Probabilities of propositions and beliefs

Start with this thesis:

1. A random belief of a random person is at least as likely true as not.

To deny (1) would be overly pessimistic, as (1) seems pretty innocuous. But I shall argue that, given two additional theses, (1) implies a substantive Principle of Credulity, namely that the mere fact that someone believes p is significant evidence for p.

Now observe this:

2. A random atomic proposition is significantly more likely false than true.

This is somewhat surprising and counterintuitive, so I shall argue for it. First, consider unary atomic propositions: the attribution of a property to an object. Now, it is plausible that most of the properties we have terms for are at least somewhat natural. And natural properties tend to have a number of relevant alternatives to them that are in some intuitive taxonomic sense "on the same level". For instance, being a horse has as relevant alternatives being an F where F ranges over all biological species, plus it has a number of alternatives whose level is harder to gauge like being an electron or being a number. Since there are lots of species, without any specific information about how many critters there are in each species, it is reasonable to think that the probability that Sam is a horse, with no specific information about Sam or horses, is quite low. Or take the way that many of the basic properties are determinates of determinables for which there is an infinite number of options: e.g., having mass exactly 17.3 kg. (I am grateful for discussions with Trent Dougherty on this point.) The low probability point is less obvious for relations. But, first, one might think that there are a lot more natural properties than relations, and so the case of properties swamps that of relations. Second, it does seem prima facie plausible that typical natural n-ary relations only relate a small subset of the n-tuples. E.g., relatively few events are related by causation. The closest to an exception is earlier than, which we would expect to relate about half of the pairs of events. But actually, it relates less than half: for The closest to an exception is earlier than, which we would expect to relate about half of the pairs of events. But actually, it relates less than half. For every pair (E,F) related by earlier than there is a pair (F,E) related by later than, but there are also pairs related by neither earlier than or later than, namely those that are simultaneous (or spacelike separated).

I now claim:

3. A random proposition is significantly more likely to be false than true.

My plausibility argument for (3) proceeds by considering the special case of the propositions that are expressed by sentences of propositional logic with natural predicates. I suspect that the claim extends to quantified sentences and even modalized ones, but right now I can only prove it for propositional logic. For the proof, we need a reasonable account of a random proposition. I shall do this by generating random grammatically correct sentences of propositional logic.

The method is this. I shall suppose that the basic connectives are "and", "or" and "not", and that we have a stock of basic predicates and names for all objects, one name per object. We first randomly choose an item from the set of basic connectives and basic predicates. If the item is an n-ary predicate P, we randomly choose a sequence (n₁,...,n_n) of n names, and write down the sentence P(n₁,...,n_n). If the item is a unary connective (i.e., "not"), we write down the connective followed by a random sentence (we recurse here). If the item is a binary connective, we write down a random sentence (recursing) in parentheses, followed by the connective, followed by another random sentence (recursing again) in parentheses. The recursion is not logically guaranteed to finish, but we can conditionalize on the recursion finishing (plus I think it's going to finish with probability one if there are enough predicates).

Now, let p₀ be the probability that a random atomic sentence is true. Let N be the number of predicates. Let p be the probability that a sentence generated by the above procedure is true. Then:

4. p=p²/(3+N)+(1−(1−p)²)/(3+N)+(1−p)/(3+N)+Np₀/(3+N).

The first term comes from the fact that we have a 1/(3+N) chance of initially generating an "and", in which case we have a probability p² of truth since both random conjuncts will have to be true; we have a 1/(3+N) chance of generating "or", and then a probability 1−(1−p)² that at least one disjunct is true; a 1/(3+N) chance of generating "not" and probability 1−p that the operand of it will be false; and then an N/(3+N) probability of generating an atomic sentence, which has probability p₀ of truth. Solving (4) for p we get:

5. p=(Np₀+1)/(N+2).

It is easy to verify that if p₀<1/2, then p<1/2. Moreover, for large N (recall that N is the number of predicates), and surely N is in fact large, p is going to converge to p₀. We can conclude to (3) in the special cases of propositions expressed by randomly generated sentences of propositional logic, and this provides significant support for (3) in general.

Now an interesting result follows from (1)-(3). The seemingly innocuous claim (1) commits us to assigning a pretty substantive amount of weight to a Principle of Credulity. For the fact that someone believes a proposition raises the probability from something that according to (3) was significantly less than 1/2 at least to 1/2. Thus the mere fact that someone believes something is significant evidence for its truth, even if it does not suffice for making the conclusion reasonable to believe.

In fact, I suspect that p₀ is very small, maybe as small as 1/100 or even much smaller. In that case, the probability of a random proposition being true might be very small. And yet we have (1). So belief is significant evidence.