SUO: Re: Re: Ontology
(Apologies to any sufficiently patient reader for the length of this).
Preface: I'm neither a Peirce acolyte nor scholar, nor one of
Russell's. I certainly acknowledge the historical
importance of what he and Whitehead did, although logic has evolved
considerably since. I admire Peirce's good points too, though I'm less
familiar with them. I'm also, BTW, not one of those six or ten who have
actually read the P.M. all the way through -- only chunks.
Thank you, Jesse. Yes, that passage seems particularly significant. I'd
already quoted some of it, and was about to quote the whole. That was the
'error' which Russell thought Peirce committed -- not taking relations
seriously enough as intensional entties, although here he defers the
question about the philosophical validity of the point. If you take
relations seriously but naively, though, you're landed in the Contradiction,
as Russell had discovered. Leading to Frege's remark about Arithmetic
See also: http://plato.stanford.edu/entries/russell-paradox/
Russell's having discovered the Contradiction, with the P.M version of Type
Theory, Russell and Whitehead were trying to mitigate the collapse of the
Fregean system, but still to maintain the reality of relations, in some
sense. Hence the 'no-class' business of the P.M.
The defects of the Logicist program were pretty soon discovered in turn, and
have been widely discussed since, of course. All of this is old news, but
it's stimulating to look at it again.
Here's a selection from the Stanford Encyclopedia of Philosophy article on
the P.M. I suppose it represents something like a 'received view'.
Principia Mathematica, the landmark work written by Alfred North Whitehead
and Bertrand Russell, and published in three volumes, in 1910, 1912 and
1913. Written as a defense of logicism (i.e., the view that mathematics is
in some significant sense reducible to logic) the book was instrumental in
developing and popularizing modern mathematical logic. It also served as a
major impetus for research in the foundations of mathematics throughout the
twentieth century. Next to Aristotle's Organon, it remains the most
influential book on logic ever written.
...Today there is not a major academic library anywhere in the world that
does not possess a copy of this landmark publication.
Significance of Principia Mathematica
Achieving Principia's main goal proved to be controversial. Primarily at
issue were the kinds of assumptions that Whitehead and Russell needed to
complete their project. Although Principia succeeded in providing detailed
derivations of many major theorems in set theory, finite and transfinite
arithmetic, and elementary measure theory, two axioms in particular were
arguably non-logical in character: the axiom of infinity and the axiom of
reducibility. The axiom of infinity in effect stated that there exists an
infinite number of objects. Thus, it made the kind of assumption that is
generally thought to be empirical rather than logical in nature. The axiom
of reducibility was introduced as a means of overcoming the not completely
satisfactory effects of the theory of types, the theory that Russell and
Whitehead used to restrict the notion of a well-formed expression, and so to
avoid paradoxes such as Russell's paradox. Although technically feasible,
many critics concluded that the axiom of reducibility was simply too ad hoc
to be justified philosophically. As a result, the question of whether
mathematics could be reduced to logic, or whether it could be reduced only
to set theory, remained open.
Despite these criticisms, Principia Mathematica proved to be remarkably
influential in at least three other ways. First, it popularized modern
mathematical logic to an extent undreamt of by its authors. By using a
*notation superior* [emph. added] in many ways to that of Frege, Whitehead
and Russell managed to convey the remarkable expressive power of modern
predicate logic in a way that previous writers had been unable to achieve.
Second, by exhibiting so clearly the deductive power of the new logic,
Whitehead and Russell were able to show how powerful the modern idea of a
could be, thus opening up new work in what was soon to be called metalogic.
Third, Principia Mathematica reaffirmed clear and interesting connections
between logicism and two main branches of traditional philosophy, namely
metaphysics and epistemology, thus initiating new and interesting work in
both these and other areas.
Thus, not only did Principia introduce a wide range of philosophically rich
notions (such as propositional function, logical construction, and type
theory), it also set the stage for the discovery of classical metatheoretic
results (such as those of Kurt Gödel and others) and initiated a tradition
of common technical work in fields as diverse as philosophy, mathematics,
linguistics, economics and computer science.
Today there remains controversy over the ultimate substantive contribution
of Principia, with some authors holding that, with the appropriate
modifications, logicism remains a feasible project. Others hold that the
philosophical and technical underpinnings of the Whitehead/Russell project
simply remain too weak or confused to be of much use to the logicist.
Interested readers are encouraged to consult Hale and Wright (2001), Quine
(1966a), Quine (1966b), Landini (1998) and Linsky (1999).
Here's another gloss:
How and whether logicism remains viable, I do not know. I think it's fairly
commonly thought that, at least as conceived of in the P.M., logicism is a
failure in important respects. In the end, Russell cataloged some of these
complaints himself, in various places.
Here also are two selections about Peirce from the Encylopedia. The whole
article remarks upon the difficulties of his life, his philosophical and
logical merits (and some potential drawbacks), and his complexity. Much in
these articles is very commendatory.
Peirce's special strength lay not so much in theorem-proving as rather in
the invention and developmental elaboration of novel systems of logical
syntax and fundamental logical concepts. He invented dozens of different
systems of logical syntax, including a syntax based on a generalization of
de Morgan's relative product operation, an algebraic syntax that mirrored
Boolean algebra to some extent, a quantifier-and-variable syntax that
(except for specific symbols) is identical to the much later
Russell-Whitehead syntax, and even two systems of two-dimensional syntax:
the entitative graphs and the existential graphs, the latter being a syntax
for logic using the mathematical apparatus of topological graph theory.
In 1870 Peirce published a long paper "Description of a Notation for the
Logic of Relatives" in which he introduced for the first time in history,
two years before Frege's Begriffschrift a complete syntax for the logic of
relations of arbitrary adicity (or arity). In this paper the notion of the
variable (though not under the name "variable") was invented, and Peirce
provided devices for negating, combining relations, and quantifying. By
1883, along with his student O. H. Mitchell, he had developed a full syntax
for quantificational logic, only a little different in specific symbols (as
was mentioned just above) from the standard Russell-Whitehead syntax that
did not appear until 1910.
[Note particularly the remark: "except for specific symbols". More on this
below. Now, due to my own deviltry, I repeat a critical note which is
sounded, but one should really read the whole thing.]
Triadism and the Universal Categories
Merely to say that Peirce was extremely fond of placing things into groups
of three, of trichotomies, and of triadic relations, would fail miserably to
do justice to the overwhelming obtrusiveness in his philosophy of the number
three. Indeed, he made the most fundamental categories of all "things" of
any sort whatsoever the categories of "Firstness," "Secondness," and
"Thirdness," and he often described "things" as being "firsts" or "seconds"
or "thirds." For example, with regard to the trichotomy "possibility,"
"actuality," and "necessity," possibility he called a first, actuality he
called a second, and necessity he called a third. Again: quality was a
first, fact was a second, and habit (or rule or law) was a third. Again:
entity was a first, relation was a second, and representation was a third.
Again: rheme (by which Peirce meant a relation of arbitrary adicity or
arity) was a first, proposition was a second, and argument was a third. The
list goes on and on. Let us refer to Peirce's penchant for describing things
in terms of trichotomies and triadic realtions as Peirce's "triadism."
If Peirce had a general rationale for his triadism, Peirce scholars have not
yet made it clear what this rationale might be. He seemed to base his
triadism on what he called "phaneroscopy," by which word he meant the mere
observation of phenomenal appearances. He regularly commented that the
phenomena just do fall into three groups and that they just do display
irreducibly triadic relations.
Although there are many examples of phenomena that do seem more or less
naturally to divide into three groups, Peirce seems to have been driven by
something more than mere examples in his insistence on applying his
categories to almost everything imaginable. Perhaps it was the influence of
Kant, whose twelve categories divide into four groups of three each. Perhaps
it was the triadic structure of the stages of thought as described by Hegel,
or even the triune commitments of orthodox Christianity (to which Peirce
seemed to subscribe). Certainly involved was Peirce's commitment to the
ineliminability of mind in nature, for Peirce closely associated the
activities of mind with a particular triadic relation that he called the
"sign" relation. (More on this topic appears in this article.) Also involved
was Peirce's so-called "reduction thesis" in logic (on which more will given
below), to which Peirce had concluded as early as 1870.
It is difficult to imagine even the most fervently devout of the passionate
admirers of Peirce, of which there are many, saying that his account (or,
more accurately, his various accounts) of the three universal categories is
(or are) clear and compelling. Yet, in almost everything Peirce wrote from
the time the categories were first introduced, they found a place. Their
analysis and an account of their general rationale, if there be such,
constitute chief problems in Peirce scholarship.
Also, here is a bit from the Stanford article, 'Peirce's Logic',
specifically on Relations:
Calculus of Relations
Peirce's calculus of relations has been criticized for remaining
unnecessarily tied to previous work on Boolean algebra and the equational
paradigm in mathematics. It has been frequently claimed that real progress
in logic was only realized in the work of Frege and later work of Peirce in
which the equational paradigm was dropped and the powerful expressive
ability of quantification theory was realized.
Nevertheless, Peirce's calculus of relations has remained a topic of
interest to this day as an alternative, algebraic approach to the logic of
relations. It has been studied by Lowenheim, Tarski and others. Lowenheim's
famous theorem was originally a result about the calculus of relations
rather than quantification theory, as it is usually presented today. Some of
the subsequent work on the calculus of relations is outlined in Maddux
----- Original Message -----
From: "Jesse Norman" <firstname.lastname@example.org>
To: <email@example.com>; "'Jay Halcomb'" <firstname.lastname@example.org>
Cc: <email@example.com>; <firstname.lastname@example.org>
Sent: Wednesday, January 14, 2004 03:39
Subject: RE: Re: Ontology
> Dear List
> The key substantive quotation from Russell's Principles of Mathematics
> appears to be this, from Ch. 2, Sec. 27:
> ---------BEGIN QUOTE---------
> The calculus of relations is a more modern subject than the calculus of
> classes. Although a few hints of it are to be found in De Morgan [fn.
> omitted], the subject was first developed by C. S. Peirce [fn.: See
> especially his articles on the Algebra of Logic, American Journal of
> Mathematics, Vols. III and VII. The subject is treated at length by
> C.S. Peirce's methods in Schroder, [Algebra der Logik], Vol. III]. A
> careful analysis of mathematical reasoning shows (as we shall find in
> the course of the present work) that types of relations are the true
> subject matter discussed, however a bad phraseology may disguise this
> fact; hence the logic of relations has a more immediate bearing on
> mathematics than that of classes or propositions, and any theoretically
> correct and adequate expression of mathematical truths is only possible
> by its means. Peirce and Schroder have realized the great importance of
> the subject, but unfortunately their methods, being based, not on Peano,
> but on the older Symbolic Logic derived (with modifications) from Boole,
> are so cumbrous and difficult that most of the applications which ought
> to be made are practically not feasible. In addition to the defects of
> the old Symbolic Logic, their method suffers technically (whether
> philosophically or not I do not at present discuss) from the fact that
> they regard a relation as essentially a class of couples, thus requiring
> elaborate formulae of summation for dealing with single relations. This
> view is derived, I think, probably unconsciously, from a philosophical
> error: it has always been customary to suppose relational propositions
> less ultimate than class-propositions (or subject-predicate
> propositions, with which class-propositions are habitually confounded),
> and this has led to a desire to treat relations as a kind of classes.
> However this may be, it was certainly from the opposite philosophical
> belief, which I derived from my friend Mr G.E. Moore [fn. omitted], that
> I was led to a different formal treatment of relations. This treatment,
> whether more philosophically correct or not, is certainly far more
> convenient and far more powerful as an engine of discovery in actual
> ---------END QUOTE---------
The following remarks from the Introduction to the P.M. I think are also
pertinent to the program initiated in the Principles, and to the
"philosophical error" mentioned above:
"There are two kinds of difficulties which arise in formal logic; one kind
arises in connection with classes and relations, and the other in connection
with descriptive functions. The poiint of the difficulty for classes and
relations, so far as it concerns classes, is that a class cannot be an
object suitable as an argument for any of its determining functions. If \a
represents a class and Phi^\a one if its determing functions [so that
a=^z(Phi z)], it is not sufficient that Phi\a be a false proposition, it
must be nonsense. Thus a certain classification of what appear to be objects
into things of essentially different types appears to be necessary. This
whole question is discussed in Chapter II, on the theory of types, and the
formal treatment in the systematic exposition which forms the main body of
this work, is guided by this discussion. The part of the systematic
exposition which is especially concerned with the theory of classes is *20,
and in this Introduction is discussed in Chapter III. It is sufficient to
note here that, in the complete treatment of *20, we have avoided the
decision as to whther a class of things has in any sense an existence as one
object. A decision of this question in either way is indifferent to our
logic... Our symbols, such as '^x(Phi x)' and Alpha and others, which
represent classes and relations, are merely defined in their use..."
This self-predicativity mentioned in this passage is directly related to the
"philosophical error" mentioned in the earlier passage. It is the "error"
of self-predicativity which was allowed in Frege, which first the Principles
and then the P.M. sought to rectify by the theory of types; in the first
case by the simple theory of types, in the latter by the
> Dr Jesse Norman
> Philosophy Department
> University College London
> Gower Street
> London WC1E 6BT
> -----Original Message-----
> From: email@example.com [mailto:firstname.lastname@example.org] On Behalf Of John
> F. Sowa
> Sent: 13 January 2004 17:58
> To: Jay Halcomb
> Cc: email@example.com; firstname.lastname@example.org;
> Subject: CG: Re: Ontology
> Since Cathy Legg already covered many of your questions,
> I'll just respond to the ones she didn't address.
> The following passage you quoted is an example of what I
> had previously called "nonsense". But I admit that is
> a loaded word. Instead, I'll use the more precise
> technical term: "lie".
Lying, again, forsooth. I suppose, then, that Whitehead was either a
co-conspirator, a dupe, or simply unconscious, since he 1)
proofed the Principles, and 2) worked with Russell on the P.M. for many
years -- yet he never remarked about this. But that Whitehead was any of
these things seems very doubtful to me.
The Principles referred several times to Peirce, and acknowledged his work.
True, Russell doesn't seem enamoured of what he refers to, although he
also says that it's important. It's also true that what he was in part
referring to included earlier works of Peirce's, which you yourself, John,
have said were not perfected.
Also the P.M. clearly acknowledges that it took much of its
symbology and usage from Peano. So does the Principles.
> BR> "In Paris, in 1900, I was impressed by the fact that, in all
> > discussions, Peano and his pupils had a precision which was
> > not possessed by others. I therefore asked him to give me his
> > works, which he did. As soon as I had mastered his notation,
> > I saw that it extended the region of mathematical precision
> > backwards towards regions which had been given over to
> > philosophical vagueness. Basing myself on him, I invented
> > a notation for relations."
> If you look at the original publications by Peano, you will
> see that Russell added *nothing* to Peano's notation. Even
> the dot notation for showing precedence, which is used
> throughout the Principia, was invented by Peano in 1889.
Here's some Peano notation:
Webster's: Notation a : the act, process, method, or an instance of
representing by a system or set of marks, signs, figures, or characters b :
a system of characters, symbols, or abbreviated expressions used in an art
or science or in mathematics or logic to express technical facts or
'Notation' is a loose term. But the term can be construed as a *method* of
writing or a *system* of symbols, not just as a *set* of symbols. As a
method or system, the term can refer either to syntactical or to semantical
concerns, and in the context of discussing logic, it might
refer to either an object language or a metalanguage.
The P.M. uses ! (predicativity), and Iota, and a variety of other symbols,
like turnstile, which aren't on my keyboard. In particular, Iota and ! are
used extensively in the P.M., in setting forth 1) Russell's theory of
descriptions, and 2) the ramified theory of types, which is the centerpiece
of the P.M. See, for example, Chapter III of the Inroduction: Incomplete
Symbols. The official definitions of much of the formal theory of the P.M.
are given in terms/symbols such as ! and Iota, and they are given in the
context of presenting the elaborate mechanics of the ramified theory of
types (including orders). See also, for example, the _definitions_ given in
the sections on Classes and Relations. -- the 'General Theory of Classes',
*20, and the 'General Theory of Relations', *21, and note how often these
symbols are introduced in various definitions, and note how many times Iota,
for example, is used in framing the definitions.
The reasonable interpretation of what Russell meant by the remark "I
invented a notation for talking about relations" is that he was referring
either to the ramified theory of types itself (conceived of as a method of
writing about relations), or alternatively he was referring to the specific
P.M. notation used in presenting the theory of types, including the use of
Iota and ! in ways not contemplated by Peano. [Recall also that the P.M.
took propositional functions as primitives, not syntactic items.]
Conclusion: in either case, 'Lied' is merely provocative and slanderous.
The Encyclopedia of Philosophy, edited by Paul Edwards, contains a nice
discussion of the P.M. under the entry for Russell, in a section on logic
and mathematics written by Prior. It ends with the remark:
"It is both easy and necessary to criticize Russell's theories concerning
the logical and semantical paradoxes, and his work in logic and the
foundations of mathematics generally, but he remains, more than any other
one person, the founder of modern logic."
Prior, BTW, was quite familiar with Peirce's work, and very laudatory of
some of it. He said, for example, that Peirce may have been the most
meticulous of logicians [in Formal Logic].
> And if you look at Peano's publications, you will see that he
> gives full credit to his predecessors. In his first publication
> on the application of logic to arithmetic in 1889, Peano says
> "The logical symbols and propositions contained in parts II,
> III, and IV, except for a few, are to be traced to the works
> of many writers, especially Boole." In a footnote, he cites
> the publications by Boole, Schroeder, Jevons, MacColl, and
> Peirce's two papers "On the Algebra of Logic", 1880 & 1885.
I'm not sure what you're actually claiming, but your evidential chain seems
quite weak so far, whatever it is: only a footnote referring to Peirce, with
the general attribution being given "to many", and it is not clear what the
footnote is attributing to Peirce, propositions or symbols. If it is
symbols, which symbols are they, specifically?
> Peano was unaware of Frege's work until he was asked to review
> a paper by Frege, which he described negatively as using an
> unreadable notation. Frege wrote a reply to Peano, which
> initiated a correspondence between them -- but Peano insisted
> that Frege translate his Begriffsschrift notation into the
> algebraic form before P. would read F's letters.
> BR> "Whitehead, fortunately, agreed as to the importance of the
> > method, and in a very short time we worked out such matters
> > as the definitions of series, cardinals, and ordinals, and
> > the reduction of logic to mathematics."
> That is an odd claim,
Actually, it's my typo. It should read: "the reduction of arithmetic to
>because the reduction of logic to
> mathematics is what Frege and Peirce had already done 20
> years earlier. What Frege and Russell were unsuccessfully
> trying to do is to reduce mathematics to logic. What W.
> and R. did in the Principia is to define mathematics in
> terms of set theory, which is today considered a part of
> mathematics, not a part of logic.
> BR> "For nearly a year, we had a rapid series of quick
> > successes. Much of the work had already been done by
> > Frege, but at first we did no know this..."
> This is one more example where Russell tries to make it
> seem that he (with a little help from Whitehead) discovered
> independently what others had accomplished 20 years earlier.
The ramified theory of types had been accomplished 20 years earlier!?!
> But Peano's publications, which Russell read, contained many
> long lists of results, which, as Peano always acknowledged,
> came from others, including Frege, Peirce, Schroeder, etc.
> Competent mathematicians, as Whitehead and Russell certainly
> were, can take a list of results and quickly reconstruct
> the proofs for themselves. That is not considered
> "groundbreaking research".
Bosh. This is exactly what is considered research in logic, and it was
certainly ground-breaking at the time -- these were the first results in
theory. We're talking about formal proofs, and a formal proof is valid only
relative to the logical system used. All of these researchers were trying to
prove the same facts of arithmetic -- all of them were trying to prove
'2+2=4' (a well-known result) and the like. No one was claiming that they'd
independently discovered such facts as that! -- but rather that they had
formal proofs of them in their various logical systems. Whitehead and
Russell were working in the P.M. system, not in the systems of Frege or
Peano or Schroder or Peirce.
> JH> No, the Principles was original, including new and careful
> > discussion of Frege's Contradiction and the beginning
> > formulations of Type Theory. The Principles, which Whitehead
> > proofed, led to the P.M.
> I agree that Russell did contribute some original points,
> such as a version of type theory and his observation of the
> contradiction in Frege's system. But the bulk of it was a
> reconstruction of "well known results" from Peano's long lists
> of formulas that he, his students, and the many predecessors
> he cites had discovered.
I repeat: Bosh. This is exactly what is considered research in logic, and it
was certainly ground-breaking at the time -- these were the first results in
type theory. We're talking about formal proofs, and a formal proof is valid
only relative to the logical system used. All of these researchers were
trying to prove the same facts of arithmetic -- all of them were trying to
prove '2+2=4' (a well-known result) and so on. No one was claiming that
they'd independently discovered such facts as that! -- but rather that they
had formal proofs of them in their various logical systems. Whitehead and
Russell were working in the P.M. system, not in the systems of Frege or
Peano or Schroder or Peirce; a proof in one is not a proof in the other.
> BR> "In June 1902, this period of honeymoon delight came to
> > an end. Cantor had a proof that there is no greatest cardinal;
> > in applying this proof to the universal class, I was led to
> > the contradiction about classes that are not members of
> > themselves. It soon became clear that this is only one of
> > an infinite class of contradictions. I wrote to Frege,
> > who replied with the utmost gravity that 'die Arithmetik
> > ist ins Schwanken geraten.'"
> Russell deserves credit for discovering a contradiction that
> Frege had overlooked. But that problem only creates a difficulty
> for Frege's set theory -- it certainly does not cause arithmetic
> to totter (Schwanken). I agree with Kronecker, Peirce, and many
> others that arithmetic is far more solid than set theory. And
> what Goedel did is the opposite of what Frege and Russell were
> trying to do: Goedel numbering models logic in arithmetic,
> which has a far more reliable foundation than logic.
What Godel did was explicitly inspired by the P.M.
> JH> Zermelo didn't publish his thoughts on Frege at the time
> > that Russell did. How is that a fault of Russell's?
> Neither Peirce nor Zermelo considered that contradiction
> to be a big deal.
Frege thought the contradiction was a blow, of course -- it _was_ a
crippling blow to his system. It's hardly accurate to say that Zermelo's
response was 'no big deal', and thereby to suggest that he merely shrugged
the whole thing off. Instead, he tackled the problem posed by the
Contradiction in another way.
I'm curious. What exactly did Peirce say about the Contradiction? How did
he propose to tackle it?
>What Zermelo published were axioms that
> avoided the contradiction. If you use ZF set theory, y
> have a simpler foundation without the theory of types.
> I was not blaming Russell.
Calling people liars is usually called 'blame'.
>I was just pointing out that
> the mainstream of logic and set theory in continental
> Europe was based on Peirce and the logicians who built on
> his work -- including, Schroeder, Peano, Hilbert, Zermelo,
> Loewenheim, Tarski, etc.
You're being vague. What does 'based on' mean, and what was based on what?
Do you simply mean the use of some symbols? The theory of types was quite
original (in details, although it had looser foreshadowings, in some sense).
Much of the most important work in logic subsequent to P.M., was explicitly
based on the theory of the P.M.
E.g., see Godel's seminal papers. See many more remarks, below and above.
>If Frege and Russell had never
> existed, the technical achievements in logic would not have
> slowed down at all -- in fact, they might have advanced
> more rapidly because Tarski had to fight an uphill battle
> against the entrenched Frege-Russellism to get model
> theory accepted.
Ah... A counterfactual -- a counter existential, in fact.
Au contraire, there is much more *evidence* (e.g., see Godel
below) that had Russell and Whitehead not existed, subsequent technical
achievements in logic would have been greatly impoverished.
Notoriously difficult stuff, counterfactuals are, but useful when we want to
be enthymematic, or even tendentious.
> These points are confirmed by the references I thought I
> had given, but they are also listed in my commentary on
> Peirce's MS 514. In any case, I add a copy at the end
> of this note.
> Some other comments:
> JH> ... I have fairly frequently heard Peirce's logic discussed
> > in academic settings. Just a short while ago, in fact, although
> > it's true that Peirce comes up less often than many others.
> > But Peirce is discussed extensively, for instance, in Kneale
> > and Kneale's Development of Logic, which is a standard historical
> > work in logic.
> Two points: (1) In 1989, there was a one-week conference
> at Harvard to celebrate Peirce's 150th birthday; that event
> helped spur a minirevival of interest in Peirce, but that
> was long after I was in college. (2) I have a high regard
> for the scholarship done in the two major histories of logic:
> the one by Kneale and Kneale, which you cited, and another
> one by Bochenski. But both of them have blind spots, partly
> caused by the harsh light cast by Russell's all-pervasive
> As examples, I'll quote some typical passage from Bochenski's
> _History of Formal Logic_:
> Page 347: In discussing quantification, Bochenski says
> "Hence it must have been developed independently of Frege
> by Mitchell (1883), Peirce (1885), and Peano (1889)."
> Those last three are independent of Frege, but they are not
> independent of one another. Mitchell was Peirce's student,
> whom Peirce graciously credited with reinterpreting P's
> own symbols Pi and Sigma as quantifiers in 1883. But P.
> had introduced Pi and Sigma in a logically equivalent form
> in 1880 without interpreting them as quantifiers.
1. Bochenski said that the *development* (of quantification) by those three
others was independent, not that the three themselves were independent of
each other. That's quite a different remark than yours. Whether he's right
or not, I don't know. Unfortunately, my Bochenski is in the woods, so I'll
have to wait to check out the context.
2. I don't know what it means to say that Pi and Sigma were logically
equivalent but were not interpreted as quantifiers. It may mean something,
of course, but I don't know what it is. What is it?
> I said, Peano cited Peirce, not Frege, as a source for
> his notation.
You're being vague again. Peano didn't cite Peirce as a source for his own
dot notation, of course, as you've already noted. And it isn't clear from
that article which notation of Peirce's he might have adopted, or even that
he adopted any notation (although he may have -- I don't know).
> Page 349: Bochenski says "Peirce's notation was adopted by
> Schroeder and today is still used in Lukasiewicz's symbolism.
> But Peano's is more widely established since its essentials
> were taken over in the Principia."
> Two problems with that point: Peano explicitly gave credit to
> Peirce and Schroeder from whom he took over the "essentials",
> but made a different choice of symbols. Russell took over
> Peano's notation unchanged -- in detail and essentials.
One problem with your point. Certainly Russell (and Whitehead too) 'took
over' Peano notation (in details and some essentials). They deliberately
adopted it in the P.M., and they very explicitly said so in many places. But
they embedded it in their own theory, with additional notation including !
and Iota. They *added* to the Peano notation, and they made a very different
use of the notation which they'd adopted in producing the ramified theory of
types -- a very different theory than HOL or FOL.
Bochenski's point is that the Peano notation became more widely used
because of the very prominence which it got in the P.M. -- precisely
because of Russell's publicity.
I don't know about the
> JS> In my first publication on conceptual graphs in 1976,
> > I was trying to combine the AI work on semantic networks
> > with a solid logic foundation, but I was not completely
> > happy with the combination.... Peirce had done it all
> > in a very elegant form, and nobody else in AI (or in
> > any of my logic courses) had noticed.
> JH> Peirce had done what 'all'?
> By "all", I meant exactly what I said in the first sentence
> of that paragraph: Establish a solid logic foundation for
> semantic networks with an elegant proof theory and a
> model-theoretic semantics. I have been citing the following
> web page with citations for those points in almost every
> note I wrote on this topic. Please look at the references:
Peirce, in his later (not earlier) works had established a system equivalent
to FOL w/identity. So a Stanford article asserts, and I'll accept it, though
I've not seen the proof myself.
Did Peirce also himself establish the completeness of this 'solid
foundation'? This wasn't done for any formal quantificational logic with
variables until Godel.
Godel, The completeness of the axioms of the functional calculus, 1930, P.
1: "Whitehead and Russell, as is well-known, constructed logic and
mathematics [in the P.M.] by basically taking certain evident propositions
as axioms and deriving the theorems of logic and mathematics from those by
means of some precisely formulated principles of inference in a purely
Godel, On formally undecidable propositions of Principia Mathematica and
related systems, 1931. "...The most comprehensive formal systems that have
been set up hitherto are the system of Principia Mathematica, on the one
hand, and Zermelo-Frankel set theory, on the other. These two systems are so
comprehensive that all methods of proof used in mathematics today are
My point in referring to these papers is the simple one that working
philosophers and logicians are still referring daily to the Godel
Incompleteness Theorem, and often tracing the connection to the P.M., just
as Godel did himself. Peirce is nowhere mentioned in these papers. However
you may regret such facts, and however much you may try to argue otherwise
using counterfactuals and slander, the remarks by Godel typify the actual
history of logic in the 20th century. What logicians read and what work they
did, had the provenance that it had, and no revision of history will make
that untrue -- except, perhaps, for some pragmatists?
> Existential Graphs
> JH> Invented the Theory of Types, for instance? Found the
> > conradiction in Frege? Invented the semantical definition
> > of truth? Proved the Completeness Theorem? Invented Godel
> > numbering? Proved the Incompleteness Theorem? None of these.
> > Shall I mention the Contradiction in Frege again?
> JH> Let me again mention Type Theory and the Contradiction in Frege.
> See Cathy's comments. As for Russell's theory of types, that
> was demolished by both Wittgenstein and Frank Ramsey to the
> point where both Russell and Whitehead admitted that it had to
> be completely rewritten for the second edition of the Principia
> -- but neither of them had the time to do it. See Ray Monk's
> biography and his citations for more detail.
Once again, the Logicist program is fairly widely considered to have failed,
for all the well-known reasons. We do, though, now have more refined
versions of Type Theory and other offshoots. Important research continues
along the paths set by Russell (and Frege, Peirce, etc.). Although, once
again, there seems to be some controversy about whether the program is in
some way still viable; at least, according to the Stanford Encylopedia.
> JS> And of course Russell referred to Peirce, but only
> > to disparage him instead of giving him credit.
> JH> Wrong. Check the originals. He did though, point out errors
> > or infelicities in Peirce, as well as commend him.
> I read a library copy of R's P. of M., so I can't check the
> citations now.
We now have the most important one before us. Russell literally pointed out
what he considered to be a "error" in Peirce.
>But R did not point out any "errors" in Peirce,
> and the "infelicities" were in Peirce's 1870 paper, which was
> was the first publication that went beyond monadic predicates.
> And R. certainly did *not* credit Peirce with inventing the
> notation he was using -- in fact, Russell lied in claiming
> that he had improved upon Peano.
If what you mean by 'improving upon Peano' is that the logicist program
failed or stalled -- well, Russell himself came to think the logicist
program in the form in which he envisaged had failed, as he acknowledged.
This is very
old news. He also hedged his bets about its success even in the early days,
in may places, BTW. But it spawned a great deal of successes, too.
So had Frege failed, before Russell. So have many others, with many
including Peirce. Such is the natural course of many or most bold programs.
Whether or not one regards Type Theory as an improvement on Peano is another
question. Improvement upon what? Upon Peano Arithmetic? But these are quite
different theories. Type Theory is intneded to provide an alternative
foundations for mathematics itself.
> JH> And, finally, the History is *delightfully* critical of many
> > philosophers. Including the pragmatists Dewey and James, BTW,
> > which I suspect galls pragmatists particularly. Again, see:
> > <http://www.sonic.net/~halcomb/Russell_Pragmatism_Power.html>
> I did follow that pointer, and I own a copy of R's _History
> of Western Philosophy_ (which I bought for $1 at a book sale).
> But my terms "travesty" and "nonsense" are mild compared to what
> professional philosophers wrote about HoWP. The following is
> from vol. 2 of Ray Monk's biography:
People love to slander Russell to this day, which shows how much he's
still on their minds.. Of course, Russell was designedly controversial.
This is also old news to everyone, I should think. Howls of outrage of one
sort or another, from many quarters, including academia, very often followed
Russell's publications. He titled one work, "Unpopular Essays". He was was
certainly scandalous in some quarters, yet the P.M. was the most
influential work in logic research for decades, as the Stanford article
Russell also won the Nobel Prize, and is equally revered in
many quarters, for many reasons. Even those who revile him concede he was a
brilliant writer; hence the uproar. He's been called a Satanist and worse.
Here's an article in which he seems to be blamed in part for Western
immorality in general:
"In the social line, Russell's so-called [sic] immoralities have regrettably
become standard operating procedures in the Western world, and legal
practice, at least in the U.S., has favored individual freedom at the
expense of social cohesion. What with computers, social and economic
algorithms, and the counterclaim [sic] that happiness comes through the lack
of constraints, we are living for better and for worse, with wisdom and
with folly, in a world that Russell helped make. "
The Monk biographies (which I haven't read, BTW) have gotten mixed reviews.
Here's a selection from a defense of Russell against Monk. It was done by a
Libertarian, I take it from reading the whole:
"However, if we move out of the darkness into the light for a moment, at
least two distinguished critics are in agreement that Ray Monk should
perhaps not have written his biography if he found the subject-matter as
distasteful as he evidently did. These and many others feel that the
personal failures of a man in life, and even gross errors of judgment,
should not be allowed to detract from the value of his work, and they are
prepared to give Russell the benefit of the doubt by seeing the positive as
well as the negative side.
Thomas Nagel for example, professor of Philosophy and Law at New York
University and author of the highly readable essay "Concealment and
Exposure," which discusses the destruction of the conventions of privacy
which the advent of the intrusive society has brought about, comments that
"Monk's relentless censoriousness about Russell's personal troubles seems
uncalled for; things can go badly wrong in any family, even that of a
tireless social commentator" and that Russell "gave incomparably more to the
world than he took from it," while Sylvia Nasar of the Columbia School of
Journalism, author of the life of John Nash, A Beautiful Mind, concludes in
her New York Times review of Monk that "unfortunately, Monk's dislike of
Russell - which he says is based largely on the fate of Russell's son and
granddaughters - and his seeming ignorance of the most basic facts about
mental illness, have skewed his judgment badly. Biographers who despise
their subjects are evidently just as much at risk of getting the story wrong
as those who worship blindly.""
And here's the Nasar link mentioned above:
Also the Thomas Nagel article: on Concealment and Exposure:
Hao Wang offers a balanced portrait of Russell in his "From Mathematics to
Philosophy", in his "Notes on Knowledge and Life: Bertrand Russell as an
Example". Among other things, he cites some of the reproaches of Russell by
Wittgenstein, Whitehead, and Santayana, and also some of the criticisms of
his life. Here are some snippets:
"In a popular article, Russell was quoted as saying that he got more fame
than many of his clever contemporaries at Cambridge by rushing into
controversies. It seems that in the pursuit of knowledge at least, polemics
and controversies tend to hamper rather than speed up progress. The purpose
of accumulation is better served by seeing what is right in someone rather
than what is wrong in him. If someone's ideas are worthless or without
content, the best treatment is silence and neglect. Polemics tend to divert
attention away from near-truth and get into a game with words. One main
difference between Wittgenstein and most contemporary academic philosophers
would seem to be the indulgence of the latter in clever, small arguments
clouded by all sort of extraneous detail.
Another aspect of Russell's way of life was his detachment from academic
institutions and and large corporations in general. In the advanced 'free'
societies today, the greatest freedom seems to be reserved for those who
have the most business acumen. As for the scientists and those who live on
words and symbols, it is practically impossible to escape
the corporations, which naturally impose restrictions to protect their own
interests. As a result, there is great pressure to conform in matters which
have remotely to do with politics of basic moral principles.
Even if one somehow manages to achieve a measure of financial independence,
there is little room to exert wide influence on public affairs by written
words and logical arguments alone. Apart from the control of the mass medai
of communication by the wrong people, there is the disadvantage of
diversification which leaves to the bewildered reader the choice from a
variety of views of very unequal
Russll partially solved this difficulty by a more or less unique combination
[Compare with the difficulties Peirce faced also.]
...Russell's influence in logic and philosophy was deep and great. He is
probably the most widely read philosopher and cited philosopher of this
century. Basic works such as Skolem's paper on free variable number theory,
Herbrand's and Godel's theses, Godel's paper on the incompletability of
arithmetic all took Principia as their point of departure. The idea of a
ramified (predicative) hierarchy plays an essential role in the study of the
foundations of set theory, in particular, the questions of independence and
relative consistency. The contrast of false and meaningless, as suggested in
the theory of types, continues to fascinate professional philosophers.
It is often said that Russell's solid contributions to human knowledge were
far smaller than his influences. The most hostile and unfair account of
Russell's achievement, which can be heard among some working logicians and
philosophers, runs more or less as follows. Many of his major ideas in logic
were anticipated by Frege and often developed more adequately by Frege: the
reduction of mathematics to logic (in paticular, the definition of natural
numbers), the invention of the propositional calculus and the
quantificational calculus, the fomrulation of the basic concepts in the
modern philosophy of logic. Even the theory of types appeard in some form in
Frege and in Schroder. Russell's paradox and the theory of descriptions
amount to two brief remarks, and the former was discovered independently by
Zermelo. The vicious-circle principle was first suggested by Richard and
Poincare. In philosophy, he was once the fashion, but has now been
supplanted by Wittgenstein; and since there is hardly any accumulation of
progress in philosophy, once out of fashion, there is practically nothing
left that can be salvaged.
That such an evaluation is entirely wrongheaded is quite obvious. The forces
which create this sort of evaluation are more complex. There is a bit of
envy and resentment of the specialist against a more or less universal
talent. There is an element of frusttration with 'scientific' philosophy and
academic philosophy in general. Perhaps more important, there is the impact
of the mathematician's criterion of originality: the search for definite and
specific innovations is something analogous to the use of the sexual act as
the sole standard for evaluating the success of a relation of love.
Moreover, there is the historian's delight in tracing anticipations and
arbitrating the distrbution of credits: a small dose is not unhealthy but
one could easily get carried away and lose sight of the broader structure.
This game reduces itself to absurdity when the conclusion is reached that
the central figure of an age, such as Russell in logic and philosophy,
*really* did nothing much. It is possible to play such a deflating game with
Many of the above comments on Russell [not those given here, but previously
in the text] as a man must be unbearably crude and in bad taste. Those who
were young in a prosperous Europe at peace (internally at any rate) are
naturally more gentle, more optimistic, and more generous than individuals
from a generation grown up in war and living a life in alien environments.
Perhaps this is why Russell seemed to have more strength, more courage, and
more hope than people whose ages are between a third and a half of his."
(Hao Wang, From Mathematics to Philosophy, P 351-2, 1974)
My own opinion of Russell, in a nutshell is this. While I disagree
with Russell more often than not, philosophically and otherwise, and while I
regard his own logicistic program as failed (so far), and while
to me his technical views more of historical interest than otherwise, still
I think the Principles and the P.M. (and much of Russell's other work)
was very, very valuable.
Something like I think of Peirce -- although differing in the details.
> RM> p. 279 HoWP "was greeted with almost universal disdain
> > by the academic philosophers who reviewed it. Even C.D. Broad,
> > an ex-pupil and admirer of Russell's who had played a hand in
> > getting him back to Cambridge, could not bring himself to
> > overlook the book's outrageous and cavalier superficialities
> > and simplifications. Nevertheless, despite its many flaws
> > (or perhaps to some extent because of them), the book became
> > a runaway best-seller and placed Russell's finances on a
> > secure footing for the rest of his life."
> Russell's other attempt to get back into serious philosophy
> was _Human Knowledge: Its Scope and Limits_, which did
> not fare any better among professional philosophers.
Some said yes, some said no.
> RM> p. 295: "He was particularly hurt by a review of _Human
> > Knowledge_ by Norman Malcolm, Wittgenstein's friend and
> > disciple, which ended by saying:
> > Anyone who feels grateful, as I do, for the splendid
> > work he did in philosophy and logic during the first
> > twenty years of this century, is likely to regard the
> > present book with considerable regret.
> > Malcolm accused Russell of slipshod language and careless
> > thought, and of not even *trying* to think through the
> > philosophical questions seriously: 'The style is jaunty and
> > bouncy and reminds me of the patter of a conjurer who wishes
> > to entertain, dazzle and bewilder the customers. I have the
> > impression that the author, after writing philosophy for so
> > many years, is not tired, but callous.'
> > Malcolm's review was especially harsh, but the view that
> > _Human Knowledge_ was inferior to Russell's earlier work was,
> > and still is, almost unanimously held by professional
> > philosophers...."
> And following is the review of HoWP:
> RM> p. 296 "He kept a copy of Malcolm's review of _Human Knowledge_,
> > which he annotated, and a copy of an even more scathing review,
> > this time of _History of Western Philosophy_, written by another
> > of Wittgensteins friends and disciples, Yorick Smythies. HoWP,
> > Smythies wrote, 'embodies what seem to me the worst features of
> > Lord Russell's previous more journalistic works, but it is of
> > poorer quality than any of these'. Severly criticizing both
> > the style and the content of the book, Smythies concluded:
> > 'I fear that Lord Russell's book will teach successfully a
> > a popular substitute for thinking and for knowledge, and that
> > it will both appeal to and stimulate slipshod thinking.'"
> Russell gave a brief summary of his prejudices in a radio
> broadcast during the time he was writing HoWP:
> "I think philosophy has suffered four misfortunes in the
> world's history -- Plato, Aristotle, Kant, and Hegel. If
> they were eliminated, philosophy would have done very well."
> (quoted by Monk, vol. 2, p. 255).
Well, there's something to be said for that opinion, in various respects,
although I'm sure Russell has said it better in many places than I ever
could. Those folks all had some very, very odd ideas at times, I would
judge. However much philosophers often like to bandy about the term 'common
sense', they're often more lacking of it than some other folks. Such is the
way of philosophers.
Try reading the HWP for an overview, if you haven't. Although he
dislikes Russell, Monk highly recommends it :)
"A History of Western Philosophy remains unchallenged as the perfect
introduction to its subject. Russell...writes with the kind of verve,
freshness and personal engagement that lesser spirits would never have
permitted themselves. This boldness, together with the astonishing breadth
of his general historical knowledge, allows him to put philosophers into
their social and cultural context... The result is exactly the kind of
philosophy that most people would like to read, but which only Russell could
possibly have written." (Source: Amazon.com)
> Hegel is more controversial, but Plato, Aristotle, and Kant are
> in most lists of the greatest philosophers of all time.
And Russell is and has been on many such lists also.
> such company, it would be an honor to be disparaged by Russell.
I recall that Russell remarked somewhere something like "I'd rather argue
with my worst enemy in philosophy, than with someone entirely ignorant of
> References that correct the histories by Bochenski and by Kneale
> and Kneale:
> Hintikka, Jaakko (1997) "The place of C. S. Peirce in the history
> of logical theory," in Brunning & Forster, eds. (1997) _The Rule
> of Reason: The Philosophy of Charles Sanders Peirce_, University
> of Toronto Press, Toronto. pp. 13-33.
> Putnam, Hilary (1982) "Peirce the Logician" Historia Mathematica
> 9:290-301, reprinted in Putnam, _Realism with a Human Face_,
> Harvard University Press, Cambridge, MA. pp. 252-260.
> Quine, Willard Van Orman (1995) "Peirce's logic," in K.L. Ketner,
> ed., _Peirce and Contemporary Thought_, Fordham University Press,
> New York. pp. 23-31.
Thanks. On the stack.
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