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#71
 
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Re: Fitch's paradox and OWA -
01-01-2010
, 12:15 PM
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That's not a change of the *semantics*. That's a change of the *syntax*. My claim is that in the possible worlds semantics, every predicate (and operator) that can vary from world to world implicitly is a function of the world. That complexity can usually be avoided because implicitly we assume that everything is talking the same world. But when you nest <> and K, it is no longer possible to make that assumption. Not without restrictions on what can be said. My point is that the knowability principle doesn't make any sense without explicit mention of possible worlds. It might make sense if we restrict the principle to propositions p that don't involve the knowability operator. But if we restrict it that way, we can't carry out Fitch's proof. |
but the proof clearly "works" and the intuition behind the proof seems
plausible enough.
Suppose that p is true, but I don't know it. Then p & ~Kp is true.
But surely, I could not know p & ~Kp. That is, I couldn't know "p is
true, but I don't know that p is true."
After all, if I know that conjunction, then I know that p is true, so
how could I know that I don't know that p is true?
The argument seems perfectly clear to me, both formally and
informally.
--
Jesse F. Hughes
"To all Leaders of the World, buy or rent the movie 'The Day
After'[...] I assure you will have a new perspective on WMDs."
-- practical advice from online petitions
#72
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Jan Hidders says... On 31 dec 2009, 18:47, stevendaryl3... (AT) yahoo (DOT) com (Daryl McCullough) wrote: From this it follows: 6. K_w'(p0) & K_w'(~K_w0(p0)) Since only true things are knowable, we have: 7. K_w'(p0) & ~K_w0(p0) That's no contradiction at all! The proposition p0 is known in one world, w', but not in another world, w0. It only becomes a contradiction when you erase the world suffixes. True, but you have now fundamentally changed the semantics of the K operator in the sense that the model theory now looks very different. You have essentially turned K from a unary operator K(p) to a binary operator K(w,p). That's not a change of the *semantics*. That's a change of the *syntax*. My claim is that in the possible worlds semantics, every predicate (and operator) that can vary from world to world implicitly is a function of the world. That complexity can usually be avoided because implicitly we assume that everything is talking the same world. But when you nest <> and K, it is no longer possible to make that assumption. Not without restrictions on what can be said. My point is that the knowability principle doesn't make any sense without explicit mention of possible worlds. |
If not, I'm a bit puzzled as to how you want to change the semantics.
It would help if you could provide a model theory to explain how you
want to change the semantics. Right now the model theory I gave
already does allow the operator K to be different in possible worlds.
So how would your semantics differ from that?
-- Jan Hidders
#73
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stevendaryl3016 (AT) yahoo (DOT) com (Daryl McCullough) writes: That's not a change of the *semantics*. That's a change of the *syntax*. My claim is that in the possible worlds semantics, every predicate (and operator) that can vary from world to world implicitly is a function of the world. That complexity can usually be avoided because implicitly we assume that everything is talking the same world. But when you nest <> and K, it is no longer possible to make that assumption. Not without restrictions on what can be said. My point is that the knowability principle doesn't make any sense without explicit mention of possible worlds. It might make sense if we restrict the principle to propositions p that don't involve the knowability operator. But if we restrict it that way, we can't carry out Fitch's proof. I haven't worked through the semantic details (at least not recently), but the proof clearly "works" and the intuition behind the proof seems plausible enough. Suppose that p is true, but I don't know it. Then p & ~Kp is true. But surely, I could not know p & ~Kp. That is, I couldn't know "p is true, but I don't know that p is true." After all, if I know that conjunction, then I know that p is true, so how could I know that I don't know that p is true? The argument seems perfectly clear to me, both formally and informally. It actually is possible for "p & ~Kp" may be true, in which case "~K(p & |
can't be known to be true, but are indeed true. This is basically
Gödel's theorem.
Fitch's proof (at least by your description) is using the proof as its
own premise. p & ~Kp can be true without knowing it, therefore you
still don't know p is true.
--
Daniel Pitts' Tech Blog: <http://virtualinfinity.net/wordpress/>
#74
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Suppose that p is true, but I don't know it. Then p & ~Kp is true. But surely, I could not know p & ~Kp. That is, I couldn't know "p is true, but I don't know that p is true." After all, if I know that conjunction, then I know that p is true, so how could I know that I don't know that p is true? The argument seems perfectly clear to me, both formally and informally. |
"knowability principle" that if something is true, then
it is possible that it is knowable. That's not a reasonable
thing to assume unless we either restrict what sort of propositions
we are talking about, or be more explicit about *who* knows what.
I don't have any problem with the proof of Fitch's paradox. It's
a valid proof, but I take it as evidence for rejecting the knowability
principle.
--
Daryl McCullough
Ithaca, NY
#75
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Explicit in the formulas? So you really do want to change the syntax? |
syntax of modal logic is inadequate to capture the intuition behind
the knowability principle.
Quote:
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If not, I'm a bit puzzled as to how you want to change the semantics. It would help if you could provide a model theory to explain how you want to change the semantics. Right now the model theory I gave already does allow the operator K to be different in possible worlds. So how would your semantics differ from that? |
quantification over possible worlds. The point about modal logic
is that it is a simpler fragment of full first-order logic, but
I think that it is not expressive enough to talk about complex
issues of necessity and knowability. Fitch's paradox shows its
limitations.
--
Daryl McCullough
Ithaca, NY
#76
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stevendaryl3... (AT) yahoo (DOT) com (Daryl McCullough) writes: That's not a change of the *semantics*. That's a change of the *syntax*. My claim is that in the possible worlds semantics, every predicate (and operator) that can vary from world to world implicitly is a function of the world. That complexity can usually be avoided because implicitly we assume that everything is talking the same world. But when you nest <> and K, it is no longer possible to make that assumption. Not without restrictions on what can be said. My point is that the knowability principle doesn't make any sense without explicit mention of possible worlds. It might make sense if we restrict the principle to propositions p that don't involve the knowability operator. But if we restrict it that way, we can't carry out Fitch's proof. I haven't worked through the semantic details (at least not recently), but the proof clearly "works" and the intuition behind the proof seems plausible enough. Suppose that p is true, but I don't know it. *Then p & ~Kp is true. But surely, I could not know p & ~Kp. *That is, I couldn't know "p is true, but I don't know that p is true." After all, if I know that conjunction, then I know that p is true, so how could I know that I don't know that p is true? The argument seems perfectly clear to me, both formally and informally. |
It did strike me that you formulated K as "I know that P". For some
reason it made me realize that it was formulated on the Stanford page
as "Somebody at some time knows that P". Under the latter
interpretation it seems now indeed a bit strange to me to require that
all facts, and specifically those of the form ~Kp, are possibly known.
I can imagine there are facts p for which we can never establish
definitively that they will not be known to somebody at some time,
except after waiting until we run out of time or persons, and by then
there will be nobody left to know this. So ~Kp might very well be both
true and unknowable.
-- Jan Hidders
#77
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Jan Hidders says... Explicit in the formulas? So you really do want to change the syntax? I'm not advocating a change in the syntax, I'm just saying that the syntax of modal logic is inadequate to capture the intuition behind the knowability principle. |
syntax?
Quote:
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If not, I'm a bit puzzled as to how you want to change the semantics. It would help if you could provide a model theory to explain how you want to change the semantics. Right now the model theory I gave already does allow the operator K to be different in possible worlds. So how would your semantics differ from that? I would just use first-order logic semantics, and allow explicit quantification over possible worlds. The point about modal logic is that it is a simpler fragment of full first-order logic, but I think that it is not expressive enough to talk about complex issues of necessity and knowability. Fitch's paradox shows its limitations. |
theory rather then FOL, but since you want to talk about possible
worlds and statements about statements, that seems more appropriate to
me anyway. The given model theory still seems to contain the paradox,
so you will want to change it. Can you show how?
-- Jan Hidders
#78
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On 2 jan, 00:14, stevendaryl3... (AT) yahoo (DOT) com (Daryl McCullough) wrote: Jan Hidders says... Explicit in the formulas? So you really do want to change the syntax? I'm not advocating a change in the syntax, I'm just saying that the syntax of modal logic is inadequate to capture the intuition behind the knowability principle. Doesn't that imply that you want to reformulate it in a different syntax? |
to assert the knowability principle, then I would formulate it in
something other than modal logic.
Quote:
|
I would just use first-order logic semantics, and allow explicit quantification over possible worlds. The point about modal logic is that it is a simpler fragment of full first-order logic, but I think that it is not expressive enough to talk about complex issues of necessity and knowability. Fitch's paradox shows its limitations. But is that not what the given model theory already does? It uses set theory rather then FOL, but since you want to talk about possible worlds and statements about statements, that seems more appropriate to me anyway. |
to allow nested instances of the knowability operator, then there
is the issue of *who* knows what. The fact that proposition p is
not known in world w1 is itself a proposition, and that proposition
can be known, but *not* in w1. Another world, w2 could know that
p is not known in w1. But you can't express that without
world indices on the knowability operator.
Now, it could be that we are not interested in what *another*
world knows about this world. So we restrict our attention to
one-world claims (all knowability operators refer to the same
world). That's fine, and in that case, the knowability principle
is just false in any nontrivial model of modal logic.
Quote:
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The given model theory still seems to contain the paradox, so you will want to change it. Can you show how? |
to formalize. The problem is that if knowability is a two-place
predicate (as opposed to an operator), then that means that
formulas have to serve double-duty: both as formulas and as
terms (that can be arguments to the knowability predicate).
In higher-order type theory, I think we can do it this way:
Introduce new types
W = the type of possible worlds
A = the type of atomic propositions
P = the type of all propositions
(the propositions are closed under the operations of
and, or, implies, negation, universal and existential
quantification)
t : W x A --> P
t(w,a) says "a is true in world w"
k : W x P --> P
k(w,p) says "p is known in world w"
Then the knowability principle could be formalized as:
forall p:P, (p -> exists w:W, k(w,p))
(any true proposition is known to be true in some world).
I think it would be a lot of work to nail down all the
details here, but my point is that the knowability
principle can be formulated in a way that isn't susceptible
to Fitch's proof.
--
Daryl McCullough
Ithaca, NY
#79
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W = the type of possible worlds A = the type of atomic propositions P = the type of all propositions |
I am not sure that propositions are types???
Let me give you the following example:
This sentence is false.
--
Vladimir Odrljin
#80
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Jan Hidders says... On 2 jan, 00:14, stevendaryl3... (AT) yahoo (DOT) com (Daryl McCullough) wrote: Jan Hidders says... Explicit in the formulas? So you really do want to change the syntax? I'm not advocating a change in the syntax, I'm just saying that the syntax of modal logic is inadequate to capture the intuition behind the knowability principle. Doesn't that imply that you want to reformulate it in a different syntax? I wouldn't say that I *want* to; I'm just saying that if I wanted to assert the knowability principle, then I would formulate it in something other than modal logic. I would just use first-order logic semantics, and allow explicit quantification over possible worlds. The point about modal logic is that it is a simpler fragment of full first-order logic, but I think that it is not expressive enough to talk about complex issues of necessity and knowability. Fitch's paradox shows its limitations. But is that not what the given model theory already does? It uses set theory rather then FOL, but since you want to talk about possible worlds and statements about statements, that seems more appropriate to me anyway. I don't think the model theory is rich enough. If you are going to allow nested instances of the knowability operator, then there is the issue of *who* knows what. The fact that proposition p is not known in world w1 is itself a proposition, and that proposition can be known, but *not* in w1. Another world, w2 could know that p is not known in w1. But you can't express that without world indices on the knowability operator. Now, it could be that we are not interested in what *another* world knows about this world. So we restrict our attention to one-world claims (all knowability operators refer to the same world). That's fine, and in that case, the knowability principle is just false in any nontrivial model of modal logic. The given model theory still seems to contain the paradox, so you will want to change it. Can you show how? Now that I think about it, it seems that it would be a mess to formalize. The problem is that if knowability is a two-place predicate (as opposed to an operator), then that means that formulas have to serve double-duty: both as formulas and as terms (that can be arguments to the knowability predicate). In higher-order type theory, I think we can do it this way: Introduce new types W = the type of possible worlds A = the type of atomic propositions P = the type of all propositions (the propositions are closed under the operations of and, or, implies, negation, universal and existential quantification) t : W x A --> P t(w,a) says "a is true in world w" k : W x P --> P k(w,p) says "p is known in world w" Then the knowability principle could be formalized as: forall p:P, (p -> exists w:W, k(w,p)) (any true proposition is known to be true in some world). I think it would be a lot of work to nail down all the details here, but my point is that the knowability principle can be formulated in a way that isn't susceptible to Fitch's proof. |
But I'm afraid I don't think that will work. The reason is that in
your logic you can still express the same things that could be
expressed in the old logic. Take for example the following proposition
in the old model theory:
(1) K(p & ~K(p))
You can still express this in your logic. You can do this by using a
predicate CW(w) that expresses that w is (equivalent to) the current
world. You can express this as follows:
(2) CW(w) =def= For all p, ( t(w,p) <-> p )
With that you can write (1) in your logic as:
(3) Forall w : W, ( CW(w) -> k(w, (p & ~k(w, p))) )
This can be done for all for all formulas in the old logic and so it
seems to me that you will still have the same paradox but written down
differently.
-- Jan Hidders
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