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#91
 
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Model != World? -
01-04-2010
, 03:44 PM
#92
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Excuse me, but I have a basic question. What is the motivation for differentiating the concepts of "World" and "Model"? |
define the truth value of a formula. In conventional logic this is
usually a model of the particular world we assume we are in. However,
in modal logic this includes the complete set of possible worlds plus
the particular world we assume we are in. Both are needed since for
basic propositions we need to inspect the actual world and the modal
operators refer also to the other possible worlds.
-- Jan Hidders
#93
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Excuse me, but I have a basic question. What is the motivation for differentiating the concepts of "World" and "Model"? |
works. (Although there may need to be certain constraints on what models of a
theory are under consideration). But the philosophical discussion of what's
possible, and what's necessary, and alternative possible worlds predates modern
model theory.
One thing that is different about modal logics is that ability to refer to
multiple possible worlds (any time you say that something is possible, you are
implicitly quantifying over possible worlds). It's not usual in model theory to
allow quantification over models in the object language (although such
quantification may take place in the metalanguage). When you consider
propositions involving modal operators, a single "possible world" is not a model
for such propositions. It's the entire structure of all possible worlds that is
being referred to by statements such as: "It is necessarily the case that X".
There is a discussion of the various uses of possible worlds here:
http://www9.georgetown.edu/faculty/a...hilThesis.html
but there is no mention of the fact that a possible world is a model of a
theory.
--
Daryl McCullough
Ithaca, NY
#94
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Excuse me, but I have a basic question. What is the motivation for differentiating the concepts of "World" and "Model"? |
signature, and interpretation function>. Now, assuming "universe =
world" we have "world != model". QED.
#95
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Excuse me, but I have a basic question. What is the motivation for differentiating the concepts of "World" and "Model"? |
modal logic:
http://drona.csa.iisc.ernet.in/~deep...odal_logic.ppt.
and for Higher-order modal logic:
http://comet.lehman.cuny.edu/fitting...ghOrdPaper.pdf
Definition for model for classical logic is:
A model is every structure S = ( A, F, R, C), where A is a non-empty
set, (A, F, C) is an algebra and R is set of relations over A.
Vladimir Odrljin
#96
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Jan Hidders says... On 3 jan, 20:55, stevendaryl3... (AT) yahoo (DOT) com (Daryl McCullough) wrote: But my claim is that you do get a contradiction for the simple reason that your logic contains the old logic. It doesn't contain the same *axioms*. In particular, I'm rejecting the "knowability principle" in favor of a variant principle that is (as far as I can see) consistent. Well, I'm not so sure. Your new variant look very similar to how the principle is formulated in my model theory. And there I got the contradiction. Well, as I said, I don't see how the proof of a contradiction could go through. The variant looks similar to your version, because I *intended* it to be the closest variant that did not lead to the contradiction. The main thing that is different is that in my variant, knowledge is about *non-modal* propositions, rather than modal propositions. The distinction is this: If I say "It is raining", that's a modal statement; it's true in some circumstances and false in others. If I say "It is raining on July 12, 2006 in New York City", then that statement is non-modal. If it is ever true, then it is always true. So my formulation of the principle of knowability is that if a *non-modal* proposition is true, then it is known in some possible world. Now, I can easily come up with statements that make this principle false, as well, using self-reference: "This statement is not known to be true in any possible world" But within the syntax that I'm suggesting, such self-reference isn't obviously possible. Let's try to make this more explicit. You have a set W of possible worlds, a set MP of modal propositions, and for each world w, a set S_w of the elements of MP true in world w. The set S_w is constrained by the following rules: 1. If Kp is in S_w, then p is in S_w (you can only know true statements) 2. And(p,q) is in S_w iff p is in S_w and q is in S_w 3. Or(p,q) is in S_w iff p is in S_w or q is in S_w. 4. Not(p) is in S_w iff p is not in S_w 5. Implies(p,q) is in S_w iff p is not in S_w or q is in S_w 6. <>p is in S_w iff for some w', p is in w' 7. []p is in S_w iff for all w', p is in w' That already looks close enough to a model theory to me. Sorry for the confusion. I'm trying to paraphrase *your* model theory. A model could be a pair (W, S) with W the set of possible worlds and S : W -> 2^F where F is the set of formulas and satisfies the rules 1-7. I strongly conjecture that those models would be isomorphic to the models in my formulation of the model theory and lead to the same formulas being true. That was my intention. Your mapping to type theory is a bit hard for me to get my head around, so I'll assume for the moment that the above is your model theory. Now, to capture this semantics in type theory, we use the following translations: 1. Introduce a type, W, of all possible worlds. 2. Introduce a type, A, of all atoms (atomic modal propositions). 3. Introduce the predicate t(w,a) saying which atoms are true in which possible worlds. 4. Introduce a predicate k(w,p) saying which propositions are known in which worlds. 5. Define MP, the type of all modal propositions, to be the type of functions from W into P. You didn't define / postulate P yet. P was already introduced in another post. It's the type of all (non-modal) propositions. If you like, you can think of a proposition as a (closed) formula. But a deeper problem is that I don't see why you let modal propositions be different propositions in different worlds. I'm trying to model facts that vary from world to world using a logic in which statements have definite truth values. It's no different from using set theory to give a semantics to modal logic. Let's take an example: Plants are green. If there are two worlds, w1 and w2, then "Plants are green in world w1" is a *different* proposition than "Plants are green in world w2". One could be false, while the other could be true. To say "It is possible that plants could be purple" is to say: "exists w:W Plants are purple in world w". The statement "Plants are green" without reference to which world you are talking about is an incomplete proposition. It becomes a proposition when you supply a world w. So it is a function from worlds to propositions. In terms of your syntax: w ||- f I would write this as f(w) Once you've made the world explicit, as is the case with w ||- f you no longer have a modal proposition, but just an ordinary proposition. Why is it not enough that their truth value can be different in different worlds? You can think of propositions as truth values, if you like. In a classical logic, there are two propositions, "true" and "false". I'm specifically using a non-classical notion of proposition, in which we *don't* identify statements that have the same boolean truth value because knowledge doesn't work that way. If I know that "Superman is 6 feet tall" that doesn't mean that I know that "Clark Kent is 6 feet tall". It also makes it hard for me to see whether this formulation is equivalent withe the above one that it is supposed to capture. 6. For each atom a, we associate a corresponding element of MP: p_a == that function f such that f(w) = t(w,a). 7. Define the operator K as follows: Kf == that function g such that g(w) = k(w,p) Kf should be Kp? Right. Look, once again, I'm formalizing the knowability principle as: forall p:P, p -> exists w:W, k(w,p) In my model theory the semantics of the formula that represented it can be formulated as: (with M being the set/class of valid models) Forall (W,w_1) in M, *forall w_2 in W, forall f in F, (W,w_2)||-f - exists w_3 in W, *(W,w_3)||-Kf Yes. I'm claiming that this is *not* a sensible formulation of the knowability principle in the case in which f itself involves the knowability operator K. If f is the formula p & ~Kp, then your principle above gives us: (W,w_2) ||- p & ~Kp - exists w_3 (W,w_3) ||- K(p & ~Kp) which is a contradiction. The problem is that the knowability principle should not (in my opinion) be about modal propositions. To give the simplest example, suppose p is true in exactly one world. Further, suppose that p is not *known* to be true in that world. In that case, it would be ridiculous to say: Since p is true in one world, then it is known to be true in another world. p *isn't* true in any world, so it can't be known to be true in any other world. But if we deal with nonmodal propositions (propositions of the form w ||- p), then we can certainly have the case that p is true only in world w1, but the *fact* that p is true in world w1 is known in world w2. If you fix W we can simplify this to: (JH-KP) forall w_2 in W, forall f in F, w_2||-f -> exists w_3 in W, w_3||-Kf Doesn't that look similar to you? Similar, but just different enough that your formulation leads to a contradiction, and mine doesn't. My two-place "knowledge" operator acts on *non-modal* propositions. In your syntax, the entire expression (w_2 ||- f) is the nonmodal proposition corresponding to my f(w_2). I would write, instead: forall w_2 in W, forall f in F, w_2 ||- f -> exists w_3 in W, w_3 ||- K(w_2 ||- f) |
explain this to me. I'm afraid I can only give a short reply now
because life and work are getting busier again.
I think I see now better your point about the fact that in different
worlds we might use the same description to refer to things that are
actually different facts. Your example being "it rains" which refers
to something different if the different worlds correspond to different
days. But I would argue that this is from the perspective of someone
who is outside the model and has some way to identify the different
worlds independent of what facts hold in them. When you are inside the
model and in a certain world the only way to distinguish them is by
looking which facts hold in them. For the rain example it could be
that in your vocabulary you can express what day it is, and then you
can distinguish the different days, but then you could have formulated
the fact that you had in mind as "it rains and it is today 5 January
2010". If the date in your world is not in your vocabulary then you
have no way of describing the differences between the "it rains"
proposition in different worlds.
For me the meaning of a proposition is in its pragmatics. If "it
rains" means that I will get wet when I go outside and I need to take
my umbrella with me, then I don't care what day it is, so it will in
that respect be the same proposition each day. Another example might
be "all mushrooms are edible" which might mean something different
when I'm in different forests, but if I have no way of knowing in
which forest I am, and if the pragmatics are the same for me (I will
eat them), then from my perspective these are the same facts.
I think I now understand also better how you want to distinguish in
your model theory between modal and nonmodal facts, and why you want
to restrict the K operator to nonmodal facts. Briefly put, since p &
~Kp is a modal fact, we then simply cannot formulate K(p & ~Kp) and
get the contradiction. Although I may not fully agree with the
philosophy behind this restriction, I agree now that this strategy,
when executed properly, could indeed avoid the contradiction.
That's it for now. As I said, it is possible that I will not be able
to reply quickly in the future, but I will certainly try to follow the
tread.
Kind regards,
-- Jan Hidders
#97
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I think I see now better your point about the fact that in different worlds we might use the same description to refer to things that are actually different facts. Your example being "it rains" which refers to something different if the different worlds correspond to different days. But I would argue that this is from the perspective of someone who is outside the model and has some way to identify the different worlds independent of what facts hold in them. |
don't explicitly talk about *specific* other worlds. However,
I think that for some uses of modal talk, we *do* have an
explicit way to characterize the other possible worlds. For
example, in a deterministic physical theory (such as Newtonian
physics), the various possible worlds are characterized by
initial conditions, which are determined by a point in phase
space.
Quote:
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When you are inside the model and in a certain world the only way to distinguish them is by looking which facts hold in them. For the rain example it could be that in your vocabulary you can express what day it is, and then you can distinguish the different days, but then you could have formulated the fact that you had in mind as "it rains and it is today 5 January 2010". If the date in your world is not in your vocabulary then you have no way of describing the differences between the "it rains" proposition in different worlds. |
language of "possible, necessary" even though there is no
way for us to know what is possible and what is necessary
(without knowledge of other possible worlds). So if you just
stick to what we can know in *this* world, it seems to me
that the only notion of "possibility" is logical consistency.
That's a very uninteresting notion of possibility.
To go beyond logical consistency, we have to have some theory
about what the other possible worlds are.
Quote:
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For me the meaning of a proposition is in its pragmatics. |
language, the "possibilities" are actually found with one
physical world (or the history of that world).
--
Daryl Mc
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