I think you want to define A?B to be true if A is true when we restrict to some neighbourhood N∈Nwsuch that A∩N is nonempty. Otherwise your later example doesn’t make sense.
Edit: Wait, I see what you mean. Fixed definition.
For Lewis, ∅?B=W for all B. In other words, the counterfactual proposition “were ϕ to occur then ψ would’ve occurred” is necessarily true if ϕ is necessarily false. For example, Lewis thinks “were 1+1=3, then Elizabeth I would’ve married” is true. This means that A∩N may be empty for all neighbourhoods N∈Nω, yet A?B is nonetheless true at w.
I think you want to define A?B to be true if A is true when we restrict to some neighbourhood N∈Nw such that A∩N is nonempty. Otherwise your later example doesn’t make sense.
Edit: Wait, I see what you mean. Fixed definition.
For Lewis,∅?B=Wfor allB. In other words, the counterfactual proposition “wereϕto occur thenψwould’ve occurred” is necessarily true ifϕis necessarily false. For example, Lewis thinks “were 1+1=3, then Elizabeth I would’ve married” is true. This means thatA∩Nmay be empty for all neighbourhoodsN∈Nω, yetA?Bis nonetheless true atw.Source: David Lewis (1973), Counterfactuals. Link:https://perso.uclouvain.be/peter.verdee/counterfactuals/lewis.pdfElaborate?