|
|
Jan Burse wrote:
> http://plato.stanford.edu/entries/peirce-logic/#TV
>
> Cite:
> "In three unnumbered pages from his unpublished notes written before
> 1910, Peirce developed what amounts to a semantics for three-valued
> logic. This is at least ten years before Emil Post's dissertation,
> which is usually cited as the origin of three-valued logic. A good
> source of information about these three pages is Fisch and Turquette
> (1966), which also includes reproductions of the three pages from
> Peirce's notes."
But see also,
plato.stanford.edu/entries/logic-manyvalued/">http://plato.stanford.edu/entries/logic-manyvalued/
Cite:
5. History of Many-Valued Logic
Many-valued logic as a separate subject was created by the Polish
logician and philosopher Lukasiewicz (1920), and developed first in
Poland. His first intention was to use a third, additional truth value
for "possible", and to model in this way the modalities "it is
necessary that" and "it is possible that". This intended application to
modal logic did not materialize. The outcome of these investigations
are, however, the Lukasiewicz systems, and a series of theoretical
results concerning these systems.
Essentially parallel to the Lukasiewicz approach, the American
mathematician Post (1921) introduced the basic idea of additional truth
degrees, and applied it to problems of the representability of
functions.
( end cite)
It appears that Lukasiewicz was working from Aristotle's start and may
or may not have been aware of Pierce's earlier work on the same
subject. It also appears that Lukasiewicz got farther, just one step
short of success. Post's "cyclical" system which is just a bit later
than that of Lukasiewicz isn't nearly as useful for reasoning.
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.
> But I doubt that three value logic is useful for reasoning.
> Because we can get away without, and still have undefined.
> For example the empty theory neither entails p nor ~p,
> so this might be interpreted as undefined.
The ability to deal consistently with that odious and uncomfortable
never-never land of the forbidden middle and the "unknown" and
"undefined" is precisely why it is useful in reasoning and has such
potentially broad application.
> The application I had for three value logic was in logics
> for computation. So A+ means the program halts with yes for
> the query A. And A- means the program halts with no for the
> query A. And ~A+&~A- means the program neither halts.
>
> So on the object level we have binary programs and not
> "truth". Respectively on the object level we have the
> "truth" of programs halting or not halting, and if halting
> then answering yes or no. It is very natural to have a
> 3 valued logic here.
>
> But it can be broken down into a two valued logic. I.e.
> A+ and A- which behave purely two valued.
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.
|
|