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/Godel%27s_theorem
http://en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem

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/Halting_problem

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/Arithmetical_hierarchy

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/Hyperarithmetical_hierarchy
http://en.wikipedia.org/wiki/Analytical_hierarchy
http://en.wikipedia.org/wiki/Projective_hierarchy

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/Saul_Kripke#Truth
http://plato.stanford.edu/entries/truth-revision/
http://en.wikipedia.org/wiki/Paraconsistent_logic

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/Ray_Solomonoff), because it assumes that
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.

--Abram Demski

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