Sunday, 22 January 2006

A Theory of Science

As you might know, I like logics. Notice that logic is in the plural. This might seem a bit strange to you. It certainly seems a bit strange to me. How can it be that there is more than one?

There are quite a number of logics at this point. We pretty much started out with Aristotelian logic, ie. the logic of syllogism which you were probably forced to study in school. Aristotelian logic was formalised and generalised during the early part of last century and this formalism has come to be called Classical Logic or often just CL.

This formalisation caused some to view with suspicion the outcome of various formal arguments. It gave rise to a more conservative 'constructive' logic which we will call Intuisionistic (or IL) whose informal interpretation is known as the Brouwer-Heyting-Kalmagorov interpretation or BHK.

Basically in Classical logic you can make proofs about things for which you can not provide examples. This also happens however in Intuisionistic logic for arguments that use the ∀ quantifier. It doesn't seem so onerous in those cases however as you can see by playing with it a bit.

We can even make things more restrictive and get Minimal Logic (or ML). Minimal logic rejects the provability of arbitrary things from Falsum. The rule is often called 'ex falso sequitur quodlibet,' or 'ex falso.'

Since then there has been a real explosion of the types of logic. There are Substructural Logics, Quantum Logic (QL), Linear Logic (LL, a pretty big fish than can even swallow CL) and a host of others.

From this the question naturally arises in my mind. Which is the right logic? As someone who writes programming languages I have a natural sympathy for IL as it leads naturally to a term calculus meaning that terms can be given back to the user that exemplify proofs. It is a natural logic to look at for the purposes of a database query language. There are however problems with it in regards to this. It is not "resource sensitive". Things change in data stores and none of the above mentioned logics provide the appropriate tools to deal with this. Linear Logic comes closest but fails to deal with sharing or ordering. Many new resource logics have been invented to deal with this but I have yet to come across something that looks to me like a suitable answer (which doesn't mean it isn't already out there!).

In science the problem may be even worse. People use some form of quasi-classical reasoning to make arguments. It seems that this might not even be the appropriate tool to use when reasoning about Quantum Mechanics. Quantum Logic has been proposed as the appropriate way to deal with Quantum quandries in some (fairly fringe) circles. So far Quantum Logic looks to me to be too anemic. Something closer to a theory that has a curry howard isomorphism with quantum computation would be more satisfying.

So what is it that makes a good logic? My personal feeling is this. A logic is a constraint framework from whence you can show various programs that are the "proofs" of the constraint apparatus are acceptable. An appropriate constraint framework is one in which constraints that apply to your system can be expressed simply with minimal work. I believe that the Classical Logic for Propositions arose as a sort of logic of the natural sciences because it was in fact a type of physics. It is a calculus in which we can present common sense notions of real things in a simple way. When we extended the apparatus to classical logic we may have gotten something that strays so far from common sense it is no longer useful (this of course is debatable, and I'm not sure how much I believe it).

Now that we have a quantum world with physics that does not function in ways that our common sense would dictate, it seems perfectly reasonable to reject the notion of classical logic in this regime. In favor of what? I think the jury is still out on this one.

As for as how to quantify what a good logic is, I'll make a couple of guesses. You want to be able to express constraint systems that apply to your realm with parsimony. You want to be able to verify and extract programs from proofs. If those two conditions are met more often for one logic than another for a particular problem, then I would deem it superior.

Of course this doesn't even go into notions of logic in ethics...

Friday, 20 January 2006

Proof Theory

Thanks to my brother I got a really cool book on proof theory called "Basic Proof Theory". It has a bunch of nice features including a from the ground up presentation of proof theory that should be relatively accesible to anyone with a background in mathematics. It demonstrates some of the connections provided by the Curry-Howard correspondance (which is my favourite part of the book) . It also describe Second order logic, which is great because I've had very little formal exposure to this. Second order logic is really beautiful since you can define all the connectives in terms of ∀, ∀2 and →. If you pun ∀ and ∀2 you have a really compact notation.

The book also forced me to learn some things I hadn't wrapped my head around. One of those was Gentzen style sequent calculus. This really turns out to be pretty easy when you have a good book describing it. I've even wrote a little sequent solver (in lisp) since I found the proofs so much fun. The first order intuisionistic sequent solver is really not terribly difficult to write. Basically I treat the proofs as goal directed starting with a sequent of the form:

⇒ F

And try to arive at leaves of the tree that all have the form:

A ⇒ A

I have already proven that 'F ⇒ F' for compound formulas F from 'A ⇒ A' so I didn't figure it was neccessary to make the solver do it. The solver currently only works with propositional formula (it solves a type theory where types are not parameteric.) but I'm interested in limited extensions though I haven't thought much about that. I imagine I quickly get something undecidable if I'm not careful.

Anyhow working with the sequent calculus got me thinking about → In the book they present the rule for R→ as such

Γ,A ⇒ Δ,B
Γ ⇒ A→B,Δ

This is a bit weird since there is nothing that goes the other direction. ie. for non of: Minimal, Intuisionistic or Classical logic do you find a rule in which you introduce a connective in the left from formulas in the right. I started looking around for something that does this and I ran into Basic Logic. I haven't read the paper yet so I can't really comment on it. I'll let you know after I'm done with it.