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{{DISPLAYTITLE:Differential Logic : Sketch 2}}

{{DISPLAYTITLE:Differential Logic}}

'''☆ Meanwhile, please see the InterSciWiki copy at [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Sketch_2 Differential Logic : Sketch 2]. ☆'''

 

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

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

'''Note.'''  ''The present Sketch is largely superseded by the article “[[Differential Logic : Introduction]]” but I have preserved it here for the sake of the remaining ideas that have yet to be absorbed elsewhere.''

'''Note.'''  ''The present Sketch is largely superseded by the article “[[Differential Logic : Introduction]]” but I have preserved it here for the sake of the remaining ideas that have yet to be absorbed elsewhere.''

'''Differential logic''' is the component of logic whose object is the description of variation — for example, the aspects of change, difference, distribution, and diversity — in [[universes of discourse]] that are subject to logical description.  In formal logic, differential logic treats the principles that govern the use of a ''differential logical calculus'', that is, a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.

'''Differential logic''' is the component of logic whose object is the description of variation — for example, the aspects of change, difference, distribution, and diversity — in [[universes of discourse]] that are subject to logical description.  In formal logic, differential logic treats the principles that govern the use of a ''differential logical calculus'', that is, a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.

A simple example of a differential logical calculus is furnished by a ''[[differential propositional calculus]]''.  A differential propositional calculus is a [[propositional calculus]] extended by a set of terms for describing aspects of change and difference, for example, processes that take place in a universe of discourse or transformations that map a source universe into a target universe.  This augments ordinary propositional calculus in the same way that the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.

A simple example of a differential logical calculus is furnished by a ''[[differential propositional calculus]]''.  A differential propositional calculus is a [[propositional calculus]] extended by a set of terms for describing aspects of change and difference, for example, processes that take place in a universe of discourse or transformations that map a source universe into a target universe.  This augments ordinary propositional calculus in the same way that the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.

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((x,~y))

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~(x~(y))

~(x~(y))

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((x)~y)~

((x)~y)~

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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''.