Changes

==Differential Logic : Series A==

==Differential Logic : Series A==

===Note 1===

===Differential Propositions===

One of the first things that you can do, once you have a really decent calculus for boolean functions or propositional logic, whatever you want to call it, is to compute the differentials of these functions or propositions.

One of the first things that you can do, once you have a really decent calculus for boolean functions or propositional logic, whatever you want to call it, is to compute the differentials of these functions or propositions.

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We have just met with the fact that the differential of the "and" is the "or" of the differentials.

We have just met with the fact that the differential of the "and" is the "or" of the differentials.

The picture illustrates the analysis of the inclusive disjunction ((''dx'')(''dy'')) into the exclusive disjunction:  ''dx''(''dy'') + ''dy''(''dx'') + ''dx dy'', a proposition that may be interpreted to say "change x or change y or both".  And this can be recognized as just what you need to do if you happen to find yourself in the center cell and desire a detailed description of ways to depart it.

The picture illustrates the analysis of the inclusive disjunction ((''dx'')(''dy'')) into the exclusive disjunction:  ''dx''(''dy'') + ''dy''(''dx'') + ''dx dy'', a proposition that may be interpreted to say "change x or change y or both".  And this can be recognized as just what you need to do if you happen to find yourself in the center cell and desire a detailed description of ways to depart it.

We have just computed what will variously be called the ''difference map'', the ''difference proposition'', or the ''local proposition'' ''Df''_''p'' for the proposition ''f''(''x'',&nbsp;''y'') = ''xy'' at the point ''p'' where ''x'' = 1 and ''y'' = 1.

 

Last time we computed what will variously be called the ''difference map'', the ''difference proposition'', or the ''local proposition'' ''Df''_''p'' for the proposition ''f''(''x'',&nbsp;''y'') = ''xy'' at the point ''p'' where ''x'' = 1 and ''y'' = 1.

In the universe ''U'' = ''X''&nbsp;&times;&nbsp;''Y'', the four propositions ''xy'', ''x''(''y''), (''x'')''y'', (''x'')(''y'') that indicate the "cells", or the smallest regions of the venn diagram, are called ''singular propositions''.  These serve as an alternative notation for naming the points <1,&nbsp;1>, <1,&nbsp;0>, <0,&nbsp;1>, <0,&nbsp;0>, respectively.

In the universe ''U'' = ''X''&nbsp;&times;&nbsp;''Y'', the four propositions ''xy'', ''x''(''y''), (''x'')''y'', (''x'')(''y'') that indicate the "cells", or the smallest regions of the venn diagram, are called ''singular propositions''.  These serve as an alternative notation for naming the points <1,&nbsp;1>, <1,&nbsp;0>, <0,&nbsp;1>, <0,&nbsp;0>, respectively.

By inspection, it is fairly easy to understand ''Df'' as telling you what you have to do from each point of ''U'' in order to change the value borne by ''f''(''x'',&nbsp;''y'').

By inspection, it is fairly easy to understand ''Df'' as telling you what you have to do from each point of ''U'' in order to change the value borne by ''f''(''x'',&nbsp;''y'').

We have been studying the action of the difference operator ''D'', also known as the ''localization operator'', on the proposition ''f''&nbsp;:&nbsp;''X''&nbsp;&times;&nbsp;''Y''&nbsp;&rarr;&nbsp;'''B''' that is commonly known as the conjunction ''xy''.  We described ''Df'' as a (first order) differential proposition, that is, a proposition of the type ''Df''&nbsp;:&nbsp;''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''dX''&nbsp;&times;&nbsp;''dY''&nbsp;&rarr;&nbsp;'''B'''.  Abstracting from the augmented venn diagram that illustrates how the ''models'', or the ''satisfying interpretations'', of ''Df'' distribute within the extended universe ''EU'' = ''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''dX''&nbsp;&times;&nbsp;''dY'', we can depict ''Df'' in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of ''U'' =  ''X''&nbsp;&times;&nbsp;''Y'' and whose arrows are labeled with the elements of ''dU'' = ''dX''&nbsp;&times;&nbsp;''dY''.

We have been studying the action of the difference operator ''D'', also known as the ''localization operator'', on the proposition ''f''&nbsp;:&nbsp;''X''&nbsp;&times;&nbsp;''Y''&nbsp;&rarr;&nbsp;'''B''' that is commonly known as the conjunction ''xy''.  We described ''Df'' as a (first order) differential proposition, that is, a proposition of the type ''Df''&nbsp;:&nbsp;''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''dX''&nbsp;&times;&nbsp;''dY''&nbsp;&rarr;&nbsp;'''B'''.  Abstracting from the augmented venn diagram that illustrates how the ''models'', or the ''satisfying interpretations'', of ''Df'' distribute within the extended universe ''EU'' = ''X''&nbsp;&times;&nbsp;''Y''&nbsp;&times;&nbsp;''dX''&nbsp;&times;&nbsp;''dY'', we can depict ''Df'' in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of ''U'' =  ''X''&nbsp;&times;&nbsp;''Y'' and whose arrows are labeled with the elements of ''dU'' = ''dX''&nbsp;&times;&nbsp;''dY''.

Any proposition worth its salt has many equivalent ways to view it, any one of which may reveal some unsuspected aspect of its meaning.  We will encounter more and more of these variant readings as we go.

Any proposition worth its salt has many equivalent ways to view it, any one of which may reveal some unsuspected aspect of its meaning.  We will encounter more and more of these variant readings as we go.

The enlargement operator ''E'', also known as the ''shift operator'', has many interesting and very useful properties in its own right, so let us not fail to observe a few of the more salient features that play out on the surface of our simple example, ''f''(''x'',&nbsp;''y'') = ''xy''.

The enlargement operator ''E'', also known as the ''shift operator'', has many interesting and very useful properties in its own right, so let us not fail to observe a few of the more salient features that play out on the surface of our simple example, ''f''(''x'',&nbsp;''y'') = ''xy''.

We may understand the enlarged proposition ''Ef'' as telling us all the different ways to reach a model of ''f'' from any point of the universe ''U''.

We may understand the enlarged proposition ''Ef'' as telling us all the different ways to reach a model of ''f'' from any point of the universe ''U''.

To broaden our experience with simple examples, let us now contemplate the sixteen functions of concrete type ''X''&nbsp;&times;&nbsp;''Y''&nbsp;&rarr;&nbsp;'''B''' and abstract type '''B'''&nbsp;&times;&nbsp;'''B'''&nbsp;&rarr;&nbsp;'''B'''.  For future reference, I will set here a few tables that detail the actions of ''E'' and ''D'' and on each of these functions, allowing us to view the results in several different ways.

To broaden our experience with simple examples, let us now contemplate the sixteen functions of concrete type ''X''&nbsp;&times;&nbsp;''Y''&nbsp;&rarr;&nbsp;'''B''' and abstract type '''B'''&nbsp;&times;&nbsp;'''B'''&nbsp;&rarr;&nbsp;'''B'''.  For future reference, I will set here a few tables that detail the actions of ''E'' and ''D'' and on each of these functions, allowing us to view the results in several different ways.

If the medium truly is the message, the blank slate is the innate idea.

If the medium truly is the message, the blank slate is the innate idea.

If you think that I linger in the realm of logical difference calculus out of sheer vacillation about getting down to the differential proper, it is probably out of a prior expectation that you derive from the art or the long-engrained practice of real analysis.  But the fact is that ordinary calculus only rushes on to the sundry orders of approximation because the strain of comprehending the full import of ''E'' and ''D'' at once whelm over its discrete and finite powers to grasp them.  But here, in the fully serene idylls of ZOL, we find ourselves fit with the compass of a wit that is all we'd ever wish to explore their effects with care.

If you think that I linger in the realm of logical difference calculus out of sheer vacillation about getting down to the differential proper, it is probably out of a prior expectation that you derive from the art or the long-engrained practice of real analysis.  But the fact is that ordinary calculus only rushes on to the sundry orders of approximation because the strain of comprehending the full import of ''E'' and ''D'' at once whelm over its discrete and finite powers to grasp them.  But here, in the fully serene idylls of ZOL, we find ourselves fit with the compass of a wit that is all we'd ever wish to explore their effects with care.

Amazing!

Amazing!

We have been contemplating functions of the type ''f''&nbsp;:&nbsp;''U''&nbsp;&rarr;&nbsp;'''B''', studying the action of the operators ''E'' and ''D'' on this family.  These functions, that we may identify for our present aims with propositions, inasmuch as they capture their abstract forms, are logical analogues of ''scalar potential fields''.  These are the sorts of fields that are so picturesquely presented in elementary calculus and physics textbooks by images of snow-covered hills and parties of skiers who trek down their slopes like least action heroes.  The analogous scene in propositional logic presents us with forms more reminiscent of plateaunic idylls, being all plains at one of two levels, the mesas of verity and falsity, as it were, with nary a niche to inhabit between them, restricting our options for a sporting gradient of downhill dynamics to just one of two, standing still on level ground or falling off a bluff.

We have been contemplating functions of the type ''f''&nbsp;:&nbsp;''U''&nbsp;&rarr;&nbsp;'''B''', studying the action of the operators ''E'' and ''D'' on this family.  These functions, that we may identify for our present aims with propositions, inasmuch as they capture their abstract forms, are logical analogues of ''scalar potential fields''.  These are the sorts of fields that are so picturesquely presented in elementary calculus and physics textbooks by images of snow-covered hills and parties of skiers who trek down their slopes like least action heroes.  The analogous scene in propositional logic presents us with forms more reminiscent of plateaunic idylls, being all plains at one of two levels, the mesas of verity and falsity, as it were, with nary a niche to inhabit between them, restricting our options for a sporting gradient of downhill dynamics to just one of two, standing still on level ground or falling off a bluff.

The same form of boundary relationship is exhibited for any cell of origin that one might elect to indicate, say, by means of the conjunction of positive and negative basis features ''u''₁&nbsp;&hellip;&nbsp;''u''_''k'', where ''u''_''j'' = ''x''_''j'' or ''u''_''j'' = (''x''_''j''), for ''j'' = 1 to ''k''.  The proposition (''u''₁,&nbsp;&hellip;,&nbsp;''u''_''k'') indicates the disjunctive region consisting of the cells that are "just next door" to the cell ''u''₁&nbsp;&hellip;&nbsp;''u''_''k''.

The same form of boundary relationship is exhibited for any cell of origin that one might elect to indicate, say, by means of the conjunction of positive and negative basis features ''u''₁&nbsp;&hellip;&nbsp;''u''_''k'', where ''u''_''j'' = ''x''_''j'' or ''u''_''j'' = (''x''_''j''), for ''j'' = 1 to ''k''.  The proposition (''u''₁,&nbsp;&hellip;,&nbsp;''u''_''k'') indicates the disjunctive region consisting of the cells that are "just next door" to the cell ''u''₁&nbsp;&hellip;&nbsp;''u''_''k''.

===Note 9===

===The Pragmatic Maxim===

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More on the pragmatic maxim as a representation principle later.

More on the pragmatic maxim as a representation principle later.

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This idea of contextual definition is basically the same as Jeremy Bentham's notion of ''paraphrasis'', a "method of accounting for fictions by explaining various purported terms away" (Quine, in Van Heijenoort, p. 216).  Today we'd call these constructions ''term models''.  This, again, is the big idea behind Schönfinkel's combinators {S, K, I}, and hence of lambda calculus, and I reckon you know where that leads.

This idea of contextual definition is basically the same as Jeremy Bentham's notion of ''paraphrasis'', a "method of accounting for fictions by explaining various purported terms away" (Quine, in Van Heijenoort, p. 216).  Today we'd call these constructions ''term models''.  This, again, is the big idea behind Schönfinkel's combinators {S, K, I}, and hence of lambda calculus, and I reckon you know where that leads.

Let me return to Peirce's early papers on the algebra of relatives to pick up the conventions that he used there, and then rewrite my account of regular representations in a way that conforms to those.

Let me return to Peirce's early papers on the algebra of relatives to pick up the conventions that he used there, and then rewrite my account of regular representations in a way that conforms to those.

I think that will serve to fix notation and set up the remainder of the account.

I think that will serve to fix notation and set up the remainder of the account.

It is common in algebra to switch around between different conventions of display, as the momentary fancy happens to strike, and I see that Peirce is no different in this sort of shiftiness than anyone else.  A changeover appears to occur especially whenever he shifts from logical contexts to algebraic contexts of application.

It is common in algebra to switch around between different conventions of display, as the momentary fancy happens to strike, and I see that Peirce is no different in this sort of shiftiness than anyone else.  A changeover appears to occur especially whenever he shifts from logical contexts to algebraic contexts of application.

So I still have a few wrinkles to iron out before I can give this story a smooth enough consistency.

So I still have a few wrinkles to iron out before I can give this story a smooth enough consistency.

Let us make up the model universe $1$ = ''A'' + ''B'' + ''C'' and the 2-adic relation ''n'' = "noter of", as when "''X'' is a data record that contains a pointer to ''Y''".  That interpretation is not important, it's just for the sake of intuition.  In general terms, the 2-adic relation ''n'' can be represented by this matrix:

Let us make up the model universe $1$ = ''A'' + ''B'' + ''C'' and the 2-adic relation ''n'' = "noter of", as when "''X'' is a data record that contains a pointer to ''Y''".  That interpretation is not important, it's just for the sake of intuition.  In general terms, the 2-adic relation ''n'' can be represented by this matrix:

This is consistent with the convention that Peirce uses in the paper "On a Class of Multiple Algebras" (CP 3.324–327).

This is consistent with the convention that Peirce uses in the paper "On a Class of Multiple Algebras" (CP 3.324–327).

We have been contemplating the virtues and the utilities of the pragmatic maxim as a standard heuristic in hermeneutics, that is, as a principle of interpretation that guides us in finding clarifying representations for a problematic corpus of symbols by means of their actions on other symbols or in terms of their effects on the syntactic contexts wherein we discover them or where we might conceive to distribute them.

We have been contemplating the virtues and the utilities of the pragmatic maxim as a standard heuristic in hermeneutics, that is, as a principle of interpretation that guides us in finding clarifying representations for a problematic corpus of symbols by means of their actions on other symbols or in terms of their effects on the syntactic contexts wherein we discover them or where we might conceive to distribute them.

I will have to leave that question as it is for now, in hopes that a solution will evolve itself in time.

I will have to leave that question as it is for now, in hopes that a solution will evolve itself in time.

===Note 15===

===Obstacles to Applying the Pragmatic Maxim===

 

'''Obstacles to Applying the Pragmatic Maxim'''

No sooner do you get a good idea and try to apply it than you find that a motley array of obstacles arise.

No sooner do you get a good idea and try to apply it than you find that a motley array of obstacles arise.

'''Obstacle 1.'''  People do not always read the instructions very carefully.  There is a tendency in readers of particular prior persuasions to blow the problem all out of proportion, to think that the maxim is meant to reveal the absolutely positive and the totally unique meaning of every preconception to which they might deign or elect to apply it.  Reading the maxim with an even minimal attention, you can see that it promises no such finality of unindexed sense, but ties what you conceive to you.  I have lately come to wonder at the tenacity of this misinterpretation.  Perhaps people reckon that nothing less would be worth their attention.  I am not sure.  I can only say the achievement of more modest goals is the sort of thing on which our daily life depends, and there can be no final end to inquiry nor any ultimate community without a continuation of life, and that means life on a day to day basis.  All of which only brings me back to the point of persisting with local meantime examples, because if we can't apply the maxim there, we can't apply it anywhere.

'''Obstacle 1.'''  People do not always read the instructions very carefully.  There is a tendency in readers of particular prior persuasions to blow the problem all out of proportion, to think that the maxim is meant to reveal the absolutely positive and the totally unique meaning of every preconception to which they might deign or elect to apply it.  Reading the maxim with an even minimal attention, you can see that it promises no such finality of unindexed sense, but ties what you conceive to you.  I have lately come to wonder at the tenacity of this misinterpretation.  Perhaps people reckon that nothing less would be worth their attention.  I am not sure.  I can only say the achievement of more modest goals is the sort of thing on which our daily life depends, and there can be no final end to inquiry nor any ultimate community without a continuation of life, and that means life on a day to day basis.  All of which only brings me back to the point of persisting with local meantime examples, because if we can't apply the maxim there, we can't apply it anywhere.

'''Obstacle 2.'''  Applying the pragmatic maxim, even with a moderate aim, can be hard.  I think that my present example, deliberately impoverished as it is, affords us with an embarassing richness of evidence of just how complex the simple can be.

'''Obstacle 2.'''  Applying the pragmatic maxim, even with a moderate aim, can be hard.  I think that my present example, deliberately impoverished as it is, affords us with an embarassing richness of evidence of just how complex the simple can be.

If the ante-rep looks the same as the post-rep, now that I'm writing them in the same dialect, that is because V₄ is abelian (commutative), and so the two representations have the very same effects on each point of their bearing.

If the ante-rep looks the same as the post-rep, now that I'm writing them in the same dialect, that is because V₄ is abelian (commutative), and so the two representations have the very same effects on each point of their bearing.

So long as we're in the neighborhood, we might as well take in some more of the sights, for instance, the smallest example of a non-abelian (non-commutative) group.  This is a group of six elements, say, ''G''&nbsp;=&nbsp;{''e'',&nbsp;''f'',&nbsp;''g'',&nbsp;''h'',&nbsp;''i'',&nbsp;''j''}, with no relation to any other employment of these six symbols being implied, of course, and it can be most easily represented as the permutation group on a set of three letters, say, ''X''&nbsp;=&nbsp;{''A'',&nbsp;''B'',&nbsp;''C''}, usually notated as ''G''&nbsp;=&nbsp;Sym(''X'') or more abstractly and briefly, as Sym(3) or ''S''₃.  Here are the permutation (= substitution) operations in Sym(''X''):

So long as we're in the neighborhood, we might as well take in some more of the sights, for instance, the smallest example of a non-abelian (non-commutative) group.  This is a group of six elements, say, ''G''&nbsp;=&nbsp;{''e'',&nbsp;''f'',&nbsp;''g'',&nbsp;''h'',&nbsp;''i'',&nbsp;''j''}, with no relation to any other employment of these six symbols being implied, of course, and it can be most easily represented as the permutation group on a set of three letters, say, ''X''&nbsp;=&nbsp;{''A'',&nbsp;''B'',&nbsp;''C''}, usually notated as ''G''&nbsp;=&nbsp;Sym(''X'') or more abstractly and briefly, as Sym(3) or ''S''₃.  Here are the permutation (= substitution) operations in Sym(''X''):

By the way, we will meet with the symmetric group ''S''₃ again when we return to take up the study of Peirce's early paper "On a Class of Multiple Algebras" (CP 3.324–327), and also his late unpublished work "The Simplest Mathematics" (1902) (CP 4.227–323), with particular reference to the section that treats of "Trichotomic Mathematics" (CP 4.307–323).

By the way, we will meet with the symmetric group ''S''₃ again when we return to take up the study of Peirce's early paper "On a Class of Multiple Algebras" (CP 3.324–327), and also his late unpublished work "The Simplest Mathematics" (1902) (CP 4.227–323), with particular reference to the section that treats of "Trichotomic Mathematics" (CP 4.307–323).

By way of collecting a short-term pay-off for all the work that we did on the regular representations of the Klein 4-group ''V''₄, let us write out as quickly as possible in "relative form" a minimal budget of representations for the symmetric group on three letters, Sym(3).  After doing the usual bit of compare and contrast among the various representations, we will have enough concrete material beneath our abstract belts to tackle a few of the presently obscur'd details of Peirce's early "Algebra + Logic" papers.

By way of collecting a short-term pay-off for all the work that we did on the regular representations of the Klein 4-group ''V''₄, let us write out as quickly as possible in "relative form" a minimal budget of representations for the symmetric group on three letters, Sym(3).  After doing the usual bit of compare and contrast among the various representations, we will have enough concrete material beneath our abstract belts to tackle a few of the presently obscur'd details of Peirce's early "Algebra + Logic" papers.

I have without stopping to think about it written out this natural representation of ''S''₃ in the style that comes most naturally to me, to wit, the "right" way, whereby an ordered pair configured as ''X'':''Y'' constitutes the turning of ''X'' into ''Y''.  It is possible that the next time we check in with CSP that we will have to adjust our sense of direction, but that will be an easy enough bridge to cross when we come to it.

I have without stopping to think about it written out this natural representation of ''S''₃ in the style that comes most naturally to me, to wit, the "right" way, whereby an ordered pair configured as ''X'':''Y'' constitutes the turning of ''X'' into ''Y''.  It is possible that the next time we check in with CSP that we will have to adjust our sense of direction, but that will be an easy enough bridge to cross when we come to it.

To construct the regular representations of ''S''₃, we pick up from the data of its operation table:

To construct the regular representations of ''S''₃, we pick up from the data of its operation table:

If the ante-rep looks different from the post-rep, it is just as it should be, as ''S''₃ is non-abelian (non-commutative), and so the two representations differ in the details of their practical effects, though, of course, being representations of the same abstract group, they must be isomorphic.

If the ante-rep looks different from the post-rep, it is just as it should be, as ''S''₃ is non-abelian (non-commutative), and so the two representations differ in the details of their practical effects, though, of course, being representations of the same abstract group, they must be isomorphic.

<blockquote>

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[http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lie.html Biographical Data for Marius Sophus Lie (1842–1899)]

[http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lie.html Biographical Data for Marius Sophus Lie (1842–1899)]

We've seen a couple of groups, ''V''₄ and ''S''₃, represented in various ways, and we've seen their representations presented in a variety of different manners.  Let us look at one other stylistic variant for presenting a representation that is frequently seen, the so-called "matrix representation" of a group.

We've seen a couple of groups, ''V''₄ and ''S''₃, represented in various ways, and we've seen their representations presented in a variety of different manners.  Let us look at one other stylistic variant for presenting a representation that is frequently seen, the so-called "matrix representation" of a group.