Gary F, List,
As you listen, perhaps, to the Dark Side of the Moon, I recommend the following
instance where Peirce reflects on his own blunder concerning the "blot". He
says:
I can make this blackened Inner Close as small as I please, at least, so long
as I can still see it there, whether with my outer eye or in my mind's eye. Can
I not make it quite invisibly small, even to my mind's eye? "No," you will say,
"for then it would not be scribed at all." You are right. Yet since confession
will be good for my soul, and since it will be well for you to learn how like
walking on smooth ice this business of reasoning about logic is -- so much so
that I have often remarked that nobody commits what is called a "logical
fallacy," or hardly ever does so, except logicians; and they are slumping into
such stuff continually -- it is my duty to [point out] this error of assuming
that, because the blackened Inner Close can be made indefinitely small,
therefore it can be struck out entirely like an infinitesimal. That led me to
say that a Cut around a Graph instance has the effect of denying it. I retract:
it only does so if the Cut enclosed also [has] a blot, however small, to
represent iconically, the blackened Inner Close. I was partly misled by the
fact that in the Conditional de inesse the Cut may be considered as denying the
contents of its Area. That is true, so long as the entire Scroll is on the
Place. But that does not prove that a single Cut, without an Inner Close, has
this effect. On the contrary, a single Cut, enclosing only A and a blank,
merely says: "If A," or "If A, then" and there stops. If what? you ask. It does
not say. "Then something follows," perhaps; but there is no assertion at all.
This can be proved, too. For if we scribe on the Phemic Sheet the Graph
expressing "If A is true, Something is true," we shall have a Scroll with A
alone in the Outer Close, and with nothing but a Blank in the Inner Close. Now
this Blank is an Iterate of the Blank-instance that is always present on the
Phemic Sheet; and this may, according to the rule, be deiterated by removing
the Blank in the inner close. This will do, what the blot would not; namely, it
will cause the collapse of the Inner Close, and thus leaves A in a single cut.
We thus see that a Graph, A, enclosed in a single Cut that contains nothing
else but a Blank has no signification that is not implied in the proposition,
"If A is true, Something is true." When I was in the twenties and had not yet
come to the full consciousness of my own gigantic powers of logical blundering,
with what scorn I used to think of Hegel's confusion of Being with Blank
Nothing, simply because it had the form of a predicate without its matter! Yet
here am I after devoting a greater number of years to the study of exact logic
than the probable number of hours that Hegel ever gave to this subject,
repeating that very identical fallacy! Be sure, Reader, that I would have
concealed the mistake from you (for vanity's sake, if for no better reason), if
it had not been "up to" me, in a way I could not evade, to expose it.
-- From "Copy T," c. 1906; one of a number of fragmentary manuscripts designed
to follow the present article.
For my part, I tend to think Peirce uses the term "blot" in an ordinary way to
refer to an instance of a relatively large spot of, say, ink on a page. Blots,
like spots on a page, are instances or replicas. Dots, as instanced by a
heavily drawn spot, on the other hand, are taken be logical symbols that are
interpreted according to a general rule (see, for example, Convention 4 of the
Beta Graphs). So, we have dots, spots and blots. How might we interpret the
philosophical meaning of a blot that fills part of a scroll in the gamma part
of the EG where there is a recto and verso side of the page? Might it be
analogous in some ways to a side of a moon on which the rays of the sun do not
directly shine?
That general idea appears to be something that Peirce explored when he
suggested that we can imagine projective rays emanating from the starting
points of inquiry, illuminating the recto side of each page in a larger book of
assertions expressed alternately in the mood of interrogatives, optatives,
imperatives and indicatives. If inquiry is honest and is governed by the goal
of seeking adequate explanations of what really is the case, then we can
imagine those rays converging as they pass through a book with an unending
multitude of pages at an ideal of truth at the end of such inquiry. As such, we
need not lose heart in the face of blunders--big or small. (see CP 6.581-7)
Yours,
--Jeff
Jeffrey Downard
Associate Professor
Department of Philosophy
Northern Arizona University
(o) 928 523-8354
________________________________
From: g...@gnusystems.ca <g...@gnusystems.ca>
Sent: Sunday, April 14, 2019 9:21 AM
To: peirce-l@list.iupui.edu
Subject: RE: [PEIRCE-L] Peirce admitted that his terminology of 1906 was bad.
John, Jon,
Much as I hate to admit it, I’ve just been on a wild goose chase trying to
check out Peirce’s cryptic remarks about his choice of “blot” in L 376. First I
managed to find images of the manuscript online (which was hard enough), just
to make sure that the transliteration from Greek was correct; and sure enough,
he wrote (at 2:15 on December 8, 1911) that he chose blot “because it is
cognate with Greek ΦΛΟΙΔ … ” — which I’m guessing is in all capitals because it
is an archaic form postulated by Greek etymologies but not occurring in modern
or even classical Greek. As to what it means, or why Peirce considered it
cognate with “blot,” I failed to find any information in L&S online or anywhere
else, including the OED. I also looked up φλἀω and found the same list of
meanings that you gave, John, which is no help at all. We would have to locate
the edition of L&S that Peirce used and find the entry on p. 1682 to solve this
mystery.
By the way, Peirce’s earlier use of “blot” (in the pseudographic sense) was not
from 1906 but from his 1903 Lowell lectures (which he said in L 376 were better
than the 1906 presentation).
About computational efficiency, yes, I know that EGs can be adapted for that
purpose, but I was referring to what Peirce wrote about EGs, which totally
ignores their adaptability to computation. (And what he wrote about “logical
machines” does not even mention EGs.)
Now I think I’ll just forget about this whole subject and put on a ΠΙΝΚ ΦΛΟΙΔ
CD.
Gary f.
-----Original Message-----
From: John F Sowa <s...@bestweb.net>
Sent: 14-Apr-19 09:15
To: peirce-l@list.iupui.edu
Subject: Re: [PEIRCE-L] Peirce admitted that his terminology of 1906 was bad.
Gary F, Jon AS,
I know what Peirce wrote about efficiency. But I also know that he was the
author of "Logical Machines" (1887) and that he urged Oscar Mitchell to
consider electrical circuits as a more efficient basis for designing logical
machines. For those reasons, Peirce has been called a pioneer in artificial
intelligence.
GF
> Computational efficiency is certainly /not/ an issue as far as
> Peirce’s design of EGs in concerned. Every time he gave an explanation
> of EGs, either in a lecture or in print, from 1898 to 1911, he
> /always/ began by saying that they were /not/ designed or intended to
> facilitate reasoning or computation, but rather to /analyze/ the
> reasoning process into /as many steps as possible/, usually
> contrasting his purposes as a logician with those of the
> mathematician, who does want to make his computations and his proofs
> as efficiently as possible.
But those comments by Peirce sound like an apology in response to some
criticisms. From today's perspective, he had no reason to apologize:
1. He was engaged in a decades-long debate with his father Benjamin,
who took pride in short and elegant proofs. The computation for
mathematical proofs could become lengthy, and Charles was very
good in doing those calculations. But he knew that each step in
mathematics would take many more steps according to his EG rules.
2. The most common logical proof procedures in his day were based
on Aristotle's rules for syllogisms. As he showed in NEM 3:168,
a one-step proof by a syllogism took six steps by the EG rules.
3. In addition to the debates with his father, he undoubtedly
met a lot of skepticism from people who had been trained in
the Aristotelian tradition and couldn't see the point of
using the many more steps required by EG rules.
4. He had not read the _Principia_ by Whitehead & Russell (1910),
whose proofs were much longer and *uglier* that the shorter,
more elegant proofs by Peirce's rules. If he had seen their
proofs and had the time to demonstrate his own methods, he
would have convinced the world of their computational power.
5. Finally, he did not know the modern notion of "derived rules
of inference". He used such rules in mathematical calculation,
but he did not realize that he could claim those complex rules
as derived rules in terms of his more basic rules of inference.
For evidence, see http://jfsowa.com/peirce/ms514.htm and search for "first
axiom". That leads to a proof of Frege's first axiom.
In Peirce-Peano notation, it's a ⊃ (b ⊃ a).
Frege assumed this axiom without proof. Note that the axiom has five symbols
other than parentheses: a, ⊃, b, ⊃, a. With Peirce's rules, the proof of this
axiom takes five steps, each of which inserts one symbol in its proper place in
the corresponding EG.
Peirce's only axiom is a blank sheet of paper. Every axiom and rule of
inference that Frege, Whitehead, and Russell assumed without proof can be
proved by Peirce's rules from a blank sheet.
Now search for "theorema". That leads to a statement of Leibniz's Praeclarum
Theorema (Splendid Theorem). In the Principia, W & R started with five axioms
and took 43 steps to prove that theorem.
But starting with Peirce's rules and a blank sheet of paper, only seven steps
are needed to prove the same theorem.
Computationally, Peirce's rules are far more efficient than Frege's rules or
Whitehead and Russell's rules. For more evidence, see the following slides:
http://jfsowa.com/talks/ppe.pdf .
Skip to slide 63 of ppe.pdf. That slide states an unsolved research problem
proposed by Larry Wos, one of the pioneers in methods of automated theorem
proving. But the solution to that problem is a corollary of two theorems about
EGs, which are stated on slide 59: the reversibility theorem and the
cut-and-paste theorem. See the reference on that slide.
In short, Peirce's rules of inference and the derived rules provable from them
are (a) psychologically realistic and (b) computationally efficient. That's
quite an achievement.
GF
> On Peirce’s choice of the term “blot”, it strikes me as bizarre to
> suggest that he mistakenly used that word instead of ‘spot’
> in L 376...
JAS
> his ethics of terminology likely constrained him to stick with
> [Clifford's terms 'spot' and 'line' for graphs], unless he had a very
> good reason to deviate from them.
I agree that Peirce would not change Clifford's terms for parts of a graph.
(Today, the words 'node' and 'arc' are commonly used, and nobody uses the words
'spot' and 'line'.)
But that passage raises some thorny issues. (See below for the only paragraph
from L 376 that mentions blots.)
1. His earlier uses of the word 'blot' were synonymous with
'pseudograph'. Why did he change the definition of the word
'blot'? Did he forget his earlier definition? Or did he decide
that the terminology of 1906 was not worth preserving? At the
beginning of L 376. he wrote that the description of 1906 "was,
on the whole, as bad as it well could be".
2. Re flao: My wife Cora has a PhD in classical philology. So I
checked her copies of Liddell & Scott. For flao, the big version
has lots of citations, but the short version has a convenient list
of senses: to crush, pound, bruise with the teeth, eat up, eat
greedily. The short version doesn't have anything for floid, but
the big version has floideo, to seethe. Can anyone guess what
Peirce intended?
3. The last sentence of that paragraph has the only examples of blots:
"The simplest blots, which are not relative, are such as `it snows',
`it thunders', etc." Those examples are medads. But since Peirce
wrote that the simplest blots are not relative, that indicates that
other blots are relative -- i.e., they are monads, dyads, triads...
Do blots differ from other predicates? If so, how? If not, what
purpose would blots serve that other predicates don't?
After the paragraph below, L 376 has two more lengthy paragraphs that move to
other topics. Then it ends abruptly in the middle of a sentence. It's
possible that some undigitized MSS may contain more information about these
blots. Or maybe not. Unless and until we get more data, there is nothing more
to say.
John
_______________________________________________________________________
The Phemic Sheet. Since the sole purpose of the Syntax I am describing is to
facilitate the anatomy, and thereby the physiology of deductive reasonings, the
reader will have anticipated the fact that no occasion has been found for
supplying it with any means of expressing mere feelings or complexes of mere
feelings, such as abound in the arts of music and of painting. Nor has any
need been found for furnishing it with means of expressing commands, not even
such as take the softened forms of requests and inquiries. We need only a mode
of indicating that what is "scribed", i.e. is marked, whether by coveting or
drawing or by a mixture of these two arts, is meant, and is not scribed for
some other purpose, as, for example, to show how it might be asserted.
Moreover, however minutely we may analyze our assertions, there will never be
the slightest need of any such fragment of meaning as that of a noun or that of
an English verb. The simplest part of speech which this syntax contemplates,
which, as scribed, I shall term a blot (a vocable I choose because it is
cognate with Greek [Gr.] floid, for which see L.& S. p.
1692, under [Gr.] flao, where many words containing this root are given; and
there are many others in all European languages] is itself an assertion. Ought
it to be an affirmation or a denial? A denial is logically the simpler,
because it implies merely that the utterer recognizes, however vaguely, some
discrepancy between the fact and the speech, while an affirmation implies that
he has examined all the implications of the latter and finds no discrepancy
with the fact. This is a circumstance to be borne in mind; but since the
denial implies recognition of the affirmation, while the affirmation is so far
from implying recognition of the denial, that one might imagine a paradisaic
state of innocence in which men never had the idea of falsity, and yet might
reason, we must admit that affirmation is psychically the simpler.
Now I think that upon this point we must prefer psychical to logical
simplicity. I therefore make the blot an affirmation. The subject of it must
for simplicity be completely indefinite. The simplest blots, which are not
relative, are such as `it snows', `it thunders', etc.
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