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.
Continue reading →

My intuitionist probability paper has appeared! I can’t tell if this is subscriber only, so maybe not everyone will be able to access it, but the website is “here”:http://projecteuclid.org/Dienst/UI/1.0/Journal?authority=euclid.ndjfl.

I’m happy the paper appeared, but I might have rather it waited a few more weeks. I’m meant to do a talk on intuitionist probability in a few weeks and I was sort of hoping the paper wouldn’t have come out so I could in complete good conscience just talk about things from the paper. (I try and stick to the policy, which I think is widespread, that it’s OK to talk about unpublished papers, but not published ones.) Hopefully hardly anyone at the talk will have actually, er, read the paper so I can get away with mostly doing that, but now I feel I better add _something_ new.

Anyway, I’m very grateful to NDJFL for printing the paper, even if their efficiency has put me in a bit of a bind.

The “papers blog”:http://www.brown.edu/Departments/Philosophy/Opp/ is posted with papers by “Daniel Dennett”:http://ase.tufts.edu/cogstud/pubpage.htm and “Kent Johnson”:http://hypatia.ss.uci.edu/lps/home/fac-staff/faculty/johnson/ headlining.

The INPC is only three days away, and I still haven’t stopped being amused by the jokes about going to Moscow for a conference. It’s just so fun to say “See you in Moscow” to friends who I will next see _in Moscow_. It all reminds me of the decade when I couldn’t read _On Denoting_ without laughing out loud at the joke about Hegelians. Anyway, despite what it says on the timetable “here”:http://www.class.uidaho.edu/inpc/7th-2004/program.html I might yet end up doing my paper in Washington not Moscow because of late-breaking adjustments. Watch this space.

One hears it said from time to time that it’s irrational to perform inductive inferences based on a single data point. Now this is sometimes irrational. For example, from the fact that Al Gore got the most votes in the last Presidential Election it would be foolish to infer that he’ll get the most votes in the next Presidential Election. But it isn’t always irrational. And this matters to some philosophical debates, and perhaps to some practical debates too.

Here’s my proof that it isn’t always irrational. Imagine on Thursday night I go and see a new movie that you’re going to go see Friday night. Friday lunchtime I tell you how the movie ended. How should you react? Most people will complain that I’ve spoiled the movie because you now know how it will end. But if induction on a single case is always bad, this is impossible. All you have is testimonial evidence of how the movie ended on a single occasion, namely Thursday night. You need to make an inferential leap to make a conclusion about how it will end Friday night. (It certainly isn’t a deductive inference because some movies have multiple endings.) That inferential leap will be induction on a single case, and will be perfectly reasonable.

That’s more or less my complete argument that induction on a single case can be perfectly rational. There is an obvious objection though. It might be argued that this isn’t _really_ induction on a single case, because it’s like underwritten by a many-case induction based on the number of previous movies that have ended the same way at multiple screenings. While that’s obviously true, it isn’t clear how much it undermines the original example.

There’s two points we could go on to debate here. First is the question of whether the inference from how the movie ended on Thursday to how it will end on Friday (the movie inference) is really an instance of induction on a single case. That looks like a relatively stale terminological debate, and I couldn’t be bothered hashing it out here. Second is the question of whether there is any way to distinguish the movie inference from what are usually taken to be bad instances of induction on a single case. This one has to be debated case by case, but I suspect the answer is in general _no_, unless there are independent reasons to dislike the particular bad instance of inductive reasoning.

Here’s a couple of illustrations of what I mean, one practical the other theoretical.

Consider the policy “Don’t start reading a blog if the first thing you read there is false.” Some might consider any application of that to be a bad instance of a single case induction – from one bad claim infer that other things the blog says are not worth reading. But just like the movie inference can be backed up by a meta-induction, this one can arguably be backed up by a claim validated by a meta-induction: that blogs which say something false the first time you open them are not worthwhile reading in the future. That claim might well be _false_, and I’m not taking sides here on whether it is true or not, but as long as the person who holds the policy believes the claim, their reasoning is no worse than the person who makes the movie inference. (Quick credit: I saw this policy defended somewhere a while ago, but Google was no help in finding where. That was more or less what inspired this post.)

Let’s take a more famous case. Why should I believe that other people have sensations? One famous answer, defended by Bertrand Russell, is that I can reason by analogy. I’m alike other people in ever so many ways, so I’m probably alike them in respect of sensations. And I know (somehow!) that I have sensations, so I know they do too.

Some people have objected that this is just induction on a single case. (E.g. Michael Rea makes that objection “here”:http://www.amazon.com/gp/reader/0199247609/ref=sib_rdr_next3_ex168/103-6891633-7112654?%5Fencoding=UTF8&keywords=ROM&p=S053&twc=14&checkSum=UUh%2B4yKxKqD3qjbFZkigaI%2BcoqguOubvU%2FiNxzZoLVA%3D#reader-page.) But it looks to me a lot like the movie inference, for it too is backed by a meta-induction. In the past, when I have tried to infer from the fact my body is a certain way to the conclusion that others are the same way, I’ve met with reasonable success. Not 100% success, but good enough for inductive purposes. Of course other people are like me in external respects – they often have two eyes, one mouth, two legs etc. But they are also like me in internal respects, at least as far as I can tell. Consider, if you aren’t too squeamish to do this, how similar the various kinds of fluids that come out of various parts of other people’s bodies are to fluids that come out of matching parts of one’s own body. X-Ray technology reveals that we are alike in even more ways than we could have previously told ‘on the inside’. So the argument _my brain states generate or constitute or correlate with phenomenal sensations, so other brains generate or constitute or correlate with phenomenal sensations_ is an instance of a schema that delivers mainly reliable instances. Just like the movie inference. And that inference can produce knowledge. So I think we can come to know about the existence of other minds with sensations on the basis of a single case, namely our own. If you don’t believe me, perhaps you don’t need to worry as much about movie spoilers as you thought you did!

I think the “Brown bloggers”:http://www.brown.edu/Departments/Philosophy/Blog/ should write about “this”:http://wilsonhellie.typepad.com/for_the_record/2004/04/stepford_stores.html. I always thought the fake barn story was meant to be a _fictional_ thought experiment!

The “papers blog”:http://www.brown.edu/Departments/Philosophy/Opp/ is posted for the day with 2 papers from the “Philosophy of Science Archive”:http://philsci-archive.pitt.edu/ being the only news.

The “papers blog”:http://www.brown.edu/Departments/Philosophy/Opp/ is posted for the day with 2 papers from the “Philosophy of Science Archive”:http://philsci-archive.pitt.edu/ being the only news.

Jonathan discusses “his favourite solution to the problem of evil”:http://www.brown.edu/Departments/Philosophy/Blog/Archives/004449.html, so I’ll take that as a reason to also discuss my favourite solution. It’s modal realism, of a sort, to the rescue.

Before I start, I should note that nothing I say here is going to be original. But what with it being Sunday night, and this being a blog not a journal, I’m not actually going to footnote anything. (UPDATE: For the proper citations, see Ned Markosian’s comment below.) Just don’t credit me for anything here – I’m merely repeating my favourite solution. Do, however, blame me for mistakes in presentation. (In general, if you want to read real scholarly work online on the problem of evil, I’d recommend not reading this blog and instead reading “Daniel Howard-Snyder’s writings”:http://www.ac.wwu.edu/~howardd/papersandbooks.html.)

OK onto the solution. Let’s assume the following metaphysical claims are all true.

* There is a class of abstract possible worlds W. (I’m not going to say what abstract and concrete amount to in any of this – on this distinction see “Gideon Rosen’s SEP entry”:http://plato.stanford.edu/entries/abstract-objects/.) In other words, weak modal realism is true.
* God cannot _change_ any of those worlds without destroying it – what happens in a world is essential to its nature.
* What God can do is make any of them that He chooses concrete. Abstract possible worlds have no moral value, but concrete worlds do have value, or disvalue if they are bad, so this choice is morally loaded.
* God’s creation is timeless, so He can’t create one and then tinker with it. For each world He faces a take-it-or-leave-it choice.

Just to be clear, I’m not saying these are true. I’m just saying they are a plausible set of views about the nature of modality and the nature of God’s powers. Note that the only ‘restriction’ on God’s powers here are of the form “God can’t do this metaphysically impossible thing”, i.e. make something lack one of its essential properties, so in that respect this isn’t meant to be a revisionary theology. (It’s revisionary metaphysics, not revisionary theology.)

If all this is true, what should God do? Well, I think He should create all and only the worlds such that it is better that they exist than that they not exist. And that will include worlds, like this one, that are not perfect but that contain more goodness than suffering. So the existence of this world as concrete entity is compatible with God’s existence, and indeed His omnipotence and benevolence.

To repair the argument from evil at this stage, the atheist has to do one of three things. First, she can argue that the metaphysics presented is implausible. She might have a point. Second, she can argue that the metaphysics doesn’t really stop God making a world concrete then tinkering with it. I sort of mean to rule that out by stipulation, but that does make the first problem somewhat worse. Third, she can argue that it would be better that the world not exist than it exist. Some days thinking about how awful things are for most animals in the wild I can almost believe this. (There’s a reason they call a really bad situation Law of the Jungle.) But I can never really believe this, and I bet even the most dedicated proponent of the Argument from Evil can’t either. So I think all the action is at the level of metaphysics, which is a nice place to locate the disagreement.

By the way, I feel like I’ve written this post before sometime, but a quick Google of the archives reveals nothing, so maybe I never got around to posting it or something. Apologies if I really am just repeating myself.

I only just found out that PPR is accessible online. The link is “here”:http://zerlina.ingentaselect.com/vl=1669820/cl=22/nw=1/rpsv/cw/ips/00318205/contp1.htm. It’s subscriber only, but if you are reading this on a university computer, or are logged in to a university network, it might work.