Is every world accessible to itself?

Question

I just realized that for the proposition "If p is necessarily true then p is true", i.e. "box p implies p", to be a tautology, we need the condition that every world is accessible to itself. That is, for every model M=(W,R) we need the reflexivity of the accessibility relation.

Here's my question:

It makes sense to assert that if a proposition is necessarily true then it is true. So every world must be accessible to itself. Because if not, then what makes sense above may not make sense anymore.

Does this mean that every world is accessible to itself? Of course there are models where reflexivity of the accessibility relation is absent. But in a philosophical point of view, I think it's safe and fair to assert that a necessary truth is a truth. And argue that every world is accessible to itself as a consequence of the above assertion. Perhaps I haven't fully understood the meaning of a world being accessible to another world. Any help is appreciated. Thanks!

Edit: I changed "p is necessarily true implies p is possibly true" to "p is necessarily true implies p is true". But I'm happy with both statements being tautologies.

Answer 1

You are correct about the relationship between □P → P and the reflexivity of the accessibility relation. As to whether you want to take this as an axiom, it depends entirely on your intended interpretation. If □ is to be interpreted as "it is necessarily true that" then □P → P holds, since, as you say, if a proposition is necessarily true then it is true.

But there are many other uses of modal logic that require other interpretations. For example, if □ is to be interpreted as "it is obligatory that" as part of a logic of obligation, then we don't want □P → P, because that would imply that everything that ought to happen does happen. Likewise, if we wish to interpret □ as "it is provable in some formal system that" then we don't want □P → P because we don't wish to assume the system is sound without proof.

Answer 2

This very much depends on your set of axioms, in other words, do your models have reflexivity.

For example,

• The class of all Kripke models is called K.
• The class of all serial Kripke models is called KD.
• The class of all reflexive Kripke models is called T.
• The class of all transitive Kripke models is called K4.
• The class of all transitive Euclidean Kripke models is called K45.
• The class of all serial transitive Euclidean Kripke models is called KD45.
• The class of all reflexive transitive Euclidean Kripke models is called S5.

These are all well studied logics, and not all of them are reflexive.

If you are studying knowledge, then it makes sense to enforce □ϕ → ϕ (or Kϕ → ϕ), i.e. if you know something, then it is the case (in other words, you cannot have false knowledge). This makes sense especially if you consider knowledge "justified true beliefs".

However, if you are studying belief (doxastic logic), then you don't necessarily want to enforce □ϕ → ϕ (or Bϕ → ϕ), i.e., it should be possibly to have false beliefs.