## Accessibility Relations Across Possible Worlds

• 10.3k
Accessibility relations seem of great import to understanding Kripke's Naming and Necessity.

I was wondering what other members think about them.

How does one 'know' epistemically when an accessibility relation obtains or is successful or not? I am mainly interested in that regards.
• 10.3k
@Banno or someone care to help me out?
• 10.3k
This is relevant. Quine was opposed to quantification of modal relations.

https://plato.stanford.edu/entries/logic-modal/#QuaModLog
• 763
How does one 'know' epistemically when an accessibility relation obtains or is successful or not? I am mainly interested in that regards.

I don't understand what you're saying hers. How does an accessibility relation obtain (that is, become actual in the real world)? An AC is just the description of which worlds can access (quantify over, basically) other worlds in a given frame using some specified system of modal logic and it's determined by the axioms of the logic. This relationship may have different properties depending on what you're investigating. S5 has the Euclidean property which S4 does not, for instance. So if two worlds can both access some specified world they can access each other.

So long as you have some frame established and specify the modal logic in use then that seems perfectly justifiable reason to say... whatever it is you're asking (not sure I understand the question).
• 10.3k
I don't understand what you're saying hers. How does an accessibility relation obtain (that is, become actual in the real world)?

Yes, if we talk about an accessibility relation of me winning the lottery in a possible world where I played the lotto, then where is the quantification of it taking place? Relative to the world where I'm stipulating such a situation?
• 763
That's not an accessibility relation obtaining, that's merely establishing what frame (the set of worlds) is being quantified over. That you're speaking about winning the lotto presumably restricts the world scope to, for example, worlds where the lottery exists, worlds where you exist, worlds where you actually play the lotto, etc. The frame is determined based on the scenarios described int he situation you gave.
• 10.3k
That's not an accessibility relation obtaining

Sorry to pester you; but, you have established a frame of reference where an accessibility relation obtains. If I didn't win the lotto then I didn't in this world; but, may have in another one. I can't assert otherwise because I'm constrained to one world only.

But, what I'm grappling with may be more succinctly described as to how does quantification work for other possible worlds?
• 763
No, a frame is just a set of worlds that share some relevant feature(s). An accessibility relation tells you which worlds can quantify over which other worlds in some particular frame. Your whole lotto thing is a perfect example, you're constraining your modal discourse to worlds where the lotto exists, you exist and in which you play the lotto. That's a frame. You can quantify over other worlds, I'm not sure what the problem is. You mentioned Quine before, but Quine's issue was about quantifying into opaque contexts and it isn't really taken too seriously anymore.

But, what I'm grappling with may be more succinctly described as to how does quantification work for other possible worlds?

The same. An accessibility relation is a feature of a modal logic, not of reality (controversial, depending on how you think reality and logic relate).
• 10.3k
No, a frame is just a set of worlds that share some relevant feature(s).

Here I go repeating myself; but, that can only depend on one world from within a stipulation of another possible world can be made.

An accessibility relation tells you which worlds can quantify over which other worlds in some particular frame.

So, yeah, I agree with that. Can you expand on this framing aspect?

Your whole lotto thing is a perfect example, you're constraining your modal discourse to worlds where the lotto exists, you exist and in which you play the lotto. That's a frame. You can quantify over other worlds, I'm not sure what the problem is.

Yes, but I don't understand how you can quantify modal relations. Sure, I can set up a frame of reference wrt. to this world relative to another world where something might have happened otherwise than in this one; but, that's the limit of what I can do. I can't say that something happened if I did nothing in this world that would conditionally restrain another world where the event could have happened otherwise.

The same. An accessibility relation is a feature of a modal logic, not of reality (controversial, depending on how you think reality and logic relate).

Ok, so this touches the crux of the issue. Basically, I understand logic to be pluralistic not unitary, so predicate logic might converge with modal logic; but, not absolutely. What do you think? The case I'm making is in regards to when do they converge or don't in this thread.
• 763
Here I go repeating myself; but, that can only depend on one world from within a stipulation of another possible world can be made.

No it doesn't, it's just a collection of worlds. It doesn't matter what world you are in, you're setting up the set of relevant worlds in the modal discourse, the actual world is not necessarily relevant and may even be excluded in some cases. Are you talking about frame conditions? (E.g. the properties of what worlds access others?)

Yes, but I don't understand how you can quantify modal relations. Sure, I can set up a frame of reference wrt. to this world relative to another world where something might have happened otherwise than in this one; but, that's the limit of what I can do. I can't say that something happened if I did nothing in this world that would conditionally restrain another world where the event could have happened otherwise.

You're not constraining worlds themselves, you're constraining the worlds that are part of the set you're speaking about basically. Once you have a frame condition and a set of worlds picked out, you just worked out the modal deductions. Your SEP link basically says this I think:

Each of the modal logic axioms we have discussed corresponds to a condition on frames in the same way. The relationship between conditions on frames and corresponding axioms is one of the central topics in the study of modal logics. Once an interpretation of the intensional operator □ has been decided on, the appropriate conditions on R can be determined to fix the corresponding notion of validity. This, in turn, allows us to select the right set of axioms for that logic.

Ok, so this touches the crux of the issue. Basically, I understand logic to be pluralistic not unitary, so predicate logic might converge with modal logic; but, not absolutely. What do you think? The case I'm making is in regards to when do they converge or don't in this thread.

Well I don't think that's really a pluralistic view of logic. Logical pluralism - which I accept- is the view that there are more than one legitimate kinds of logic or more than one correct logic. Standard, normal modal logics essentially add on to classical predicate logic, and even have a translation scheme to understand modal statements in classical form, so I don't think it's really entirely different. A real test would be if you accept both classical and intuitionistic logic, because those logics actually disagree with each other about what arguments are valid or not.
• 10.3k
Are you talking about frame conditions? (E.g. the properties of what worlds access others?)

• 10.3k
Here are some of the responses I got from Stack Exchange:

https://philosophy.stackexchange.com/questions/58946/when-are-accessibility-relations-satisfied/58954#58954

bold
italic
underline
strike
code
quote
ulist
image
url
mention
reveal