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Fitch's paradox and OWA

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  #41  
Old   
Daryl McCullough
 
Posts: n/a

Default Re: Fitch's paradox and OWA - 12-31-2009 , 01:22 PM






Marshall says...
Quote:
On Dec 31, 7:10=A0am, stevendaryl3... (AT) yahoo (DOT) com (Daryl McCullough)
wrote:

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
No, I don't think they did that. What they did was
to assume K(p & ~K(p)), and show that that leads to
a contradiction. That's a proof of ~K(p & ~K(p)).
So we have |- ~K(p & ~K(p)). Then we can apply the
rule "If |- f, then |- [] f" to conclude
[]~K(p & ~K(p))

--
Daryl McCullough
Ithaca, NY


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  #42  
Old   
Nam Nguyen
 
Posts: n/a

Default Re: Fitch's paradox and OWA - 12-31-2009 , 02:29 PM






Marshall wrote:
Quote:
On Dec 30, 8:16 pm, Barb Knox <s... (AT) sig (DOT) below> wrote:
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.
Godel didn't show any of the 2 you've mentioned.

Quote:
Are you saying those are equivalent?
If I'm the one answering this question then "No": defining a model of a formal
system is not the same as demonstrating anything about a formal system that's
supposed to be about the model. Naturally.

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?
Yes: The part "nat := 0 | succ nat" isn't syntactical. [In the context
of FOL, being syntactical is being part of a FOL language/formula which
this part doesn't seem to be].
Quote:
It certainly seems to me that the above is fully syntactic,
and is a complete definition of basic arithmetic.
That's *not* the canonical knowledge of arithmetic: what happens to the usual
syntactical symbol '<', in your "complete definition"?

Quote:
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.
Setting aside the missing "<", what you've defined up there is
*in no way* conforming with the _FOL definition of a model_ which
the naturals is supposed to be collectively. For example, what's
the set of 2-tuples that would correspond to your '+'?

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.
First order undecidable formulas are in a different class than those
that aren't model-able, aren't truth assigned-able.

I asked you before:

"(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)?"

Now if you let (1') be defined as:

(1') df= (1) /\ A1 /\ A2 /\ ... A9

where A1 - A9 are Q's axioms (a la Shoenfield). Tell us, Marshall, what models or
what kinds of models that you think you could assign 'true' or 'false' to
(1')? If you really can't - which I don't think you can - then don't you at
least think of the possibility that there are arithmetic statements that can't
be true or false?

Why is it that a statement has to be true or false while _there's no way_ to
assign a truth value to it any way? Other than we might have grown up accustomed
to it, what kind of reasoning is that?

Ok I might sound a bit rhetorical here. But can you technically answer my question
about (1')?

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  #43  
Old   
Nam Nguyen
 
Posts: n/a

Default Re: Fitch's paradox and OWA - 12-31-2009 , 03:07 PM



Daryl McCullough wrote:
Quote:
Nam Nguyen says...
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.
So, are you with me that there could be statements that are neither true or false,
on the ground that we can't assign a truth value to them; i.e., on the ground
what we've _intuitively perceived_ as the "natural numbers" is _not adequate_ for
us to say they are true or false?

Quote:
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.
But that's _not_ my point! The statements I have in mind are the ones that
can _not_ be assigned true or false, in the first place! Do you see that they
aren't of the same kind of statements you've alluded to?

Quote:
--
Daryl McCullough
Ithaca, NY

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  #44  
Old   
Barb Knox
 
Posts: n/a

Default Re: Fitch's paradox and OWA - 12-31-2009 , 03:08 PM



In article
<a3f061ed-3838-4be9-b73a-836141dc640f (AT) u7g2000yqm (DOT) googlegroups.com>,
Marshall <marshall.spight (AT) gmail (DOT) com> wrote:

Quote:
On Dec 30, 8:16*pm, Barb Knox <s... (AT) sig (DOT) below> wrote:
*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?
Yes, in this context. Since we are finite beings we need to use finite
systems.


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.
This is the usual first-order initial-algebra definition, and with the
addition of "succ x = succ y -> x=y" and an induction schema gives
first-order Peano Arithmetic. First-order logic is indeed formal (i.e.,
syntactic) in that all inferencing activities consist of finite
operations on finite strings. But, via Goedel and others, the Peano
axioms do NOT fully characterise the natural numbers N. N is indeed a
model (the Standard Model) which satisfies these axioms, but there are
also *non-standard models* which satisfy these axioms -- these models
contain infinite elements in addition to the usual naturals.

You can get some of the flavour of non-standard models by considering
the following non-standard model for just succ, where every element has
a unique successor and predecessor:

0, 1, 2, ... ..., w-2, w-1, w, w+1, w+2, ...


So, we can readily produce purely formal systems that are satisfied by
N, but all of them (as far as I know) are also satisfied by other,
non-standard, models. Try as we might, those pesky infinite
non-standard integers keep cropping up. That is the sense in which I
mean that we apparently can not formally fully characterise N.

(Note that we similarly cannot formally define "finite", so the dodge of
saying something like "the naturals are defined by the Peano axioms plus
the restriction that everything is finite" can not be expressed purely
formally.)


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,
I am not an expert in that field, but I believe that almost all of real
analysis can be reconstructed using just computable numbers, e.g. the
work of Bishop.

Quote:
or suggest that statements
that are undecidable one way or the other are somehow
neither true nor false. What they are is undecidable.
They are true or false in any *particular* model. Since we apparently
cannot formally pin down arithmetic to have just one particular model
(the Standard one) then there will always be some arithmetic statements,
the undecidable ones, which are true in some models and false in others.
Thus it is unreasonable to say that an undecidable statement is simply
"true" or "false" -- we need to specify a particular model, almost
always the Standard one, which we can not fully characterise formally.

This doesn't prevent doing interesting number theory, but it is at least
somewhat bothersome from a foundational perspective.

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  #45  
Old   
Marshall
 
Posts: n/a

Default Re: Fitch's paradox and OWA - 12-31-2009 , 03:53 PM



On Dec 31, 11:22*am, stevendaryl3... (AT) yahoo (DOT) com (Daryl McCullough)
wrote:
Quote:
Marshall says...



On Dec 31, 7:10=A0am, stevendaryl3... (AT) yahoo (DOT) com (Daryl McCullough)
wrote:
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

No, I don't think they did that. What they did was
to assume K(p & ~K(p)), and show that that leads to
a contradiction. That's a proof of ~K(p & ~K(p)).
So we have |- ~K(p & ~K(p)). Then we can apply the
rule "If |- f, then |- [] f" to conclude
[]~K(p & ~K(p))
Certainly steps 4 - 9 constitute an RAA proof with
the assumption being K(p & ~K(p)).

However what I was referring to was specifically
how they get from step 7 to step 8 within that
RAA proof. Your response does not seem to
address that particular issue.

Are your comfortable with how step 8 is
obtained from step 7 via Rule C as described
on this page?

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

It's entirely possible that I misunderstand
Jan Hidder's point, or rule C, or something
else entirely, however I would like to at
least feel that we were discussing the same
step in the proof.


Marshall

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  #46  
Old   
Marshall
 
Posts: n/a

Default Re: Fitch's paradox and OWA - 12-31-2009 , 05:27 PM



On Dec 31, 1:08*pm, Barb Knox <Barb... (AT) LivingHistory (DOT) co.uk> wrote:
Quote:
*Marshall <marshall.spi... (AT) gmail (DOT) com> wrote:
On Dec 30, 8:16 pm, Barb Knox <s... (AT) sig (DOT) below> wrote:
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?

Yes, in this context. *Since we are finite beings we need to use finite
systems.
I have no disagreement with the point about finiteness, but I
don't see how that point leads to saying that a theory is
the same thing as a definition. That is rather tantamount to
saying that theories are all there are, and that's just not
true. There are things such as computational models,
for examples. It seems entirely appropriate to me to
use a computational model as the definition of something,
which is why I gave a computational model of the naturals
as a definition.

Perhaps worse, if it's not possible to have a definition of
anything, then I don't see how you can have any
theories, either. Theory of what? If you have no
definition, I don't see how you can even claim to
have an object under discussion.


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.

This is the usual first-order initial-algebra definition, and with the
addition of "succ x = succ y -> x=y" and an induction schema gives
first-order Peano Arithmetic.
Small points:

First of all, I claim "succ x = succ y -> x=y" is necessarily
the case via the definition of =.

Secondly, I claim we don't need to explicitly add any
induction schema, because induction on the naturals
in this case is merely a special case of structural
induction, which is itself merely a special case of
case analysis on the constructors for nat, and case
analysis is always available, as it were.

These are perhaps just quibbles.


Quote:
*First-order logic is indeed formal (i.e.,
syntactic) in that all inferencing activities consist of finite
operations on finite strings. *But, via Goedel and others, the Peano
axioms do NOT fully characterise the natural numbers N. *N is indeed a
model (the Standard Model) which satisfies these axioms, but there are
also *non-standard models* which satisfy these axioms -- these models
contain infinite elements in addition to the usual naturals.

You can get some of the flavour of non-standard models by considering
the following non-standard model for just succ, where every element has
a unique successor and predecessor:

* * 0, 1, 2, ... *..., w-2, w-1, w, w+1, w+2, ...

So, we can readily produce purely formal systems that are satisfied by
N, but all of them (as far as I know) are also satisfied by other,
non-standard, models. *Try as we might, those pesky infinite
non-standard integers keep cropping up. *That is the sense in which I
mean that we apparently can not formally fully characterise N.
I can see how your above set could be a model for PA, but
I don't see how it's supposed to be something that conforms
to the definition I gave.

For one thing, addition on the naturals is supposed to be total.
What is the result of "2 + w" under my definition of +? It does
not terminate, because you have introduced elements with
infinite descending deconstruction. That my addition operator
is total over (nat, nat) is provable; if there is some value
for which it is not total that value must therefor not
belong to nat.

For another thing, my definition doesn't have any "w" in
it, so you don't get to insert them in to the process.
We are supposed to be being syntactical here; recall
that you wanted to keep out second order logic and
set theory, so no "w".

Perhaps most importantly, I defined "nat" as those
things that are constructed via one of the two
specified constructors. Your w-elements are not
so constructed, so they cannot meet the definition
I gave.

I have noticed in the past that logicians and set
theorists don't necessarily buy the idea that
the universe consists only of those objects that
can be constructed using explicitly defined
construction rules. I am rather inclined to say
"tough," but perhaps I'll get better results if
I just say that's fine, but anything that isn't so
constructed isn't a natural, by definition.


Quote:
(Note that we similarly cannot formally define "finite", so the dodge of
saying something like "the naturals are defined by the Peano axioms plus
the restriction that everything is finite" can not be expressed purely
formally.)
It seems to me that syntax is necessarily finite, but again
this is perhaps just a quibble.


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,

I am not an expert in that field, but I believe that almost all of real
analysis can be reconstructed using just computable numbers, e.g. the
work of Bishop.
I'd accept "almost all" but note that "almost all" isn't the
same as "all". For example, the order relation on computable
numbers is not itself computable, sadly. Also isn't it the
case that the least-upper-bound property is lost if we
limit ourselves to computables?

Regardless, the bigger issue, it seems to me, is
that any such system is going be be distinctly more
complex than the reals, and that complexity has a
nontrivial cost.


Quote:
or suggest that statements
that are undecidable one way or the other are somehow
neither true nor false. What they are is undecidable.

They are true or false in any *particular* model. *Since we apparently
cannot formally pin down arithmetic to have just one particular model
(the Standard one) then there will always be some arithmetic statements,
the undecidable ones, which are true in some models and false in others. *
Even if we can pin it down, we still have statements that we don't
know if they are true or false. It might require an infinite amount
of computation to decide. Or just more than we will ever have.


Quote:
Thus it is unreasonable to say that an undecidable statement is simply
"true" or "false" -- we need to specify a particular model, almost
always the Standard one, which we can not fully characterise formally.
Sure, but whatever those statements do evaluate to, we can
narrow it down to one of two possibilities, even if we can't narrow
it any further.


Quote:
This doesn't prevent doing interesting number theory, but it is at least
somewhat bothersome from a foundational perspective.
I agree that it is bothersome!


Marshall

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  #47  
Old   
Nam Nguyen
 
Posts: n/a

Default Re: Fitch's paradox and OWA - 12-31-2009 , 05:40 PM



Barb Knox wrote:
Quote:
In article
a3f061ed-3838-4be9-b73a-836141dc640f...oglegroups.com>,
Marshall <marshall.spight (AT) gmail (DOT) com> wrote:

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

Yes, in this context. Since we are finite beings we need to use finite
systems.


(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.

This is the usual first-order initial-algebra definition, and with the
addition of "succ x = succ y -> x=y" and an induction schema gives
first-order Peano Arithmetic. First-order logic is indeed formal (i.e.,
syntactic) in that all inferencing activities consist of finite
operations on finite strings. But, via Goedel and others, the Peano
axioms do NOT fully characterise the natural numbers N. N is indeed a
model (the Standard Model) which satisfies these axioms, but there are
also *non-standard models* which satisfy these axioms -- these models
contain infinite elements in addition to the usual naturals.

You can get some of the flavour of non-standard models by considering
the following non-standard model for just succ, where every element has
a unique successor and predecessor:

0, 1, 2, ... ..., w-2, w-1, w, w+1, w+2, ...


So, we can readily produce purely formal systems that are satisfied by
N, but all of them (as far as I know) are also satisfied by other,
non-standard, models. Try as we might, those pesky infinite
non-standard integers keep cropping up. That is the sense in which I
mean that we apparently can not formally fully characterise N.

(Note that we similarly cannot formally define "finite", so the dodge of
saying something like "the naturals are defined by the Peano axioms plus
the restriction that everything is finite" can not be expressed purely
formally.)


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,

I am not an expert in that field, but I believe that almost all of real
analysis can be reconstructed using just computable numbers, e.g. the
work of Bishop.

or suggest that statements
that are undecidable one way or the other are somehow
neither true nor false. What they are is undecidable.

They are true or false in any *particular* model. Since we apparently
cannot formally pin down arithmetic to have just one particular model
(the Standard one) then there will always be some arithmetic statements,
the undecidable ones, which are true in some models and false in others.
Agree. The question - and the heart of my argument - is whether or not there
exists a formula F such that it's impossible to know/assert a truth value
in the collection K of _all_ arithmetic models: K = {the standard one, the
non-standard ones}? I've argued that there exist such statements.


Quote:
Thus it is unreasonable to say that an undecidable statement is simply
"true" or "false" -- we need to specify a particular model, almost
always the Standard one, which we can not fully characterise formally.

This doesn't prevent doing interesting number theory, but it is at least
somewhat bothersome from a foundational perspective.
Arguably, FOL isn't just for number theories and so there's always a possibility
the existences of such formulas might shed some light about FOL systems that
we've largely ignored: e.g. systems that have infinite number of logical symbols,
some of which might represent isomorphic - but different - operations.

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  #48  
Old   
Marshall
 
Posts: n/a

Default Re: Fitch's paradox and OWA - 12-31-2009 , 05:52 PM



On Dec 31, 12:29*pm, Nam Nguyen <namducngu... (AT) shaw (DOT) ca> wrote:
Quote:
Marshall wrote:
On Dec 30, 8:16 pm, Barb Knox <s... (AT) sig (DOT) below> wrote:
*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.

Godel didn't show any of the 2 you've mentioned.
"Any effectively generated theory capable of expressing
elementary arithmetic cannot be both consistent and complete.
In particular, for any consistent, effectively generated formal
theory that proves certain basic arithmetic truths, there is an
arithmetical statement that is true, but not provable in the theory."

So there cannot be a complete finite theory of basic arithmetic.


Quote:
Are you saying those are equivalent?

If I'm the one answering this question then "No": defining a model of a formal
system is not the same as demonstrating anything about a formal system that's
supposed to be about the model. Naturally.
Well we agree on one thing. That's unusual.

Quote:
It certainly seems to me that the above is fully syntactic,
and is a complete definition of basic arithmetic.

That's *not* the canonical knowledge of arithmetic: what happens to the usual
syntactical symbol '<', in your "complete definition"?
It's easy to extend this with <.


Quote:
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.

Setting aside the missing "<", what you've defined up there is
*in no way* conforming with the _FOL definition of a model_ which
the naturals is supposed to be collectively. For example, what's
the set of 2-tuples that would correspond to your '+'?
The goal was to provide a syntactic definition of the
naturals, which I did. The goal was not to provide
a FOL model. Nonetheless it's pretty easy to
get there from here. For example:

{((x, y), z) | x+y=z}


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.

First order undecidable formulas are in a different class than those
that aren't model-able, aren't truth assigned-able.

I asked you before:

* *"(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)?"
You keep assuming that the mere fact that a sentence is
undecidable means that it has some definite truth value
that is not one of {true, false}. Apparently you just take
this as a given. I, however, regard it as a false statement.


Quote:
Now if you let (1') be defined as:

(1') df= (1) /\ A1 /\ A2 /\ ... A9

where A1 - A9 are Q's axioms (a la Shoenfield). Tell us, Marshall, what models or
what kinds of models that you think you could assign 'true' or 'false' to
(1')? If you really can't - which I don't think you can - then don't you at
least think of the possibility that there are arithmetic statements that can't
be true or false?
I suppose anything is possible, in some vague, New-Age sort of
way. I suppose if someone were to supply some convincing
argument as to why there must be some third possibility,
I would at least consider it.

However, I have yet to hear any convincing argument
in favor of there being a third possibility. The mere fact
of a decision being hard, even infinitely hard, does not
suggest to me the existence of some third truth value
for a sentence to have.


Quote:
Why is it that a statement has to be true or false while _there's no way_to
assign a truth value to it any way? Other than we might have grown up accustomed
to it, what kind of reasoning is that?

Ok I might sound a bit rhetorical here. But can you technically answer myquestion
about (1')?
It seems to me that the definitions of the various things we
are talking about necessitate that a statement is either
true or false. The definition does not admit to the existence
of any third possibility. That some statements are undecidable
does not alter the definition of the terms the statements
were made with; the definitions remain as they were.
Thus every statement must have one of the two truth
values, by definition.

Now, if you want to make some new system to evaluate
statements in, that could certainly be defined with more
than the usual two possibilities. But that wouldn't be the
usual basic arithmetic; it'd be something new.

Although I don't consider reasoning by analogy to the
real world to be a great technique, it is at least suggestive
that there are real-world statements that we can
narrow down to few possibilities but cannot narrow
down to one. For example, Mr. McCullough's coin-and-
railroad story. We could even further say we were
close enough to see the coin landed definitely on
one side, but we weren't close enough to say
which side it was.


Marshall

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Marshall
 
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Default Re: Fitch's paradox and OWA - 12-31-2009 , 05:58 PM



On Dec 31, 3:40*pm, Nam Nguyen <namducngu... (AT) shaw (DOT) ca> wrote:
Quote:
Barb Knox wrote:

They are true or false in any *particular* model. *Since we apparently
cannot formally pin down arithmetic to have just one particular model
(the Standard one) then there will always be some arithmetic statements,
the undecidable ones, which are true in some models and false in others.. *

Agree. The question - and the heart of my argument - is whether or not there
exists a formula F such that it's impossible to know/assert a truth value
in the collection K of _all_ arithmetic models: K = {the standard one, the
non-standard ones}? I've argued that there exist such statements.
Why would the existence of such statements imply that there
are truth values other than true or false?


Marshall

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  #50  
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Nam Nguyen
 
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Default Re: Fitch's paradox and OWA - 12-31-2009 , 06:03 PM






Marshall wrote:
Quote:
On Dec 31, 1:08 pm, Barb Knox <Barb... (AT) LivingHistory (DOT) co.uk> wrote:
Marshall <marshall.spi... (AT) gmail (DOT) com> wrote:
On Dec 30, 8:16 pm, Barb Knox <s... (AT) sig (DOT) below> wrote:
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?
Yes, in this context. Since we are finite beings we need to use finite
systems.

I have no disagreement with the point about finiteness, but I
don't see how that point leads to saying that a theory is
the same thing as a definition. That is rather tantamount to
saying that theories are all there are, and that's just not
true. There are things such as computational models,
for examples. It seems entirely appropriate to me to
use a computational model as the definition of something,
which is why I gave a computational model of the naturals
as a definition.
You seemed to have confused between the FOL definition of models of formal
systems in general and constructing a _specific_ model _candidate_. In defining
the naturals, say, from computational model ... or whatever, you're just
defining what the naturals be. It's still your onerous to prove/demonstrate
this definition of the naturals would meet the definition of a model for,
say Q, PA, .... So far, have you or any human beings successfully demonstrated
so, without being circular? Of course not.

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