List, John, Jerry and Jon,
LEM presents one of the three basic misassuptions in modern logic. For
all I know CSP and Brouwer came to similar conclusions independently.
They also offered their grounds and conclusions very differently.
There was a deep change in math and locic during and after the centuries
1500-1600.
Arabic influence, for starters. Latinization of ancient greek
philosophical heritage.
Modal logic was self-evident for Plato, Aristoteles etc.
Modern (especially formal) logic is just feebly trying to recover and
gather together the remants after LEM & the other two mispremisses.
Kirsti
John F Sowa kirjoitti 11.10.2017 09:20:
Jerry LRC, Jon AS, List,
Jerry
[JFS] Since a contradiction is always false, a contradiction
implies everything.
Everything? While this assertion is widely repeated in
the literature, I think it is highly problematic.
It's widely repeated because it is a fundamental assumption
of most versions of formal logic -- i.e., of every logic that
assumes the Law of Excluded Middle (LEM).
But it is indeed problematic. Brouwer, for example, rejected
LEM for intuitionistic logic.
And even for systems that are based on LEM, nobody actually claims
that everything has been proved. Instead, they recognize that there
is a mistake somewhere, and they start searching for it.
Jon
[JFS] For modal logic, there are three options:
necessary, possible, and contingent (not necessary and not
impossible).
Did you mean to say necessary, impossible, and contingent?
Yes. I wrote that too hastily. "not impossible" is a synonym
for "possible". For the three options, I should have written
necessary, impossible, and contingent (possible and not necessary).
But after I sent that note, I did some googling, which led me
to the article "Peirce and Brouwer" by Conor Mayo-Wilson:
http://mayowilson.org/Papers/Peirce_Brouwer.pdf
Some excerpts:
page 1
In his 1908 "The Unreliability of the Logical Principles" Brouwer
rejected the law of excluded middle (LEM)...
Five years earlier, Peirce had reached similar conclusions...
p. 2
Peirce and Brouwer's common rejection of LEM is not simply a
coincidence, but rather, stems from a deep underlying similarity
in their respective philosophical analyses of the continuum.
p. 3
Peirce and Brouwer seemed to have no knowledge of each other's work.
Brouwer might have learned of Peirce's ideas on semiotics in the
1920's through his association with Lady Welby... However, the two
most likely worked independently...
Fernando Zalamea also discusses Peirce and Brouwer in connection
with the continuum. But he doesn't mention Lady Welby:
http://uberty.org/wp-content/uploads/2015/07/Zalamea-Peirces-Continuum.pdf
In any case, these sources indicate that Peirce began to reconsider
his ideas about LEM around the same time as the Lowell lectures.
His thoughts about the continuum seem to be the original reason.
But by 1909, his thoughts led to 3-valued logic and a new way
of representing and describing existential graphs.
John
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