|From:||Chris Menzel <chris.menzel@xxxxxxxxx>|
|Date:||Fri, 13 Jul 2012 13:21:19 -0500|
On Wed, Jul 11, 2012 at 5:58 PM, John F Sowa <sowa@xxxxxxxxxxx> wrote:
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. ;-)
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
I don't understand the relevance of this remark. Perhaps we are talking past one another.
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.
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.
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