Saul Kripke is speaking the language of modal logic when he uses the operator "possibly" in Naming and Necessity. He appears to have in mind counterfactual situations (i.e., counterfactual conditionals) or possible worlds such that if P is logically or modally possible in one or more possible world, then there is a counterfactual situation [a possible world] in which P is evidently not logically or modally impossible. In other words, it could have been the case (possibly) that mental states obtain without brain states. It also might have been the case that my arms were white instead of "black." One Kripkean example of possibility addressing this issue is that there is a counterfactual situation (call it W1) in which the first Postmaster General is not identical with the inventor of bifocals. He doesn't come right out and say that he is using "possibility" thus; however, Naming and Necessity
turns on counterfactual situations and possible worlds. And I now know that Kripke basically ignores the question
about whether something might have existed or not.
Concerning "necessity," Kripke writes, "Thus the identity of pain with the stimulation of C-fibers, if true, must be necessary" (N & N, 149).
What does he mean by "necessary" here? Hasker makes a
distinction between logical (i.e. conceptual)
necessity and metaphysical necessity, noting that
statements such as the one above are metaphysically
necessary, meaning that they are true in all possible
worlds. Kripke also asserts that "This table is not
made of ice," IF TRUE, is necessarily true or true in
all possible worlds (i.e. counterfactual situations).
so necessity evidently means "true in all
counterfactual situations or possible worlds."
For "rigid," Kripke simply writes that the RD or "rigid designator"
(which is apparently stipulative) names the same object in
all possible worlds. The RD, Nixon, names the same
object in W1, W2, W3 . . .
Now here is what I also found on a website article written by
<<[Tyler] Burge then identifies Kripke's essays as the first "account of names [in terms of] a theory of necessity. He counted names as 'rigid designators' - expressions that maintained a certain constancy of reference through variation in the possible worlds by reference to which modal sentences might be evaluated" [1992: 25]. However, these ideas about names and rigid designators were not presented first by Kripke, but earlier presented by Plantinga , Føllesdal  and Marcus .>>
Smith thinks that Kripke posits a similarity of
identity between logical and metaphysical necessity.
Nevertheless, he still defines it [i.e. necessity] as "true in all