Fitch's paradox and OWA

#31

Nam Nguyen says...

Quote:

Daryl McCullough wrote:
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.

That was my point. There can be statements that are true, but which
we will never know that they are true. There can also be statements
that are true, but which we have no way of ever knowing that they are
true. For example, I flip a coin, and before I see whether it lands
heads up or tails up, it is run over by train, smashing it into a
flat, smooth chip of metal. Now, there is no way of ever knowing
whether it was heads-up or tails-up. But it is possible that
"It was heads-up before it was smashed" is true.

Statements can be true even if there is no way to ever know that they
are true.

--
Daryl McCullough
Ithaca, NY

#32

Nam Nguyen says...

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.

Well, it is certainly *possible* that "Plato was left-handed" is a statement
that is both true and unknowable (at this late date).

--
Daryl McCullough
Ithaca, NY

#33

On Dec 30, 11:15*pm, Bob Badour <bbad... (AT) pei (DOT) sympatico.ca> 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.

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.

Bob,

Nam is a kook; you can safely ignore anything he says.

Marshall

PS. Ah, the years of history! Too bad no one on sci.logic will get it.

#34

Okay, I've thought about it a little more, and I have come to
the conclusion that Fitch's paradox is invalid. Or perhaps the
statement of the knowability principle is wrong.

Here's the proof of the contradiction:

1. (Knowability principle) For all p: p -> <> K(p)

where <>Phi means "Phi is possibly true" and K(Phi) means
"Phi is known".

2. (Non-omniscience principle) For some p: p & ~K(p)

3. Letting p0 be the true but unknown proposition, we have
p0 & ~K(p0)

4. From 1&3, we have <>K(p0 & ~K(p0))

At this point, let me switch to possible world semantics: <> Phi
means "Phi is true in some world". So let's switch to the world
in which K(p0 & ~K(p0)) is true. In that world we have

5. K(p0 & ~K(p0))

From this it follows:

6. K(p0) & K(~K(p0))

But only true things are knowable, so from K(~K(p0)) it
follows that ~K(p0). So we have

7. K(p0) & ~K(p0)

which is a contradiction.

The mistake becomes clearer if we explicitly introduce
possible worlds. Let's use w ||- Phi to mean "Phi is true
in world w" and K_w(Phi) to mean "Phi is known in world
w". Let's introduce w0 to mean "our world". Then the
proof becomes the following:

1. (Knowability principle) for all p: (w0 ||- p) -> exists w, K_w(p)

In other words, if p is true in our world, then there exists another
world in which p is knowable.

2. (Non-omniscience principle) for some p: w0 ||- p & ~K_w0(p)

3. Introducing the constant p0 for this unknown proposition, we
have: w0 ||- p0 & ~K_w0(p0)

4. From 1&3, we have exists w, K_w(p0 & ~K_w0(p0))

5. Letting w' be a name for some world making the existential true,
we have: K_w'(p0 & ~K_w0(p0))

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.

--
Daryl McCullough
Ithaca, NY

#35

On Dec 31, 7:10*am, stevendaryl3... (AT) yahoo (DOT) com (Daryl McCullough)
wrote:

Quote:

Jan Hidders says...

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?

He didn't say that there was an explicitly stated rule of
that form. He said that in step 8 of the derivation, they
use a rule that was explicitly stated as
If |- f then |- []f
but they use it *as if* the rule was
f |- []f

Reading that page, it looks like what he is saying accurately
describes the step taken, but I know very little about
modal logic.

Quote:

I don't think that's a sensible modal logic rule.

That's his point, as I understand it.

Marshall

#36

On Dec 31, 12:18*am, Nam Nguyen <namducngu... (AT) shaw (DOT) ca> wrote:

Quote:

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,

Your point is still wrong.

Quote:

which means that collectively the naturals isn't a _complete_ model of Q or its
extensions.

Your conclusion is also still wrong, unsurprisingly.

Marshall

#37

On Dec 30, 8:16*pm, Barb Knox <s... (AT) sig (DOT) below> wrote:

Quote:

*Marshall <marshall.spi... (AT) gmail (DOT) com> wrote:

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.

I was more under the impression that Goedel showed there
was no complete finite theory of them, rather than no
way to define them. Are you saying those are equivalent?

Quote:

*(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.)

Here's a possible definition:

nat := 0 | succ nat

x + 0 = x
x + succ y = succ x+y

x * 0 = 0
x * succ y = x + (x * y)

Is there some way this definition is not fully syntactic?
It uses no quantifying over predicates, so it can't be
using second order logic.

It certainly seems to me that the above is fully syntactic,
and is a complete definition of basic arithmetic. Are
there statements that are true of this definition that
can't be captured by any finite theory? Sure there
are, but that has nothing to do with whether it's
a proper syntactic definition. To say it's not a syntactic
definition, you have to point out something about
it that's not syntactic, or not correct as a model
of the naturals.

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.

While I have sympathy for that position, I don't think it's
tenable in the long run. Or anyway, it's not tenable to go
from "of little value" to suggesting that we should, say,
not attend to the real numbers because of the existence
of uncomputable numbers, or suggest that statements
that are undecidable one way or the other are somehow
neither true nor false. What they are is undecidable.

Marshall

#38

Marshall wrote:

Quote:

On Dec 31, 12:18 am, Nam Nguyen <namducngu... (AT) shaw (DOT) ca> wrote:
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,

Your point is still wrong.

Why? Are you saying all formulas (written in the language of arithmetic) must
have to be truth-definable? Do you have a reason so? Or are you just saying
that - as usual it seems?

Quote:

which means that collectively the naturals isn't a _complete_ model of Q or its
extensions.

Your conclusion is also still wrong, unsurprisingly.

What isn't unsurprising is your "refute" does have any technical details
to back it up.

Sigh! Does every technical debate have to be personal fight of sort to you?

#39

Nam Nguyen wrote:

Quote:

Marshall wrote:
On Dec 31, 12:18 am, Nam Nguyen <namducngu... (AT) shaw (DOT) ca> wrote:
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,

Your point is still wrong.

Why? Are you saying all formulas (written in the language of arithmetic)
must
have to be truth-definable? Do you have a reason so? Or are you just saying
that - as usual it seems?

which means that collectively the naturals isn't a _complete_ model
of Q or its
extensions.

Your conclusion is also still wrong, unsurprisingly.

What isn't unsurprising is your "refute" does have any technical details
to back it up.

I meant "What is unsurprising ..."

Quote:

Sigh! Does every technical debate have to be personal fight of sort to you?

#40

Nam Nguyen wrote:

Quote:

Marshall wrote:
On Dec 31, 12:18 am, Nam Nguyen <namducngu... (AT) shaw (DOT) ca> wrote:
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,

Your point is still wrong.

Why? Are you saying all formulas (written in the language of arithmetic)
must
have to be truth-definable? Do you have a reason so? Or are you just saying
that - as usual it seems?

which means that collectively the naturals isn't a _complete_ model
of Q or its
extensions.

Your conclusion is also still wrong, unsurprisingly.

What isn't unsurprising is your "refute" does have any technical details
to back it up.

I do hate typo; and here's the correct version:

"What is unsurprising is your "refute" doesn't have any technical details
to back it up."

Quote:

Sigh! Does every technical debate have to be personal fight of sort to you?