Gary R and list,
Being a non-mathematician myself, I’ve been drafting an introduction to
Peirce’s EGs for the likes of us. I have the current draft online now here: EG
introduction (gnusystems.ca) <https://gnusystems.ca/TS/draftEGs.htm> It
includes many links both to Peirce’s own introduction to EGs (Lowell Lecture 2
of 1903) and to other (later) Peirce quotations.
Of course I’d welcome comments and suggestions (even from mathematicians!).
Gary f.
From: peirce-l-requ...@list.iupui.edu <peirce-l-requ...@list.iupui.edu> On
Behalf Of Gary Richmond
Sent: 18-May-21 23:58
John, Jon, List,
JAS: Peirce anticipates aspects of the formal system that is now known as
intuitionistic logic--e.g., defining negation as the implication of falsity...
JFS: Nobody knows what Peirce would have said about the less dogmatic treatment
of intuitionistic logic by Heyting and others, including Oostra. But it's
doubtful that he would have approved of the drastic reduction in acceptable
mathematical theories. He was always highly suspicious of any attempt to block
the way of inquiry.
"Dogmatic treatment of intuitionistic logic, etc."?? How in the world is
"intuitionistic logic--e.g., defining negation as the implication of
falsity..." (emphasis added) a "drastic reduction in acceptable mathematical
theories"? Acceptable to whom? You? Who else?
And, really, it seems fairly over the top to describe it as "an attempt to
block the way of inquiry?" Certainly your saying so does not make it so. Please
explain how this "blocks the way of inquiry" for folk like me who are
apparently radically deficient in mathematics and logic so simply can't see it
as such.
As JAS wrote:
JAS: . . . Peirce seeks to account for objective or ontological
indeterminacy--there are some propositions that are neither true nor false
because reality is indeterminate. Years later, he briefly explores the logical
implications of this in his "Logic Notebook."
CSP: Triadic Logic is that logic which, though not rejecting entirely the
Principle of Excluded Middle, nevertheless recognizes that every proposition, S
is P, is either true, or false, or else S has a lower mode of being such that
it can neither be determinately P, nor determinately not-P, but is at the limit
between P and not P. (R 339:515[344r], 1909)
JAS: Intuitionistic logic also recognizes this limit but does not assign a
third truth value to it, which is what Peirce goes on to propose. Accordingly,
in today's terminology, what he [Peirce] is primarily rejecting in these
passages is not the (so-called) law of excluded middle ("either a proposition
or its negation is true"), but rather the principle of bivalence ("every
proposition is either true or false") [emphasis added].
In a word: Who is actually being 'dogmatic' here?
Best,
Gary R
“Let everything happen to you
Beauty and terror
Just keep going
No feeling is final”
― Rainer Maria Rilke
Gary Richmond
Philosophy and Critical Thinking
Communication Studies
LaGuardia College of the City University of New York
On Tue, May 18, 2021 at 10:59 PM John F. Sowa <s...@bestweb.net
<mailto:s...@bestweb.net> > wrote:
Jon AS,
I'm writing an article about Peirce's writings on logic in 1911, which I'll
post to P-list soon. And I'm glad that we can agree on that point.
JFS> However, Peirce and Brouwer were on opposite sides of fundamental issues
about the nature of mathematics. ... In general, Brouwer's assumptions were
diametrically opposed to Peirce's fundamental assumptions.
JAS> I wholeheartedly agree.
But in R670. Peirce definitively rejected the idea of defining negation in
terms of implication and falsity. I show that in my article.
JAS> Peirce anticipates aspects of the formal system that is now known as
intuitionistic logic--e.g., defining negation as the implication of falsity...
Nobody knows what Peirce would have said about the less dogmatic treatment of
intuitionistic logic by Heyting and others, including Oostra. But it's
doubtful that he would have approved of the drastic reduction in acceptable
mathematical theories. He was always highly suspicious of any attempt to block
the way of inquiry.
John_ _ _ _ _ _ _
_ _ _ _ _ _ _ _ _ _
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