logical systems (Query about symbolic logic)

Jim Farmelant farmelantj at juno.com
Mon Jul 7 19:17:11 MDT 2003



On Mon, 07 Jul 2003 15:52:45 -0400 Les Schaffer <schaffer at optonline.net>
writes:
> 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"].

Was Zwicky's logic a three-value logic like Reichenbach's?
In that case we might have an issue of priority over who developed it
first. Reichenbach's 1944 book *Philosophic Foundations of
Quantum Mechanics* makes no mention of Zwicky.

>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 remember years ago reading some papers by Patrick
Heelan on quantum lattice logics.  Heelan is a Jesuit
priest/physicist/philosopher.

>
> 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."

The argument is not too different from the arguments that
the philosopher Susan Haack(see her book
*Deviant Logic, Fuzzy Logic*) has made against deviant
or alternative logics.  She contends that they cannot
be formulated or expressed without relying upon good
old fashion classical logic.  And like Quine, she believes
that just about anything that is expressible in terms of
alternative logics can be expressed in terms of classical
logic.

>
>
> 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?

Well Priest's point is that classical logic is explosive, in
that if we try to assert both A and not-A simultaneously,
then their conjunction will logically (via material implication)
imply anything.  Priest contends that certain scientific
theories (such as QM) are not explosive in his sense,
and so therefore dialetheism and paraconsistent logics
can be quite legitimately applied to such theories.

>
> 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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