On Sep 25, 2008, at 1:54 AM, John F. Sowa wrote: Pat, On the technical issues about model theory, naive physics, and methods of deductive reasoning, we have no disagreements. PH> The goal of semantics is NOT to determine which sentences are true and which false. It is simply to DEFINE what this distinction
means.
I agree that for a *theory* of truth, a formal specification need not be computable.
In most cases, it cannot be computable. It doesn't even make sense to talk about part of the real world being 'computable'. That may be sufficient for pure mathematics,
but not for building robots, answering questions, or designing
airplanes.
WRONG. It has nothing to do with the subject-matter. Now, let me say at once that of course semantics alone is not 'sufficient' for doing anything. But your obviously intended conclusion, that these 'practical' topics require a different semantics from that used in describing truth in pure mathematics, is just plain mistaken. The same semantic account applies in all these cases. Building an airplane needs a lot more than just semantics or just mathematics or just calculation, of course: but it does require semantics, and the semantics that it requires isn't a different kind of semantics. The old logical positivists said that the meaning
of a sentence is its method of verification. Popper emphasized
falsification. But either way, they considered testing truth
values to be an integral part of a theory of meaning.
Maybe they did. However, IMO they were mistaken. And it is certainly a lot conceptually clearer if one makes a clear distinction between theories of truth and theories of verification/falsification. If your position is that semantics is inherently concerned with verification, you should state this position clearly, as it is a highly controversial and extreme position, and not one that most ontology engineers (or even philosophers) would subscribe to, I suspect. (And if you really do hold this view, I cannot understand how you can use formal logical methods. What do you take their semantics to be?) For any practical application, some computable method is needed to
show (a) the starting assumptions are true in terms of the model,
(b) all observed values are consistent with the computed values,
and (c) reasonable values are computed for the unknowns of interest.
Semantics does not talk about 'starting assumptions' or 'observed values' or 'unknowns of interest'. Whatever you are talking about here is not semantics. PH> 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 recognize that you were doing naive physics with deduction in FOL. But mainstream physics and engineering are based on continuous math with little or no use of FOL.
Its not usually formulated this way, but continuous mathematics CAN be expressed in FOL, though some of it requires 2OL to be fully expressive enough. There's nothing inherently "non-continuous" about FOL. PH> 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.
That may be true in principle,
It is true. I have no idea what kind of 'principle/other' contrast you are appealing to here. but only with uncountable models
that get down to the level of individual points in space.
? No, with whatever models are needed to model the continuity in question. The commonest textbook notions of limits and continuity are written in terms of interval sizes (the familiar delta-epsilon definitions), which are straightforward to write in logic, in fact they are essentially already in logic in the textbooks. There is no conceptual strain involved in using logic to express all this stuff.
Trying
to map a differential equation to that level and compute its truth
value for some continuous model is decidedly nontrivial.
Im not talking about computing anything. Why do you have this fixation with computing? We are arguing about semantics. PH> 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.
The engineers are certainly thinking about the truth of their assumptions and the results they calculate.
And when they do, I bet that they are implicitly using Tarskian thinking. They do whenever they think about what words like 'and' , 'every' and 'implies' mean, or when they figure out that, say, this beam should touch that surface, according to the calculations. That is, that a relationship (touching) should hold true between two things: pure FOL/Tarski. PH> But the fact remains that semantics is about the representation --> reality relationship, just as linguistics is about language.
Fine. And the engineers are using differential equations to do the representation, they are mapping them to reality, and they are using that mapping to determine the truth (i.e., accuracy) of their work.
Which is exactly what Tarskian semantics would say they are doing. PH> 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.
As I said in my previous note, there are two parts to Tarski's evaluation method:
Wait. Tarskian semantics does not describe an 'evaluation method'. It defines truth (in an interpretation) of complex sentences in terms of that of their component sentences. That is not a 'method'. 1. The part that takes a complex sentence s, with arbitrary
combinations of quantifiers and Boolean operators, and
reduces it to very simple sentences. In Tarski's case,
those simple sentences are atoms that apply a single
relation R to a tuple of values (V1,...,Vn).
2. The part that checks the truth of the simple sentences;
for example, it checks whether the tuple (V1,...,Vn) is
in the extension of R.
You have described it, misleadingly, as though it were a method or algorithm, but ignoring that, OK. What I suggested in my previous note is to do part #1 along
the same lines as Tarski's, but to allow more general types
of sentences; e.g., differential equations.
I don't know why you think this doesn't already apply to differential equations. Any equation is an atomic sentence, and as long as you can unpack the mathematical notation into some series of function applications (which is textbook stuff for differential calculus, cf. MathML) then Tarski applies to it directly. Tarskian methods do have their limitations, as we know: modalities require them to be extended, they don't handle nonmonotonic reasoning, etc.,, but conventional mathematics has always been straightforward. Then part #2
would evaluate such an equation in terms of the model.
Following the standard rules for semantics of functional terms, yes, exactly. PH> Applied mathematics makes no mention of "engineering models" or of how to determine "truth"...
By the way, what exactly is a traditional "engineering model",
according to you? Can you give a general characterization of
these things?
An example would be a boundary-value problem: 1. Given a bounded connected region R of space (2 or more dimensions) with some function F defined by some partial differential equation Q at all points of R. 2. Let B be the boundary of R with the values of F specified by some known function G at all points of B. 3. Compute the values of F at all interior points of R.
That is a problem. I asked for an example of an engineering model.
For this example, the only Boolean operators are implicit conjunctions. The task to be done is to solve Q for all the interior points of R. If Q is a typical equation of a well-studied kind and the boundary B is not too complex, the solution should be relatively straightforward. PH> 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.
I won't make a sweeping statement about all of applied math, but I would say that this is one typical example. The methods for solving Q can compute values of F for all points of interest. If sensors are placed at the corresponding points in the real world, the computed values of F could be compared to the measured values to determine whether they are equal within the expected error bounds. But note that evaluating a differential equation is quite different from comparing tuples. I would be happy to extend Tarskian semantics to allow such math, but it is a bit more than ordinary "Tarskian semantics, pure and simple".
Doing that math isn't itself Tarskian semantics, because its not semantics at all. But if you were to try to analyze the semantics involved in describing that mathematical doing - accounting for how the mathematical steps were logically valid, say - then Tarskian semantics is all the semantics you would need. You don't need some other semantic theory as well as, or instead of, Tarski.
Pat
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