# When are Accessibility Relations satisfied?

We can only "measure" (quantify) counterfactuals by an accessibility relation to our own world. Therefore how can we assert something as necessarily true in all possible world's if quantification of modal relations (counterfactuals) is/are restricted to only our world?

• Explain your question, what exactly do you mean by "counterfactuals," what do you mean by "quantify," is it the usual existential and universal or modal quantifier? Why are you calling modal relations "counterfactual," not all modal relations are "counterfactual" if we use the usual meaning of the term "counterfactual." Why are you saying all modal relations are restricted to only our world? what do you mean by "our world?" Are you working within a kripkean frame-work? Dec 16, 2018 at 3:00
• Counterfactuals mean what they are. A possible state of affairs. Yes, I adopt the usual meaning of "quantify" here. The rest of the questions are a matter of semantics that is well established. Dec 16, 2018 at 3:02
• Not really well established, but contextual. Are you working within a kripkean system? Do you know how QML works? I am asking so I can answer the question. Dec 16, 2018 at 3:04
• "Do you know how QML works?" I don't. Dec 16, 2018 at 3:05

We can't, unless we use a modal logic in which it doesn't matter which world you're at when evaluating the truth of modal sentences:

S5 (with the additional assumption that there's only one equivalence class of worlds).

• So, framing conditionals also fails too? Dec 16, 2018 at 17:54

First of all I want to state that Quantification over possible worlds has some problems: both logical and philosophical. So take this as a mere exposition.

Let A be the set of all worlds accessable to A* where A* is the actual world in A. Let C be the set of all constants {c1,c2,...,cn}. Let R be the set of all relations such that {P1^n, P^2^n,...Pm^n} n>=0. Let V be the set of all variables {x1,x2,...xn}

Model M is

M=A

D: non-empty domain of objects

R: accessability relation on W×W

V_M is a valuation funtion of the model such that

1) If cm∈C, then V(cm)∈D

2) If Pm^n∈R and n=0, then V(Pm)∈Powersetof(W)

3) If P is a predicate of power n not equal to zero, then V(Pm) is a relation from W to powerset (D1×D2×...Dm)

Let Φ be the formula you are speaking about, then to say □Φ in all possible worlds is to say:

∀Wn∈W, □Φ is true (M) in Wn.