ONT Re: Differential Logic A -- Discussion
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DLOG A. Discussion Note 17
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Let me see if I can still remember how it came about that
the pragmatic maxim emerged in the context of pouring the
foundations for differential logic.
I was looking at sets of propositions, conceived as sets of functions
of the form f : B^k -> B. Then I contemplated the action of a couple
of operators on these sets of propositions, operators that were meant
as logical analogues of the usual finite difference operators E and D.
Of course, E and D refer to whole parameterized families of operators,
where "E" denotes the enlargement or shift operators, and "D" denotes
the difference or delta operators. I wrote all this out for the case
of k = 2 starting at this point:
Re: Differential Logic A6. http://suo.ieee.org/ontology/msg05368.html
Table 4. E(f) Expanded Over Differential Features {dx, dy}
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| | | | | | |
| | f | T_11 f | T_10 f | T_01 f | T_00 f |
| | | | | | |
| | | Ef| dx dy | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | x y | x (y) | (x) y | (x)(y) |
| | | | | | |
| f_2 | (x) y | x (y) | x y | (x)(y) | (x) y |
| | | | | | |
| f_4 | x (y) | (x) y | (x)(y) | x y | x (y) |
| | | | | | |
| f_8 | x y | (x)(y) | (x) y | x (y) | x y |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | x | x | (x) | (x) |
| | | | | | |
| f_12 | x | (x) | (x) | x | x |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | (x, y) | ((x, y)) | ((x, y)) | (x, y) |
| | | | | | |
| f_9 | ((x, y)) | ((x, y)) | (x, y) | (x, y) | ((x, y)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | y | (y) | y | (y) |
| | | | | | |
| f_10 | y | (y) | y | (y) | y |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((x)(y)) | ((x) y) | (x (y)) | (x y) |
| | | | | | |
| f_11 | (x (y)) | ((x) y) | ((x)(y)) | (x y) | (x (y)) |
| | | | | | |
| f_13 | ((x) y) | (x (y)) | (x y) | ((x)(y)) | ((x) y) |
| | | | | | |
| f_14 | ((x)(y)) | (x y) | (x (y)) | ((x) y) | ((x)(y)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | (()) | (()) | (()) | (()) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | |
| Fixed Point Total | 4 | 4 | 4 | 16 |
| | | | | |
o-------------------o------------o------------o------------o------------o
In this context, the set of shift operators acts as a mathematical group
on the set of propositions, splitting it up into disjoint and exhaustive
subsets that are technically known as "group-reduced equivalence classes",
but that are rather more commonly known as "orbits". Generally speaking,
all of the elements in the same orbit have some common property that is
said to be "preserved" by the action of the group.
For example, consider the case for k = 2 that is presented in Table 4.
The set of shift operators, redescribed as a set of 4 transformations,
T_ij in {T_00, T_01, T_10, T_11}, form a mathematical group that acts
on the set of 16 propositions, in such a way that the 16 propositions
are partitioned into exactly 7 orbits. Looking over the propositions
in each orbit we can see that the members of the same orbit all have
similar geometric "shapes" when viewed as figures in a venn diagram.
Why is this important?
Here are some reasons that come to mind:
1. Group invariants afford us with one of the ways that invariant properties
of objects, actual and formal, commonly come to be recognized in practice.
2. Group actions, by gathering together "birds of a feather", that is,
objects that have some order of similar structure, or isomorphisms,
into common orbits, act to reduce the complexity of the underlying
domain in ways that may also reduce the computational complexity
of working with objects in that domain.
3. Group representations, that is, the representations of groups that we get
by considering them as sets of operators on sets of relatively concrete
objects, are prime examples of how to form operational definitions of
abstruse concepts. In this way they provide us with useful guidance
as to how we might apply the pragmatic maxim in more general cases.
For example, Peirce gave operational representations of concepts
like "truth" and "falsity" by means of the same sort of tactic.
Now, you must not expect the last word on defining a difficult
concept to come from such an isolated form of representation,
but even the smallest exemplar of the concept can make its
contribution to dispelling its more problematic mysteries.
Jon Awbrey
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http://www.cs.bsu.edu/homepages/mighty/history.html
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