Certainly model theory is a general theory of interpretation of
axiomatic set theory, but as such it is not committed to any
extra-systematic objects, only to the sets themselves as defined by the
axioms. Rather, model theory studies the mathematical structures by
examining first-order sentences true of those structures and the sets
definable in those structures by first-order formulas. So, model theory
is neither more nor less than the structure and sets that are definable
within an axiomatic theory.
In other words, we require extra-logical individuals to be extensional.
And since, according to Russell, and van Heijenoort, as I said in my
previous post, there are no individuals in the classical Boole-Schröder
calculus, that system would, again according JvH, be intensional rather
than extensional. Since we have a universe of discourse in Aristotle,
De Morgan, Boole, et al,m rather than THE UNIVERSE (i.e. the universal
domain), "individuals" in their logic are merely representatives of a
class that is given by definition, rather than an element of a set or a
collection of individuals with a specified property. To employ
Husserl's terminology (for example in his debates with Voigt) in order
to avoid our contemporary expectations regarding the meaning and
implications of "intensional" and "extensional", if that would help
clarify matters, JvH would have said, as Husserl did w.r.t. Schröder's
Algebra der Logik (again, had he employed the alternative terminology)
the logics of Aristotle, Boole, et al are *conceptual* [a
Begriffskalkul or Folgerungscalcul] rather than *contentual* [a
Inhaltslogik].
One final point. So far as I recall, I did not say that "model theory
necessarily intensional"; in any case, I know that I did not say
*necessarily*, although it can (and should) be inferred that JvH would,
as I noted, consider the classical Boole-Schroder logic to be (again in
Husserlian terms, if you prefer, a Folgerungslogik, or intensional,
rather than a an Inhaltslogik, or extensional.
Irving
----- Message from michael...@comcast.net ---------
Date: Wed, 7 Dec 2011 10:39:51 -0500
From: "Michael J. DeLaurentis" <michael...@comcast.net>
Reply-To: "Michael J. DeLaurentis" <michael...@comcast.net>
Subject: RE: [peirce-l] "On the Paradigm of Experience Appropriate for
Semiotic"
To: 'Jon Awbrey' <jawb...@att.net>, PEIRCE-L@LISTSERV.IUPUI.EDU
In what sense is model theory necessarily intensional? In standard modern
usage, a model simply extensionally assigns interpretations [individuals and
sets, sometimes ordered] to categories of symbols in the object language.
Where's the intensionality? [Leave aside for the moment modal/opaque
contexts.]
-----Original Message-----
From: C S Peirce discussion list [mailto:PEIRCE-L@LISTSERV.IUPUI.EDU] On
Behalf Of Jon Awbrey
Sent: Wednesday, December 07, 2011 9:40 AM
To: PEIRCE-L@LISTSERV.IUPUI.EDU
Subject: Re: [peirce-l] "On the Paradigm of Experience Appropriate for
Semiotic"
* Comments on the Peirce List slow reading of Joseph Ransdell,
"On the Paradigm of Experience Appropriate for Semiotic",
http://www.cspeirce.com/menu/library/aboutcsp/ransdell/paradigm.htm
IA: Once again, there is a complex of related dichotomies that van
Heijenoort
applied to distinguish the Aristotelian-Boolean stream (of which Peirce
was
a part, according to Van) from the Fregean, including logic as
calculus/logic
as language, model-theoretic (or intensional)/set-theoretic, (or
extensional,
so as to include both Russell's use of set theory and Frege's
course-of-values
semantic), syntactic/semantic, and, finally, relativism/absolutism.
This has been coming pretty thick and fast, so let me see if I can sift it
out.
Aristotelian-Boolean . | Fregean
logic as calculus .... | logic as language model-theoretic ...... |
set-theoretic intensional .......... | extensional syntactic ............ |
semantic relative ............. | absolute
Did you intend to align things that way?
Or did you intend them as coordinate axes?
Jon
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Irving H. Anellis
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Peirce Edition, Institute for American Thought
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USA
URL: http://www.irvinganellis.info
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