The Problem of Logical Omniscience

 

            The problem of logical omniscience is this. How is it possible for a person to believe some sentence but not believe other sentences which are logically equivalent to it or logical consequences of it? It’s clear enough that this is possible, since we do after all have disciplines like logic, mathematics, and computer science which largely aim to fill in the logical consequences of certain axioms which are already believed. Also, people seem to be able to believe logical contradictions, as when putting down the wrong answer on a math test.

            To some people this issue of how we can fail to believe the consequences of, and indeed believe things incompatible with, our other beliefs feels like a paradox while to others it is merely a constraint on what any correct account of belief should be like. It offers particularly dire problems for those who model a person’s belief state as something like a probability function over different possible worlds. For, of course, logically equivalent statements correspond to the same set of possible worlds.

            However, in this little note I aim not to assess the severity of the problem or the chances of a solution but to say how I think it should be solved.

            The solution I want to advocate depends on (and I think provides us with some addition reason for favoring) a certain very general and relatively popular style of philosophy of mind. According to this school of philosophy of mind, having a belief that there is an apple on the table is not about having an experience which feels a certain way or of having a certain kind of item stored in one’s brain. Rather it is about having a suitable pattern of dispositions to say and do and maybe also experience certain things (e.g. to say yes to ‘is there an apple on the table’ if you speak English and to reach down to the table if you are feeling hungry and to be surprised if the thing on the table wasn’t grown on a tree). If we think of a person as having a sort of web of interacting tendencies to do, say, feel and the like then having a belief is a matter of there being a certain kind of pattern in the web.

            Now let’s think about mathematical beliefs in particular. The statements in question probably won’t have any very direct connection to actions other than saying certain sentences or writing certain things down. But they will have a connection to other beliefs and statements. So we can make out the difference between someone who just says a certain sentence and someone who understand and believes that sentences by noting a kind of swath of other logically or practically related sentences which they are disposed to assent to as well. This swath might be broader or narrower for different people but a certain minium of relevant connections is required before we attribute a person who is disposed to say something a belief.

Since I am not trying to state or argue for a particular theory of mind within this school but rather to say how generally how I think the problem of logical omniscience should be resolved, I won’t go into much detail about how exactly the distance measure implied in the picture above (to believe that P a person needs to have some portion of the dispositions to speak and act which are ‘close’ to that proposition) should work. I just want to note that a) by talking about believers having a swath or closely related dispositions to act I mean that they should have a suitably large number of dispositions which are closely related, not that they should have a large fraction of the dispositions which are closely related (e.g. they don’t need to be disposed to assert P in every human language) and b) that one can imagine finding creatures which drew certain kinds of inferences more quickly and others more slowly than us and a different idea of which other sentences are ‘close’ to a given one might be appropriate for attributing these creatures beliefs.

With all this in place the answer I want to propose to the problem of logical omniscience is simple. A creature has a certain belief if they have a sufficient number of suitably related dispositions to act, speak, experience etc. We simply need to say that different logical consequences of the same belief will be different ‘distances’ from that belief (at least for humans) (think about how we say ‘the proof is immediate’ or ‘the requisite argument is long and involved) in the sense relevant to belief attribution. Thus a person can count as having a belief even if they don’t assent to (and hence come even close to believing) certain of its more distant logical consequences. And someone can believe inconsistent propositions provided that these propositions are suitably distant from one another. This distance allows them to assent to and act on a suitable swath of things around each of the inconsistent beliefs.

The invocation of relations of ‘distance’ or ‘immediacy’ in the explanation of how we can have inconsistent beliefs also has the virtue of matching intuitions about cases in which one cannot have inconsistent beliefs. For example it is hard to imagine anything that would count as a monolingual English speaker having both the belief that it is raining and the belief that it is not raining. One feels that the best they could do would be to behave randomly, saying that it is raining and then denying it, heading out doors with an umbrella one moment but advising others not to bother the next. But this seems better described as not qualifying as having either belief than as having both. The answer I have just advocated to the problem of logical omniscience gives an explanation of how this could be. The problem here is that the inconsistent beliefs in question are too ‘close’ to each other, so that it is not possible to find a suitable swath of related dispositions to act around each proposition such that each both swaths can be jointly realized. If believing that its raining requires doing something like taking an umbrella when you go out and saying that it is raining and not saying that it isn’t raining, and saying that it is not raining requires not taking an umbrella, not saying that it is raining, and saying that it isn’t raining than the impossibility of believing this logically inconsistent pair, will fall out of the logical impossibility of realizing the relevant dispositions to act.