Pat, (01)
On the technical issues about model theory, naive physics, and
methods of deductive reasoning, we have no disagreements. (02)
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. (03)
I agree that for a *theory* of truth, a formal specification need
not be computable. That may be sufficient for pure mathematics,
but not for building robots, answering questions, or designing
airplanes. 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. (04)
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. (05)
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. (06)
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. (07)
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. (08)
That may be true in principle, but only with uncountable models
that get down to the level of individual points in space. Trying
to map a differential equation to that level and compute its truth
value for some continuous model is decidedly nontrivial. (09)
PH> The naive physics stuff didn't set out to solve such problems. (010)
I'm sorry I mentioned the naive physics work in the same note
with engineering physics. But the mainstream of physics and
engineering supports major industries with continuous math.
(Mathematica is widely used by physicists and engineers.) (011)
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. (012)
The engineers are certainly thinking about the truth of their
assumptions and the results they calculate. (013)
PH> But the fact remains that semantics is about the representation
> --> reality relationship, just as linguistics is about language. (014)
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. (015)
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. (016)
As I said in my previous note, there are two parts to Tarski's
evaluation method: (017)
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). (018)
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. (019)
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. Then part #2
would evaluate such an equation in terms of the model. (020)
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? (021)
An example would be a boundary-value problem: (022)
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. (023)
2. Let B be the boundary of R with the values of F specified
by some known function G at all points of B. (024)
3. Compute the values of F at all interior points of R. (025)
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. (026)
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. (027)
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. (028)
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". (029)
John (030)
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