It is clear that when we use the phrase “It is possible that…” it is not in all cases used to express mere alethic possibility, that is, “It is logically possible that p.” [◊P] Other times it is used to express what is called epistemic possibility, that is, “For all we (or I) know p might be true.”. It preliminarily seems like a good idea to explicate this as “It is compatible with everything we know that p is true and that p is false.”.1 But this is an improper explication as pointed out in Possible Worlds.2
Consider the example of Goldbach’s Conjecture (GC), that is, that every even number greater than 2 is the sum of two prime numbers.3 A mathematician might say that it is possible that (GC) is true. If we explicate that as suggested above, then we get that (GC) and not-(GC) is consistent with everything we know. We may formalize this explication as:
(∀P)(EP↔◊[P∧(∀n)Q1∧Q2∧Q3∧…∧Qn∧([∀Q][KQ])]) where “EP” means “P is epistemically possible”, “KQ” means “Q is known”.4
However, since (GC) is a mathematical proposition, then it is either necessarily true, or necessarily false. If it is necessarily true, then it’s negation is not consistent with everything we know. All necessary falsehoods are inconsistent with any proposition.5 If (GC) is false, then (GC) is necessarily false, and, thus it is not consistent with everything we know. If (GC) is true, then it is necessarily true, but then the claim that it is false is necessarily false and thus not consistent with everything we know. I note that this objection applies when one deals with non-contingent propositions.
The authors of Possible Worlds suggest instead that epistemic possibility should be explicated without alethic terms at all. They suggest the plain explication of: We (or I) do not know that (GC), and we do not know that not-(GC).
1Simplifying here. It is possible to formulate it without assuming bivalence.
2N. Swartz, R. Bradley, 1979, pp. 229-230.
3Some examples: 4 is the sum of 1 and 3. 6 is the sum of 3 and 3. 8 is the sum of 5 and 3. Etc. http://en.wikipedia.org/wiki/Goldbach%27s_Conjecture
4This is a bit complicated because it uses propositions as variables and propositions are written in the upper case in formalizations. It is to be read as: For all propositions, that P is epistemically possible is logically equivalent with that it is logically possible that (P and Q1 and Q2 and Q3 etc, and that for all Q’s, Q is known.
5To say that two propositions are consistent is to say that they are both true in some possible world, but a necessary falsehood is not true in any possible world, thus, it is not true together with any other proposition in any possible world. Hence, it is not consistent with any proposition. More about this in chapter 1 of Possible Worlds.