Your answer reveals two kinds of misunderstandings, one about my old liquids work, the other, more relevant to this forum, about the purpose of semantics. I'll try to keep them separate.
On Sep 23, 2008, at 11:07 PM, John F. Sowa wrote:
Pat,
I agree that Tarski's evaluation function is a formal construct.
Whether or not one applies that function to a sentence s to get
a value of T or F is a separate issue. But in the case of finite
models (such as the tables of a relational DB for a closed world),
computational methods logically equivalent to Tarski's are used
to answer questions.
Two points here (which of course we have argued about interminably in the past.) First, finite models are rare: so rare, in fact, that IMO they can be effectively ignored for practical ontology work. Any formalism that uses numerals, for example, has no finite models. CLIF has no finite models because its semantics recognizes all character strings as entities. Finite models are handy for describing limited, "closed" worlds such as database tables, and a powerful basis for determining decideability of limited logics, but not to serve as models for an entire ontology; at best they will be finite parts of larger models.
But more seriously, the computational methods you refer to can be described also, and IMO more usefully, as processes of inference. They are typically performed by matching and searching, neither of which have anything directly to do with Tarski. While some inferences under some conditions can be described in semantic terms, as 'building a model', nothing is gained by this kind of re-description, and it only works in very restricted circumstances.
JFS>> I agree that in principle one might conceive of using a Tarski
style of model theory to relate statements about liquids to the
world.
PH> wait a second. "In principle", "might conceive"? But (of course)
that is exactly what is done by the 'ontology of liquids', since
that ontology is phrased in FO logic, which has a Tarskian semantics.
So it certainly can be done and indeed has been done...
I agree that you assumed a Tarski-style model for your ontology, which
is very good of its kind. But most engineering work with fluids uses
continuous math, such as differential equations.
True, but entirely beside the point. That old liquids work wasn't intended to reproduce or support "engineering work": it was an exercise in 'naive physics', trying to capture in an ontology the pre-formal common-sense intuitions about the everyday behavior of liquids, based on non-quantitative facts which cannot support any continuous mathematics. You don't need partial differential equations to understand that if you up-end an open cup of coffee, it will spill. So this observation isn't relevant either to that work or to Tarski.
BTW, as Im sure you know but other readers may not, the subsequent development of 'qualitative reasoning' systems systems shows that a great deal of highly practical technology can be more usefully based on some very simple pre-calculus math, deliberately eschewing continuous mathematics. Continuous mathematics is useful when you have enough information available to use it, and when the questions being asked require it, but it can be a hindrance in other circumstances. Not that this has anything at all to do with Tarskian semantics, which applies to continuous mathematical language just as well as it does to any other.
JFS>> But in practice, the exercise of evaluating Tarski's function
to determine whether any particular statement about liquids is true
or false in terms of the world is never done.
PH> Im not sure what you are talking about. Tarskian semantics does
not require any functions to be 'evaluated', whatever that means.
Its not a computational theory.
I agree. But if it is to have any kind of relevance to practical
applications, those functions must be evaluated on actual sentences.
NO. They
never need to be evaluated. In fact, it doesn't make sense to even talk of evaluating them. They aren't the kind of thing that
can be evaluated, for the most part. They are part of a theory of truth. The corresponding computations are processes of inference, not of semantic evaluation. The relevance of semantics to practical applications comes from the completeness theorem, basically.
PH> Take a statement from the liquids ontology, a simple one such as the
axiom that lists the possible ways that liquids can occupy space. This
talks of liquids and spaces and ways that the liquid can occupy the
spaces, and it - the axiom - has a perfectly conventional Tarskian
semantics. What is "not being done" here, according to you?
Solving the kinds of problems that are routinely done with the
continuous methods of fluid mechanics.
Then your entire post is a non-sequiteur. The naive physics stuff didn't set out to solve such problems. You might as well observe that it does not do French cookery. But in any case, suppose that one did set out to solve these problems, using continuous mathematics: what would that have to do with Tarskian semantics?
PH> Not "evaluating" truth. Semantics is concerned with a deeper,
prior issue: defining truth in the first place. It is probably
good to do this before trying to evaluate it, IMO.
I'll admit that there may be good reasons why the truth value for
a given sentence cannot be determined. But a fundamental *goal*
of any semantic theory is to determine whether sentences are true
or false.
WRONG. The goal of semantics is NOT to
determine which sentences are true and which false. It is simply to DEFINE what this distinction means. Determining actual truth is what courts do, by sifting evidence and gathering testimony. It is a topic entirely beyond the scope of this forum.
It really is centrally important to understand this very basic point, if we are to have meaningful conversations about semantic issues.
I don't have much sympathy for "deep" semantic theories
that cannot be used in practice.
I will try to avoid using the word "deep", with or without scare quotes. For my part, I have very little sympathy for anti-intellectual innuendo.
PH> Now, one might claim that Tarski's theory is inadequate because
it doesn't apply to the case of continuous math (which is false,
but one could claim it); but it simply makes no sense to claim
(as you here seem to) that semantics is a different way of
describing the world than that used by continuous mathematics.
No. I am claiming that Tarski's evaluation function is useless
for the differential equations used by physicists and engineers.
I have no idea what this is supposed to mean. "Useless for" in what sense? It certainly
applies to differential equations. Now, it may well be that engineers using calculus are not thinking actively about semantics, any more than someone speaking English is thinking actively about linguistics. But the fact remains that semantics is about the representation--> reality relationship, just as linguistics is about language.
Whether the representation is concerned with discrete or continuous aspects of reality is irrelevant. In fact, I still don't understand why you have brought this entire topic up, what point you think is being made here. It seems as though you are presuming that there is some kind of contrast or tension involved between continuous mathematics and Tarskian semantics, but I am at a loss to understand what you think this contrast amounts to.
And I would claim that the continuous methods for determining the
truth of such equations are just as "semantic" as Tarski's function.
Garbage. This is merely mis-using the term "semantic" and creating confusion. Continuous mathematics is no more or less semantic (or "semantic") than any other part of applied mathematics. Semantics is the science of explaining how
any representational language - which includes continuous math, of course - relates to the world it sets out to describe. The truth of continuous equations isn't defined or determined by continuous mathematics itself. The same semantics that determines truth does so for discrete and continuous equations.
As an example, consider describing time, and writing axioms which distinguish discrete from continuous time. It takes second-order logic to do it (itself an semantic insight of some importance) but it can be done in a single representational language. Which of course has a Tarskian semantics.
I admit that a collection of differential equations could be
considered a conjunction to which Tarski's method might be used.
Don't keep speaking in the subjunctive voice. It
is so used. In fact, I know of no other semantic framework that can be applied here.
But I suggest the following approach, which is what physicists
and engineers have been using informally for centuries:
Everyone has been using Tarskian semantics
informally for centuries. That is the point of Tarski's theory: it simply puts onto a rigorous footing the intuitive ideas that everyone had been using previously. (Or, to be more exact, a subset of them that turns out to be of wide utility.)
1. For any equation q, use the traditional methods of applied
mathematics to determine the truth value of q in terms of
a traditional *engineering model*.
There are no such 'traditional methods'. Applied mathematics makes no mention of "engineering models" or of how to determine "truth". I went through tyears of solid mathematics education, everything from algebraic topology to quantum electrodynamics, and nobody even once mentioned any of these notions. If you had mentioned "truth" to any of my professors they would have told you to go talk to a philosopher.
By the way, what exactly is a traditional "engineering model", according to you? Can you give a general characterization of these things?
2. For any statement s that combines equations with quantifiers
and Boolean operators, use a Tarski-style method to reduce s
to equations (as in #1) instead of atoms (as in Tarski models
with a set of entities and a set of relations over them).
Equations ARE atomic sentences, albeit of a restricted form. There is no contrast here.
3. After s has been reduced to equations, each of them can be
evaluated in terms of the domain to determine a value T or F.
This method is every bit as "semantic" as anything that Tarski
ever did
This IS Tarskian semantics, John, pure and simple. Its not an alternative to Tarski. All you have done here is recommend that all of applied mathematics be reduced to equations.
, and it is compatible with the practices of physicists
and engineers since the time of Newton and Leibniz.
They don't work
entirely in terms of equations. Try reading Newton's Principia, it makes for illuminating reading. Or Einstein's own very readable account of special relativity, with hardly an equation in it anywhere..
Furthermore,
it avoids any claim that the world consists of tuples.
No, it also assumes this. An equation T = S is true just when whatever T denotes is the same as what S denotes. So we need to say what it is that S and T denote in the world in question, which means that we need to have a set of entities and a mapping from terms to that set. And that is a tuple, in
exactly the sense used previously.
I'm not saying that Tarski's evaluation function is bad for what
it can do. But I would claim that for many kinds of statements,
there are better methods for determining their truth in terms
of the world.
AFAIK, there aren't ANY other methods. Everything you suggest above is simply applying Tarskian ideas to a restricted formalism (pure equations) but isn't an
alternative to Tarski's programme.
Pat
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