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Directory:Jon Awbrey/Papers/Differential Logic (view source)
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You may be wondering what happened to the announced subject of ''Differential Logic'', and if you think that we have been taking a slight ''excursion'' — to use my favorite euphemism for ''digression'' — my reply to the charge of a scenic rout would need to be both "yes and no". What happened was this. At the sign-post marked by Sigil 7, we made the observation that the shift operators <math>\operatorname{E}_{ij}\!</math> form a transformation group that acts on the propositions of the form <math>f : \mathbb{B}^2 \to \mathbb{B}.</math> Now group theory is a very attractive subject, but it did not really have the effect of drawing us so far off our initial course as you may at first think. For one thing, groups, in particular, the groups that have come to be named after the Norwegian mathematician Marius Sophus Lie, have turned out to be of critical utility in the solution of differential equations. For another thing, group operations afford us examples of triadic relations that have been extremely well-studied over the years, and this provides us with quite a bit of guidance in the study of sign relations, another class of triadic relations of significance for logical studies, in our brief acquaintance with which we have scarcely even started to break the ice. Finally, I could hardly avoid taking up the connection between group representations, 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'', and if you think that we have been taking a slight ''excursion'' — to use my favorite euphemism for ''digression'' — my reply to the charge of a scenic rout would need to be both "yes and no". What happened was this. At the sign-post marked by Sigil 7, we made the observation that the shift operators <math>\operatorname{E}_{ij}\!</math> form a transformation group that acts on the propositions of the form <math>f : \mathbb{B}^2 \to \mathbb{B}.</math> Now group theory is a very attractive subject, but it did not really have the effect of drawing us so far off our initial course as you may at first think. For one thing, groups, in particular, the groups that have come to be named after the Norwegian mathematician Marius Sophus Lie, have turned out to be of critical utility in the solution of differential equations. For another thing, group operations afford us examples of triadic relations that have been extremely well-studied over the years, and this provides us with quite a bit of guidance in the study of sign relations, another class of triadic relations of significance for logical studies, in our brief acquaintance with which we have scarcely even started to break the ice. Finally, I could hardly avoid taking up the connection between group representations, a very generic class of logical models, and the all-important pragmatic maxim.
* [http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Lie.html Biographical Data for Marius Sophus Lie (1842–1899)]
* [http://www-history.mcs.st-andrews.ac.uk/Biographies/Lie.html Biographical Data for Marius Sophus Lie (1842–1899)]
==Note 23==
==Note 23==