David Lewis believed that possible worlds are real and their existence are similar to actual world. Any world is causally and spatio-temporally disconnected from other worlds. David Lewis was an empiricist and I want to know how this view might be consistent with empiricism if there is no epistemic access to other possible worlds?
David Lewis (IEP), a great contemporary philosophical mind, had empirical beliefs and certainly, like most professional contemporary philosophers, had a naturalized epistemology. But was he an empiricist? In one sense but not the other.
An EMPIRICIST1 as you have written it is one who has empirical beliefs, such as science is a necessary instrument for philosophical discourse. But EMPIRICIST2 appeals to a historical philosopher such as the British Empiricists like David Hume and John Locke who had a very specific set of views.
So, David Lewis believed in science and the role of the senses in determining truth. He, like his mentor Quine, place a primacy on empirical methods, no doubt. The question is better phrased, how does a someone with a naturalized epistemology make a claim or stake out a position that, strictly speaking, isn't supported by science. Obviously modal realism is one such position. I think modal realism is rubbish, so I'm not familiar enough with Lewis's defense of his thesis, but here's a PhilSE question that seems to get into it. His argument does begin with the obvious experience that possibility seems to exist, and thus requires some form of explanation, and they might find refuge in the idea that arguments about the "ultimate nature of reality" was shaken with the Copenhagen interpretation. Wave-particle duality and an absence of hidden variables are pretty exotic to common experience. The notion there are other universes isn't that far off, and in fact seems to be quite a popular belief these days, though Sabine Hofstetter does a good job of trying to keep the science real by exposing MWI as pseudoscientific (YT).
EDIT IN RE: COMMENT
as we don't have direct access to some scientific entities such as quark, they are useful theoretical notions and David Lewis may answer in this way. Do you agree with me?
Of course, we believe in things we don't have direct access to all of the time. Some access is temporarily unavailable (I haven't seen your new car yet), and some of it is available indirectly through complex apparatuses (Gravitational waves exist because of LIGO). Who is to say that we might not find access to these other worlds eventually?
From the Encyclopedia of Philosophy's entry "Modality, Philosophy and Metaphysics of" (Vol. 6, pg. 287):
Lewis (1986) made quite clear that the case for genuine modal realism was a philosophical inference to the best explanation, not a single silver bullet-like argument. He claimed that when all things were considered, his theory possessed the best balance of theoretical virtues and vices."
So, what is that argument? Like all arguments made by too-much-thinkers, it's probably complex. According to the IEP's article on Modal Metaphysics regarding it:
[The claims of existence without location] could mislead, however, in suggesting that Lewisian worlds are a type of abstract object, akin to universals or sets. Realists about abstracta sometimes say that their objects lack a location, despite the fact they exist. However, Lewis concedes at least three senses in which his worlds qualify as “concrete.” First, note that if sets and universals are counted as abstract, then a contrast can be with individuals or particulars. In that case, Lewisian worlds qualify as non-abstract or “concrete,” since they are particulars. (But, note that a concrete world can be home to abstract objects all the same.) Second, the abstract/concrete distinction sometimes concerns whether an object has spatio-temporal dimensions. Yet here too, since Lewis’ worlds are spatio-temporal kinds of entities, they qualify as “concrete.” Finally, Lewis recognizes that some things might be abstract in the sense of being an “abstraction,” that is, they might be the kind of entity represented by an incomplete or gappy description. (An example would be “the Average American”). In line with Kripke, however, Lewis accepts that each possible world is described by the sentences in some maximally consistent set—and the set would describe the world completely. So worlds are concrete by this criterion also.
Remember, philosophy is often a speculative enterprise. Long before electrons were confirmed as charge carriers, there were speculations of particles that carried charge. It is the job of science to confirm the physical, but the part of science that puts forth new conjectures is itself, strictly speaking much more metaphysical because it consists of examining the ontology of the science in question. There is much merit in a physicalist skepticism that demands all things be proven to exist by causal interaction, but the basic activity of imagining new things is what allows us to find those things to be proven in the first place. Sorting out the ontological statuses can even be confusing given the physics. Is gravity real? What about the Higgs-Boson field? Tough to say, even for professional physicists.
Science also has to admit that the speculation of multiple worlds, while lacking evidence of existence, is itself an inductive truth, not a deductive one, and thus uncertain by definition. Scientific discovery is subject to monotonic logic and defeasible reasoning. Maybe the folks at CERN will open up a portal to another universe one of these days!
The answer depends upon what you mean by the words believed, possible worlds and real.
If you believe that somewhere, in different universes presumably, there are countless worlds that exist in the way that Earth itself exists, in which all possible combinations of events are played out, then your beliefs are not consistent with what I take the word empiricism to mean.
On the other hand, if you believe possible worlds to mean hypothetical alternative configurations of reality, then they are like many other conceptual tools that can be used to build theories .