Last time I discussed the logic of necessity and possibility under basically two interpretations. First we considered the possibility that □φ should be interpreted as claiming that φ was a tautology, or, in other words, that it can be proved without appeal to any additional axioms. This is a very simple way to interpret possibility and is also, in my opinion, very intuitive. The other way discussed to interpret the logic of necessity was in terms of sets of possible worlds. For our almost purely logical purposes it was sufficient to understand a possible world as a set of facts. □φ was then determined simply by considering whether φ held in each possible world belonging to the set of possible worlds.
Although these two approaches to possibility are conceptually very different they have in common a central idea, namely that statements involving claims of necessity or possibility are to have their truth values definitively determined in an objective and purely formal way on the basis of the totality of all facts. This is similar to the way the truth of a claim such as ∀xPx, which is determined to be true or false based on whether P holds of every object under consideration. But despite the fact that □ and ◊ are part of a logical notation we don’t have to interpret them in such a logically cut and dried way. Specifically we can interpret them in a way which is not settled based on the totality of facts, but rather the facts as known from a specific viewpoint. Instead of saying that □φ means that φ is provable without any axioms we can instead understand it as indicating that we possess a proof of φ. And thus since ◊φ = ~□~φ it (◊) indicates that we do not have a proof of ~φ, but is silent about whether we have a proof of φ or not. Or, in more natural terms, ◊φ states that φ is an open possibility, that we have not been able to rule it out yet. I would call such an interpretation epistemic possibility.
In such an interpretation of necessity and possibility many fewer things are necessary and more are simply possible then were under the purely logical interpretation. Of course in the logical interpretation there were a number of sentences (actually an infinite number) for which there was neither proof nor disproof. If G is one of those sentences the only thing we could claim about it would be that ◊G and ◊~G. But for every other φ we could assert □φ or □~φ. However, if we are working with the epistemic possibility interpretation then there only a finite number such φ which we can assert □φ or □~φ of. This follows from the simple fact that as beings of finite capacity we can only have seen a finite number of proofs (even collectively, if we wish to interpret what has been demonstrated as a collective exercise), and since it requires a proof to assert □φ obviously we can only make a finite number of such assertions. And so necessity is clearly much more sparse than it was under the logical possibility or possible worlds interpretations.
This may make epistemic possibility seem aesthetically unattractive. Why should we bother with this interpretation, or try to understand ordinary language claims about necessity and possibility through it? Well one reason might be that it is the right interpretation to understand these claims through, but I will put that aside for the moment. Consider instead how this interpretation can deal with nested possibility quantifiers. In our previous two interpretations nested quantifiers were, for the most part, redundant. If the decision as to whether a statement is necessary or just possible is determined by the complete body of facts then surely that claim about possibility is itself necessarily true, given that we have just shown that it followed definitely from the complete body of facts. However, this is not the case with epistemic possibility. A statement involving epistemic possibility says something about our possession of a proof (and not, it should be noted, whether a proof exists). And thus a claim about the epistemic possibility of such a statement is about whether we have a proof about whether can or can’t have a certain proof. Thus adding such quantifiers genuinely expresses a new idea. Let’s consider each of the four possibilities: □□φ says that we have a proof that we have a proof of φ, which is rather trivial although hard to establish (we must prove that our proof is without error); □◊φ says that we have a proof that we can’t disprove φ (prove ~φ), we might say this about G (and ~G), mentioned earlier, because it states that there is something that we cannot in principle prove about them, even though we aren’t able to make a positive claim; ◊□φ states that we do not have a proof stating that we can’t prove φ, or, in other words, that proving φ hasn’t been ruled out; and, finally, ◊◊φ asserts that we do not have a proof that we can disprove φ, which is to say very little positively except that certain proofs are not off the table yet.
Obviously we wouldn’t expect to come across such nested quantifiers except in the context of proof theory, or possibly other meta-theoretical inquiries. For example, a theory about black holes may rule out the possibility of determining anything about what is happening inside them. Thus we may, on the basis of that, claim that □◊ψ, where ψ is some statement about what is happening inside the black hole. Or, in normal language, that we have demonstrated the impossibility of ruling out certain claims about what is happening inside the black hole (due to our inability to look inside). Nested quantifiers might also occur when we wish to qualify our ability to reason. Let us say that we have deduced ~□φ, that φ cannot be logically demonstrated. But we might still concede that ◊□φ, that we admit that the process we reached that conclusion by was fallible, although in ways we cannot immediately detect, and thus at the level where we are considering our proofs themselves we cannot rule out the possibility that our proof was wrong. Of course such qualifications don’t add anything to our inquiries, they simply concede that we are imperfect.