Thanks for your comments. Re (1), I'm not doing modal metaphysics. I'm not intending possible worlds to be taken as anything other than a logical tool - maybe I should've been more clear about that. But yes, you're right that we don't have to use "possible worlds" (however we interpret them), and I should've pointed that out. Re (2), unfortunately, there will be a degree of notational laziness in these videos! Re (3), I have a video precisely about that (ML 1.3).
Let's say that on room 0's table, you see the formula "Nec(p v ~p)". What this means is that in *every* room whose door is open, the formula "(p v ~p)" will be on its table. Do you see how this works? When I say stuff like "w1 is accessible from w0", just imagine that w0 and w1 are rooms, that you're in w0, and that there's a door that opens to w1.
Now suppose that in each room, there's a table, and on each table, there are lots of little bits of paper. These bits of paper have formulas written on them. Suppose, on room 0's table, you see the formula "Pos(p & q)". What this means is that in at least one (maybe more) of the rooms whose door is open, the formula "(p & q)" will be on its table.
Instead of worlds, imagine you have a bunch of rooms. You're in room 0. You can see a bunch of doors - to rooms 1, 2, 3 and so on. Some of these doors are open and some of them are locked shut. Let's say the door to room 1 is open. In that case, we say you can *access* room 1; room 1 is *accessible from* room 0.
Okay, so I'm 11:37 in and if I'm getting this right (sorry if you clarify later), The choice of R doesn't actually restrict the number of worlds the model considers, it just groups them up and it's up to you which group you wish to consider. So in your example, it seems like one way you could define R is that w0Rw1 if and only if they have identical truth assignments on all propositions relating to our world's laws of nature. This seems to me an equivalence relation whereby each equivalence class contains all possible worlds that agree on a certain set of physical laws, so it's not necessarily the case that R restricts the possible worlds we consider, it just groups up different worlds into different groups in which different propositions are considered possible. Am I on the right track here or am I way off?
Very helpful, as usual. Many thanks, Kane. One quick question pertaining to the relationship between a proposition's necessity in a given world relative to that world's access to other worlds. Shortly before the 13:00 minute mark you discuss this showing that indeed the necessity of p in w0 obtains iff p holds in all worlds to which p has access (w1, w2,...w3). So far so good. However, even if p obtains in all those worlds accessible from w0, is it not the case that such could be simply a contingent circumstance, I.e., p is obtaining only contingently, not necessarily in those worlds? And, if such is the case, does it not remain possible that p would still not hold in w0? Perhaps another way of putting the question is this: Does the mode of a proposition in a given world pertain to the issue of the same proposition's mode in worlds that can access that world?
After watching all your videos about modal logic will I be able to understand Kripke's completeness theorem? If not, could you tell me how much and what else I should study to reach a proper level of knowledge in modal logic to understand such a theorem?
What does “access” mean, doesn’t it mean “to enter”? So if w0 and w1 do have a logical relationship, why is that called “accessibility” and not something else? What does one mean when they say w1 is “accessible from w0”?
Doesn't what you propose at the end, concerning necessity, imply that w0Rw0 = 1? In other words, that the system you're using for discussing necessity is here reflexive in terms of accessibility?
Good! However, a few suggestions: 1. you should mention how members of W don't have to be 'possible worlds'.. they could be information states say of a computer or sets or linguistic entities. Its one thing to specify the semantics of modal logics and another to do 'modal metaphysics'. 2. in your definitions of necessity and possibility you should use w and w' instead of w0 and w1. 3. perhaps remark about how R in K is unreflexive
However, you do have to be careful with this analogy because it doesn't follow from "w1 is accessible from w0" that "w0 is accessible from w1". So, imagine that each room has two sets of doors - one set being those through which people from other rooms can enter, and the other being those through which people in the room can leave. For room 1, the door allowing people from room 0 to enter might be open, while the door allowing people in room 1 to exit to room 0 might be locked.
I've heard about p zombies that are considered to fit one of these categories: - ontologically possibles - ontologically not possibles And one of these: - conceivable - not conceivable Can you explain this?
Great video, but I have a problem and possible improvement in mind. If it is considered that w0 accesses to w1 and observes that it is not necessarily p there, it should be able to modify its judgement in about the necessity of p. I think the direction of flow of information is more important in this case rather accessability of the worlds. Yet it could be better defined what accesaability means in this sense, is it only empirical observation that whether it is p or not p, for example. Yet also considering that possible worlds are only mental conceptions or tools, this discussion becomes further blurry.
In the last example, ~p is true in world 1 and world 1 only access world 2 and p is true in world 2. so p is true in every world which world 1 access hence p is necessary in world 1 by definition. but p is false in world 1. contradiction. right?
How does one decide, in System K, between which possible worlds relation R obtains? E.g. what determines that the possible in which pigs fly is accessible from the actual world, but that the possible word where pigs orbit the sun is not accessible (or is it?)? Is this just something that the formal system has to presuppose as given, or is R definable in some other terms?
If w0 accesses w1 and w1 accesses w2, then doesn't w0 access w2, or, w0Rw2? I'm asking because if this is the case, then wouldn't a_w0(nec(p)) = 0 in the top right diagram on the last slide, since w0Rw2 and a_w2 = 0?
Suppose at w0 p = 1, and at w1 p = 0. Is w0Rw1 = 1? Surely not, because p = 1 rules out p = 0. But every possible world is different from every other in at least one respect. So no world accesses any other.
