Helmut, Jeffrey, Jon A, Clark, list,
HR
Not every triadic relation is categorically thirdness. But which are?
That's a good question. Some basic principles:
1. For each of Peirce's categories, there is a characteristic question:
Firstness: What is it? What kind of mark?
Secondness: How does it relate or react to something?
Thirdness: Why does it relate or react to something?
Answer: (1) quality; (2) relation or reaction; (3) mediation.
2. Peirce used the term 'triad', not 'triadic relation'. You can
represent a verb, such as 'give' as either a triadic relation
or as a nominalized node attached to three dyadic relations.
See the attached give.gif, which shows two ways of mapping the
sentence "Sue gives a child a book" to a graph -- a conceptual
graph, which may be mapped to an existential graphs or to an
algebraic formula.
3. As give.gif illustrates, a triad in a graph cannot be eliminated
by using dyadic relations. But a graph with a tetrad, pentad, ...
can always be mapped to and from a graph that has no nodes with
more than 3 links.
4. A non-degenerate triad always involves something law-like,
intentional, or mental. That implies some mind or quasi-mind
that does the interpretation. Nominalists don't like that way
of talking.
5. Strict nominalists, such as Carnap, deny that abstract entities
exist -- that includes all forms of Thirdness: laws, intentions,
goals, purposes, or habits. That is why Carnap insisted that all
laws of science are nothing more than summaries of observations.
HR
Is it reasonable to say that a relation has an intension and an
extension, the intension is firstness, and the extension secondness
(of the relation, which is secondness)?
As Church pointed out, the intension is a rule of correspondence
that determines the extension. That rule is a law-like entity,
which is a kind of Thirdness. Note, by the way, that multiple,
independently defined rules may specify the same extension.
Two functions or relations may differ in intension , but be
identical in extension.
JBD
why [do] nominalists such as J.S. Mill and Nelson Goodman strongly
prefer extensional systems--and have significant reservations about
using intensional systems in philosophy
All definitions by intension imply some law-like rule. That's why
Carnap insisted that laws of science are "summaries of observations".
He would never say "law of nature" -- because (a) the phrase implies
that there exists something called nature, and (b) it also implies
that nature has something called laws.
JA
an extensional definition of a 2-place relation ... can be generalized
to k-place relations and then beyond the finite arity case... But
there is nothing remotely nominal going on here, as the definition
invokes sets of tuples. Sets and tuples are the very sorts of
abstract objects nominal thinkers would eschew
Church used sets to define extensions because he had no reason to
avoid sets. But Lesniewski and other nominalists replaced set
theory with mereology. If you have a set of N elements, you have
N+1 entities, which consist of the N elements plus an abstract set.
But with mereology, a collection of N elements does not imply
the existence of anything more than the N elements. Therefore,
a mereologist could say that the elements of an N-tuple are parts
of a whole called an N-tuple. They would not consider the whole
as something distinct from the sum of its parts.
JA
I have never found going on about Firstness Secondness Thirdness
all that useful in any practical situation.
If you think at those terms as a count of links in a graph, they
don't explain much. It's better to look at the three kinds of
questions summarized at the top of this note.
JA
until you venture to say exactly *which* monadic, dyadic, or triadic
predicate you have in mind, you haven't really said that much at all.
CG
Glad I’m not alone in thinking that.
I agree that we need to look at specific cases. For examples of the
kinds of Thirdness that nominalists deliberately ignore, see Section 2
(pp. 3 to 8) of http://www.jfsowa.com/pubs/signproc.pdf .
The title of that section is "A Static, Lifeless, Purposeless World".
Some excerpts:
p. 4: Einstein criticized Russell's "Angst vor der Metaphysik"
and said “Mach was a good experimental physicist but a miserable
philosopher”; he made “a catalog not a system.”
p. 5: When introducing Russell for his William James Lectures
at Harvard, Whitehead said “This is my friend Bertrand Russell.
Bertie thinks that I am muddleheaded, but then I think that he
is simpleminded” (Lucas 1989:111). That remark is consistent with
a statement attributed to Russell: “I’d rather be narrow minded
than vague and wooly” (Kuntz 1984:50).
p. 5: While reviewing Quine’s Word and Object, [Rescher] was struck
by the absence of any discussion of events, processes, actions, and
change. After reviewing the literature, Rescher (1962) realized that
Quine’s static views were endemic in the tradition: “The ontological
doctrine whose too readily granted credentials I propose to revoke
consists of several connected tenets, the first fundamental, the rest
derivative..."
pp. 7-8: In his _Logische Aufbau_, Carnap noted that the "construction"
(Aufbau) of the sign relation from physical objects “is more difficult
than any of the other relations which we have hitherto undertaken.”
The remainder of signproc.pdf has more examples that show how Peirce's
categories could support a broader foundation.
John
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