Changes

{{DISPLAYTITLE:Differential Logic : Sketch 2}}

{{DISPLAYTITLE:Differential Logic}}

<font color="red" size="3">'''Note. The ever-sucky MathJerx parser has made hash of the text below.  See the InterSciWiki copy at [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Sketch_2 Differential Logic : Sketch 2].'''</font>

<font color="red" size="3">'''Note. The ever-sucky MathJerx parser has made hash of the text below.  See the InterSciWiki copy at [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Sketch_2 Differential Logic : Sketch 2].'''</font>

|}

|}

<math>\begin{array}{rcccccc}

<math>\begin{array}{rcccccc}

f

f

& + & (x) & \cdot & (y) & \cdot & ~~\mathrm{d}x~~\mathrm{d}y~~

& + & (x) & \cdot & (y) & \cdot & ~~\mathrm{d}x~~\mathrm{d}y~~

\end{array}\!</math>

\end{array}\!</math>

Any proposition worth its salt can be analyzed from many different points of view, any one of which has the potential to reveal an unsuspected aspect of the proposition's meaning.  We will encounter more and more of these alternative readings as we go.

Any proposition worth its salt can be analyzed from many different points of view, any one of which has the potential to reveal an unsuspected aspect of the proposition's meaning.  We will encounter more and more of these alternative readings as we go.

((x,~y))

((x,~y))

\\[4pt]

\\[4pt]

22:03, 8 December 2014 (UTC)y~~

16:08, 11 December 2014 (UTC)y~~

\\[4pt]

\\[4pt]

~(x~(y))

~(x~(y))

\\[4pt]

\\[4pt]

~~x22:03, 8 December 2014 (UTC)

~~x16:08, 11 December 2014 (UTC)

\\[4pt]

\\[4pt]

((x)~y)~

((x)~y)~

<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>

<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" | &mdash; Charles Sanders Peirce, "Issues of Pragmaticism", CP 5.438

| align="right" | &mdash; Charles Sanders Peirce, "Issues of Pragmaticism", [CP 5.438]

|}

|}

{| align="center" cellspacing="10"

{| align="center" cellspacing="10"

|

| align="center" |

<math>\begin{matrix}

<math>\begin{matrix}

\mathrm{G}

\mathrm{G}

And so, by expanding effects, we get:

And so, by expanding effects, we get:

<math>\begin{matrix}

<math>\begin{matrix}

\mathrm{G}

\mathrm{G}

& + & \mathrm{h}:\mathrm{e}

& + & \mathrm{h}:\mathrm{e}

\end{matrix}\!</math>

\end{matrix}\!</math>

More on the pragmatic maxim as a representation principle later.

More on the pragmatic maxim as a representation principle later.

|}

|}

So, for example, let us suppose that we have the small universe <math>\{ \mathrm{A}, \mathrm{B}, \mathrm{C} \},\!</math> and the 2-adic relation <math>\mathit{m} = {}^{\backprime\backprime}\, \text{mover of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 22:03, 8 December 2014 (UTC)}\, {}^{\prime\prime}\!</math> that is represented by the following matrix:

So, for example, let us suppose that we have the small universe <math>\{ \mathrm{A}, \mathrm{B}, \mathrm{C} \},\!</math> and the 2-adic relation <math>\mathit{m} = {}^{\backprime\backprime}\, \text{mover of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 16:08, 11 December 2014 (UTC)}\, {}^{\prime\prime}\!</math> that is represented by the following matrix:

{| align="center" cellspacing="10"

{| align="center" cellspacing="10"

Back to our current subinstance, the example in support of our first example.  I will try to reconstruct it in a less confusing way.

Back to our current subinstance, the example in support of our first example.  I will try to reconstruct it in a less confusing way.

Consider the universe of discourse <math>\mathbf{1} = \mathrm{A} + \mathrm{B} + \mathrm{C}\!</math> and the 2-adic relation <math>\mathit{n} = {}^{\backprime\backprime}\, \text{noder of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 22:03, 8 December 2014 (UTC)}\, {}^{\prime\prime},\!</math> as when "<math>X\!</math> is a data record that contains a pointer to <math>Y\!</math>".  That interpretation is not important, it's just for the sake of intuition.  In general terms, the 2-adic relation <math>n\!</math> can be represented by this matrix:

Consider the universe of discourse <math>\mathbf{1} = \mathrm{A} + \mathrm{B} + \mathrm{C}\!</math> and the 2-adic relation <math>\mathit{n} = {}^{\backprime\backprime}\, \text{noder of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 16:08, 11 December 2014 (UTC)}\, {}^{\prime\prime},\!</math> as when "<math>X\!</math> is a data record that contains a pointer to <math>Y\!</math>".  That interpretation is not important, it's just for the sake of intuition.  In general terms, the 2-adic relation <math>n\!</math> can be represented by this matrix:

{| align="center" cellspacing="10"

{| align="center" cellspacing="10"

|}

|}

Recognizing <math>\mathit{1} = \mathrm{A}\!:\!\mathrm{A} + \mathrm{B}\!:\!\mathrm{B} + \mathrm{C}\!:\!\mathrm{C}\!</math> to be the identity transformation, the 2-adic relation <math>\mathit{n} = {}^{\backprime\backprime}\, \text{noder of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 22:03, 8 December 2014 (UTC)}\, {}^{\prime\prime}\!</math> may be represented by an element <math>\mathit{1} + \mathrm{A}\!:\!\mathrm{B} + \mathrm{B}\!:\!\mathrm{C} + \mathrm{C}\!:\!\mathrm{A}\!</math> of the so-called ''group ring'', all of which just makes this element a special sort of linear transformation.

Recognizing <math>\mathit{1} = \mathrm{A}\!:\!\mathrm{A} + \mathrm{B}\!:\!\mathrm{B} + \mathrm{C}\!:\!\mathrm{C}\!</math> to be the identity transformation, the 2-adic relation <math>\mathit{n} = {}^{\backprime\backprime}\, \text{noder of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 16:08, 11 December 2014 (UTC)}\, {}^{\prime\prime}\!</math> may be represented by an element <math>\mathit{1} + \mathrm{A}\!:\!\mathrm{B} + \mathrm{B}\!:\!\mathrm{C} + \mathrm{C}\!:\!\mathrm{A}\!</math> of the so-called ''group ring'', all of which just makes this element a special sort of linear transformation.

Up to this point, we're still reading the elementary relatives of the form <math>I:J\!</math> in the way that Peirce reads them in logical contexts: <math>I\!</math> is the relate, <math>J\!</math> is the correlate, and in our current example we read <math>I:J,\!</math> or more exactly, <math>\mathit{n}_{ij} = 1,\!</math> to say that <math>I\!</math> is a noder of <math>J.\!</math>  This is the mode of reading that we call ''multiplying on the left''.

Up to this point, we're still reading the elementary relatives of the form <math>I:J\!</math> in the way that Peirce reads them in logical contexts: <math>I\!</math> is the relate, <math>J\!</math> is the correlate, and in our current example we read <math>I:J,\!</math> or more exactly, <math>\mathit{n}_{ij} = 1,\!</math> to say that <math>I\!</math> is a noder of <math>J.\!</math>  This is the mode of reading that we call ''multiplying on the left''.