Re: The L3 Revolution

galathaea wrote:

much of lukasiewicz early work
was heavily indebted to aristotle

he would tell his students
(or so i was told)
that when a brilliant man has left so much thought exposed
study it
for there may be many things still undiscovered

and when you disagree
it is better to debate brilliant dead men
than no one at all

I can agree with that.

Post's "cyclical" system which is just a bit later
than that of Lukasiewicz isn't nearly as useful for reasoning.

i would disagree with this

I haven't seen what it's useful for. My thought was something that
reduces to classical logic if you eliminate the 3rd value, and Post's
cyclical negation doesn't do that.

his system
provides a layered refinement
of the logics between the boolean and the heyting
and the structural distinctions
all have natural functional interpretations in decision theory

I'm not familiar at all with decision theory, so I can't comment on
that.

It's the comment that the intended application to modal logic did not
materialize that intrigues me. Why didn't it and where is this
discussed? In one of the references in the article I cited, I think
it's Rosser and Turquette, but it could be Ackermann or Rescher,
there is mention of an objection posed by Gonseth, but this objection
as quoted is answerable. Lewis was aware of Lukasiewicz' approach,
discussed it, and rejected it, but his objections are also answerable.
I've not been able to get easy access to the older philosophical
journals where this might have been further discussed.

the one i have seen who most developed the modal interpretation
was not active in the logic community

hans reichenbach's three-valued quantum logic
was lukasiewicz with a different implication

in his papers and books
the modal interpretation is clear from the reading
recapitulating much of your page

just not in a symbolism familiar to logicians

others with more background
have returned and revised reichenbach
often returning to lukasiewicz
though to his infinite truth-valued logics

see for example
jaroslaw pykacz


I did look at Reichenbach's logic when I was searching throug MVL to
see whether anyone had done everything I had already, and whether my
system could include any of the other 3-valued systems.

http://www.iep.utm.edu/r/reichenb.htm

I thought his "alternative implication" was too permissive, did evil
things to the contrapositive and biconditional, and could be defined as
~[]P v Q in my system if it were needed. It looked to be too narrowly
focused on application to quantum theory and wasn't sufficiently
general-purpose for me. I already had his & and v, the "standard
implication", and the standard and alternative "equivalences". I
didn't see a good use for the cyclical negation, the "complete"
negation, or the quasi-implication.



The modal operators of L3 have the same effect of limiting uncertainty
and containing it within the more tractable 2-valued approach. That's
partly why I don't use the constant U, because then uncertainty would
tend to proliferate.

what objection do you have
against this proliferation

Partly, the first steps into this realm go through the "logic" of
irrational and obstinate denial, and proceed further through complexity
to confusion. Each new 3-valued function that can be defined in terms
of others has its own unique set of properties that has to be
investigated and explored. This is a back door into the full range of
three-valued systems as discussed, partially, at:

http://xyzzy.freeshell.org/trinary/

I'm more interested in keeping my focus on the extension of classical
logic.


if you were to take a separateness axiom of truth values
(any reasonably justifiable one
like the tendency you comment on)
and were led to a countable infinity of uncertain values

what would be the down-side
philosophically?

I don't know about philosophically. I do think it's easier and more
practical to deal with one uncertain value than a countible infinity of
them.

.

Relevant Pages

  • Re: The L3 Revolution
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  • Re: Modal Logic
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  • Re: Modal Logic
    ... Lewis and Langford discussed Lukasiewicz 3-valued logic but rejected ... and multi-valued logics and modal logics ... Lukasiewicz logic *is* a bit odd, but if one wants deal with the ...
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