If we know nothing about them, the statements could equally be true or false, and positively or negatively dependent. The same argument which makes us assume 50% probability to individual statements would also make us assume independence between statements. The possibilities cancel out, so to speak.
Though more subtle, I feel that the 50% prior for “individual statements” is also wrong, actually; it’s not even clear a priori which statements are “individual” – just figuring that out seems to require a quite refined model about the world.
Ludwig Wittgenstein tried to solve this problem in an a priori fashion with a theory of “logical atomism”. So without a refined model of the world. In the Tractatus he postulated that there must be “atomic” propositions. For example, the proposition that Bob is a bachelor is clearly not atomic (but complex) because it can be decomposed into the proposition that Bob is a man and that Bob is unmarried. And those are arguably themselves sort of complex statements, since the concept of a man or of marriage themselves allow for definitions from simpler terms. But at some point, we hit undefinable, primitive terms, presumably those which refer directly to particular sense data or the like. Wittgenstein then argued that these atomic propositions have to be regarded as being independent and having probability 1⁄2.
Or more precisely, he came up with the concept of truth tables, and counted the fraction of the rows in which the conditions of the truth tables are satisfied. Each row he assumed to be a priori equally likely when only atomic propositions are involved. So for atomic propositions P and Q, the complex proposition “P and Q” has probability 1⁄4 (only one out of four rows makes this proposition true, namely “true and true”), and the complex proposition “P or Q” has probability 3⁄4 (three out of four rows in the truth table make a disjunction true: all except “false or false”).
This turns out to be equivalent to assuming that all atomic propositions have a) probability 1⁄2 and are b) independent of each other.
However, logical atomism was later broadly abandoned for various reasons. One is that it is hard to define what an atomic proposition is. For example, I can’t assume that “this particular spot in my visual field is blue” is atomic. Because it is incompatible with the statement “this particular spot in my visual field is yellow”. The same spot can’t be both blue and yellow, even though that wouldn’t be a logical contradiction. The two statements are therefore not independent, so they can’t be atomic. But it is hard to think of anything more primitive than qualia predicates like colors.
If we know nothing about them, the statements could equally be true or false, and positively or negatively dependent. The same argument which makes us assume 50% probability to individual statements would also make us assume independence between statements. The possibilities cancel out, so to speak.
Though more subtle, I feel that the 50% prior for “individual statements” is also wrong, actually; it’s not even clear a priori which statements are “individual” – just figuring that out seems to require a quite refined model about the world.
Ludwig Wittgenstein tried to solve this problem in an a priori fashion with a theory of “logical atomism”. So without a refined model of the world. In the Tractatus he postulated that there must be “atomic” propositions. For example, the proposition that Bob is a bachelor is clearly not atomic (but complex) because it can be decomposed into the proposition that Bob is a man and that Bob is unmarried. And those are arguably themselves sort of complex statements, since the concept of a man or of marriage themselves allow for definitions from simpler terms. But at some point, we hit undefinable, primitive terms, presumably those which refer directly to particular sense data or the like. Wittgenstein then argued that these atomic propositions have to be regarded as being independent and having probability 1⁄2.
Or more precisely, he came up with the concept of truth tables, and counted the fraction of the rows in which the conditions of the truth tables are satisfied. Each row he assumed to be a priori equally likely when only atomic propositions are involved. So for atomic propositions P and Q, the complex proposition “P and Q” has probability 1⁄4 (only one out of four rows makes this proposition true, namely “true and true”), and the complex proposition “P or Q” has probability 3⁄4 (three out of four rows in the truth table make a disjunction true: all except “false or false”).
This turns out to be equivalent to assuming that all atomic propositions have a) probability 1⁄2 and are b) independent of each other.
However, logical atomism was later broadly abandoned for various reasons. One is that it is hard to define what an atomic proposition is. For example, I can’t assume that “this particular spot in my visual field is blue” is atomic. Because it is incompatible with the statement “this particular spot in my visual field is yellow”. The same spot can’t be both blue and yellow, even though that wouldn’t be a logical contradiction. The two statements are therefore not independent, so they can’t be atomic. But it is hard to think of anything more primitive than qualia predicates like colors.