Tuesday, December 16, 2008

Tom's Algebra of Logic and James's Boundary Calculus

I have been toying around with ways to represent propositional-logic forms of Carroll's literal diagrams in a notation that could potentially allow for clarification of complicated instances (many terms) and yet would still preserve something of the diagrammatic character of the diagrams. And the most promising was a version of notation borrowed from Jeffrey James's version of boundary mathematics (PDF). Such a notation has three key features:

instance ()
abstract []
inverse <>

Out of these you can build number systems. But you can also do any sort of Boolean algebra with them, which means you can do standard propositional logic. So, for instance, this can be the disjunction (p v q):

(pq)

This would then be conjunction:

([p][q])

To negate anything, we take its inverse, which, as noted above, is done with angle brackets:

<p>

Then rules for doing things with these brackets make it possible to handle all the transformations of standard propositional logic.

In toying with this a bit, I realized that it seemed a bit familiar; and indeed it was. It's closely related to Tom's algebra of logic. Here are Tom's types applied to propositional logic with their boundary counterparts:

[+a+b] : (ab)
[+a-b] : (a<b>)
[-a+b] : (<a>b)
[-a-b] : (<a><b>)
(+a+b) : ([a][b])
(+a-b) : ([a][<b>])
(-a+b) : ([<a>][b])
(-a-b) : ([<a>][<b>])

The brackets are reversed, but of course that's arbitrary. Of course, one would expect similarities in notations; but it never would have occurred to me offhand that Tom's algebra of logic was in the same family as boundary mathematics and so it was a bit of a surprise. (Surprise, I find, is almost the essence of logic.) But it means that one should be able to adapt boundary mathematics to do categorical syllogisms; and, vice versa, Tom's algebra of logic to do a hefty amount of mathematics. A smarter person wouldn't have been surprised by that, either; but I am very slow about these things. Something for me to think about, in both directions.

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