SUO: Differential Logic
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
SUO Working Group,
One way to think about mappings between different ontologies, and also about
ontologies that develop through time -- the two problems are intimately related --
is in terms of transformations from universe of discourse to universe of discourse,
the sort of thing that one is naturally tempted to call a "transformation of discourse".
Re-starting from the ground up, as experience constantly teaches that we must,
we may contemplate simple propositions and unanalyzed predications of the sort
that one finds pictured in euler-venn diagrams and that one computes in terms
of bits and boolean functions.
So, working at the basement level, a mapping between two different universes of discourse,
which may of course only be two different ways of describing the same universe of discourse,
could be written F : U -> V, and a transformation that describes the changes that occur in
a single universe of discourse, which may of course come in the corresponding varieties of
"alias" and "alibi" flavors, could be written F : U -> U.
Transformations like these can be very complex things to think about --
for instance, one may be thinking of a neuroid system of formal neurons
that carry one bit each, and so one's universe is a state space U that is
isomorphic to B^n, where B = {0, 1} and n is roughly 10^10, give or take --
so we usually end up having to approach such creatures, the transformations
F : U -> U and F : U1 -> U2, in series of increasing orders of approximation.
That is what differential calculus is all about. A "derivative" or a "differential"
of a transformation F is a "locally linear approximation" to F. In many ways, one
can think of differentiation as an operation that takes the global description of
a transformation and distributes the information into locally relevant forms.
These days, differential calculus and differential geometry are carried out in terms
of a thing called the "tangent functor", which is the category theoretic expression
of what we do when we take derivatives. A "functor" W is a "mapping of maps" or
a "transformation of transformations", so W takes a map F : X -> Y into another
map WF : WX -> WY. Roughly speaking then, the particular sort of functor that
we will soon know and love as the "tangent functor" T is one that takes the
map F : X -> Y and gives back the locally relevant version TF : TX -> TY.
Cranking the analogy for logic produces the subject of "differential logic",
which has been my pursuit for a decade or two. I am almost done serializing
one of my more coherent, but also more detailed, papers on the subject, and I
have appended the outline of links so far.
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Differential Logic and Dynamic Systems
0. Purpose
1. Review and Transition
D01. http://suo.ieee.org/ontology/msg04799.html
2. Functional Conception of Propositional Calculus
D02. http://suo.ieee.org/ontology/msg04800.html
D03. http://suo.ieee.org/ontology/msg04801.html
D04. http://suo.ieee.org/ontology/msg04802.html
D05. http://suo.ieee.org/ontology/msg04803.html
D06. http://suo.ieee.org/ontology/msg04804.html
D07. http://suo.ieee.org/ontology/msg04805.html
D08. http://suo.ieee.org/ontology/msg04806.html
D09. http://suo.ieee.org/ontology/msg04807.html
D10. http://suo.ieee.org/ontology/msg04808.html
D11. http://suo.ieee.org/ontology/msg04809.html
3. Differential Extension of Propositional Calculus
D12. http://suo.ieee.org/ontology/msg04810.html
D13. http://suo.ieee.org/ontology/msg04811.html
D14. http://suo.ieee.org/ontology/msg04812.html
D15. http://suo.ieee.org/ontology/msg04813.html
D16. http://suo.ieee.org/ontology/msg04814.html
D17. http://suo.ieee.org/ontology/msg04815.html
4. Back to the Beginning: Some Exemplary Universes
D18. http://suo.ieee.org/ontology/msg04816.html
D19. http://suo.ieee.org/ontology/msg04817.html
D20. http://suo.ieee.org/ontology/msg04818.html
D21. http://suo.ieee.org/ontology/msg04819.html
D22. http://suo.ieee.org/ontology/msg04820.html
D23. http://suo.ieee.org/ontology/msg04821.html
D24. http://suo.ieee.org/ontology/msg04822.html
5. Transformations of Discourse
D25. http://suo.ieee.org/ontology/msg04823.html
5.1. Foreshadowing Transformations: Extensions and Projections of Discourse
D26. http://suo.ieee.org/ontology/msg04824.html
D27. http://suo.ieee.org/ontology/msg04825.html
5.2. Thematization of Functions: And a Declaration of Independence for Variables
D28. http://suo.ieee.org/ontology/msg04826.html
D29. http://suo.ieee.org/ontology/msg04827.html
D30. http://suo.ieee.org/ontology/msg04828.html
D31. http://suo.ieee.org/ontology/msg04829.html
D32. http://suo.ieee.org/ontology/msg04830.html
D33. http://suo.ieee.org/ontology/msg04832.html
5.3. Propositional Transformations
D34. http://suo.ieee.org/ontology/msg04833.html
D35. http://suo.ieee.org/ontology/msg04834.html
D36. http://suo.ieee.org/ontology/msg04835.html
5.4. Analytic Expansions: Operators and Functors
D37. http://suo.ieee.org/ontology/msg04836.html
D38. http://suo.ieee.org/ontology/msg04837.html
D39. http://suo.ieee.org/ontology/msg04838.html
D40. http://suo.ieee.org/ontology/msg04839.html
D41. http://suo.ieee.org/ontology/msg04840.html
D42. http://suo.ieee.org/ontology/msg04841.html
D43. http://suo.ieee.org/ontology/msg04842.html
D44. http://suo.ieee.org/ontology/msg04843.html
D45. http://suo.ieee.org/ontology/msg04844.html
5.5. Transformations of Type B^2 -> B^1
D46. http://suo.ieee.org/ontology/msg04845.html
D47. http://suo.ieee.org/ontology/msg04846.html
D48. http://suo.ieee.org/ontology/msg04847.html
D49. http://suo.ieee.org/ontology/msg04848.html
D50. http://suo.ieee.org/ontology/msg04849.html
D51. http://suo.ieee.org/ontology/msg04850.html
D52. http://suo.ieee.org/ontology/msg04851.html
D53. http://suo.ieee.org/ontology/msg04852.html
D54. http://suo.ieee.org/ontology/msg04853.html
D55. http://suo.ieee.org/ontology/msg04854.html
D56. http://suo.ieee.org/ontology/msg04855.html
D57. http://suo.ieee.org/ontology/msg04856.html
D58. http://suo.ieee.org/ontology/msg04857.html
D59. http://suo.ieee.org/ontology/msg04858.html
D60. http://suo.ieee.org/ontology/msg04859.html
D61. http://suo.ieee.org/ontology/msg04860.html
D62. http://suo.ieee.org/ontology/msg04861.html
D63. http://suo.ieee.org/ontology/msg04862.html
D64. http://suo.ieee.org/ontology/msg04863.html
D65. http://suo.ieee.org/ontology/msg04864.html
D66. http://suo.ieee.org/ontology/msg04865.html
D67. http://suo.ieee.org/ontology/msg04866.html
D68. http://suo.ieee.org/ontology/msg04867.html
D69. http://suo.ieee.org/ontology/msg04868.html
D70. http://suo.ieee.org/ontology/msg04869.html
5.6. Taking Aim at Higher Dimensional Targets
D71. http://suo.ieee.org/ontology/msg04870.html
D72. http://suo.ieee.org/ontology/msg04871.html
5.7. Transformations of Type B^2 -> B^2
D73. http://suo.ieee.org/ontology/msg04872.html
D74. http://suo.ieee.org/ontology/msg04873.html
D75. http://suo.ieee.org/ontology/msg04874.html
D76. http://suo.ieee.org/ontology/msg04875.html
D77. http://suo.ieee.org/ontology/msg04876.html
D78. http://suo.ieee.org/ontology/msg04877.html
D79. http://suo.ieee.org/ontology/msg04878.html
D80. http://suo.ieee.org/ontology/msg04879.html
D81. http://suo.ieee.org/ontology/msg04880.html
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o