Changes

==Casual introduction==

==Casual introduction==

==Formal development==

==Formal development==

==Expository examples==

==Expository examples==

==Differential Logic : First Approach==

==Differential Logic : First Approach==

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This table is abstractly the same as, or isomorphic to, the versions with the ''E''_''ij'' operators and the ''T''_''ij'' transformations that we discussed earlier.  That is to say, the story is the same — only the names have been changed.  An abstract group can have a multitude of significantly and superficially different representations.  Even after we have long forgotten the details of the particular representation that we may have come in with, there are species of concrete representations, called the ''regular representations'', that are always readily available, as they can be generated from the mere data of the abstract operation table itself.

This table is abstractly the same as, or isomorphic to, the versions with the ''E''_''ij'' operators and the ''T''_''ij'' transformations that we discussed earlier.  That is to say, the story is the same - only the names have been changed.  An abstract group can have a multitude of significantly and superficially different representations.  Even after we have long forgotten the details of the particular representation that we may have come in with, there are species of concrete representations, called the ''regular representations'', that are always readily available, as they can be generated from the mere data of the abstract operation table itself.

For example, select a group element from the top margin of the Table, and "consider its effects" on each of the group elements as they are listed along the left margin.  We may record these effects as Peirce usually did, as a logical "aggregate" of elementary dyadic relatives, that is to say, a disjunction or a logical sum whose terms represent the ordered pairs of <input : output> transactions that are produced by each group element in turn.  This yields what is usually known as one of the ''regular representations'' of the group, specifically, the ''first'', the ''post-'', or the ''right'' regular representation.  It has long been conventional to organize the terms in the form of a matrix:

For example, select a group element from the top margin of the Table, and "consider its effects" on each of the group elements as they are listed along the left margin.  We may record these effects as Peirce usually did, as a logical "aggregate" of elementary dyadic relatives, that is to say, a disjunction or a logical sum whose terms represent the ordered pairs of <input : output> transactions that are produced by each group element in turn.  This yields what is usually known as one of the ''regular representations'' of the group, specifically, the ''first'', the ''post-'', or the ''right'' regular representation.  It has long been conventional to organize the terms in the form of a matrix:

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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.

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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.

You may be wondering what happened to the announced subject of "Differential Logic".  If you think that we have been taking a slight excursion my reply to the charge of a scenic rout would be both "yes and no".  What happened was this.  We chanced to make the observation that the shift operators E_''ij'' form a transformation group that acts on the set of propositions of the form ''f''&nbsp;:&nbsp;'''B'''²&nbsp;&rarr;&nbsp;'''B'''.  Group theory is a very attractive subject, but it did not have the effect of drawing us so far off our initial course as one might at first think.  For one thing, groups, in particular, the special family of groups that have come to be named after the Norwegian mathematician Marius Sophus Lie, turn out to be of critical importance in the solution of differential equations.  For another thing, group operations afford us examples of 3-adic relations that have been extremely well-studied over the years, and thus they supply us with no small bit of guidance in the study of sign relations, another class of 3-adic relations that have significance for logical studies, in our brief acquaintance with which we have scarcely even begun to break the ice.  Finally, I could not resist taking up the connection between group representations, which constitute a very generic class of logical models, and the all-important pragmatic maxim.

You may be wondering what happened to the announced subject of "Differential Logic".  If you think that we have been taking a slight excursion my reply to the charge of a scenic rout would be both "yes and no".  What happened was this.  We chanced to make the observation that the shift operators E_''ij'' form a transformation group that acts on the set of propositions of the form ''f''&nbsp;:&nbsp;'''B'''²&nbsp;&rarr;&nbsp;'''B'''.  Group theory is a very attractive subject, but it did not have the effect of drawing us so far off our initial course as one might at first think.  For one thing, groups, in particular, the special family of groups that have come to be named after the Norwegian mathematician Marius Sophus Lie, turn out to be of critical importance in the solution of differential equations.  For another thing, group operations afford us examples of 3-adic relations that have been extremely well-studied over the years, and thus they supply us with no small bit of guidance in the study of sign relations, another class of 3-adic relations that have significance for logical studies, in our brief acquaintance with which we have scarcely even begun to break the ice.  Finally, I could not resist taking up the connection between group representations, which constitute a very generic class of logical models, and the all-important pragmatic maxim.

[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.

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Then we rewrote these permutations — since they are functions ''f''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;''X'' they can also be recognized as 2-adic relations ''f''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''X'' — in "relative form", in effect, in the manner to which Peirce would have made us accustomed had he been given a relative half-a-chance:

Then we rewrote these permutations - since they are functions ''f''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;''X'' they can also be recognized as 2-adic relations ''f''&nbsp;&sube;&nbsp;''X''&nbsp;&times;&nbsp;''X'' - in "relative form", in effect, in the manner to which Peirce would have made us accustomed had he been given a relative half-a-chance:

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* Edelman, Gerald M. (1988), ''Topobiology : An Introduction to Molecular Embryology'', Basic Books, New York, NY.

* Edelman, Gerald M. (1988), ''Topobiology : An Introduction to Molecular Embryology'', Basic Books, New York, NY.

* Leibniz, Gottfried Wilhelm, Freiherr von, ''Theodicy : Essays on the Goodness of God, The Freedom of Man, and The Origin of Evil'', Austin Farrer (ed.), E.M. Huggard (trans.), based on C.J. Gerhardt (ed.), ''Collected Philosophical Works'', 1875–1890, Routledge and Kegan Paul, London, UK, 1951.  Reprinted, Open Court, La Salle, IL, 1985.

* Leibniz, Gottfried Wilhelm, Freiherr von, ''Theodicy : Essays on the Goodness of God, The Freedom of Man, and The Origin of Evil'', Austin Farrer (ed.), E.M. Huggard (trans.), based on C.J. Gerhardt (ed.), ''Collected Philosophical Works'', 1875-1890, Routledge and Kegan Paul, London, UK, 1951.  Reprinted, Open Court, La Salle, IL, 1985.

* McClelland, James L., and Rumelhart, David E. (1988), ''Explorations in Parallel Distributed Processing : A Handbook of Models, Programs, and Exercises'', MIT Press, Cambridge, MA.

* McClelland, James L., and Rumelhart, David E. (1988), ''Explorations in Parallel Distributed Processing : A Handbook of Models, Programs, and Exercises'', MIT Press, Cambridge, MA.