# Re: The seven step-Mathematical preliminaries

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Le 12-juin-09, à 08:28, A. Wolf a écrit :```
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>
>> There, they call arithmetic soundness what me (and many logician)
>> call
>> "soundness", when they refer to theories about numbers.  Like
>> Mendelson I
>> prefer to use the term logically valid, to what you call soundness.
>
> I may have misstated myself, but the wiki article you pointed me to
> agrees
> with what I tried to say:

That wiki article is not so good.

> A logical system is sound if every provable
> statement is valid.

... meaning "true in all models of the theory".  OK.

Like a "tautology", for example  "P OR NOT P"  is true is all models
(those with P true, and those with P false).

A propositional calculus is sound if it proves only tautologies (true
in all models),
and complete if it proves all tautologies, that is: all the
propositions which are true in all models.

The intuitive idea is that a reasoning is valid if its truth status
does not depend on the way we interpret it.

> Validity is not the same as soundness.

Logicians from different fields use terms in different ways. In
provability logic and in recursion theory, soundness means often
"arithmetical soundness".
For example, in recursion theory,  theory will be said Sigma_2 sound
when all Sigma_2 propositions proved in the theory are true ... in the
usual model (N,+,*).

> There are valid
> arguments that are unsound.  For example, if I say "x is not equal
> to x,
> therefore there are no more than five natural numbers", this is a
> valid
> (i.e., logically true) argument.  But it's also an unsound argument,
> because
> there is no interpretation where x is not equal to x.

Here most, if not all logicians, would disagree. Both in classical and
intuitionistic logic, To deduce any proposition from a falsity is
always a valid argument. Nobody will say that an argument is non valid
because its premise are absurd. Except in the "relevance logic" field.
Well, they will say that the reasoning is not ... relevant.

>
> What you're calling soundness I would call omega-consistent, but I
> see from
> the article that this is sometimes called "arithmetical soundness".

Soundness is a semantical notion. By the Tarski phenomenon such a
notion cannot be even just defined or expressed in the theory. That is
why Gödel makes the terrible effort for not using such a notion, which
was considered a bit controversial in those days.
Omega-consistency, like consistency, can be defined in a purely
syntactical way, and is much weaker than soundness.

>
>> The word "true" alone has no meaning. It refers always to a model,
>> or to a
>> collection of models.
>
> One could make the same argument about the symbol "=" not having any
> meaning
> outside of a model, but "true" has a standard meaning in logic, one
> that is
> often used interchangeably with "valid" (a stronger property).  The
> general
> "true" means "true under any interpretation".

That is validity for me. But let us not debate on vocabulary,
especially before making a bit more logic.

>
>> Oh, you mean a definition of natural number such that the model
>> would be
>> finite in scope. This is non sense for me. Pace Torgny.
>
> Nonsense for me too, apart from the philisophical musings.

OK.

>
>> Well, there is just no categorical first order definition of the
>> finite
>> sets of natural numbers.  And second order definition, assumes  the
>> notion
>> of infinite set.
>
> I'm not sure what you mean here.  Of course there is no categorical
> first-order theory of N.
>

We agree then. For the others, a theory is categorical if all its
model are isomorphic. In a sense, such a theory succeeds in capturing
completely its semantics. By well know theorems, such theories are
very rare, and in fact, when effective, very poor and very exceptional.

Bruno

http://iridia.ulb.ac.be/~marchal/

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