Please help me refine a dissertation topic in mathematical logic

[wigglywoogly]

I'm looking to write a dissertation in the field of logic (for a philosophy degree).

I'm deeply interested in logic, but whenever I consider the material beyond my courses it becomes pretty daunting. I'm reasonably familiar with:

*First Order Logic
*Set Theory and ZFC
*Cantor's Diagonal Argument
*Church/Turing Thesis
*Godel / Tarski / Lob limitative results (I find some of this pretty tricky!)
*Probability Theory
*Utility Theory (Von Neumann / Morgenstern etc.)
*Formal Semantics

I would really like to talk about infinity / the continuum / aleph-null; but beyond waxing lyrical about how cool they are, I'm not exactly sure what I'd say.

I don't know whether I'll end up choosing something too hard. Hopefully this isn't too vague a question. Please spare me the 'do what you feel' and 'you should know for yourself' answers.

 

[MarneMath]

So, I'm really unclear what exactly you want and how much mathematics you know. I think discussing any of this in a meaningful way would be highly improbable without a reasonable level of mathematical maturity. I also don't know what exactly you wish to do. Do you want to just give a overview on the topic, add new insights or argue for or against something?

Either way, a good place to start with any research is to look at journals that related to your interest. Perhaps the journal of mathematical logic may help or the journal of Symbolic logic.

 

[wigglywoogly]

Don't worry about my maths, my maths is solid.

I'm a 3rd year undergrad logic student, about to go into 4th year. I don't really feel capable of adding new insights, but I don't think any of my peers are in much of a position to do this either.

Arguing for/against would be nice but first I need a topic.

Thanks for your journal recommendations I'm going to chek them out.

 

[Stephen Tashi]

At what level have you studied probability theory? - from the measure theoretic point of view?

 

[wigglywoogly]

At what level have you studied probability theory? - from the measure theoretic point of view?

Yes I know what a normalized measure is. I have studied probability from the axioms upwards. Including the philosophical concerns of Baysianism vs. Frequentism.

 

[verty]

Yes I know what a normalized measure is. I have studied probability from the axioms upwards. Including the philosophical concerns of Baysianism vs. Frequentism.

An idea I have is to discuss/compare probability and fuzzy logic. Are they compatible? Are they unrelated? What is needed to make them compatible? How easy is translating from one to the other? What would probability's ideal logic look like?

Researching this could be quite interesting.

-edit-
Actually, I think this won't work because "fuzzy logic" needs to be well defined but there doesn't seem to be much coherence to what I read now about it. One author says this, another says that, to try to bring it together is going to be too much of a task, I think. Here is a quote from the wikipedia page:

Many statisticians are persuaded by the work of Bruno de Finetti that only one kind of mathematical uncertainty is needed and thus fuzzy logic is unnecessary. On the other hand, Bart Kosko argues that probability is a subtheory of fuzzy logic, as probability only handles one kind of uncertainty. He also claims to have proven a derivation of Bayes' theorem from the concept of fuzzy subsethood. Lotfi A. Zadeh argues that fuzzy logic is different in character from probability, and is not a replacement for it. He fuzzified probability to fuzzy probability and also generalized it to what is called possibility theory.

But I was hoping that probability is just probability, there really shouldn't be so many variants. Oh well.

 

[wigglywoogly]

An idea I have is to discuss/compare probability and fuzzy logic. Are they compatible? Are they unrelated? What is needed to make them compatible? How easy is translating from one to the other? What would probability's ideal logic look like?

Researching this could be quite interesting.

-edit-
Actually, I think this won't work because "fuzzy logic" needs to be well defined but there doesn't seem to be much coherence to what I read now about it. One author says this, another says that, to try to bring it together is going to be too much of a task, I think. Here is a quote from the wikipedia page:



But I was hoping that probability is just probability, there really shouldn't be so many variants. Oh well.

Interesting. I did ask one of my logic teachers about this very thing a while back but he seemed to suggest it would be a bit of a dead end.

Comparing 'probability' and 'fuzzy logic' may be a bit of an ill-posed question, in any case.

fuzzy logic is a system of logic, whereas probability is a branch of mathematics that sits on top of whichever system of logic we have used to build arithmatic (Q, Peano, whatever).

 

[verty]

Comparing 'probability' and 'fuzzy logic' may be a bit of an ill-posed question, in any case.

fuzzy logic is a system of logic, whereas probability is a branch of mathematics that sits on top of whichever system of logic we have used to build arithmatic (Q, Peano, whatever).

My viewpoint on this is pretty much in line with Carnap's paper: Empiricism, Semantics and Ontology. Let's leave it at that.