Indistinguishable, the above source seems to contradict something you said, if I understood you correctly, that in S5 every world is accessible to every other world.
Requiring the accessibility relation to be reflexive, transitive and symmetric is to require that it be an equivalence relation. This isn’t the same as saying that every world is accessible from every other. But it is to say that the class of worlds is split up into classes within which every world is accessible from every other; and there is no access between these classes. S5, the system that results, is in many ways the most intuitive of the modal systems, and is the closest to the naive ideas with which we started.
Actually, I mean write them out in narrative English. But I must say, your “start” is making some things begin to click for me. I can’t tell you how much I appreciate your assistance with this. I will owe you mightily.
I was just about to post about what you just said to Indistinguishable. He and I both have been presuming, at least apparently, that in an S5 model every world is related to every other world. But as you have just discovered, that’s not true.
In my own case, I just plain forgot. For some reason, when people mention S5, they generally sort of automatically think of a model where every world is related to every other world. But that is not strictly speaking required to satisfy S5.
In fact, even a model in which each world relates only to itself satisfies S5.
But for some reason, the “image” that comes to mind when you say “S5” is the one in which every world relates to every other. This seems to be a matter of practice and tradition rather than logic.
There are two relations, relation(1) and relation(2). There is also a particular designated world W. These can be described together as follows.
W, and only W, relates(1) to worlds.
Two worlds are related(2) only if W is related(1) to each of them.
That’s using my “version 2” of the first relation. “Version 1” would go something like this: W, and only W, relates(1) to worlds, and W relates(1) to all worlds.
I don’t like how that guy just flat out denies that epistemic possibility can be talked about using model-theoretic tools. (What he says is it can’t be captured using talk of “possible worlds” but I can’t understand how that is a significant claim unless he means you can’t deal with it model-theoretically.) That’s just not true. Granted, some “epistemically possible” worlds will contain contradictions, and "possible worlds’ are usually thought of as containing no contradictions. But that’s just on the “canonical model.” (And others relevantly like it.) You can build other kinds of models. Plenty of people have worked on paraconsistent models, for example, in which you have worlds which contain contradictions. (For example, Graham Preist has worked on models which include “impossible worlds”* for dealing with things like, for example, fictional entitites.) You can build the model in such a way that contradiction doesn’t “explode” and make everything true like it would in a more standard kind of model.
He’s right that people don’t agree about how to understand epistemic possibility. But to claim that it flat out can’t be handled model-theoretically seems far too strong. In fact, its in dealing with things like epistemic possibility (and fictional possibility, and conceptual possibility, and so on) that model theoretic tools might be made really interesting and useful. IMO. [/soapbox]
-FrL-
*This invites a question I like to ask: How can we be so sure the actual world is a possible world?
Most people’s first reaction is to say “Isn’t it just possible by definition” but if you think about it, that can’t be an adequate answer. One shouldn’t be able to force a substantive claim to be true just by virtue of the definition of terms.
Yes, it does, actually. I’ll wait a bit before cheering, but it seems offhand to satisfy the model fairly well. We have the transitivity of worlds through and only through W and the universality of W itself.
I couldn’t agree more. I’ve often said that if we can define things into truth, then we can prove that pigs fly by defining fly to mean “wallow in mud”. At the very least, we have to start from a premise (which in this case, I guess, would be the B axiom: A -> <>A).
I perhaps should have made it more clear, but I actually remarked on this in post #19.
That is to say, if one limits one’s attention solely to models in which every world is accessible to every other world, then the corresponding logic will be S5. But, also, if one limits one’s attention to models where accessibility is an equivalence relation but where it is not necessarily the case that every world can access every other, one still gets the corresponding logic of S5.
In a sense, the equivalence relation characterization is better, in that it is more general. However, sometimes, just because it is easier to state, I have described S5 as the logic where every world can access every other, and I apologize for not making the details of this point clearer.
BTW I guess the relations I described can be combined into one as follows:
wRv -> ((w == W) or (WRw &WRv))
in other words,
One world w is related to another world v only if either w is W or W is related to both w and v.
I think that amounts to the same as what I offered before.
To be honest, the disjunctive nature of the solution I’ve offered (reflected in the existence of two accessibility relations in the one case or in the disjunctive nature of the single relation in the second case) makes me uncomfortable. It feels ad hoc. But my feelings may be misplaced.
It sounds to me like you want to model the following situation:
There is a real world, which contains a set of true statements
There are also a set of entities, each of which has a set of statements, representing what that entity knows and/or believes about the real world
I would not use modal logic to model something like that. I would probably use some sort of fuzzy logic. Each entity ascribes a level of belief to its own statements. I am not sure where I would go with this, though.
How about a model with worlds, accessibility relations, and all that, but wherein the holding of the accessibility relation is itself fuzzy. The relation holds “to a degree” in some sense.
How would this be reflected in the interpretation of the modal operators? I guess they would have to be fuzzy as well. But how would this work? Since a world’s relating to other worlds will have different amounts of fuzziness in different cases, will you just have to maybe average out the levels of fuzziness found in the holdings of the relation in order to get the level of fuzziness of truth of the modal statement? Maybe?
Oh dammit, I just got confused about something elementary. Far from the first time.
Okay, to my recollection, the axiom that gives you a reflexive frame is A -> A.
But here’s a model in which A -> A holds as an axiom:
There are two worlds, W and V. WRV, and VRW. (But not-WRW and not-VRV)
In W, A and A are both true. (And so, A -> A is true a forteriori.)
The same in V: A and A are both true.
This all is consistent, I think. A is true in W, W is related (only to) V, and A is true in V. Right. And the same goes for the other direction.
So it looks like A -> A is true in all the worlds in the model. Doesn’t that make it an axiom of the model? But then, if it’s an axiom, shouldn’t the frame be reflexive? But it’s not.
What have I forgotten?
-FrL-
(Sorry for the hijack, Lib, but this hardly seemed worth starting a new thread.
What you’ve forgotten is that when people say “The axiom scheme A -> A corresponds to a reflexive frame”, “The axiom scheme A -> A corresponds to a transitive frame”, and so on, they don’t mean that any set up of possible worlds and truth-assignments at those worlds which satisfies the axiom scheme also satisfies the frame condition. Rather, what they mean is the following:
Take a set up of possible worlds and an accessibility relation on it; call this a frame. Note that it does not have any truth-assignments yet (in the sense of assignments of true and false to primitive statements at each world). If a statement would be true in every world of a frame under every truth-assignment, we will say it is valid in that frame.
A system of axioms corresponds to a frame condition when the class of statements provable from those axioms is the same as the class of statements which are valid in all frames satisfying that frame condition.
Thus, A -> A corresponds to reflexive frames in the sense that what can be proved from A -> A is precisely the same as what holds in every world of every reflexive frame under every truth-assignment.
Your supposed counterexample is not really a counterexample, because it breaks once we change the truth-assignment.
Incidentally, with that in mind, we can see now how it can be the case that S5 both corresponds to “The accessibility relation is an equivalence relation” and “The accessibility relation holds between all worlds”. The statements which hold under all truth-assignments in every world of every frame in which accessibility is an equivalence relation are all the same as the statements which hold under all truth-assignments in every world of every frame in which accessibility holds between all worlds. And this class of statements is the theorems of S5 (taken as a propositional logic, that is; I’ve declined to specify above how frames should deal with the intermingling of individuals and possible worlds [whether the same individuals need to exist at all possible worlds, or if existence of an individual at a world is closed under accessibility, or what. The different choices here result in different S5 systems of first-order logic]).
And about each other, though some of that information might be “tainted” by internal interpretation. For example, suppose that in world v, it is true that “John wears a blue shirt.” Likewise, it is true in world w that “John wears a blue shirt.” But suppose that in world v, it is true that “Blue shirts are ugly” but in world w, it is true that “Blue shirts are beautiful”. And further, suppose that “ugly” in each world carries a different connotative power, such that ugly in v is just a description, but in w, ugly is an insult on top of a description. So even if “Blue shirts are ugly” is true in w, there is more information because “Ugly is an insult” is also true. One can see how entangled communication would become and how much w and v would be prone to misunderstand one another in a full language just from this simple three-level example.
Well…again, it seems to me that your model has some underlying notion of “truth”, and then a layer of “interpretation” that holds for each agent (I really do not like the “world” terminology here). “John is wearing a blue shirt” is true and that truth does not vary. “John is wearing an ugly shirt” is true for some of the agents but not for others; some of the agents are absolutely sure of that, and some are only partially sure of it.
I think fuzzy logic is a good tool to get from statements like “The temperature is 30 degress F today” to “Sure is cold today.”
