The short form of the question is this:
Is there a model of modal propositional calculus that gives the modal operators the meanings
◻φ — φ is valid, or always true
◊φ = ¬◻¬φ — φ is satisfiable, or can be true
?
The long version, with some background, is that I was trying to put together an informal, short, simplified explanation of modal logic for non-mathematicians. I made a mistake when trying to remember the semantics of S5, where I can essentially drop the relation between possibles worlds. The resulting semantics was not S5, but did seem to be consistent.
My approach and notation might not be standard.
I started with propositional logic with a minimal expressively complete set of operators:
syntax:
𝔓 = some countable set of letters
p ∈ L for p ∈ 𝔓
¬χ ∈ L for χ ∈ L
(φ → ψ) ∈ L for φ ∈ L, ψ ∈ L
with other connectives as abbreviations; for example.
⊤ ≝ (p → p)
⊥ ≝ ¬⊤
semantics (only the 'weak' semantics):
⊨φ if ‖φ‖(F)=1 for all interpretations F : 𝔓 → 2
where
‖φ‖ : (𝔓 → 2) → 2
‖a‖(F) = F(a) when a ∈ 𝔓
‖¬φ‖(F) = 1 − ‖φ‖(F)
‖(φ → ψ)‖(F) = max { 1 − ‖φ‖(F), ‖ψ‖(F) }
Then I added the modal operator to the syntax:
◻χ ∈ L for χ ∈ L
◊φ ≝ ¬◻¬φ
What I then should have done was introduced possible worlds, and the essentially S5 Kripke semantics:
possible worlds, W some (countably infinite) set, e.g. ℕ
interpretation now a map also from W : W → (𝔓 → 2)
for any interpretation F:
‖φ‖ : (W → (𝔓 → 2)) → 2
‖a‖(F)(w) = F(w)(a)
‖¬φ‖(F)(w) = 1 − ‖φ‖(F)(w)
‖(φ → ψ)‖(F)(w) = max { 1 − ‖φ‖(F)(w), ‖ψ‖(F)(w) }
‖◻φ‖(F)(w) = min { ‖φ‖(F)(t) | for all t∈W }
⊨φ if ∀F ∀w ∙ ‖φ‖(F)(w) = 1
But, I tried to short-cut the introduction of worlds, and did this instead:
I left interpretations as in the basic propositional calculus : 𝔓 → 2
I left these:
‖a‖(F) = F(a)
‖¬φ‖(F) = 1 − ‖φ‖(F)
‖(φ → ψ)‖(F) = max { 1 − ‖φ‖(F), ‖ψ‖(F) }
⊨φ if ∀F ∙ ‖φ‖(F) = 1
and I added:
‖◻φ‖(F) = min { ‖φ‖(G) | for all G }
That last step is non-standard, and a deviation from S5.
In both S5 and my 'broken' semantics, we have
⊨ ⊤
⊨ ◻⊤
⊭ ⊥
⊭ ◻⊥
⊭ p
but only in my 'broken' semantics do we have
⊨ ◊p
For S5 this would be wrong, as not all possible world/interpretation combinations have that p holds somewhere; i.e. there is an interpretation where p is false in all worlds.
But, on the other hand, reading this as "there is a propositional interpretation where p is true" does not seem an unreasonable one for study. It doesn't seem to be obviously trivial, or obviously inconsistent.
I think my semantics is like S5, but with the additional property that the worlds must cover all possible interpretations. More accurately, as that set would not be countable, I think it's enough to cover all possible interpretations with all-but-finitely-many propositions true and also all-but-finitely-many propositions false.
I've searched online, and have looked through some books (including Chellas), but can't find anything like this semantics discussed.
Is this familiar to you?
Many thanks, Rob.
