Tuesday, March 25, 2008

0 and 1

There is an interesting discussion going on at FQI about noncognitivism and logic; I show up a few times in the comments thread. One of my arguments there is that propositional logic is simply an interpreted algebra. Thus none of its features have any necessary relation to the things that most interest philosophers in using it -- truth values. You can have a system of reasoning that has exactly the same structure in which nothing has a truth value. All you have to do is find another way to interpret the 0's and 1's of the algebra. And it is not difficult to find other workable interpretations; computer scientists use a different interpretation of the same algebra for some of the things they do. You can have topological interpretations, about which I know very little; and, of course, you can just take the algebra at face value. I'm sure there are many other viable interpretations. And in all of these the logical structure is preserved. So you don't need truth values to have the same inferences that you get in propositional logic; you just need some viable interpretation of 0 and 1.

I've recently come to think that this is a fundamentally important fact that we philosophers need to take more trouble to remember than we do -- we get so caught up in truth functions that we forget that what they really are is a mathematical structure that's convenient (at least for some purposes) for modelling true and false; the 'true' and the 'false' do no significant logical work. They are, I would suggest, epiphenomenal; or, perhaps more accurately, bridging concepts that facilitate application of the mathematical structure to model something that's not itself mathematical. In any case, there is nothing sacred about the way we do things; and nobody's proven that there isn't some fantastically better way to do it, waiting to be discovered. But discovering it requires recognizing that the world is bigger than our ordinary tools would suggest.

That I have become convinced of this is an extraordinary pain, because having been taught in the modern school systems, my mathematical background is extraordinarily patchwork, piecemeal, and undeveloped; and thus it hurts to have to just go back to basics and learn on my own all the things I should have been taught about (say) rings and fields but no one ever, ever mentioned to me. I expect many long years of making many stupid mistakes in the process. But as far as I can see, there's no helping it.

There's an article at the SEP, by the way, on the mathematics of Boolean algebras. It is singularly unilluminating to anyone who might actually be ignorant enough to need it; so I recommend you find some other introduction. Actually, Wikipedia usually isn't a bad place to start; it usually has decent links and not-overly-misleading explanations. Perhaps there are lots of computer science majors who edit Wikipedia rather than sleeping....

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