Gary R and Jon,
I agree that Jon has done a lot of good work in analyzing
Lane's book and digging through Peirce's MSS to relate RL
and CSP. I appreciate his work, and I thank him for it.
But the way Jon is using Peirce's logic is not the way Peirce
defined it. What I'm about to say may sound like nitpicking,
but Peirce was the supreme artist of nitpicking. He insisted
that his logic be used with absolute precision. Since Peirce
is no longer with us, somebody has to pick the nits.
We should also respect Peirce's ethics of terminology. For a
21st c. audience, his ethics would require us to use whatever
terminology has become standard in the 21st c. When I pick nits
or suggest changes in terminology, I have several goals:
1. Be constructive: help clarify the discussions in a way
that is as accessible as possible for a 21st c. reader.
2. Be faithful to Peirce: prefer verbatim selections from
his writings and state precise equivalences for any of his
terminology that should be updated for a modern audience.
3. For teaching students and to clarify ambiguities for anybody,
every logical point should be stated in two ways: (1) EGs
and (2) Peirce-Peano algebra.
Point #3 is required by Peirce's ethics of terminology. I agree
with Peirce that EGs are "the logic of the future". But a good
teacher always begins with something familiar to the students
before leading them to any recommended improvements. And the
overwhelming majority of people who have taken any course in
logic have been taught Peirce-Peano algebra, but not EGs.
I won't go through the details of Jon's note of July 13 (most
of which I agree with or sympathize with). But I believe it's
necessary to pick the nits of the following quotation:
CSP: Truth belongs exclusively to propositions. A proposition has
a subject (or set of subjects) and a predicate. The subject is a
sign; the predicate is a sign; and the proposition is a sign that
the predicate is a sign of that of which the subject is a sign.
If it be so, it is true. (CP 5.553, EP 2:379; 1906)
JAS: As helpfully diagrammed by Existential Graphs, every assertion
is the attribution of general concepts (Spots) by means of
continuous predicates (Pegs) to indefinite individuals (Lines
of Identity).
First point: All modern logicians and their students would agree
with Peirce's first sentence above. But they would say that the
second sentence is a special case of an atom, and they would add
that a general proposition links one or more atoms with Boolean
operators (AND, OR, NOT, IF...) and uses proper names or quantified
variables to represent the subjects.
If a 21st c. logician said that to Peirce, he would instantly
agree and say that the modern term 'atom' represents what he
called an indivisible EG. He would then show how his EGs map
to and from modern notations (predicate calculus and the many
variations implemented in digital computers).
Next point: Peirce would not agree with Jon's sentence above.
His first objection would be that 'assertion' is not a synonym
for 'proposition'. A proposition is an ens rationis, but an
assertion is a "speech act" (to use Austin's term) that adds
a claim that the proposition happens to be true.
For Peirce, a proposition is a type in the trichotomy of
mark/token/type. The same type may be expressed in an open-
ended variety of tokens in different versions of natural languages
and formal logics. And different tokens may be expressed in an
open-ended variety of spoken, written, printed, or signed ways.
And any token may be used in an open-ended variety of speech
acts: asserting, judging, mentioning, discussing, claiming,
doubting, hoping, fearing, wishing, lying, persuading...
Another extremely important point is that any proposition p
expressed in Peirce's algebra of 1885 may be mapped to and
from an EG that expresses exactly the same proposition p.
The converse is not true because Peirce's Gamma graphs can
express propositions that cannot be expressed in his 1885
notation. However, there are modern extensions to predicate
calculus that can express Gamma graphs. For some discussion
of the issues, see http://jfsowa.com/pubs/5qelogic.pdf
The equivalence in expressive power of EGs and the algebraic
forms implies that syntactic differences have no semantic
meaning whatsoever. Therefore, any syntactic feature that is
lost in translation in either direction is *meaningless*.
Re spot, peg, and line: The words 'spot' and 'line' were
Cayley's choices for naming the two kinds of parts in a graph.
But graph theory was invented and reinvented many times by
mathematicians working in different branches. Euler was the
first with his solution to the problem of the Seven Bridges
of Königsberg. His words were 'island' and 'bridge'. Other
mathematicians used different pairs: vertex/edge, node/arc,
node/link, point/line.
Nobody today uses the word 'spot'. And Peirce himself rarely
drew an actual spot in his examples of EGs. He just attached
lines of identity to the words that named his predicates. I
believe that's the reason why he avoided the word 'spot' in
his 1911 intro to EGs. See http://jfsowa.com/peirce/eg1911.pdf
Re peg: The number of pegs of a predicate corresponds to the
number of blanks used in defining the predicate by blanking out
names in a sentence that states a proposition. In the algebra,
a predicate is represented by a name such as Q followed by a list
of names or expressions. A triadic predicate, for example,
could be represented in the algebra as Q(x1,x2,x3).
As Peirce said in eg1911.pdf (NEM 3:164) "Individual graphs usually
carry 'pegs', which are positions on their periphery appropriated
to denote, each one of them, one of the subjects of the graph."
In short, a peg is a *position* on the periphery of an EG predicate
that maps to a *position* in the list of the algebraic predicate.
The phrase "appropriated to denote" is unclear, but the translation
to the algebraic form shows exactly what Peirce intended. He never
said nor implied that a peg is a continuous predicate.
Re "indefinite individuals (Lines of Identity)": A line of identity
maps to an existentially quantified variable in the algebra. The EG
cat—on—mat maps to (∃x)(∃y)(cat(x) & mat(y) & on(x,y)). The English
sentence "A cat is on a mat", the existential graph, and the algebra
are indefinite in exactly the same way: They just say that there is
an indefinite cat and an indefinite mat in a relationship named 'on.
JAS
[Incidentally, Lane observes in chapter 6 that Peirce's "idea of
vagueness is quite different from the contemporary one" (p. 139);
accordingly, I wonder if it would be more perspicuous to employ
the term indefinite rather than vague when referring to Peirce's idea.]
No. The sentence "A cat is on a mat" is indefinite about the
referents of the two subjects, but neither Peirce nor any modern
logician would say that it's vague.
Peirce wrote "The vague might be defined as that to which the
principle of contradiction does not apply" (CP 5.505, see below).
The sentence "A cat is on a mat" is not vague because it must
be either true or false. It can't be both or neither.
For more examples and discussion, see "What is the source of
fuzziness?": http://jfsowa.com/pubs/fuzzy.pdf
John
_____________________________________________________________________
Peirce (CP 5.505)
Logicians have too much neglected the study of vagueness, not
suspecting the important part it plays in mathematical thought.
It is the antithetical analogue of generality. A sign is objectively
general, in so far as, leaving its effective interpretation
indeterminate, it surrenders to the interpreter the right of
completing the determination for himself. "Man is mortal." "What
man?" "Any man you like." A sign is objectively vague, in so far
as, leaving its interpretation more or less indeterminate, it
reserves for some other possible sign or experience the function
of completing the determination. "This month," says the almanac-
oracle, "a great event is to happen." "What event?" "Oh, we shall
see. The almanac doesn't tell that." The general might be defined
as that to which the principle of excluded middle does not apply.
A triangle in general is not isosceles nor equilateral; nor is a
triangle in general scalene. The vague might be defined as that to
which the principle of contradiction does not apply. For it is
false neither that an animal (in a vague sense) is male, nor that
an animal is female.
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