I recently joined the everything list. It looks to be some good philosophical fun. The list practice is to submit a "joining post" that reviews intellectual background. I am replicating mine here.

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Hi everyone!

My name is Abram Demski. My interest, when it comes to this list, is:

what is the correct logic, the logic that can refer to (and reason

about) any mathematical structure? The logic that can define

everything definable? If every possible universe exists, then of

course we've got to ask which universes are possible. As someone

mentioned recently, a sensible approach is to take the logically

consistent ones. So, I'm asking: in what logic?

I am also interested in issues of specifying a probability

distribution over these probabilities, and what such a probability

distribution really means. Again there was some recent discussion on

this... I was very tempted to comment, but I wanted to lurk a while to

get the idea of the group before posting my join post.

Following is my view on what the big questions are when it comes to

specifying the correct logic.

The first two big puzzles are presented to us by Godel's

incompleteness theorem and Tarski's undefinability theorem. The way I

see it, Godel's theorem presents a "little" puzzle, which points us in

the direction of the "big" puzzle presented by Tarski's theorem.

http://en.wikipedia.org/wiki/

http://en.wikipedia.org/wiki/

The little puzzle is this: Godel's theorem tells us that any

sufficiently strong logic does not have a complete set of deduction

rules; the axioms will fail to capture all truths about the logical

entities we're trying to talk about. But if these entities cannot be

completely axiomized, then in what sense are they well-defined? How is

logic logical, if it is doomed to be incompletely specified? One way

out here is to say that numbers (which happen to be the logical

entities that Godel showed were doomed to incompleteness, though of

course the incompleteness theorem has since been generalized to other

domains) really are incompletely specified: the axioms are incomplete

in that they fail to prove every sentence about numbers either true or

false, but they are complete in that the ones they miss are in some

sense actually not specified by our notion of number. I don't like

this answer, because it is equivalent to saying that the halting

problem really has no answer in the cases where it is undecidable.

http://en.wikipedia.org/wiki/

Instead, I prefer to say that while decidable facts correspond to

finite computations, undecidable facts simply correspond to infinite

computations; so, there is still a well-defined procedure for deciding

them, it simply takes too long for us to complete. For the case of

number theory, this can be formalized with the arithmetical hierarchy:

http://en.wikipedia.org/wiki/

Essentially, each new quantifier amounts to a potentially infinite

number of cases we need to check. There are similar hierarchies for

broader domains:

http://en.wikipedia.org/wiki/

http://en.wikipedia.org/wiki/

http://en.wikipedia.org/wiki/

This brings us to the "big" puzzle. To specify the logic an refer to

any structure I want, I need to define the largest of these

hierarchies: a hierarchy that includes all truths of mathematics.

Unfortunately, Tarski's undefinability theorem presents a roadblock to

this project: If I can use logic L to define a hierarchy H, then H

will necessarily fail to include all truths of L. To describe the

hierarchy of truths for L, I will always need a more powerful language

L+1. Tarski proved this under some broad assumptions; since Tarski's

theorem completely blocks my project, it appears I need to examine

these assumptions and reject some of them.

I am, of course, not the first to pursue such a goal. There is an

abundant literature on theories of truth. From what I've seen, the

important potential solutions are Kripke's fixed-points, revision

theories, and paraconsistent theories:

http://en.wikipedia.org/wiki/

http://plato.stanford.edu/

http://en.wikipedia.org/wiki/

All of these solutions create reference gaps: they define a language L

that can talk about all of its truths, and therefore could construct

its own hierarchy in one sense, but in addition to simple true and

false more complicated truth-states are admitted that the language

cannot properly refer to. For Kripke's theory, we are unable to talk

about the sentences that are neither-true-nor-false. For revision

theories, we are unable to talk about which sentences have unstable

truth values or multiple stable truth values. In paraconsistent logic,

we are able to refer to sentences that are both-true-and-false, but we

can't state within the language that a statement is *only* true or

*only* false (to my knowledge; paraconsistent theory is not my strong

suit). So using these three theories, if we want a hierarchy that

defines all the truth value *combinations* within L, we're still out

of luck.

As I said, I'm also interested in the notion of probability. I

disagree with Solomonoff's universal distribution

(http://en.wikipedia.org/wiki/

the universe is computable. I cannot say whether the universe we

actually live in is computable or not; however, I argue that,

regardless, an uncomputable universe is at least conceivable, even if

it has a low credibility. So, a universal probability distribution

should include that possibility.

I also want to know exactly what it means to measure a probability. I

think use of subjective probabilities is OK; a probability can reflect

a state of belief. But, I think the reason that this is an effective

way of reasoning is because these subjective probabilities tend to

converge to the "true" probabilities as we gain experience. It seems

to me that this "true probability" needs to be a frequency. It also

seems to me that this would be meaningful even in universes that

actually happened to have totally deterministic physics-- so by a

"true probability" I don't mean to imply a physically random outcome,

though I don't mean to rule it out either (like uncomputable

universes, I think it should be admitted as possible).

Well, I think that is about it. For now.

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