends on a particular theorem in the logic of relatives.  If it weren't for that, there would hardly be any reason to single out three.
+The practical use of Peirce's categories is simply to organize our thoughts about what sorts of formal models are demanded by a material situation, for instance, a domain of phenomena from atoms to biology to culture.  To say that &ldquo;''k''-ness&rdquo; is involved in a phenomenon is simply to say that we need ''k''-adic relations to model it adequately, and that the phenomenon itself appears to demand nothing less.  Aside from this, Peirce's realization that ''k''-ness for ''k'' = 1, 2, 3 affords us with a sufficient basis for all that we need to model is a formal fact that depends on a particular theorem in the logic of relatives.  If it weren't for that, there would hardly be any reason to single out three.
 In order to discuss the various forms of iconicity that might be involved in the application of Peirce's logical graphs and their kind to the object domain of logic itself, we will need to bring out two or three ''categories of structured individuals'' (COSIs), depending on how one counts.  These are called the ''object domain'', the ''sign domain'', and the ''interpretant sign domain'', which may be written <math>{O, S, I},\!</math> respectively, or <math>{X, Y, Z},\!</math> respectively, depending on the style that fits the current frame of discussion.

@@ Line 90: / Line 90: @@
 Notice that, with rooted trees like these, drawing the arrows is optional, since singling out a unique node as the root induces a unique orientation on all the edges of the tree, ''up'' being the same as ''away from the root''.
-We have treated in some detail various forms of the initial equation or logical axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~(~)~)} = \quad {}^{\prime\prime}.\!</math>&nbsp; For the sake of comparison, let's record the plane-embedded and topological dual forms of the axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(~)(~)} = \texttt{(~)} {}^{\prime\prime}.\!</math>
+We have treated in some detail various forms of the initial equation or logical axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(} ~ \texttt{(} ~ \texttt{)} ~ \texttt{)} = \quad {}^{\prime\prime}.~\!</math>&nbsp; For the sake of comparison, let's record the plane-embedded and topological dual forms of the axiom that is formulated in string form as <math>{}^{\backprime\backprime} \texttt{(} ~ \texttt{)(} ~ \texttt{)} = \texttt{(} ~ \texttt{)} {}^{\prime\prime}.~\!</math>
 First the plane-embedded maps:

@@ Line 70: / Line 70: @@
 |}
-This portrays the pointer <math>\mathit{index}_{\,0}\!</math> as the address of a record that contains the following data:
+This portrays the pointer <math>\mathit{index}_{\,0}~\!</math> as the address of a record that contains the following data:
 {| align="center" cellpadding="10"
-| <math>\mathit{datum}_{\,1}, \mathit{datum}_{\,2}, \mathit{datum}_{\,3}, \ldots,\!</math> and so on.
+| <math>\mathit{datum}_{\,1}, \mathit{datum}_{\,2}, \mathit{datum}_{\,3}, \ldots,~\!</math> and so on.
 |}