Author: Ralph E. Kenyon, Jr. (diogenes)
Monday, May 21, 2007  08:38 am

A scientific argument or "proof" is "rigorous" if it can be represented as a sequence of small steps where every step proceeds directly from previous steps (or from an assumption) using specific valid rules of inference  provided that the set of assumptions is minimal (mutually independent) and consistent.

Author: Ralph E. Kenyon, Jr. (diogenes)
Monday, May 21, 2007  09:51 pm

David wrote An argument must also achieve "corespondence". In [other] words, it must be supported by experience and/or predicted effects whose existance cannot be attributed to other causes. This is not a "scientific argument or proof"; it is empiricism. See The Philosophy of Karl Popper, Heraclitus? or Xenophanes?, and define:empiricism. I do not use "argument" as in the metaphor "argument is war"; for that I use "dispute". Because of my mathematics and philosophy background, I only use "argument" in the "argument is proof" metaphorical sense.

Author: Ralph E. Kenyon, Jr. (diogenes)
Tuesday, May 22, 2007  07:47 am

Thomas, The example of Newtonian addition of velocities showed that the theory incorrectly predicted the result. By modus tolens, logic "proved" that the theory was incorrect. The "theory" was a map that had been used, but once it had been "falsified" by contrary emprirical observations, it needed to be revised. A revision, relativity, which includes, to put it simply, the assumption that the speed of light is always a constant relative to the observer, was missing from the old map, but has been added to the new map. We "infer" and "project" that this portion of the theory, this assumption, has a "corresponding" "structure" in what is going on, and we label that abstraction "a characteristic of light". The deductive process in logic using the language of mathematics to describe, account for, and predict our observation abstractions is strictly truth preserving. So I would sharply distinguish Logic: the methods of deriving conclusions strictly consistent with assumptions. Mathematics: a language for describing relations. Science: a process of using logic and mathematics to build theories that account for our observations. Natural language: How we express all the above as well as anything else in verbal terms. Natural language in neither consistent nor mathematical, although subsets of it can be either or both when carefully formulated.

Author: Ralph E. Kenyon, Jr. (diogenes)
Tuesday, May 22, 2007  11:52 am

I never used "good" with respect to arguments. I used "rigorous" and "strictly truth preserving". These are lower level abstractions as compared to the high level and ambigous "good".

Author: Ralph E. Kenyon, Jr. (diogenes)
Tuesday, May 22, 2007  12:48 pm

Tarski gave a definition for "truth" that Korzybski allued to in Science and Sanity. In the four levels or perspectives on the use of language, the veracity of models, effectiveness of maps, etc., (and "arguments" supporting, proposing, delineating, etc., maps, models, etc.,) depends on (1) Correct grammar, (2) Valid logic (strictly "truth" preserving), and (3) Fixed semantics (unambiguous reference). The (4) understanding and use of models depends on the semantic reactions of individuals to all the above. If you don't understand or know how to use "truth" and "true" in the context of logic, semantics, and Tarki's model theory, well, begin here.

Author: Ralph E. Kenyon, Jr. (diogenes)
Tuesday, May 22, 2007  12:59 pm

I would call "using the language of mathematics to describe, account for, and predict our observation abstractions" "applying" mathematics"  in physics, in the stock market (much less rigorous), etc.

Author: Ralph E. Kenyon, Jr. (diogenes)
Tuesday, May 22, 2007  04:19 pm

Tarski, of course, did not introduce levels of abstraction; Bertrand Russell did in the Theory of Types in 1903, in direct response to his discovery, known as Russell's Paradox, of the contradiction in Frege's work. Tarski was apparently the first to explicitly formalize the distinction between levels in a logic of semantics in his 1933 work (not translated into English until 1956}, but which Korzybski made references to. It was Russell, however, whom Korzybski cited extensively by including a reprint of a 1928 article about it as appendix II of Science and Sanity. Tarski's work may be considered pioneering in metamath and model theory. It uses the level distinction from the theory of types. Related is the fact that Gödel proved that any system strong enough to contain arithmetic would contain statements, the truth of which cannot be decided on the basis of the axioms of the system. It requires a "higher" level system to show the truth of such statements. (1931). Tarski knew Gödel, and the method used in Gödel's incompleteness theorem builds numerical metamathematical constructs. I can see a relation there leading to generalizing those constructs and formalizing the "language barrier" between levels.
