Re: Fitch's paradox and OWA

On Dec 31, 12:29 pm, Nam Nguyen <namducngu..._at_shaw.ca> wrote:
> Marshall wrote:
> > On Dec 30, 8:16 pm, Barb Knox <s..._at_sig.below> wrote:
> >>  Marshall <marshall.spi..._at_gmail.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.

> > 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.

> > 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 <.

> > 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}

> >>> 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.

> 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.

> 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')?

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 Received on Fri Jan 01 2010 - 00:52:34 CET