[Inquiry] Peirce's Rules Of Inference

[Jon Awbrey]

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PROI.  Note 4

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Response to message by Frithjof Dau on the CG List:

Frithjof & All,

I will have to save the bits about P=?=NP for when I have
the leisure to arrange my ducks more serially, but let me
skip to the question about normal forms and try to tackle
it the easiest possible way, by way of concrete examples.

I have come to view a proof as an example of a semiotic process,
a sequence of transformations "relieved against the background"
of a triadic sign relation that comes with the formal system in
view.  But there are many different types of proof, and clearly
not all semioses are proofs.

For instance, I notice that John Sowa often gives us examples of
proofs that start out with the sign that we call a blank SA, read
as "true" under the existential interpretation, and then transforms
it under the permissions of Peirce's rules to the sign that we want
to prove is a theorem.  I forget whether he likes equational (2-way)
or implicational (1-way) transformations the best, but I found myself
being led early on to maintain reversible transformations for as long
as it was possible to do so during the progress of a proof, as these
preserve all of the information that is borne by the orginal sign.

Incidentally, somewhere in Peirce's manuscripts, that I was long ago
reading a microfilm copy of for my undergrad work, he comments on
this very difference between equational and modus ponens styles
of semioses and he says something to the effect that if he had
to start from scratch all over again that he would probably
emphasize the equational, information-preserving type of
process more than he had done. (I'd greatly appreciate
it if anybody knows this passage -- I think he uses
an example of a boy with a milk-pail (?)).

Well, I'm rambling, but I have to break here anyway ...

Jon Awbrey

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