Saturday, November 7, 2009

The Problem of Logical Omniscience and Inferential Role

I just looked over a very old thing I wrote about the problem of logical omniscience. The problem of Logical Omniscience is: How can you count as believing one thing, while not believing (or even explicitly rejecting) something logically equivalent?

I suggested that propositions have certain preferred inferential roles, and that you count as believing that P to the extend that you are disposed to make enough of these preferred inferences, quickly and confidently enough.

So for example, someone can believe that a function is Turing computable but not that it's recursive, even though these two statements are provably equivalent, because they might be willing to make enough of the characteristic inferences associated with Turing computability, but not those for recursive-ness. (The characteristic inferences for " Turing computable" would be those that people call "immediate" from the definition of Turing computability, and ditto for the -different- characteristic inferences for recursive).

This is interesting because:
1. The characteristic inferences associated with a proposition/word will NOT supervene on the inferences which that proposition/word justifies. Since Turing computability and recursive-ness are probably equivalent, the very same inferences are JUSTIFIED for each one of them. But "This function is Turing computable" and "This function is recursive" need to have different characteristic inferences, to explain how you can know one but not the other.

2. Given (1), if you want to attach meanings to individual words, these meanings should not only include things like sense and reference which help build up the truth conditions for sentences involving that word, but also something like characteristic inferences, which helps you chose when to attribute someone a belief involving this word, rather than another which word would always contribute in exactly the same way to the truth conditions of any sentence.

2. It's commonly said that aliens would have the same math as us. If this means that they wouldn't disagree with us about math that sounds right. But if it means that they would (before contact with humans) believe literally the same propositions as we do, I don't think so.

For, think about all the many different notions we could define which would be equivalent to Turing computability, but have different characteristic inferences. If you buy the above, each of these notions corresponds to a slightly different thought. Thus for the aliens to believe the exact same mathematical claims as we do, they would have to have the same definitions/mathematical concepts. But it's much less clear whether aliens would have the same aesthetic sense guiding what definitions they made/mathematical concepts they came up with. For example, I'm much more convinced that aliens would accept topology than that they would have come up with it. I mean, just think about the different kinds of math developed just by humans in different eras and countries.

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