Fitch's paradox and OWA

#21

On Dec 30, 7:51*pm, stevendaryl3... (AT) yahoo (DOT) com (Daryl McCullough)
wrote:

Quote:

Nam Nguyen says...
Daryl McCullough wrote:
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.

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

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.

I believe Nam is roughly of the opinion that if we can't know which
one of {true, false} a sentence is, then we have no basis for saying
it must be one or the other. I seem to recall being less than
completely
clear on that point myself sometime in the past, in re the halting
problem, and getting a sound sci.logic thrashing by some guy
as a result. His name was Darren McColor, or anyway it was
something like that. Boy was I embarrassed!

Marshall

#22

In article
<e3cb76b6-8d77-4a92-bd71-7cd6e163d061 (AT) k17g2000yqh (DOT) googlegroups.com>,
Marshall <marshall.spight (AT) gmail (DOT) com> wrote:

Quote:

On Dec 30, 6:22*pm, Nam Nguyen <namducngu... (AT) shaw (DOT) ca> wrote:

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.

By the nature of the construction of predicate logic, every arithmetic
formula must be either true or false in the standard model of the
natural numbers.

But, we have no satisfactory way to fully characterise that standard
model! We all think we know what the natural numbers are, but Goedel
showed that there is no first-order way to define them, and I don't know
of *any* purely formal (i.e., syntactic) way to do do. (The usual ways
to define them are not fully syntactic, but rely on "the full semantics"
of 2nd-order logic, or "a standard model" of set theory, both of which
are more complicated than just relying on "the Standard Model" of
arithmetic in the first place.)

So, we can say we have a fully-pinned-down notion of arithmetical truth,
but only in terms of a background (the Standard Model) which we can't
fully pin down.

Quote:

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.

Speaking philosophically (since I'm posting from sci.philoisophy.tech),
entities which in some sense exist but are thoroughly inaccessible seem
to be of little value. This applies to the truth values of any
statements which can never be known to be true or false.

Quote:

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

--
---------------------------

Quote:

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B B aa rrr b |
BBB a a r bbb | Quidquid latine dictum sit,
B B a a r b b | altum videtur.
BBB aa a r bbb |
-----------------------------

#23

Marshall wrote:

Quote:

On Dec 30, 6:22 pm, Nam Nguyen <namducngu... (AT) shaw (DOT) ca> wrote:
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.

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.

It _would_ be a virtue, yes, but only, as you said, "_If_ it's actually
the case"! But is it?

Quote:

Do you have any reason to believe that there exist statements
of arithmetic that *don't* fall in to one of those two categories?

Yes. There are statements written in the lanaguage of arithmetic that
no one could possibly assign a truth value to them. For example:

(1) There are infinite counter examples of GC.

Tell me what you'd even suspect as a road-map to assign true or false to (1)?

Quote:

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.

Similarly as in provably-undecidable case (though not identical), there's
a 3rd scenario: you can't assign arithmetic truth or falsehood a a certain
formula, and in which case the formula is neither true or false! (Of course
in such case you could assume it's true or false - but not both - at will.)

Quote:

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

Of course. But I've also cited reasons.

#24

Marshall says...

Quote:

I believe Nam is roughly of the opinion that if we can't know which
one of {true, false} a sentence is, then we have no basis for saying
it must be one or the other.

But typically, for some statements such as "The Greek philosopher
Plato was left-handed" I don't know whether the statement is true
or not, and I also don't know whether anyone else knows whether it
is true or not, and I don't know whether it is *possible*, at this
late date, to find out whether it is true or not. But surely, it's
either true or false, right?

--
Daryl McCullough
Ithaca, NY

#25

Daryl McCullough wrote:

Quote:

Marshall says...

I believe Nam is roughly of the opinion that if we can't know which
one of {true, false} a sentence is, then we have no basis for saying
it must be one or the other.

But typically, for some statements such as "The Greek philosopher
Plato was left-handed" I don't know whether the statement is true
or not, and I also don't know whether anyone else knows whether it
is true or not, and I don't know whether it is *possible*, at this
late date, to find out whether it is true or not.

Quote:

But surely, it's either true or false, right?

No. Not surely. Since by our assumption here is nobody would know about
his handed-ness, his nervous system to both arms might not have functioned
at all to begin with and hence whether or not he was left-handed is moot
and is not-truth assignable. As well, there are people are strong equally
on both arms and therefore handed-ness is not applicable to them.

#26

Daryl McCullough wrote:

Quote:

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

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.

Don't want to beat a dead horse so to speak but not knowing a truth because
its proof (knowledge) is _finitely_ larger than what one can possibly know
is *not* the same as not knowing a truth value because the statement is not
*genuinely* truth-assigned-able. The "sand in the world" being an even number
example above is of the 1st kind: not the 2nd kind.

#27

Nam Nguyen wrote:

Quote:

Daryl McCullough wrote:

Marshall says...

I believe Nam is roughly of the opinion that if we can't know which
one of {true, false} a sentence is, then we have no basis for saying
it must be one or the other.

But typically, for some statements such as "The Greek philosopher
Plato was left-handed" I don't know whether the statement is true
or not, and I also don't know whether anyone else knows whether it
is true or not, and I don't know whether it is *possible*, at this
late date, to find out whether it is true or not.

But surely, it's either true or false, right?

No. Not surely. Since by our assumption here is nobody would know about
his handed-ness, his nervous system to both arms might not have functioned
at all to begin with and hence whether or not he was left-handed is moot
and is not-truth assignable. As well, there are people are strong equally
on both arms and therefore handed-ness is not applicable to them.

The term is ambidextrous and ambidextrous is not left-handed so the
predicate would be false if that were the case.

It doesn't get tricky until handedness is equally strong in both arms
but not for the same things like a person who writes left-handed but
shoots right-handed etc.

--
is there something in it for them, like maybe bailouts, if they can
panic us into doing something politically to cover them?

November 19, 2007 - John S Bolton

http://tinyurl.com/y9e4vxh

#28

Bob Badour wrote:

Quote:

Nam Nguyen wrote:

Daryl McCullough wrote:

Marshall says...

I believe Nam is roughly of the opinion that if we can't know which
one of {true, false} a sentence is, then we have no basis for saying
it must be one or the other.

But typically, for some statements such as "The Greek philosopher
Plato was left-handed" I don't know whether the statement is true
or not, and I also don't know whether anyone else knows whether it
is true or not, and I don't know whether it is *possible*, at this
late date, to find out whether it is true or not.

But surely, it's either true or false, right?

No. Not surely. Since by our assumption here is nobody would know about
his handed-ness, his nervous system to both arms might not have
functioned
at all to begin with and hence whether or not he was left-handed is moot
and is not-truth assignable. As well, there are people are strong equally
on both arms and therefore handed-ness is not applicable to them.

The term is ambidextrous and ambidextrous is not left-handed so the
predicate would be false if that were the case.

The _analogy_ was under the assumption that we'd logically live under a binary
world where the negation of "left-handed" is "right-handed". I don't think we
were arguing about precise meanings of biological/physiological matters.

My point still stands: if it's _impossible_ (as opposed to just being difficult)
to assign truth values to a formula then the formula is neither true nor false,
which means that collectively the naturals isn't a _complete_ model of Q or its
extensions.

Quote:

It doesn't get tricky until handedness is equally strong in both arms
but not for the same things like a person who writes left-handed but
shoots right-handed etc.

#29

On 31 dec, 01:07, stevendaryl3... (AT) yahoo (DOT) com (Daryl McCullough) wrote:

Quote:

Jan Hidders says...

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.

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).

My apologies. Everywhere where I wrote (W,w) |- f I actually meant
(W,w) ||- f.

So what I wanted to say with the above is the following. You are of
course right that what (C) really says is:

(C) if |- f then |- []f

And, assuming that for all f it holds that |- f iff ||- f, this is in
fact confirmed by the model theory. However, in the inference process
of the paradox as described on the Stanford page the rule is used as
if it says f |- []f or |- f->[]f, and that would have the much
stronger model-theoretic meaning that I described.

Their reasoning can be simplified to this:

(1) p & ~Kp (assumption, for arbitrary variable p)
(2) <>Kp (from (1) using KP)
(3) []~Kp (from (1) using (C))
(4) ~<>Kp (from (3) using (D)
(5) ~(p & ~Kp) (from (1) and contradicting (2) and (4))
(6) Forall p (~(p & ~Kp)) (forall introduction)
(7) Forall p (p -> Kp) (propositional reasoning)

The error in the reasoning is caused by the omission of |- before each
formula. If you add that, it is clear that at step (5) it is concluded
erroneously that |- ~(p & ~Kp) but it should have said that "it is not
true that |- (p & ~Kp)", which is of course not the same thing.

-- Jan Hidders

#30

Jan Hidders says...

Quote:

So what I wanted to say with the above is the following. You are of
course right that what (C) really says is:

(C) if |- f then |- []f

And, assuming that for all f it holds that |- f iff ||- f, this is in
fact confirmed by the model theory. However, in the inference process
of the paradox as described on the Stanford page the rule is used as
if it says f |- []f or |- f->[]f, and that would have the much
stronger model-theoretic meaning that I described.

I don't see a rule saying f |- []f. Where did you see that?

I don't think that's a sensible modal logic rule. That is
essentially saying that there is no difference between
f and []f. (Usually, the accessibility relation on worlds
is set up so that []f -> f. So if we add f -> []f, then
f and []f are logically equivalent.)

Quote:

Their reasoning can be simplified to this:

(1) p & ~Kp (assumption, for arbitrary variable p)
(2) <>Kp (from (1) using KP)
(3) []~Kp (from (1) using (C))

No, we don't have |- ~Kp. We only have (W,w) ||- ~Kp.
So we can't conclude |- []~Kp.

--
Daryl McCullough
Ithaca, NY