Some modal systems and characteristic axioms in ascending levels of strength:
K: L(p->q) ->(Lp->Lq)
T: Lp->p (reflexivity)
B: p->LMp (symmetry)
S4: Lp->LLp (transitivity)
S5: Mp->LMp (Since K, T, B, S4 are all valid in S5, the accessibility relation in S5 is reflexive, symmetric, and transitive.)
Where "->" is the material conditional, "L" is "it is necessary that", and "M" is "it is possible that."
In S5, the accessibility relation is unrestricted—every possible world is possible relative to every other possible world.
______
Two definitions—Lewis will dispute these. They are from Plantinga, (an actualist about possible worlds):
An object x has a property p essentially iff x has p in every world in which x exists.
A proposition p is true in a world W iff were W actual p would be true.
From Lewis's counterpart theory (Lewis, the concretist about possible worlds):
An object x possibly has a property p iff there is a possible world W in which a counterpart of x has p.
______
Note the Lewis semantics for counterfactuals:
p>q iff a world in which p&q is true is closer to the actual world than any world in which p&~q
Note also the reduction of intensionalist talk to extensionalist talk (e.g. take propositions to be sets of worlds, properties to be sets of possible objects that have a certain quality, relations to be sets of n-tuples (35-36). Both Lewis and Stalnaker do this. But this runs into the problem of having "2+2=4" and "every object is green or non-green" express the same proposition. (Also, necessarily equivalent predicates express the same property.)
"The problem of transworld identity" "What makes it the case that object x in world W1 is identical with y in W2"? (39)
Note that the argument against Leibniz's Law on p 37 from an object in different worlds having different properties parallels the argument against Leibniz's Law from an object having different properties at different times. The solutions will be analogous, too—either world-index all properties, or world-index Leibniz's Law.
A worldbound individual doesn't exist in any other possible world and hence (argues Plantinga) has all its properties essentially.
Chisholm: Take an object x and an object y and gradually flip their properties across worlds to where y has all the properties x started with and conversely. This shows there is a problem with transworld identity—how do we individuate these objects?
R: But there is a property x has that y can't have, namely its haecceity—being x or being identical with x.
Loux calls Lewis a possibilist, but he's really not, not if a possibilist is someone who thinks there are nonactual objects. Lewis thinks that every object is actual relative to the world in which it exists (in the way that every object is here—relative to its own surroundings). Meinong, on the other hand, is the quintessential possibilist. Lewis just thinks that nonactual objects exist in other concrete possible worlds that are spatiotemporally unrelated to us.
Contrast this with actualists like Plantinga, who takes possible worlds to be maximal states of affairs and objects existing in other worlds will be given a paraphrase (see above). Adams takes them to be sets of propositions. (So Plantinga's worlds are something like giant conjunctions that contain every atomic state of affairs or its negation, and Adams' every proposition or its negation.)
Plantinga and Adams are unable to analyze away modality (recall this from McGinn): e.g. a proposition p is true iff it's true in some possible world. Lewis thinks that he can analyze away modality in quantifying over all worlds.