Directory talk:Jon Awbrey/Papers/Differential Logic : Introduction - Revision history

Jon Awbrey: /* Logical Cacti */

2015-12-02T18:38:45Z

Logical Cacti

Jon Awbrey: cleanup workspace

2015-12-01T22:56:33Z

cleanup workspace

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Jon Awbrey: /* Operational Representation */ + a few more paragraphs

2015-12-01T22:24:04Z

Operational Representation: + a few more paragraphs

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Jon Awbrey: /* Operational Representation */ + a few more paragraphs

2015-12-01T22:04:48Z

Operational Representation: + a few more paragraphs

Jon Awbrey: /* Operational Representation */ don't know why but inserting " " after ":" at the problem point seems to fix it

2015-12-01T21:54:46Z

Operational Representation: don't know why but inserting " " after ":" at the problem point seems to fix it

Jon Awbrey: /* Operational Representation */ + next clause

2015-12-01T16:00:18Z

Operational Representation: + next clause

Jon Awbrey: /* Operational Representation */ add text up to next problem area

2015-11-30T21:30:08Z

Operational Representation: add text up to next problem area

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Jon Awbrey: /* Operational Representation */ + table + 2 paragraphs

2015-11-30T21:12:17Z

Operational Representation: + table + 2 paragraphs

Jon Awbrey: /* Operational Representation */ + next 3 paragraphs

2015-11-30T21:08:43Z

Operational Representation: + next 3 paragraphs

Jon Awbrey: + first part of ==Operational Representation==

2015-11-30T20:42:11Z

+ first part of ==Operational Representation==

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← Older revision		Revision as of 18:38, 2 December 2015
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−	==Logical Cacti==	+	==Work Area • Logical Cacti==

	; Theme One Program — Logical Cacti		; Theme One Program — Logical Cacti

← Older revision		Revision as of 22:04, 1 December 2015
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	Up to this point, we are still reading the elementary relatives of the form <math>i\!:\!j\!</math> in the way that Peirce read them in logical contexts:  <math>i\!</math> is the relate, <math>j\!</math> is the correlate, and in our current example <math>i\!:\!j,\!</math> or more exactly, <math>m_{ij} = 1,\!</math> is taken to say that <math>i\!</math> is a marker for <math>j.\!</math>  This is the mode of reading that we call “multiplying on the left”.		Up to this point, we are still reading the elementary relatives of the form <math>i\!:\!j\!</math> in the way that Peirce read them in logical contexts:  <math>i\!</math> is the relate, <math>j\!</math> is the correlate, and in our current example <math>i\!:\!j,\!</math> or more exactly, <math>m_{ij} = 1,\!</math> is taken to say that <math>i\!</math> is a marker for <math>j.\!</math>  This is the mode of reading that we call “multiplying on the left”.
		+
		+	In the algebraic, permutational, or transformational contexts of application, however, Peirce converts to the alternative mode of reading, although still calling <math>i\!</math> the relate and <math>j\!</math> the correlate, the elementary relative <math>i\!:\!j\!</math> now means that <math>i\!</math> gets changed into <math>j.\!</math> In this scheme of reading, the transformation <math>a\!:\!b + b\!:\!c + c\!:\!a\!</math> is a permutation of the aggregate <math>\mathbf{1} = a + b + c,\!</math> or what we would now call the set <math>\{ a, b, c \},\!</math> in particular, it is the permutation that is otherwise notated as follows:
		+
		+	{\| align="center" cellpadding="6"
		+	\|
		+	<math>\begin{Bmatrix}
		+	a & b & c
		+	\\
		+	b & c & a
		+	\end{Bmatrix}\!</math>
		+	\|}
		+
		+	This is consistent with the convention that Peirce uses in the paper “On a Class of Multiple Algebras” (CP 3.324–327).
		+
		+	We've been exploring the applications of a certain technique for clarifying abstruse concepts, a rough-cut version of the pragmatic maxim that I've been accustomed to refer to as the ''operationalization'' of ideas. The basic idea is to replace the question of ''What it is'', which modest people comprehend is far beyond their powers to answer definitively any time soon, with the question of ''What it does'', which most people know at least a modicum about.
		+
		+	In the case of regular representations of groups we found a non-plussing surplus of answers to sort our way through. So let us track back one more time to see if we can learn any lessons that might carry over to more realistic cases.
		+
		+	Here is is the operation table of <math>V_4\!</math> once again:
		+
		+	<br>
		+
		+	{\| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
		+	\|+ <math>\text{Klein Four-Group}~ V_4\!</math>
		+	\|- style="height:50px"
		+	\| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" \|
		+	<math>\cdot\!</math>
		+	\| width="22%" style="border-bottom:1px solid black" \|
		+	<math>\mathrm{e}\!</math>
		+	\| width="22%" style="border-bottom:1px solid black" \|
		+	<math>\mathrm{f}\!</math>
		+	\| width="22%" style="border-bottom:1px solid black" \|
		+	<math>\mathrm{g}\!</math>
		+	\| width="22%" style="border-bottom:1px solid black" \|
		+	<math>\mathrm{h}\!</math>
		+	\|- style="height:50px"
		+	\| style="border-right:1px solid black" \| <math>\mathrm{e}\!</math>
		+	\| <math>\mathrm{e}\!</math>
		+	\| <math>\mathrm{f}\!</math>
		+	\| <math>\mathrm{g}\!</math>
		+	\| <math>\mathrm{h}\!</math>
		+	\|- style="height:50px"
		+	\| style="border-right:1px solid black" \| <math>\mathrm{f}\!</math>
		+	\| <math>\mathrm{f}\!</math>
		+	\| <math>\mathrm{e}\!</math>
		+	\| <math>\mathrm{h}\!</math>
		+	\| <math>\mathrm{g}\!</math>
		+	\|- style="height:50px"
		+	\| style="border-right:1px solid black" \| <math>\mathrm{g}\!</math>
		+	\| <math>\mathrm{g}\!</math>
		+	\| <math>\mathrm{h}\!</math>
		+	\| <math>\mathrm{e}\!</math>
		+	\| <math>\mathrm{f}\!</math>
		+	\|- style="height:50px"
		+	\| style="border-right:1px solid black" \| <math>\mathrm{h}\!</math>
		+	\| <math>\mathrm{h}\!</math>
		+	\| <math>\mathrm{g}\!</math>
		+	\| <math>\mathrm{f}\!</math>
		+	\| <math>\mathrm{e}\!</math>
		+	\|}
		+
		+	<br>

	==Logical Cacti==		==Logical Cacti==

@@ Line 2,314: / Line 2,314: @@
 |-
+|
-<math>\begin{array}{c}
+<math>\begin{matrix}
 f_{1}
 \\[4pt]
@@ Line 2,322: / Line 2,322: @@
 \\[4pt]
 f_8
-\end{array}\!</math>
+\end{matrix}\!</math>
+|
 <math>\begin{matrix}
@@ Line 2,901: / Line 2,901: @@
 Recognizing that <math>a\!:\!a + b\!:\!b + c\!:\!c\!</math> is the identity transformation otherwise known as <math>\mathit{1},\!</math> the 2-adic relative term <math>m = {}^{\backprime\backprime}\, \text{marker for}\, \underline{~ ~ ~}\, {}^{\prime\prime}~\!</math> can be parsed as an element <math>\mathit{1} + a\!:\!b + b\!:\!c + c\!:\!a\!</math> of the so-called ''group ring'', all of which makes this element just a special sort of linear transformation.
-Up to this point, we are still reading the elementary relatives of the form <math>i\!:\!j</math> in the way that Peirce read them in logical contexts:
+Up to this point, we are still reading the elementary relatives of the form <math>i\!:\!j\!</math> in the way that Peirce read them in logical contexts:&nbsp; <math>i\!</math> is the relate, <math>j\!</math> is the correlate, and in our current example <math>i\!:\!j,\!</math> or more exactly, <math>m_{ij} = 1,\!</math> is taken to say that <math>i\!</math> is a marker for <math>j.\!</math>&nbsp; This is the mode of reading that we call &ldquo;multiplying on the left&rdquo;.
 ==Logical Cacti==

@@ Line 2,314: / Line 2,314: @@
 |-
+|
-<math>\begin{matrix}
+<math>\begin{array}{c}
-f_1
+f_{1}
 \\[4pt]
 f_2
@@ Line 2,322: / Line 2,322: @@
 \\[4pt]
 f_8
-\end{matrix}\!</math>
+\end{array}\!</math>
+|
 <math>\begin{matrix}
@@ Line 2,900: / Line 2,900: @@
 Recognizing that <math>a\!:\!a + b\!:\!b + c\!:\!c\!</math> is the identity transformation otherwise known as <math>\mathit{1},\!</math> the 2-adic relative term <math>m = {}^{\backprime\backprime}\, \text{marker for}\, \underline{~ ~ ~}\, {}^{\prime\prime}~\!</math> can be parsed as an element <math>\mathit{1} + a\!:\!b + b\!:\!c + c\!:\!a\!</math> of the so-called ''group ring'', all of which makes this element just a special sort of linear transformation.
 ==Logical Cacti==

← Older revision		Revision as of 21:12, 30 November 2015
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	<br>		<br>
		+
		+	{\| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
		+	\|- style="height:50px"
		+	\| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" \|
		+	<math>\cdot\!</math>
		+	\| width="22%" style="border-bottom:1px solid black" \|
		+	<math>\mathrm{e}\!</math>
		+	\| width="22%" style="border-bottom:1px solid black" \|
		+	<math>\mathrm{f}\!</math>
		+	\| width="22%" style="border-bottom:1px solid black" \|
		+	<math>\mathrm{g}\!</math>
		+	\| width="22%" style="border-bottom:1px solid black" \|
		+	<math>\mathrm{h}\!</math>
		+	\|- style="height:50px"
		+	\| style="border-right:1px solid black" \| <math>\mathrm{e}\!</math>
		+	\| <math>\mathrm{e}\!</math>
		+	\| <math>\mathrm{f}\!</math>
		+	\| <math>\mathrm{g}\!</math>
		+	\| <math>\mathrm{h}\!</math>
		+	\|- style="height:50px"
		+	\| style="border-right:1px solid black" \| <math>\mathrm{f}\!</math>
		+	\| <math>\mathrm{f}\!</math>
		+	\| <math>\mathrm{e}\!</math>
		+	\| <math>\mathrm{h}\!</math>
		+	\| <math>\mathrm{g}\!</math>
		+	\|- style="height:50px"
		+	\| style="border-right:1px solid black" \| <math>\mathrm{g}\!</math>
		+	\| <math>\mathrm{g}\!</math>
		+	\| <math>\mathrm{h}\!</math>
		+	\| <math>\mathrm{e}\!</math>
		+	\| <math>\mathrm{f}\!</math>
		+	\|- style="height:50px"
		+	\| style="border-right:1px solid black" \| <math>\mathrm{h}\!</math>
		+	\| <math>\mathrm{h}\!</math>
		+	\| <math>\mathrm{g}\!</math>
		+	\| <math>\mathrm{f}\!</math>
		+	\| <math>\mathrm{e}\!</math>
		+	\|}
		+
		+	<br>
		+
		+	This table is abstractly the same as, or isomorphic to, the versions with the <math>\mathrm{E}_{ij}\!</math> operators and the <math>\mathrm{T}_{ij}\!</math> transformations that we took up earlier. That is to say, the story is the same, only the names have been changed. An abstract group can have a variety of significantly and superficially different representations. But even after we have long forgotten the details of any particular representation there is a type of concrete representations, called ''regular representations'', that are always readily available, as they can be generated from the mere data of the abstract operation table itself.
		+
		+	To see how a regular representation is constructed from the abstract operation table, select a group element from the top margin of the Table, and “consider its effects” on each of the group elements as they are listed along the left margin. We may record these effects as Peirce usually did, as a ''logical aggregate'' of elementary dyadic relatives, that is, as a logical disjunction or boolean sum whose terms represent the ordered pairs of <math>\mathrm{input} : \mathrm{output}\!</math> transactions that are produced by each group element in turn. This forms one of the two possible ''regular representations'' of the group, in this case the one that is called the ''post-regular representation'' or the ''right regular representation''. It has long been conventional to organize the terms of this logical aggregate in the form of a matrix:

	==Logical Cacti==		==Logical Cacti==

← Older revision		Revision as of 21:08, 30 November 2015
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	More concretely viewed, the group as a whole pushes the set of sixteen propositions around in such a way that they fall into seven natural classes, called ''orbits''. One says that the orbits are preserved by the action of the group. There is an ''Orbit Lemma'' of immense utility to “those who count” which, depending on your upbringing, you may associate with the names of Burnside, Cauchy, Frobenius, or some subset or superset of these three, vouching that the number of orbits is equal to the mean number of fixed points, in other words, the total number of points (in our case, propositions) that are left unmoved by the separate operations, divided by the order of the group. In this instance, <math>\mathrm{T}_{00}\!</math> operates as the group identity, fixing all 16 propositions, while the other three group elements fix 4 propositions each, and so we get:   Number of Orbits  =  (4 + 4 + 4 + 16) ÷ 4  =  7.   Amazing!		More concretely viewed, the group as a whole pushes the set of sixteen propositions around in such a way that they fall into seven natural classes, called ''orbits''. One says that the orbits are preserved by the action of the group. There is an ''Orbit Lemma'' of immense utility to “those who count” which, depending on your upbringing, you may associate with the names of Burnside, Cauchy, Frobenius, or some subset or superset of these three, vouching that the number of orbits is equal to the mean number of fixed points, in other words, the total number of points (in our case, propositions) that are left unmoved by the separate operations, divided by the order of the group. In this instance, <math>\mathrm{T}_{00}\!</math> operates as the group identity, fixing all 16 propositions, while the other three group elements fix 4 propositions each, and so we get:   Number of Orbits  =  (4 + 4 + 4 + 16) ÷ 4  =  7.   Amazing!
		+
		+	{\| align="center" cellpadding="0" cellspacing="0" width="90%"
		+	\|
		+	<p>Consider what effects that might ''conceivably'' have practical bearings you ''conceive'' the objects of your ''conception'' to have. Then, your ''conception'' of those effects is the whole of your ''conception'' of the object.</p>
		+	\|-
		+	\| align="right" \| — Charles Sanders Peirce, “Issues of Pragmaticism”, (CP 5.438)
		+	\|}
		+
		+	One other subject that it would be opportune to mention at this point, while we have an object example of a mathematical group fresh in mind, is the relationship between the pragmatic maxim and what are commonly known in mathematics as ''representation principles''. As it turns out, with regard to its formal characteristics, the pragmatic maxim unites the aspects of a representation principle with the attributes of what would ordinarily be known as a ''closure principle''. We will consider the form of closure that is invoked by the pragmatic maxim on another occasion, focusing here and now on the topic of group representations.
		+
		+	Let us return to the example of the ''four-group'' <math>V_4.\!</math> We encountered this group in one of its concrete representations, namely, as a ''transformation group'' that acts on a set of objects, in this case a set of sixteen functions or propositions. Forgetting about the set of objects that the group transforms among themselves, we may take the abstract view of the group's operational structure, for example, in the form of the group operation table copied here:
		+
		+	<br>

	==Logical Cacti==		==Logical Cacti==