Thanks for this elaboration, Irving. I wasn't clear whether JvH's use of "intensional(ity)" was tantamount to what we today would commonly characterize as "semantics." If I understand, in a sense, it does, with the twist that "extensional" calculi, with their single, fixed, universe of discourse and referents, have a built-in, presupposed, if not explicitly stated semantics, while "intensional" calculi require interpretation and allow, potentially anyway, for a range of interpretations [enter Skolem - Lowenheim].
It struck me that, for JvH, "intensional" calculi seemed to require/allow a range of interpretations and "extensional" calculi to have their semantics built into the object language (perhaps unstated, by presupposition of a single, fixed universe of discourse and referents). -----Original Message----- From: C S Peirce discussion list [mailto:PEIRCE-L@LISTSERV.IUPUI.EDU] On Behalf Of Irving Sent: Wednesday, December 07, 2011 6:24 PM To: PEIRCE-L@LISTSERV.IUPUI.EDU Subject: Re: [peirce-l] "On the Paradigm of Experience Appropriate for Semiotic" 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 > > CC: Arisbe, Inquiry, Peirce List > > -- > > facebook page: https://www.facebook.com/JonnyCache > inquiry list: http://stderr.org/pipermail/inquiry/ > mwb: http://www.mywikibiz.com/Directory:Jon_Awbrey > knol profile: http://knol.google.com/k/Jon-Awbrey# > oeiswiki: http://www.oeis.org/wiki/User:Jon_Awbrey > polmic: www.policymic.com/profiles/1110/Jon-Awbrey > > ---------------------------------------------------------------------------- > ----- > You are receiving this message because you are subscribed to the PEIRCE-L > listserv. To remove yourself from this list, send a message to > lists...@listserv.iupui.edu with the line "SIGNOFF PEIRCE-L" in the body of > the message. To post a message to the list, send it to > PEIRCE-L@LISTSERV.IUPUI.EDU > ----- > No virus found in this message. > Checked by AVG - www.avg.com > Version: 2012.0.1873 / Virus Database: 2102/4665 - Release Date: 12/07/11 > > ---------------------------------------------------------------------------- ----- > You are receiving this message because you are subscribed to the > PEIRCE-L listserv. To remove yourself from this list, send a message > to lists...@listserv.iupui.edu with the line "SIGNOFF PEIRCE-L" in > the body of the message. To post a message to the list, send it to > PEIRCE-L@LISTSERV.IUPUI.EDU > ----- End message from michael...@comcast.net ----- Irving H. Anellis Visiting Research Associate Peirce Edition, Institute for American Thought 902 W. New York St. Indiana University-Purdue University at Indianapolis Indianapolis, IN 46202-5159 USA URL: http://www.irvinganellis.info ---------------------------------------------------------------------------- ----- You are receiving this message because you are subscribed to the PEIRCE-L listserv. To remove yourself from this list, send a message to lists...@listserv.iupui.edu with the line "SIGNOFF PEIRCE-L" in the body of the message. To post a message to the list, send it to PEIRCE-L@LISTSERV.IUPUI.EDU ----- No virus found in this message. Checked by AVG - www.avg.com Version: 2012.0.1873 / Virus Database: 2102/4665 - Release Date: 12/07/11 --------------------------------------------------------------------------------- You are receiving this message because you are subscribed to the PEIRCE-L listserv. To remove yourself from this list, send a message to lists...@listserv.iupui.edu with the line "SIGNOFF PEIRCE-L" in the body of the message. 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