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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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| height="100px" | [[Image:Rooted Edge.jpg|20px]]

| height="100px" | [[Image:Rooted Edge.jpg|20px]]

| <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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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: