On Sep 16, 2008, at 12:15 PM, Chris Menzel wrote:
On Tue, 16 Sep 2008, John F. Sowa wrote:
A Tarski-style model is a set of entities and a set of
relations among those entities. Those entities and relations are
*approximate* representations of aspects of the world according to
Sure. Every model leaves out information that is found in the piece of
the world that the model represents.
Represents? Ouch. If there is a 'represents' relationship between models and reality, then all our axiomatic ontologies must be given a two-stage semantics, in which model theory describes the first stage of interpretation, yielding a new kind of 'representation' which then needs another, presumably different, semantic theory to relate it to actuality. Not only has this project never been undertaken to completion, I don't think its needed.
I've heard you make a stronger
claim, though, namely that Tarski-style models can't contain real world
objects like people, tables, cabbages, etc. That is completely false.
But they do not have an existence in the world that is independent of
the ontology that we use to characterize them and identify particular
Well, yeah, I mean, you do have to have people, tables, and cabbages in
your ontology to include them in the domain of a model. You seem to be
wanting to say something deeper, but I'm not seeing it.
We have argued about that point before, and logicians and philosophers
have taken many different positions on it. Although I do not believe
that the real world *is* a model, it is at least conceivable that some
people might make that claim.
It might be conceivable, but I don't know how anyone could *plausibly*
make that claim. By definition a model is a mathematical entity of a
certain sort, usually an ordered n-tuple of some ilk. The real world,
whatever it is, isn't an n-tuple.
Well now, lets engage on this question. Because it seems to me that both of you are falling into a trap here, one that treats mathematical language as making a kind of restricted ontological commitment because it speaks about 'mathematical stuff' (like tuples and relations and sets). I think this is a mistake, and the cause of a lot of confusion about the proper role of formal semantic descriptions. Mathematical language is simply a particular precise language for talking about the actual world. If an engineer uses differential equations to analyze the stresses in a suspension bridge cables, all that mathematics she is using is about - is being used to refer to - very definite real-world things: bridges, cables, stresses, forces. Its not being used to refer to an approximate representation of the bridge, but to the actual bridge. (It may itself BE an approximate representation of the bridge, of course, but that does not imply that it REFERS TO such an approximation: in fact, quite the contrary, if it is a descriptive representation of the bridge of any kind, then it must refer to the bridge itself.) If someone objects: but such a description will always be incomplete, always omit some aspect of reality; then the reply is: yes, of course: so what? An incomplete description of something can still be of that complete thing. Semantics is not about being exhaustive, it is an analysis of what descriptions are, or can be, about.
I take all this as kind of obvious when we are talking about bridges. But exactly the same considerations apply when we are using (a different, but still a) mathematical language to talk about semantics. The fact that Tarskian/Kripkean structures are described in the mathematical language of set theory does not mean that these structures must be 'made of mathematics' and therefore cannot be made of reality. That is exactly the same reasoning that would prohibit the use of calculus to engineers because calculus is about mathematical limits (or maybe infinitesimals) and hence not about bridges or forces. These semantic structures are described using set theory, which since its inception has been understood to be a general theory of collections of anything. So there really are remarkably few limitations on what the structures being described are 'really' made of. Which is of course exactly what we would hope for a general semantic theory.
Now, I am pretty sure that Chris agrees with all the above. But let me ask: why, then, is it implausible to claim that the real world is an n-tuple? After all, what does this mean? It means only that the real world exhibits a structure which can be mathematically described in terms of n-tuples. While I will concede that such a description might lack a certain Wordsworthian quality, its not at all obvious that it is necessarily wrong. (In fact, its so minimal that its almost certainly correct in some sense.) And thats all it means to say that world "is" an n-tuple. Of course this doesn't mean that the world is nothing but an n-tuple, whatever that could mean. Even pure mathematical language doesn't make claims like that: an Abelian group might also be a commutative field. Such 'extra' structure is never ruled out by any mathematical description.
But to claim that some imaginary possible worlds have an objective
existence that can determine truth or falsity in our formalisms is
beyond my "Will to Believe".
Whoa, complete change of subject there...
By the way, my own stance on possible worlds is that a possible world is one way the real world might have been. Whether or not these things really, actually, exist in a solid Lewisian sense is of course debateable, but for ontology the question is irrelevant: we cannot manage to think about plans and outcomes and the future without considering such things, so for ontology purposes they must be considered to exist, even if we want to be much more careful when we do metaphysics. If we ever do do metaphysics; for myself, I would be quite happy to forswear metaphysics entirely, and stick to ontology.
What is the signature of the language? Does it have binary
relations in it? Etc. . It's not enough to say 'use FOL' or
some such reply. That does not determine a set of sentences.
My answer is that I think of Kripke-style possible worlds as
a collection of Tarski-style sets of entities and relations.
That is a reasonable, if somewhat informal, way to think of them.
Those relations determine the signature and ontology.
Hang on -- as noted already, in Kripke *structures* there are no
languages involved at all. And when you do add a language, there is
nothing about the Kripke structure that determines anything about the
I'll confess that if you think those possible worlds have an
independent existence in a Platonic heaven, I cannot imagine any
formalizable way to map them into a Dunn-style model.
No one has made any such claim. The discussion here about converting
between Dunn and Kripke models has been entirely mathematical in nature.
So my claim reduces to this:
1. A Kripke-style model in which each "possible world" is
a Tarski-style model (with a set of entities D and a
set of relations R over D) can be mapped to a unique
I gave what I believe is a simple counterexample to this claim. Did you
miss that post?
2. But I do not know how to define or even think coherently
about a Kripke-style model that consists of possible worlds
that have an objective existence independent of any particular
ontology that would determine the Tarski-style pairs (D,R).
There is no problem at all if you accept the objective existence of
possible worlds. Of course, perhaps you don't. (I don't, except as
mathematical entities of a certain sort that don't really deserve the
If you claim that there is a meaningful notion of a Kripke model that
is independent of any ontology, then I'd like to know how you think it
could be mapped to anything computable or formalizable.
The notion of a Kripke model requires only that there be sets. But any
additional entities you add to your ontology beyond sets can themselves
in the domains of those models.
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