Re: [ontolog-forum] Truth

[Chris Menzel <chris.menzel@xxxxxxxxx>]

On Wed, Jul 11, 2012 at 5:58 PM, John F Sowa <sowa@xxxxxxxxxxx> wrote:

Chris,

Our disagreement is not so much about names, but about the nature of
models in model theory.  In the Tractatus, Wittgenstein assumed that
the real world was the model:  "The world is everything that is the
case."  Then atomic sentences have a one-to-one mapping to facts.

But in his transitional period, he abandoned that mapping of
sentences to facts.  Instead, he mapped theories (Satzsysteme)
to the world.  From his _Philosophical Remarks_ of 1929-30:

> The Satzsystem is like a ruler laid against reality. An entire
> system of propositions is now compared to reality, not a single
> proposition.

In model-theoretic terms, LW's "ruler" is a Tarski-style model that
is composed of surrogates that represent individuals.  Then that
model is treated as an approximation to the reality.  For a diagram
that illustrates that mapping, see Slide 23 of

    http://www.jfsowa.com/talks/dynolang.pdf

Yes, sure that is a very familiar sort of picture and a familiar understanding of models. In both the model theoretic and the engineering senses, models are of course approximations to the real world because they omit information deemed unimportant. But, of course, that doesn't mean that a semantical model can't be exactly correct as far as it goes. BTW, it would be great if in the future you numbered your slides. ;-)

 

> Then they don't have a naming relation in their language. At best they
> have some shadow thereof whose meaning is not encoded in the formalism
> but which is interpreted pragmatically by the users, much the way box
> and arrow diagrams can be useful to a select group of modelers who
> understand the unstated semantic conventions of their diagrams.

Absolutely!  That is all we can ever have in our computers -- or in
our brains.

I was a bit careless in talking about a naming relation in the language.What I meant of course was a predicate whose intended semantics is a relation between names and the things they name. But your enthusiastic "Absolutely!" suggests agreement that I don't see. It is just a fact that there are languages with naming predicates — indeed, through the magic of Gödel coding, any theory containing a bit of number theory can construct such a predicate. I am talking about languages like that, in which semantical predicates like "Names" or "True" with associated formal properties are explicitly part of the language, so that you can explicitly say such things as "Names('Aristotle',Aristotle)" or "True('∀xFx')" and draw out important logical consequences of such assetions. That latter point is the kicker — you can introduce what you call semantical predicates and pretend they have a certain intended meaning, but if those meanings are not encoded in axioms and are not reflected in the semantics, they are like box/arrow formalisms with no actual meanings and no logical consequences — and no real dangers. Formal languages with genuine semantic predicates do exist and they can be implemented in our computers and represented in our brains.

 

We have shadows in the computer, and technicolor pictures
in the brain.  They're called "mental models", and the neural evidence
for them shows that the cognitive scientists were right all along.

> But that's not a logical solution to the problem.

Of course not.  It's impossible to have a logical solution to an
empirical problem.  A theory of context is an empirical theory
about language use.

I don't understand the relevance of this remark. Perhaps we are talking past one another.

 

> There is nothing in the logic  of the DB to ensure that the intended
> semantic relation holds. It is  simply understood by the users.

The best that any database can do is to accept whatever we tell it.
And that is all that people can do for anything they have not been
able to observe for themselves.

JFS

> Database administrators have probably faced more such examples
> than philosophers have dreamed of.

> I highly doubt it.

I recommend Bill Kent's book _Data and Reality_, not as a book on logic
or philosophy.  But it is loaded with good examples that illustrate
the kinds of issues that DB administrators (or practical ontologists)
run into.

And I recommend R. L. Martin's classic Recent Essays on Truth and the Liar Paradox and the recent book Axiomatic Theories of Truth by Volker Halbach for understanding the issues I'm referring to.

 

If you have any examples of paradoxes that cannot be solved as well or

better in terms of the diagram in my slide 23, I would really like to
see it.  I don't believe that you can find any, but I would very much
like to know of any that test the hypothesis.

But that diagram doesn't even broach the issue of paradox, let alone solve it. Here's one way into the welter of problems: Add a truth predicate to your language. Now axiomatize the predicate so it captures its intended meaning. Here's a start, where "[φ]" is a name for the sentence φ: True([φ]) ↔ φ.  Oops. Liar Paradox (and any of a dozen related paradoxes). What now?

That's the problem. It's ridiculously hard. Of course, your answer is always the Tarski hierarchy. That might work in limited circumstances for some applications. But the Tarski hierarchy doesn't solve the problem, it simply avoids it by restricting the languages in the hierarchy so that paradoxes can't even be expressed. That's like avoiding illness by enclosing yourself in a sterile bubble. It might keep you healthy but it puts severe limits on what you can do.

-chris

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