nl-kr-request@CS.ROCHESTER.EDU (NL-KR Moderator Brad Miller) (10/20/87)
NL-KR Digest (10/20/87 00:26:05) Volume 3 Number 36
Today's Topics:
Re: The definite article
Re: Indescribably Delicious (author: Berke)
----------------------------------------------------------------------
Date: Wed, 7 Oct 87 05:07 EDT
From: Claus Tondering <ct@dde.uucp>
Subject: Re: The definite article
In article <2063@kitty.UUCP>, larry@kitty.UUCP (Larry Lippman) writes:
> Something I recently read in a periodical caused me to give some
> thought to the requirement for definite and indefinite articles. It caused
> me to pose my own question: Is there any real need for the definite and
> indefinite article in English?
> After some reflection, I came to a tentative answer: NO.
The New Testament gives a good example of the need for a definite
article. In John's Gospel (I don't remember the exact chapter and
verse) Jesus says: "I am the way, the truth, and the life." By this
he meant the only way, the only truth, so that there is no other way
to God except through Jesus.
Now, in a Russian translation of the Bible the verse is presented without
the definite article, because Russian has none. In Russian the verse
becomes: "I am way, truth, and life", which may either mean the same thing
as the English translation, or it may mean "I am a way, a truth, and a life",
which does not exclude the existence of other ways to God.
So, you see, the definite article has quite significant theological con-
sequences here. Fortunately, both Greek and Aramaic (Jesus' own language)
had a definite article!
--
Claus Tondering
Dansk Data Elektronik A/S, Herlev, Denmark
E-mail: ct@dde.uucp or ...!uunet!mcvax!diku!dde!ct
------------------------------
Date: Fri, 9 Oct 87 10:01 EDT
From: Joe Chapman <joe@haddock.ISC.COM>
Subject: Re: The definite article
I'm surprised no one has mentioned ancient Greek in this discussion.
Originally the Greek article (ho, e, to) was a demonstrative; it's
used as such in Homer. Sometime in the 5th century it began to be
used as a definite article. [For a grand tour of all of this, look up
"ho" in Liddell-Scott. It takes up *pages*. Absolute chloroform.]
Incidentally, there's an argument that the development of the definite
article and the abstractions it makes expressible (especially with
infinitives: einai=to be; to einai=being) helped to begin the
development of philosophy in Greece. This usage is very common in
Parmenides and Plato, for example.
I have heard the assertion that Greek was the first Indo-European
language to develop the indefinite article, and that therefore it was
far ahead of other IE languages in its ability to express philo-
sophical concepts; I can't cite a reference for the first notion, nor
do I know enough Sanskrit to rule on the second, so take it as you will.
Joe Chapman
harvard!ima!joe
------------------------------
Date: Sat, 10 Oct 87 00:21 EDT
From: Larry Lippman <larry@kitty.UUCP>
Subject: Re: The definite article
In article <1987Oct6.215107.14061@sq.uucp>, msb@sq.uucp (Mark Brader) writes:
> For instance, when I used to travel regularly by train between Kitchener
> and Toronto, I would often overhear railway people referring to train 665
> as "that #665", with no special implication. Similarly, it's common for
> sports fans to say [usually to non-fans like me :-)], "How about those
> Blue Jays?" [Well, it WAS common until this week, anyway... :-)]
>
> This suggests that we may eventually lose the distinction between "that"
> and "the" (wild conjecture: the forms will eventually exist side by side
> varying according to the initial sound of the following word!), and have to
> evolve a new word for "that".
Okay, now I'm going to pick on you Canadians. :-)
I have various Canadian friends, and I seem to notice they often
use phrases which have no definite or indefinite article. Like, "This
morning Metro [referring to Toronto] traffic was atrocious."
Since we live less than 50 air miles from Hamilton or Toronto,
my wife often watches Canadian television. On occasion, I will watch
Canadian news just to observe the Canadian viewpoints of U. S. events.
I am struck by phrases which are devoid of articles, a common example of
which is: "Following his accident, Mr. Jones is in hospital resting
comfortably." "in hospital" - obviously, no article.
Is it my imagination, or is the definite article disappearing in
Canada at a greater rate than in the U.S.? If so, why?
<> Larry Lippman @ Recognition Research Corp., Clarence, New York
<> UUCP: {allegra|ames|boulder|decvax|rutgers|watmath}!sunybcs!kitty!larry
<> VOICE: 716/688-1231 {hplabs|ihnp4|mtune|seismo|utzoo}!/
<> FAX: 716/741-9635 {G1,G2,G3 modes} "Have you hugged your cat today?"
------------------------------
Date: Sat, 10 Oct 87 03:04 EDT
From: wales@CS.UCLA.EDU
Subject: Re: The definite article
In article <2101@kitty.UUCP> larry@kitty.UUCP (Larry Lippman) writes:
> I have various Canadian friends, and I seem to notice they often
>use phrases which have no definite or indefinite article. Like, "This
>morning Metro [referring to Toronto] traffic was atrocious."
> Since we live less than 50 air miles from Hamilton or Toronto,
>my wife often watches Canadian television. On occasion, I will watch
>Canadian news just to observe the Canadian viewpoints of U. S. events.
>I am struck by phrases which are devoid of articles, a common example
>of which is: "Following his accident, Mr. Jones is in hospital resting
>comfortably." "in hospital" - obviously, no article.
> Is it my imagination, or is the definite article disappearing in
>Canada at a greater rate than in the U.S.? If so, why?
No, I think there are other explanations for what you are observing.
Regarding your first example, I think this is simply an instance of
"Metro traffic" being thought of as an abstract, general entity. I,
out here in Los Angeles, could say "This morning, traffic on the San
Diego Freeway was bumper-to-bumper" -- without having to say "*the*
traffic" -- and it would sound perfectly all right.
As for the second example, there are a handful of fixed expressions of
this type (in all forms of English) where the definite article is not
customarily used. For example: "in school"; "in church"; "in bed".
"In hospital" is perfectly idiomatic British English -- and this partic-
ular Briticism is still the norm in Canada as well (though not in the
States, where we can only say "in *the* hospital").
As far as I can tell (and, although I am an American, I have developed
a fairly high level of familiarity with Canadian English), there is no
more of a general tendency in Canada for a wholesale disappearance of
the definite article than there is in the US.
-- Rich Wales // UCLA Computer Science Department // +1 213-825-5683
3531 Boelter Hall // Los Angeles, California 90024-1596 // USA
wales@CS.UCLA.EDU ...!(ucbvax,rutgers)!ucla-cs!wales
"Sir, there is a multilegged creature crawling on your shoulder."
------------------------------
Date: Sat, 10 Oct 87 12:14 EDT
From: Creative Business Decisions <Q2816@pucc.Princeton.EDU>
Subject: Re: The definite article
In article <2101@kitty.UUCP>, larry@kitty.UUCP (Larry Lippman) writes:
> Okay, now I'm going to pick on you Canadians. :-)
> I have various Canadian friends, and I seem to notice they often
>use phrases which have no definite or indefinite article. Like, "This
>morning Metro [referring to Toronto] traffic was atrocious."
"Freeway traffic was atrocious this morning."
"Rush hour traffic was ... "
"New Jersey traffic ... "
I don't see anything there that isn't idiomatically American.
> Since we live less than 50 air miles from Hamilton or Toronto,
>my wife often watches Canadian television. On occasion, I will watch
>Canadian news just to observe the Canadian viewpoints of U. S. events.
>I am struck by phrases which are devoid of articles, a common example of
>which is: "Following his accident, Mr. Jones is in hospital resting
>comfortably." "in hospital" - obviously, no article.
"In hospital" has been standard British usage for ages. We say,
"Hospitalized" or "in the hospital" instead. On the other hand,
we say "in jail," just as they do. (Well, ok, they say, "in gaol.")
> Is it my imagination, or is the definite article disappearing in
>Canada at a greater rate than in the U.S.? If so, why?
I don't think these changes are anthing new. Nor do I think there's
much evolution of them.
Roger Lustig (Q2816@PUCC)
BRING BASEBALL BACK TO WASHINGTON!
------------------------------
Date: Sat, 10 Oct 87 14:52 EDT
From: Samuel B. Bassett <samlb@well.UUCP>
Subject: Re: The definite article
In article <2101@kitty.UUCP> larry@kitty.UUCP (Larry Lippman) writes:
> . . . "in hospital" - obviously, no article.
> Is it my imagination, or is the definite article disappearing in
>Canada at a greater rate than in the U.S.? If so, why?
"in hospital" is a Britishism -- so it's not necessarily the Canadians'
fault, tho' I _have_ heard a lot of Canadian speech that lacks the definite
article, too . . .
Canadian comments?
--
Sam'l Bassett -- Semantic Engineering for fun & profit.
34 Oakland Ave., San Anselmo CA 94960; DDD: (415) 454-7282
UUCP: {hplabs,ptsfa,lll-crg}!well!samlb; Internet: samlb@well.uucp
Compuserve: 71735,1776; WU Easylink ESL 6284-3034; MCI SBassett
------------------------------
Date: Sat, 10 Oct 87 19:53 EDT
From: Max Hauser <max@eros.uucp>
Subject: Re: The definite article
In article <4188@well.UUCP> samlb@well.UUCP (Samuel B. Bassett) writes:
>
> "in hospital" is a Britishism -- so it's not necessarily the Canadians'
>fault, tho' I _have_ heard a lot of Canadian speech that lacks the definite
>article, too . . .
But this is not peculiar to Canada; the New England states of the US are
also fond of it. When I was in Boston 1979-81 I heard a lot of
"She was taken to hospital"
"We are going to town meeting"
This was in small towns inland of Boston, which always had a lot of
town meetings. They were in vogue.
I will not venture to repeat what I heard further north in New England
simply because it's not the same without regional accent, but it was
similar.
While I'm on the subject, I remember that everything around Boston
seems to be called Somebody Memorial Something, as I remarked in
correspondence back home to the provinces at the time. For example,
the Boston Pops performed outdoor concerts in a band shell called the
Hatch Memorial Shell. As Arthur Fiedler had recently died, they
offered a Fiedler Memorial Concert. A local college radio station
in the Walker Memorial Building petitioned for call letters alluding
to their location. Et sic de similibus.
Max Hauser / max@eros.berkeley.edu / ...{!decvax}!ucbvax!eros!max
State University at the Democratic Republic of Berzerkeley
"Warning: You are entering a nuclear-free zone. Possession or
discharge of nuclear weapons within city limits may be subject
to police citation."
------------------------------
Date: Tue, 13 Oct 87 22:52 EDT
From: Mark Brader <msb@sq.uucp>
Subject: Re: The definite article
I think Rich Wales is right in his analysis of the phrases Larry Lippman
quoted. [Now if someone could explain why Americans need a "the" in the
middle of "in hospital", but not in the middle of "in bed"... :-)]
As further evidence for the non-disappearance of "the" in Canada, I point
out that I bank at The Royal Bank of Canada, whereas an American might bank
at Bank of America with no "the". Even where the name itself does not
include a "the", I find it idiomatic to say that I do not bank, though I
once did, at the Bank of Montreal. (The ads for the latter never include
the article, so there is evidently some variation of usage here. But Rich
tells me that there is no such variation in U.S. usage.)
Mark Brader "Not looking like Pascal is not a language deficiency!"
utzoo!sq!msb, msb@sq.com -- Doug Gwyn
------------------------------
Date: Sat, 10 Oct 87 02:58 EDT
From: berke@CS.UCLA.EDU
Subject: Re: Indescribably Delicious (author: Berke)
(First, appologies to those who requested the Naming and Knowledge
paper. There were more requests than I expected and many were from
Europe and Asia. I was away at the Artificial Life conference for
a week, and will send copies as soon as I scrounge up the postage!
Thank you for your replies and requests, Pete.)
Now, in reply to:
steves@cs.qmc.ac.uk (Stephen Sommerville)
in article 1437 of Newsgroup: sci.lang
Subject: Re: Indescribably Delicious (Berke)
Date: 14 Sep 87 15:57:44 GMT
Please change the name in the (author) part of the subject line if
you follow-up to my postings. Sometimes before I "kill" subjects,
I like to know if certain people have responded, and so would prefer
to see author's names in the subject line. My summarizer '=' command
lists just the subject of articles, not the authors, so I put my
name in the subject line, hoping others will.
I have no solutions to offer to the purported conundrum
of how to interpret phrases like "indescribably A", since I
suspect such phrases only appear puzzling to one with an overly
simple notion of 'meaning'
It is true that in your article you do not offer any solutions. Perhaps
you could explain a better notion of 'meaning' than the overly simple one
you attribute to me. You have not been presented with my notion of
'meaning', but I would appreciate any substantive solution to any
problem with any simple theory of meaning.
'Indescribably delicious' is not a conundrum. It is a name for a concept
that cannot be expressed in words.
Fregean 'concepts' are certainly not what Church (in "Logic of
Sense and Denotation" or "The Need for Abstract Entities in Semantics")
meant by "sense". Frege also has the notion of "Sense (Sinn)" in
"Ueber Sinn und Bedeuting", as
contrasted with the term "Begriff (concept)".
You are completely wrong, but the confusion is partly my fault. Church
translates the German 'Begriff' as 'propositional function'. He uses the
word 'concept' to refer to the entities that can serve as senses of names.
If you don't believe me, ask him.
In ordinary circumstances, the sense of a name is a concept of an object
that the name names. Church has tried to remain as true to both Frege
and Russell's terminologies, as many ideas in both are parallel. Church
probably translates 'Begriffe' as 'propositional functions' because Frege's
Begriffe resemble Russell's propositional functions.
I believe that 'Begriff' literally translates to 'concept', but this is
only one of the many misfortunes plaguing translations of Frege.
> required by the assumption that names name things <
is precisely to repeat the mistake against which
Frege was arguing in "Ueber Sinn und Bedeutung"
- one which thereby manufactures the puzzle with
which the article opens over "The Morning Star = The Evening
Star".
Frege was commited to what Wittgenstein called an "Augustinian" theory
of language, that there are some objects that names name.
If you review a copy of "Uber Sinn und Bedeutung," you will see
that it opens with a discussion of 'A=B'. Venus comes in pages
later after Bucephalus, the moon through a telescope, Odysseus in
Ithaca, and 5 being a prime number. "The mistake" against which
Frege was arguing in "Uber Sinn..." was not the assumption that
names name things, but that that is all that they do. I think
Wittgensteing successfully illustrated that the "Augustinian"
assumption has problems. But it is precisely Frege's insistance
that names name things that pushes him to try to explain how
sometimes what they name appears to vary, i.e., in indirect contexts
(ungerade Rade).
I'm afraid that equating Fregean concepts with Carnap's
"intensions" (or "intensionalities") won't do either. This is a
very well known Carnapian blunder in interpreting Frege -
First, let me say that we should both be so lucky to blunder as Carnap.
>When we use a word, we usually (purport to) denote an object and
>express a concept. There are problems with this. The main one
>is called the 'paradox of the name relation by Church', the
>'antinomy of the name relation' by Carnap. It was discovered
>by Frege when he asked "How can A=B, if true, differ in content
>from A=A?"
Sorry, this is just plain wrong. The antinomy of the name
relation as cited by Carnap in "Meaning and Necessity" (p.133+)
concerns the failure of intersubstitutivity of terms in modal
(or any other kind of 'intensional' (sic!) context). For
example, though "Necessarily (9 > 7)" and "The number of
planets = 9" are both true, "Necessarily (the number of planets
> 7)" is false. The example from Frege concerning "A=B" is not
an antinomy - but an illustration of the need to distinguish
sense from reference ("Ueber Sinn und Bedeutung", p. 1). It is
confusing to conflate problems of semantically interpreting
identity statements with that of failures of extensionality and
intersubstitutivity in modal contexts!
Excuse me. I did not mean that "How can A=B..." is an example of the
paradox of the name relation. If you look at your copy of "Uber Sinn
und Bedeutung" you'll see that it starts with this question. Frege
then says that it is a problematic question if you take the position,
as he says he did in "Beggriffschrift," that equality is a relation
between objects. He then takes the position that equality is a
relation between names, that they "denote" [my usage] the same object.
He then discusses problems with this, one of which has to do with what
you seem to have a handle on calling "failures of extensionality and
intersubstitutivity in modal contexts," (what I think I call the Paradox
of the Name Relation). If you object to my original wording of this,
how about: "Frege discovered the paradox of the name relation, though
he did not call it that. He
leads into his discussion of the problem with "How can A=B..."
I believe what you are calling 'reference' Church calls 'denotation' the
relation of a name to the object of which it is a name. A name names,
or denotes, its denotation. Frege's word for this, 'Bedeutung' is literally
translated as 'meaning'. Church follows Mill in calling this relationship
'denotation'.
>When we use a word, we usually (purport to) denote an object
and express a concept . . . <
This simply restates a version of the "'Fido'-Fido" theory -
The first clause, that names are used to denote, point to, name, objects
is the central tenet of an "Augustinian" theory of language. That
names also express concepts is Church's way of saying Frege's idea that
names have senses. Mill thought names had connotations
in addition to denotations. Similarly, though not exactly, Frege (1879)
thought names had "Sinn" in addition to "Bedeutung."
Though Mill's work predates Frege's, Frege seems as unaware of Mill
as the rest of the world was of Frege before Russell popularized
Frege's logic and writings. Church chose
Mill's word for denotation, Frege's word 'sense' for connotation.
I think because he agrees more with Frege, but 'Bedeutung' is too
confusing to simply translate into English as 'meaning'. By 1879
when Frege published Uber Sinn und Bedeutung, he was already committed
to using 'Bedeuten' for the relationship some of us have come to call
denotation, but which I believe from your writing that you call reference.
I don't know why you call it 'reference'.
It is common in modern philosophy. Sometimes it seems to be based
on Russell's usage of the word 'reference'. At times Russell did, but
Russell changed his mind several times about what names did. Church
is most committed to Russell's first edition of Principia Mathematica
(1905). Except for "On Denoting" also 1905. Post-1905 Russell is
not exactly precise on naming, nor does it use consistent terminology
and punctuation.
>It is commonly thought that Russell's theory of
>descriptions solves this paradox, but it does not. Russell's
>theory of meaning requires intensionalities as does Frege's.
This is misleading. The notion of "intensionality" derives from
Carnap's work, dating at the earliest from the 1930's. Russell
wrote "On Denoting" (which sets out the theory of descriptions)
in 1905, whilst Frege recognised the need for "concepts" as the
referents of predicates by about 1890. Neither would have
accepted the imputation they 'require' intensions (as meanings
of descriptions/ predicates).
The word 'intension' may come from Carnap. The concept of a "triangle
of meaning" is common to early Russell, Frege, Mill, and many others
back at least to Aristotle. I think you mean to speak of "Begriffe" as the
denotations of predicate symbols. Perhaps you can see from this why
Church translates 'Begriff' as 'propositional function' rather
than as 'concept'. Frege's
theory certainly requires intensional objects, senses of names.
The belief that Russell's theory of descriptions allows the
elimination of intensional/abstract entities seems to be inherent
in much modern work. I'm thinking immediately of Barwise and Perry's
Situation work, but I'm sure we can find other examples. Russell's
theory still needs propositional functions to be taken in intension
rather than in extension (that is, two propositional functions,
P and Q can still be unequal even if they are true of the same
objects. If propositional
functions are taken in extension, as Russell suggests in the second
edition to Principia, the paradox of the name relation can be
produced in his theory. Church has shown this much the way Russell
showed that "Russell's paradox" could be produced in Frege's theory.
If it hasn't been
published yet, it should be. Russell addresses this issue in
Appendix C, yet his proposal to block the semantic paradoxes
is vague, and he makes many mistatements in the Appendix C.
About the only vaguely correct claim, here, is
that Frege was openly committed to the need for abstract
entities in a proper semantics of language. Russell disagreed,
but could not articulate a non-conceptualist semantics until he
came to accept Wittgenstein's Tractarian doctrines of Logical
Atomism (by about 1917). (This is not an endorsement of logical
atomism, nor an endorsement of Russell's version of
Wittgenstein's Tractatus - both of which are problematic. It
mislocates Russell's theory of incomplete symbols (of which the
theory of descriptions is a part) to suppose it concerns the
"paradox of naming". Russell's concern in 1905 was to overcome
the paradoxes of set theory (including his own). As is noted in
his Autobiography, the theory of descriptions was his first
clue to how to overcome them - not a revamping of Frege's
sense/reference distinction!)
Let's observe that Wittgenstein whole-heartedly rejects his
Tractatus in his later work, e.g., Philosophical Investigations.
>Russell too
>insisted on the formal expression of logic, and wholly adopted Frege's
>language. Russell linearized it - Frege's was two-dimensional. Russell added
>ramified type theory to avoid certain paradoxes.
This is the worst misinterpretation of all. To claim that the logic of
"Principia Mathematica" is just a linearised version of Frege's
"Grundgesetze", with ramified type theory tacked on as an
afterthought, entirely misrepresents both works.
Prior to his becoming aware of Frege's work, I believe that Russell used a
notation due to Peano, which featured, among other things, the use of a
backwards 'C' for implication. Russell translated Frege's two-dimensional
notation into a linear notation. How can you deny that? Do you actually
deny that Russell used Frege's logic in Principia? Do you think he
independently came up with quantification? The words "is just a" and
"tacked on" are yours. The language of Frege wholy adopted by Russell is
quantificational logic, invented, as far as I can tell, by Frege in
Begriffschrift. Don't you think you're being a bit sensitive and casually
insulting here? The "Grundgesetze" was not mentioned by me. Russell
did pursue the logicist program. "Grundgesetze" was Frege's attempt.
Principia was Russell's. Frege's was prior. Russell continued his work,
but needed a type theory. Frege's variables were "universal" in that
they ranged over all individuals. Russell saw the need for a type theory
to block the paradoxes.
The logic of "Principia Mathematica"(First
Edition, 1910 - 1913)
Though Principia was first published in 1910, the first volume, mostly
written by Russell, was actually written prior to the actual publication
date, and so counts in Church's "Russell's 1905 and before." May I quote
from the preface?
"In the matter of notation, we have as far as possible followed Peano,
supplementing his notation, when necessary, by that of Frege of by that
of Schroder... In all questions of logical analysis, our chief debt
is to Frege. Where we differ from him, it is largely because the
contradictions showed that he, in common with all other logicians
ancient and modern, had allowed some error to creep into his premisses;
but appart from the contradictions, it would have been almost impossible
to detect this error."
Ramified Type Theory does resolve the paradoxes,
but at an unacceptable cost to Russell's programme to
establish the identity of logic and mathematics. What makes
this such a misinterpretation is, of course, the author's ingenuous lack
of familiarity with the paradoxes ("certain paradoxes"). It was
Russell's discovery of the paradox which bears his name IN the
logic of Frege's "Grundgesetze" which prompted his ten-year
effort to solve this (and the other related vicious-circle
paradoxes). That ten-year project is what motivated "Principia
Mathematica" (> a mere linearised version of Frege < ???).
You do not know
me, nor do you know Russell or Frege, but you are very free with ascribing
certain intentions and motivations to us. Dr. Paul Penfield of MIT once
recommended to me that I never publically ascribe motivations to anyone
since they are very hard to be right about. I tend not to use 'of course'
in my writing because it is usually an abbreviation for "now I really
can't back up what I am saying."
Ramified Type Theory is needed only to resolve the semantic paradoxes,
one of which is the paradox of the name relation. For the set theoretic
paradoxes, a simple type theory will do. Related "simple" options
exist, say in Zermelo-Frankel set theory.
What Wittgenstein et al objected to in ramified type theory was not the
ramification of types, but the Axioms of reducibility.
Developing math from logical definitions was only one of the tasks
rendered impossible by ramified types. The Axioms of reducibility
were attacked as wildly and absurdly
counter-intuitive. The real problem in obtaining math from logic alone,
in my humble opinion, which I have
entirely borrowed from Church is in the matter of impredicative definition
which is needed to prove the Induction postulate of Peano (unless you
as do intuitionists, take natural numbers as granted) and to prove
the least upper bound theorem. Church also describes the axiom of
infinity as an "embarassment" to the logicist program to produce math
entirely from definitions in logic.
There is considerably more of a very confused/ing nature in the
rest of Berke's article - but my patience is exhausted. Beware the
enthusiastic Hegelian who sees all things as related to
everything else (Frege to Wittgenstein, Russell to Husserl) -
they are often simply molding history to their own purposes!
Steve Sommerville
I definitely see Russell related to Frege, and Wittgenstein related to Husserl.
Who are you to tell people to beware of me? I think it is not your patience
that is exhausted but your understanding, your manners, and a certain
willingness to try to understand someone who is speaking or writing
that you don't know.
Since you imply my ignorance of the paradoxes, I'll follow this
note with part of one I previously posted to the net on the paradoxes. It is
mostly Church's words, since, compared to you, I'm relatively naive
on such matters.
I have no solutions to offer
is how you started your note. You have pointed out some things that
a complete discussion of such matters needs, such as Church translating
'Begriff' as 'propositional function' to maintain maximum compatability
with Russell's terminology. For that I thank you.
I would however, be interested in any solutions that you do have to
offer when you get them.
Sincerely,
Peter Berke
(note on paradoxes follows)
4. Here's an abbreviation of what Alonzo Church has to say on
this matter in the Dictionary of Philosophy, edited by D. D.
Runes, Littlefied, Adams, & Co., publishers, 1980 edition in
paperback) under the heading: "Paradoxes, logical." The first
paragraph is severely abbreviated and I introduced the
terminology of adjectives "applying" to themselves.
Grelling's paradox (1908) distinguishes adjectives that apply to
themselves (called 'autological') from those that don't (called
'heterological'). Autological adjectives include 'polysyllabic'.
Heterological adjectives include 'new' and 'alive'.
[Church, presumably explaining Grelling, considers adjectives to
denote (point to, stand for) properties. Thus he says "Let us
distinguish adjectives - i.e., words denoting properties - as
autological or heterological according as they do or do not have
the property which they denote (in particular, adjectives
denoting properties which cannot belong to words at all will be
heterological)." Whether adjectives denote properties as was
held commonly in an earlier era of logic is an intense question
best avoided at the moment. So I substituted the terminology of
adjectives applying to themselves, as I feel it is essentially
what he meant but does not introduce the tangential issue. Keep
this in mind when we discuss the resolution of the paradoxes.]
"Paradox arises when we ask whether 'heterological' is
autological or heterological.
"That paradoxes of this kind could be relevant to mathematics
first became clear in connection with the greatest ordinal
number, published by Burali-Forti in 1897, and the paradox of the
greatest cardinal number, published by Russell in 1903. The
first of these had been discovered by Cantor in 1895, and
communicated to Hilbert in 1896, and both are mentioned in
Cantor's correspondence with Dedekind of 1899, but were never
published by Cantor.
"From the paradox of the greatest cardinal number Russell
extracted the simpler paradox concerning the class t of all
classes x such that not x in x. (Is it true or not that t is in
t?) Russell communicated this simplified form of the paradox of
the to Frege in 1902 and published it in 1903. The same paradox
was discovered independently by Zermelo before 1903 but not
published."
"Numerous solutions of these paradoxes have been proposed.
Russell's solution of the paradoxes is embodied what is now know
as the ramified theory of types, published by him in 1908, and
afterwards, made the basis of Principia Mathematica. Another
solution is the simple theory of types. This was proposed as a
modification of the ramified theory of types by Chwistek in 1921
and Ramsey in 1926, and adopted by Carnap in 1929. Another
solution is the Zermelo set theory, proposed by Zermelo in 1908,
but since considerably modified and improved."
"Unlike the ramified theory of types, the simple theory of types
and the Zermelo set theory both require the distinction (first
made by Ramsey) between the paradoxes which involve use of the
name relation, or the semantical concept of truth, and those
which do not. The paradoxes of the first kind (Epimenides,
Grelling's, Koenig's, Richard's) are solved by the supposition
that notations for the name relation and for truth (having the
requisite formal properties) do not occur in the logistic system
set up - and in principle, it is held, ought not to occur. The
paradoxes of the second kind (Burali-Forti's, Russell's) are
solved in each case in another way."
------------------------------
End of NL-KR Digest
*******************