Gary F, Mary L, Kirsti, Jerry LRC, and list,
In 1911, Peirce presented his clearest and simplest version of EGs.
He explained the essentials in just 8 pages of NEM (3:162 to 169).
I believe that it is his final preferred version, and I'll use it
for explaining issues about the more complex 1903 version.
Gary
[Mary's] question about the “blot” has me thinking again about
“the two peculiar graphs” which are “the blank place which asserts
only what is already well-understood between us to be true, and
the blot which asserts something well understood to be false”
Kirsti,
instead of warning against confusing SPOT, DOT and BLOT, it would
have been most interesting to hear how they are related.
In his 1911 terminology, Peirce did not use the words 'spot', 'dot',
or 'blot'. Instead, a spot is just a very short line of identity.
The line represents an existential quantifier, and there is no
reason to distinguish long lines from short lines (spots).
He used the word 'peg' instead of 'dot'. Each relation has zero
or more pegs, to which lines of identity may be attached.
He also shaded negative areas (nested in an odd number of negations)
and left positive areas unshaded (nested in an even number, zero or
more, negations). A blot is just a shaded area that contains
nothing but a blank.
Gary
[The blank place and the blot] are peculiar in several ways,
and each is in some sense the opposite of the other.
Each is the negation of the other. The blank place is unshaded,
and the blot is a shaded blank.
Gary
For instance, the blank cannot be erased, but any graph can be
added to it on the sheet of assertion; while the blot can be
erased, but nothing can be added to it, because it “fills up
its area.”
One reason why the "the blank place" is "peculiar" is that Peirce
had talked about it in two different ways. He called the sheet
of assertion the universe of discourse when it contains all the
EGs that Graphist and Grapheus agree is true.
But the blank, by itself, is true before anything is asserted.
In modern terminology, the blank is Peirce's only axiom. Any EG
that can be proved without any other assumptions is a theorem.
In 1911, Peirce clarified that issues by using two distinct terms:
'the universe' and 'a sheet of paper'. The sheet is no longer
identified with the universe, and there is no reason why one
couldn't or shouldn't shade a blank area of a sheet.
Gary, quoting Peirce
[A blot] "fills up its area."
In 1911, Peirce no longer used this metaphor. With the rules
of 1903 or 1911, a blot or a shaded blank implies every graph.
To prove that any graph g can be proved from it:
1. Start with a sheet of paper that contains a shaded blank.
2. By the rule of insertion in a shaded area, insert the graph
for not-g inside the shaded area. All the shaded areas of not-g
then become unshaded, and the unshaded areas become shaded.
3. The resulting graph consists of g in an unshaded area that is
surrounded by a shaded ring that represents a double negation.
4. Finally, erase the double negation to derive g.
Another important point: In 1911, Peirce allowed any word, not
just verbs, to be the name of a relation. From NEM, page 3.162:
Every word makes an assertion. Thus ——man means "There is a man"
in whatever universe the whole sheet refers to.
The dash before "man" is the "line of identity."
This EG is Peirce's first example in 1911. And note that he begins
with a Beta graph. In fact, he does not even mention the distinction
between Alpha and Beta. The same rules of inference apply to both.
For Peirce's version of 1911 with my commentary, see
http://jfsowa.com/peirce/ms514.htm
Jerry,
CSP’s genius [etc.] make it difficult for anyone to project his
thoughts into rarefied logical, mathematical, scientific or
philosophical atmospheres.
Yes. He wrote volumes of insights that we still need to explore.
But you can't put words in his mouth. If you can't find where he
stated something explicitly, you can't claim him as the source.
Note my discussion above. Every one of my claims is based on
something that Peirce explicitly wrote.
John
