To elaborate: S is necessary if and only if S is true in all possible worlds. But we have these two ways of assessing the truth of S at each world W. We can take either the subjunctive reading - "If W had been the case, S would have been the case" - or else the indicative reading: "If W is the case, S is the case." If the former holds for all worlds W, then S is subjunctively necessary. If the latter holds for all worlds W, then S is indicatively necessary. But these two types of necessity can diverge.
For example, consider a world W where the clear drinkable liquid in lakes and rivers is XYZ rather than H2O. Considered counterfactually, we would describe this as a world without water. After all, water just is H2O (given the assumption that this identity holds in the actual world). This identity is "metaphysically necessary" - there is no possible world, considered counterfactually, where the identity is broken.
But let us instead consider world W as actual. If we accept the hypothesis that the clear drinkable liquid in our actual lakes and rivers is XYZ rather than H2O, then we rationally should conclude that water is XYZ, not H2O. So given that W is a possible world, a "way the world might be", there seems a straightforward sense in which it is possible for water to not be H2O. It is indicatively, but not subjunctively, possible.
We combine the two dimensions by assessing subjunctive possibility relative to a presupposed indicative possibility. For example, given the hypothesis that actually water is XYZ, it follows that a world with H2O in place of XYZ would be a world without water. After all, identities hold of subjunctive necessity. As Kripke's Naming and Necessity taught us, whether cats are animals or demons is a matter for empirical discovery -- but whichever they turn out to be, they are that essentially. If some other entity were of a different nature to our actual cats (whatever that nature might be), then it would not be a cat. A key insight of two-dimensionalism is to turn this presupposed 'actuality' into a variable that can itself range over various possible worlds. (To reiterate: this forms the 'indicative' sense of possibility, i.e. possible worlds considered as actual.)
Importantly, this sense of indicative or "conceptual" possibility is no mere pseudo-possibility. Indeed, it doesn't invoke a distinct modal space at all. On this explication of the 2-D framework, we do not contrast "metaphysically possible worlds" and "conceptually possible worlds". Rather, there's just one space of possible worlds, and the difference between 'metaphysical' and 'conceptual' possibilities merely lies in how we describe the worlds. (As already explained, we will describe them differently depending on whether we consider them as actual or counterfactual.) The full importance of this point will be brought out in subsequent posts, applying the 2-D framework to illuminate foundational issues in the philosophy of mind and meta-ethics.
I've hinted at the centrality of semantical considerations to the 2-D framework. My next post will spell this out more clearly by delving into some philosophy of language.
* = [Note that 'deep epistemic' possibility shouldn't be confused with the more superficial notion according to which anything we don't know to be false counts as "epistemically possible". Suppose a young math student doesn't yet know that there is no greatest prime number. Then the existence of such a number is an "epistemic possibility" for him, in the weak sense. Further, if we factor in experiential knowledge, then the claim that "I exist" is arguably "epistemically necessary" in the strict sense. But neither of these judgments hold true of indicative modality. There are possible worlds, considered as actual, where I do not exist. (Any uncentered world will do.) But there are none containing a greatest prime number. As these cases illustrate, Chalmers' notion of "deep" epistemic necessity abstracts away from a subject's actual knowledge and contingent cognitive limitations. It instead concerns what is a priori knowable on ideal rational reflection. On some views, this could depart from 'indicative necessity' as defined in this post. But I will save such complications for another day.]