logical systems (Query about symbolic logic)

Les Schaffer schaffer at optonline.net
Mon Jul 7 13:52:45 MDT 2003


Jim Farmelant, from last week (Jurriaan, i'll get to your stuff too!):

on three-valued logic:

> That sort of thing goes at least as far back as to Hans Reichenbach,
> the physicist turned logical empiricst philosopher.

i learned the other day that F. Zwicky (Caltech astronomer) was the
first to suggest alternative logic be applied to QM [that from Eves
"Foundations and Fundamental Concepts of Mathematics"]. i didnt mean
to suggest Sudbery was the first to discuss this. if you check the
reference you see he elaborates on further work in the area: so-called
lattice theories, which are mathematical generalizations of projective
geometry.

i think you might like Sudbery's book. He is a mathematician with a
good feel for QM, and devotes a full chapter to quantum metaphysics
and interpretations, giving all the various alternatives an
interesting review.

His take on quantum logics i __will__ shortly post on my web site:

   http://folks.astrian.net/godzilla/qm-logic.html

but a key quote here:

  "The analogy between logic and geometry is superficial. It is
  possible to formulate non-Euclidean geometry without using or
  mentioning Euclidean geometry, but it is not possible to formulate
  quantum logic without using classical logic (in the
  meta-theory). Thus the solutions to the problems of measurement and
  inseparability are cheap; they depend on a selective ban on the
  distributive law. If quantum logic were consistently adopted as a
  logic in the true sense of the word (i.e. a method of reasoning), it
  would involve reconstructing the whole of mathematics - a herculean
  and probably impossible task."


which brings me back to an issue i raised earlier, which is, how do
these attacks on classical logic produce something __new__.

> In QM both the wave and particle interpretations involve
> descriptions of interphenomena.  Neither description can escape the
> introduction of causal anomalies.  One of his examples of this is in
> electron diffraction experiments.  Individual flashes are said to
> introduce a causal anomaly into the wave description, while the
> diffraction pattern introduces a causal anomaly into the particle
> description.  Neither the particle interpretation nor the wave
> interpretation can provide us with a normal description, according
> to Reichenbach.

check out Heisenberg's "Physical Principles of the Quantum Theory". he
gives a masterly presentation of his and Bohr's notion that quantum
uncertainty means only that there is something lacking in both the
wave and particle pictures.

now back to your earlier post on explosiveness:

> Classical logic is explosive because if we have both A and not-A
> then B is always entailed regardless of its own truth value.

i've been thinking about this some more. i guess my problem with the
"problem" of explosive behavior is with the logical truth table for
implication:

   A      B      A --> B
   ---------------------
   T      T         T
   T      F         F
   F      T         T
   F      F         T

which is a hard one to really appreciate.  only the second line makes
real sense.  From Eves, bottom p. 346:

  The deifnition of implication in logic is a controversial matter,
  and other analyses of the nature of implication have been
  proposed. Implication as defined above [Les: the truth table i show
  above] is called _material implication_. C.I. Lewis has introduced a
  concept called strict implication, which seems more nearly to
  correspond to the relation holding when a conclusion is said to be
  deductible from premises, but as yet there is no definition of
  implication dependent on propositional structure that has won
  general acceptance among logicians.


so my question is, how much does the notion of explosiveness depend on
material implication? and can you give a "real-world" example of the
problem of explosiveness?

then we can move on to Priest's dialetheism and dialectical logic.

M Kline [Mathematical Thought from Ancient to Modern Times, Vol I]
makes the argument that classical logic first appeared amongst the
Greeks after their hearts were broken upon discovery of irrational
numbers. the thinking was to be "more careful" in mathematical
derivations so that we "don't get burned again".

i like this operational characterization of mathematics, more than any
metaphysical discussions about truth and logic. and of course, its all
funny, because the greeks were wrong to be so negative about
irrationals. as Kline takes pains to point out, the greeks were SO
anal about their geometric derivations, they missed out on a lot of
good stuff.

les schaffer






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