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The problem is known, both Huges e Cresswell (p.275) and Fitting e Mendelson (102) mention it.

Example of the problem: we have a formula: □ (P(x) v ¬P(x)) that is true in the world w under the valuation s that assign to the variable the object o that is part of the domain of the world w. For the interpretation of □ we should have that (P(x) v ¬P(x)) is true under s in every world v accessible from w. But how to deal with the fact that the object o may not exists in other worlds v?

I see that we can assume gap in true value and say that in that world v where o is not in the domain of v the formula in neither true neither false.

Also we can use an expanding domain model, where Dw ≤ Dv, but then we have that CBF is valid. (both are not very good)

The question is: what is the “standard” solution to avoid this problem in QML with varying domain? Someone can explain it to me?

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  • There are several proposed responses. The issue relates to the distinction between possibilism and actualism. Possibilists accept that there exist things that are not actual; actualists hold that only actual things exist. There are multiple variations of each. Timothy Williamson prefers to distinguish between what he calls necessitism and contingentism: he defends the view that everything that exists does so necessarily. Kripke has a system under which the Converse Barcan Formula is invalid and P(x) is false at worlds where an object does not exist.
    – Bumble
    Oct 5, 2023 at 17:41
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    There is more detail in this article. plato.stanford.edu/entries/possibilism-actualism
    – Bumble
    Oct 5, 2023 at 17:41
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    QML is bivalent thus your speculation about truth-gap with shrinking domain has no basis thus P(o) is naturally assumed to be false at its every accessible contracting world v, hence CBF is invalid while BF is valid which is worrisome. So with a simple varying domain semantics it's hard to be totally at peace for both BF and CBF, there's no easy alternative. Kripke proposed KQML with arbitrary varying domain adding some free logic for convenience along with the downside that the semantics of classical quantifies have to be rectified according to the actual domain of a generic world... Oct 5, 2023 at 23:48
  • i was referring more to this statement on Fitting and Mendelson pag. 102 : "The other approach, which is the one we follow, is to say that even though v(x) might not exist in the domain associated with w, it does exist under alternative circumstances we are willing to consider, and consequently talk about v(x) is meaningful. Then at w, either the property P is true of v(x) or is false of it, and in any event, P(x) V -,P(x) holds. As we said, this is the approach we follow. " This seems good, but i think i dont trully understand what they are proposing. Oct 7, 2023 at 10:09
  • The alternative circumstances they are willing to consider where v(x) is meaningful must be the well known constant domain approach aka SQML which is technically much more straightforward but the downside is it’s incompatible with natural languages and both BF and CBF are valid which is not good as you say… Oct 9, 2023 at 1:28

1 Answer 1

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See Nino Cocchiarella & Max Freund, Modal Logic. An Introduction to Its Syntax and Semantics (2008, Oxford UP), Ch.8.6 Actualist-Possibilist Secondary Semantics, page 169-on:

As represented by a model (indexed by a given language), the notion of a possible world involves a single comprehensive domain or universe of discourse. [...] One consequence of the fact that every possible world (of a given logical space) consists of the same totality of objects is the logical truth in the primary semantics of the Carnap-Barcan formula and its converse.

Convention: Hereafter, we shall understand by a class of worlds an ordered pair <A, e>, where A is a class of worlds in the original sense and e is a function with A as domain and such that for all a ∈ A, e(a) ⊆ Ua. We will understand e(a) to be the set of objects actually existing in the model (possible world) a and continue to take Ua to be the domain or universe of discourse of A, except that now that domain is understood ontologically to be the set of possibilia of a. In addition, we will retain the satisfaction clauses of definition 526 [page 165: the definition of satisfaction for quantified formulas] above except for replacing A by <A, e> throughout and applying each of the clauses to all of the formulas and not just the standard formulas of the language in question.

This means that, in addition to the standard quantifiers, ranging over the full domain of possibilia, there is a new pair of e-quantifiers ranging over only on the existing objects.

Here is the satisfaction clause for the e-universal quantifier:

if ϕ is a formula of L, then ass satisfies ∀xϕ in <A, e> at a iff for all d ∈ e(a), ass(d/x) satisfies ϕ in <A, e> at a.

With this, we have two possibilities: either (i) we consider the unmodified notion of logical truth, where the assignments range over the full domain of possibilia, in which case the formula P(x) v ¬P(x) is satisfied, or (ii) we consider only "actualist" assignments, in which case the formula is not satisfied by ass(d/x) if object d does not exist in world a.

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