From: mutation@dorsai.com (Florian Lengyel)
Subject: Re: Extensionalism (was re: Russell's Paradox)
Organization: The Dorsai Embassy +1.718.729.5018
X-Newsreader: Tin 1.1 PL4
Date: Wed, 18 Nov 1992 07:27:09 GMT

holmes@garnet.idbsu.edu (Randall Holmes) writes:
: In article <1992Nov14.012610.8929@tamsun.tamu.edu> cmenzel@tamsun.tamu.edu (Chris Menzel) writes:
: >In article <1992Nov3.092753.19804@dorsai.com> mutation@dorsai.com (Florian Lengyel) writes:
: >>>>In article <1992Oct29.212223.20343@tamsun.tamu.edu>
: >>>>cmenzel@tamsun.tamu.edu (Chris Menzel) writes:
: >>>
: >>>>>In article <1992Oct26.170430.14372@guinness.idbsu.edu>
: >>>>>holmes@garnet.idbsu.edu (Randall Holmes) writes:
: >>>
: >>>RH:
: >>>>>>_Contra_ Harter (miscellaneous quibbles):
: >>>>>>
: >>>>>>The Good Russell may have admitted the possibility of "intensions",
: >>>>>>but the Exalted Quine has liberated us from the influence of those
: >>>>>>"creatures of darkness".  Extensions are adequate for all admissable
: >>>>>>purposes.
: >>>
: >>>CM:
: >>>>>Really, Randall?!  Could you then please give me an extensional
: >>>>>analysis of the sentences in the following arguments that explains why
: >>>>>the first is valid but the second is not:
: >>>>>
: >>>>>1)  John believes there are spies.
: >>>>>    Mary believes whatever John does.
: >>>>>    Therefore, Mary believes there are spies.
: >>>>>
: >>>>>2)  John believes that 2+3=5.
: >>>>>    Necessarily, 2+3=5 iff first-order logic is undecidable.
: >>>>>    Therefore, John believes that first-order logic is undecidable.
: >>>>>
: >>>>>Or is a purpose admissible just in case it is amenable to
: >>>>>extensionalist analysis?
: >>
: >>As for Chris Menzel's second argument, it is not clear to me
: >>how to construe belief, that is, how one ought to parse "John believes
: >>that 2+3=5" in order to compute valid inferences from it. There is more
: >>going on here than the substitution of equal strings, but I would
: >>consider the invalidity of the string substitution to be a failure
: >>to parse "John believes that 2+3=5" , rather than an indication that 
: >>the invalidity of the argument cannot be explained via extensional analysis.
: >
: >Sorry not to have been able to respond sooner.  My point was not to
: >assert that the argument's invalidity *cannot* be explained
: >extensionally; it was a challenge to Randall and other extensionalists
: >simply to provide such a framework that would enable us to represent
: >the logical form of the arguments in such a way as to provide cogent
: >analyses of their validity or invalidity, as the case may be.  Their
: >inability to provide such a framework suggests the inadequacy of
: >extensionalism as a general philosophical ontology.
: >
A philosopher of my aquaintance once told me that the closest approximation
to an understanding of intentionality that we have so far is provided by
Kripke frame semantics: the intension of a sentence is its extension in
all possible worlds.

Although I have given the subject of Kripke semantics only fleeting
consideration (I am preoccupied with other part of mathematics, but
I will rectify this situation sooner rather than later if I can be
persuaded) this definition would appear to give an extensional
account of the intensions of sentences.

CM
: >For friends of intensionality such a framework is ready to hand
: >(though not to say without puzzles of its own).  That-clauses are
: >grammatical singular terms that denote propositions, which possess an
: >internal structure akin to the grammatical structure of the
: >that-clauses that denote them.  Sentences of the form "s believes
: >that-A" are simple two place predications involving an object and a
: >proposition.  Thus, the first argument above is of the form
: >
: >  Bel(j,p)
: >  (x)(Bel(j,x) --> Bel(m,x))
: >  ---
: >  Bel(m,p)
: >
: >which is a simple, logically valid argument in first-order logic.  The
: >second argument needs a little more apparatus to represent properly.
: >For any formula A, let [A] be a singular term that denotes the
: >proposition expressed by A.  (There are several intensional logics
: >around that formalize these ideas.)  Given this, the second argument
: >is of the form
: >
: >  Bel(j,[A])
: >  [](A <--> B)
: >  ---
: >  Bel(j,[B]
: >  
: >Here the intuitive invalidity of the argument indicates that the logic
: >of intensional abstracts like [A] does not permit general substitution
: >of equivalent formulas, since in general these will express different
: >propositions.  Granted, this doesn't really *explain* the invalidity
: >of the argument in the way the above analysis explains the validity of
: >the first argument, but it does provide us with a clear framework for
: >representing the form of the argument and encoding the logic of the
: >terms involved in a rigorous fashion.
: 
RH
: But the extensionalist does have a simple recourse; let proposition =
: sentence throughout.
: 

Bel(j, X) fails to be a truth function of its second argument: it appears
to require somewhat more than a two valued second argument. So it looks
as though taking the [] operator to be the identity function will be
a flop. I suppose taking the second argument to be the "extension in
all possible worlds" would work, presumably a function from worlds
to sentences. Before I continue I must confess to a overwhelming sense
of self-indulgence ... I must return to commutative algebra.

Indulgently,
-- 
MUTATION@DORSAI.COM
FLORIAN@NEOTERIC.COM
Newsgroups: sci.philosophy.meta