Logical omniscience: beyond fragmentation and impossible worlds
Standard Bayesianism assumes that agents are "logically omniscient": they assign credence 1 to all necessary truths. This seems wrong.
But we need to be clear about the data (as Stalnaker has emphasized, for example in 1991 and 1999). It's not a datum that we don't give credence 1 to all necessary truths. "Credence" is a theoretical term. The data that cast doubt on logical omniscience are things like this:
- I utter "I don't know the 12th digit of pi".
- I utter "I'm more confident that the 12th digit of pi is greater than 3 than that it's smaller".
- I complain that the cashier gave me the wrong change when in fact I miscounted.
- I make a stupid move in a chess game.
The attitude reports may seem decisive: if "I don't know the 12th digit of pi" is true, then surely any theory according to which I know (or I'm sure about) the 12th digit of pi is false!
But that's too quick. The mechanics of attitude reports is complicated and misleading. In particular, attitude reports can be metalinguistic. (See Mates 1950.) I might say, "I don't know if it's warmer than 50 degrees Fahrenheit", even though I'm walking around in a T-shirt. What I'm unsure about isn't really the temperature: I'm confident that it's more than 10 degrees Celsius (=50 degrees Fahrenheit). What I'm unsure about are the conventions for the Fahrenheit scale; I don't know what temperature is picked out by "50 degrees Fahrenheit".
Let p be the hypothesis about the world that's expressed by "it's warmer than 50 degrees Fahrenheit". The same hypothesis is expressed by "it's warmer than 10 degrees Celsius". I know that this hypothesis is true. But I don't know that it's expressed by "it's warmer than 50 degrees Fahrenheit". And oddly, I can express this by saying "I don't know that it's warmer than 50 degrees Fahrenheit".
Some apparent failures of logical omniscience can be explained away in this fashion. If I don't know what "dodecahedron" means, I might say "I don't know if a dodecahedron has 12 faces", even though it's a necessary truth that a dodecahedron has 12 faces. My apparent ignorance of a necessary truth is really ignorance of a contingent truth about words.
But this isn't the whole story. Many apparent failures of logical omniscience have a different character. These are the hard cases. They all have the puzzling feature that there's a sense in which I actually know the necessary truths of which I'm ignorant.
A maths example. An integer is divisible by 3 iff the sum of its decimal digits is divisible by 3. I recently explained why this is true to my 10-year-old daughter, starting with an arbitrary two-digit integer, ab: We know that this means a*10 + b. And a*10 = a*9 + a. So ab = a*9 + (a + b). Obviously, a*9 is always divisible by 3. So if we divide a*9 + (a + b) by 3, we get the same remainder as if we just divide a+b.
Nowhere in this explanation did I give her any genuinely new information, and not just because what I said is necessarily true. Even re-interpreted metalinguistically, my daughter already knew everything I told her. At no point could she have said "oh, I didn't know that". My explanation only helped her make explicit what she knew all along.
A chess example might be clearer, because there's no need to invoke the metalinguistic reinterpretation.
I'm a poor chess player. Suppose I make a move that allows my opponent to checkmate me in two moves. I didn't see that coming: I didn't know that if I make this move, I can be checkmated in two moves. But why not? You could have explained to me: "If you make this move, your opponent can move their bishop here; then you have to do either A or B; in the first case, they can checkmate you by doing X; in the second case, they can checkmate you by doing Y." None of this would be news to me. I certainly knew that they could move their bishop to the relevant place. And so on. Your explanation helps me see what I knew all along.
Intuitively, it's clear what's going on in cases like these. The subject has all the required information. But they haven't "put it together"; they haven't converted it into the form that's needed to act on it, verbally or nonverbally.
To understand this kind of failure, we need a model of how agents store and retrieve information, and how this affects their choices. That's the heart of the problem of logical (non-)omniscience. Discussions of the problem often ignore it.
Many authors rest content with modelling the surface data. They try to come up with formal revisions of Bayesianism that allow assigning nontrivial probabilities to necessary truths, perhaps by allowing for "impossible worlds". I don't find this terribly helpful. When I didn't realize that my chess move leads to an easy checkmate, did I assign positive credence to an impossible situation in which my move was safe? What kind of situation would this be? Is it a situation where my opponent can't move their bishop? Or where that isn't a threat to which I have to respond with A or B? And so on. No answer looks plausible. And no answer reflects that fact that my problem is one of access, not ignorance.
Fragmentation models of logical nonomniscience, going back to Lewis 1982, look more promising. These models are inspired by the idea that the subject in a hard case of logical nonomniscience fails to "put together" all the information they have. The information is stored in different "fragments" of their cognitive system. But that's still a crude simplification. In a typical example, the subject is perfectly able to put together two or three of the relevant facts, and to act on the result, but they fail to put together seven or eight pieces of information. The fragmentation models I know of don't account for that.
In general, the problem isn't just about combining pieces of information. It's about retrieving information that's needed in a specific format from a database that stores swaths of information in a variety of formats. This may fail for all sorts of reasons. Conversion between formats may be too hard. The relevant part of the database may be poorly indexed. Or whatever.
One challenge here is to find a good level of abstraction. Standard Bayesianism assumes that everything is instantly accessible. A more realistic model would drop this assumption, ideally without committing to details about the agent's cognitive architecture. (We shouldn't assume, for example, that information is stored in sentential form.)
It's not obvious if this challenge can be met. Sometimes tricky data can be accommodated by tweaking our models. But sometimes the data reflect "lower-level" facts that have no simple, systematic characterisation at the level we've interested in.