There’s some first thoughts on how to do probability theory in Łukasiewicz’s 3-valued logic below the fold, but since they are more diary style notes-to-self than actually something written up for public consumption, I don’t want anyone to take them too seriously. But some days I like using my web diary as my diary, so I use it as a depository for first thoughts.

So I had an idea for solving the little problem noted in the post below. It’s sort of a tacit claim of my paper that it’s really dead easy to generalise what I said there to other logics beyond intuitionist logic. And the only formal feature of intuitionist logic that I appeal to is the fact that it doesn’t validate excluded middle, I should be able to knock up something using other non-LEM logics without too much hassle, right? And since I left the abstract kinda vague so a more general talk about non-classical probability will fly, if everything here works I’m set.

Well, that’s the theory. And if what I’ve drawn on the back of this envelope is any guide, it looks like it won’t be too hard to draw a probability theory for the Kleene 3-valued logic. I have to tinker with the ‘boundary axioms’. I had axioms that all theorems are probability 1 and all antitheorems probability 0 and used that to ensure that all propositions have probability between 0 and 1. But that only follows if (a) the logic allows weakening and (b) there are theorems and antitheorems. Well we’re OK for (a), but (b) is a problem in the Kleene logic. But that isn’t too much problem – just stipulate that probabilities are between 0 and 1 and I think we get what we want.

The Łukasiewicz logic is a little harder. The axioms I give at the bottom of page 2 of the “NDJFL paper”:http://projecteuclid.org/Dienst/Repository/1.0/Disseminate/euclid.ndjfl/1082637809/body/pdfview are all sound with respect to the intended semantics, but they aren’t I think complete. What’s the intended semantics? Just a generalisation of the standard classical semantics. Just as classical probability functions are defined as measures on classes of (2-valued) assignment functions, Łukasiewicz probability functions should be measures on classes of (3-valued) assignment functions. Or so I think anyway. But that semantics validates, inter alia, the following inference.

bq. Pr(A) = 0
Pr(~A) = 0
Pr(B) = 0
Pr(~B) = 0
Therefore Pr(A -> B) = 0

And that, er, doesn’t follow from the axioms that I’ve given there. So I need a new axiom to deal with the fact that conditionals with ‘middle-valued’ antecedents and consequents are true. And I need it to be sound and to complete the axioms. And ideally I’d figure it out yesterday.

If you see me on a cross-country flight later this week tearing pages off a notepad while making incomprehensible noises about how I’m cursed by the inexpressibility of indeterminacy, well just try to ignore me. I’ll be better as soon as I figure out the axiom I need. It actually shouldn’t be too hard as soon as I start looking at the problem clearly.

UPDATE: I think I have it.

Pr(A -> B) = 1 – Pr(A v ~B) + Pr(A & B) + Pr(~A & ~B).

That’s sound, and I think it’s probably sufficient for completeness. No proof yet though.

SECOND UPDATE: Ugh, that wasn’t necessary at all. In fact the whole post may be mistaken. The axiom I introduced follows from the previous axioms, as follows.

In the 3-value Łukasiewicz logic, (A -> B) v (A v ~B) is a theorem. So it gets probability 1. And (A -> B) & (A v ~B) is equivalent to (A & B) v (~A & ~B). So just applying axiom 4, we get

Pr(A -> B) + Pr(A v ~B) = Pr((A -> B) v (A v ~B)) + Pr((A -> B) & (A v ~B))
= 1 + Pr((A & B) v (~A & ~B))
= 1 + Pr(A & B) + Pr(~A & ~B)

From which Pr(A -> B) = 1 – Pr(A v ~B) + Pr(A & B) + Pr(~A & ~B) follows immediately.

None of this helps with the hard question of how to deal with the continuum-valued case, but I’m leaving that until I’m more awake.