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{{DISPLAYTITLE:Differential Logic : Introduction}}

{{DISPLAYTITLE:Differential Logic : Introduction}}

'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''

'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''

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| <math>\texttt{(~)}\!</math>

| <math>\texttt{(}~\texttt{)}\!</math>

| <math>\mathrm{false}\!</math>

| <math>\mathrm{false}\!</math>

| <math>0\!</math>

| <math>0\!</math>

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The simplest expression for logical truth is the empty word, usually denoted by <math>\boldsymbol\varepsilon\!</math> or <math>\lambda\!</math> in formal languages, where it forms the identity element for concatenation.  To make it visible in context, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((~))} {}^{\prime\prime},\!</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.\!</math>  Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> may be used for [[exclusive disjunction]].  For example, we have the following paraphrases of algebraic expressions by means of parenthesized expressions:

The simplest expression for logical truth is the empty word, usually denoted by <math>\boldsymbol\varepsilon\!</math> or <math>\lambda\!</math> in formal languages, where it forms the identity element for concatenation.  To make it visible in context, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((}~\texttt{))} {}^{\prime\prime},\!</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.\!</math>  Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> may be used for [[exclusive disjunction]].  For example, we have the following paraphrases of algebraic expressions by means of parenthesized expressions:

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It happens that there are just two possible groups of 4 elements.  One is the cyclic group <math>Z_4\!</math> (from German ''Zyklus''), which this is not.  The other is the Klein four-group <math>V_4\!</math> (from German ''Vier''), which this is.

It happens that there are just two possible groups of 4 elements.  One is the cyclic group <math>Z_4\!</math> (from German ''Zyklus''), which this is not.  The other is the Klein four-group <math>V_4\!</math> (from German ''Vier''), which this is.

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 &ldquo;those who count&rdquo; 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: <math>\text{Number of orbits}~ = (4 + 4 + 4 + 16) \div 4 = 7.\!</math>  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 &ldquo;those who count&rdquo; 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: &nbsp; Number of Orbits &nbsp;=&nbsp; (4 + 4 + 4 + 16) &divide; 4 &nbsp;=&nbsp; 7. &nbsp; Amazing!

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For example, given the set <math>X = \{ a, b, c \},\!</math> suppose that we have the 2-adic relative term <math>\mathit{m} = {}^{\backprime\backprime}\, \text{marker for}\, \underline{~ ~ ~}\, {}^{\prime\prime}\!</math> and the associated 2-adic relation <math>M \subseteq X \times X,\!</math> the general pattern of whose common structure is represented by the following matrix:

For example, given the set <math>X = \{ a, b, c \},\!</math> suppose that we have the 2-adic relative term <math>\mathit{m} = {}^{\backprime\backprime}\, \text{marker for}\, \underline{~ ~ ~}\, {}^{\prime\prime}~\!</math> and the associated 2-adic relation <math>M \subseteq X \times X,\!</math> the general pattern of whose common structure is represented by the following matrix:

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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.

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: <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 &ldquo;multiplying on the left&rdquo;.

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;.

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:

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: