Nilone wrote:

I don't understand the paradox.

http://plato.stanford.edu/entries/fitch-paradox/

explains:

suppose that

(KP) all truths are knowable, i.e. can be known by somebody at some time

and

(NonO) not all truths are known now

then

(1) there is an unknown truth p

and then

(2) p is true and unknown is itself a truth

and hence, by KP,

(3) (p is true and unknown) can be known by somebody at some time

"However, it can be shown independently that it is impossible
to know this conjunction. Line 3 is false."

I'm eager to see that demonstration. Clearly, if p is unknown (1),
then so is the truth than p is true and unknown (2), but KP doesn't
contradict that - all it says is that p may be known, perhaps at some
other time, and indeed, at that same time, the statement that p was true
but unknown will also be known to be true.

They seem to mess up the scoping of K and P, simplifying their language
until paradoxes become inevitable; the paradox results from that,
as far as I can see. In any case, the paradox depends
on the exact formalization on K and P, which isn't given.
Once again, I'm left with the feeling that these Stanford guys
could use some field experience in database design.

--
Reinier

"Reinier Post" <rp (AT) raampje (DOT) lan> wrote

Your addition of 'now' to (NonO) is the cause of your confusion. Go back
and re-read what you cited.


If K is the epistemic operator meaning 'it is known by someone at some time
that,' then not K would have to deny that, meaning 'it is not known by
anyone at any time that,' so with that in mind....

(KP) forall p(p implies possibly Kp): all truths are knowable by somebody at
some time.

(NonO) exists p(p and not Kp): there is a truth that is not known by anybody
at any time.

These are contradictory. If all truths are knowable by somebody at some
time then there can't be a truth that is not known by anybody at any time.

On Dec 26, 4:36*am, Jan Hidders <hidd... (AT) gmail (DOT) com> wrote:
From plato.stanford.edu:

"As for the knowability proof itself, there continues to be no
consensus on whether and where it goes wrong."

So ... you gonna publish that? Give those modal logic guys
a kick in the pants? :-)


Marshall

On 26 dec, 22:39, Marshall <marshall.spi... (AT) gmail (DOT) com> wrote:
Only after checking carefully whether I'm right. :-) Unfortunately it
turns out that I was wrong. But in the process I did find what I think
is really the problem with the paradox.

You can "check" the reasoning by writing down a model theory for the
logic and see if all the inferences are actually valid there. In this
case the model theory is a bit more complicated than usual because we
are reasoning about possible worlds. Basically a model will now look
like a pair (W,w) where W is the set of all possible worlds and w is
the world we are in right now. In normal propositional logic a single
world is described by the set of true atomic propositions. In the
setting of the paradox it also contains propositions of the form Kf
where f is some formula, which say that f is known in that world. So a
world could for example be {a, b, Ka} where a and b are true, but only
a is actually known. A complete model could for example be ({w1, w2,
w3}, w1) where w1 = {a,b,Ka}, w2 = {a,Ka} and w3 = {b,Kb}.

For normal logic operators the model theory is straightforward:
- (W,w) |- f1 & f2 iff (W,w) |- f1 and (W,w) |- f2
- (W,w) |- ~f1 iff not (W,w) |- f1
etc.

For the basic proposition and the K-facts:
- (W,w) |- a iff a in w
- (W,w) |- Kf iff Kf in w

For the modal operators:
- (W,w) |- []f iff (W,w') |- f for all w' in W
- (W,w) |- <>f iff (W,w') |- f for some w' in W

It can be verified that in this model theory all the inference steps
that are used in the paradox are always valid. For the assumptions
(A), (B), (C) and (D) on the Stanford page it holds that (D) follows
from the model theory. So we can question the validity of (A), (B) and
(C). As it turns out (I will not explain that here) you don't need (A)
and (B) to get the result of the paradox, so I'm going to focus on
(C).

If we reformulate the meaning of (C) in the model theory we get:

(model-C) If (W,w) |- f then (W,w) |- []f.

Given the semantics of []f this is equivalent with:

(model-C') If (W,w) |- f then (W,w') |- f for all w' in W.

Note that in particular this will hold for f's that are basic
propositions or negations of basic propositions. Since the basic
propositions hold for (W,w') iff they are elements of w', and their
negation only holds if they are not elements of w', it follows that w
and w' must always contain exactly the same elements, and therefore in
fact the same world. In other words, (C) says effectively that there
is always only one possible world. Knowing this, it is not surprising
we get such unintuitive results, because it means that everything that
is possible, i.e., true in one of the possible worlds, is actually
necessary, i.e., true in all possible worlds. This also explains how,
from the assumption that every true fact is possibly known, we can
come to the conclusion that every true fact is necessarily known.

Conclusion: axiom (C) is not a modest modal assumption at all, and in
fact quite absurd.

-- Jan Hidders

On 26 dec, 22:39, Marshall <marshall.spi... (AT) gmail (DOT) com> wrote:
Only after checking carefully whether I'm right. :-) Unfortunately it
turns out that I was wrong. But in the process I did find what I think
is really the problem with the paradox.

You can "check" the reasoning by writing down a model theory for the
logic and see if all the inferences are actually valid there. In this
case the model theory is a bit more complicated than usual because we
are reasoning about possible worlds.

Basically a model will now look like a pair (W,w) where W is the set
of all possible worlds and w is the world we are in right now which
must of course be an element of W. In normal propositional logic a
single world is described by the set of true atomic propositions. In
the setting of the paradox it also contains propositions of the form
Kf where f is some formula, which say that f is known in that world. I
will call those atomic propositions and Kf propositions collectively
basic propositions.

So a world could for example be {a, b, Ka} where a and b are true, but
only a is actually known. A complete model could for example be ({w1,
w2, w3}, w1) where w1 = {a,b,Ka}, w2 = {a,Ka} and w3 = {b,Kb}.

For normal logic operators the model theory is straightforward:
- (W,w) |- f1 & f2 iff (W,w) |- f1 and (W,w) |- f2
- (W,w) |- ~f1 iff not (W,w) |- f1
etc.

For the basic proposition and the K-facts:
- (W,w) |- a iff a in w
- (W,w) |- Kf iff Kf in w

For the modal operators:
- (W,w) |- []f iff (W,w') |- f for all w' in W
- (W,w) |- <>f iff (W,w') |- f for some w' in W

It can be verified that in this model theory all the inference steps
used in the paradox are always valid. For the assumptions (A), (B),
(C) and (D) on the Stanford page <http://plato.stanford.edu/entries/
fitch-paradox/> it holds that (D) is always valid in the above model
theory. So we can question the validity of (A), (B) and (C). As it
turns out (I will not explain that here) you don't need (A) and (B)
to get the result of the paradox, so I'm going to focus on (C).

If we reformulate the meaning of (C) in the model theory we get:

(mC) If (W,w) |- f then (W,w) |- []f.

Given the semantics of []f this is equivalent with:

(mC') If (W,w) |- f then (W,w') |- f for all w' in W.

Note that in particular this will hold for f's that are basic
propositions or negations of basic propositions. Since the basic
propositions hold for (W,w') iff they are elements of w', and their
negation only holds if they are not elements of w', it follows that w
and w' must always contain exactly the same elements, and therefore in
fact be the same world. In other words, (C) says effectively that
there is always only one possible world. Knowing this, it is not
surprising we get such unintuitive results, because it means that
everything that is possible, i.e., true in one of the possible worlds,
is actually necessary, i.e., true in all possible worlds. This also
explains how, from the assumption that every truth is possibly known,
we can come to the conclusion that every truth is necessarily known.

Conclusion: axiom (C) is not a modest modal assumption at all, and in
fact quite absurd.

-- Jan Hidders

Jan Hidders says...

I don't think that that is correct. Rule (C) says that
if p is a *theorem* (that is, p is provable) then it is
necessarily true (and so is true in all worlds).

In Kripke semantics, we distinguish between what is true
in one world and what is provable. So you should be writing


(W,w) ||- f

to mean f is true in world w (where W is the set of all possible
worlds) instead of

(W,w) |- f

I'm not sure what the latter would mean.

--
Daryl McCullough
Ithaca, NY

By the way, I haven't thought about it a huge amount, but I
don't have any problems with the paradox, because I don't
accept the premise: Every true proposition is potentially knowable.
It seems to me that sufficiently complex true propositions may never
be known. Certainly there are candidate mathematical truths, such
as Goldbach's conjecture, that we have no idea how to ever prove,
so it seems plausible (to me) that we may never come to know that
they are true.

--
Daryl McCullough
Ithaca, NY

Daryl McCullough wrote:
But how can we know it's true in the first place, when its being true
can't be known?

Let me add more to what you've said.

One of the shortcomings of modern mathematical logic is that it assumes
every single formula written in the language of arithmetic "must be"
arithmetically either true or false.

There is a class of formulas (written in the language) whose arithmetic
truths or falsehoods can't be established as a matter of principle. [The
existence of this class could be demonstrated]. GC and the formula "There
are infinite counter examples of GC" are candidates of being in such class.

On Dec 30, 6:22*pm, Nam Nguyen <namducngu... (AT) shaw (DOT) ca> wrote:
If it's actually the case (that every statement of basic arithmetic
is either true or false) then it's not a shortcoming to say so.
On the contrary, that would be a virtue.

Do you have any reason to believe that there exist statements
of arithmetic that *don't* fall in to one of those two categories?
Note that not being able to know which one it is is not the same
thing as it actually being something other than true or false.

(I'm guessing you actually disagree with that last sentence,
though.)


Marshall

Nam Nguyen says...
I didn't say that we can *know* it is true. That's my point---something
can be true without anyone knowing that it is true. It might be true,
for example, that there is an even number of grains of sand in the world, but we
may never find that out. Is e^pi rational? We may never find out.

--
Daryl McCullough
Ithaca, NY