On 7/11/2018 5:40 PM, Jerry LR Chandler wrote:
[JFS] Every Alpha and Beta EG can be mapped to a statement
in the first-order subset of his algebra of 1885.
Frankly, it is very risky to make such conceptual leaps,
especially with CSP wandering rhetoric!
I was not making a conceptual leap. I was stating a theorem:
Every Alpha and Beta EG can be mapped to and from a logically
equivalent statement in first-order logic in Frege's notation,
Peirce's FOL subset of 1885, or the FOL subset of the Principia
Mathematica.
And a "bivalent radicle": -O-O-S-O-O-
Those diagrams show the limits of chemical theory in his day.
Why do you think these two examples, both based on inorganic
ionic radicals, are the limits of chemistry in his day?
I'll retract my statement about "chemical theory in his day".
But Peirce's example -O-O-S-O-O- is a rather poor hypothesis
about the structure of the sulfate ion.
In contrast with the rather primitive chemistry, I believe
that many of Peirce's logical ideas are still at the forefront
of the latest research. For example, note slides 45 & 46 of
http://jfsowa.com/talks/egintro.pdfthis is SO wide an assertion
... can you extend your views of graphs to Tarsi’s meta-languages
and Lesniewski’s part-whole relations in terms of medads monads,
dyads,… ?
I was not claiming that Peirce had anticipated *everything* in
20th c logic, but I am claiming that his proof theory is still
at the forefront of current research.
I cited problem #24 from a book by a pioneer in computational
methods for theorem proving. See the reference below.
Are the words (medad monad, dyads, …) anything more than the
count of omissions of proper names from a sentence?
This gets into the issues about intensions and extensions.
If you just consider extensional issues (truth conditions and
logical equivalence), the answer is no: medad, monad, dyad...
are just words for the number of blanks or lambda variables.
But as Church said, two functions are equivalent by extension
iff they have the same values for all arguments. But two
functions that are equivalent by extension may have very
different rules for computing the values -- that would imply
that the are different by intension.
For example, there are many proof methods for FOL, and they
all determine exactly the same set of provable statements.
But some of them are more efficient than others (take fewer
steps). Others are more systematic (easier to find a proof).
And others are pedagogically better (easier to teach).
Should those issues be considered part of the meaning?
John
____________________________________________________________________
Wos, Larry (1988) _Automated Reasoning: 33 Basic Research Problems_,
Englewood Cliffs, NJ: Prentice Hall.
Problem #24
Is there a mapping between clause representation and natural-
deduction representation (and corresponding inference rules and
strategies) that causes reasoning programs based respectively on
the two approaches or paradigms to attack a given assignment in
an essentially identical fashion?
For an outline of the proof, see slides 42 to 46 of
http://jfsowa.com/talks/egintro.pdf
For the details, see Section 6 of
http://jfsowa.com/pubs/egtut.pdf
Nobody was able to solve this problem with any of the proof
procedures that had been developed during the past century.
But with Peirce's methods, it was a simple exercise.
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