Ex(~x=x), counterpart theory, and contingent identity
mitch
In another post I suggested that John Correy's ideas about non-reflexive
identity might have a "rational" formulation with respect to something
called counterpart theory.
(I think "categoreal" would be the right term here--in my humble
opinion, the apparent self-contradictions associated with John's
intuitions arise from vagueness and ambiguity associated with standard
presuppositions rather than "irrational" error on his part. For the
record, Langholm's investigation of determinability and
indeterminability in first-order contexts includes incoherent formulas.
Exclusion negation is not informationally well-behaved.)
The link,
http://www.sussex.ac.uk//Users/muralir/kct_final.pdf
is an article on counterpart theory in which the modal formula,
<>((x=x) & ~(x=x))
is discussed as well as what is actually done in counterpart theory to
exclude such an incoherent result.
:-)
mitch
John
The tie-in with contingent identity is as asserted by (1).
(1) AxAy(x=y -> (N(x=x & y=y) <-> N(x=y)))
That is, identical(x,y) are necessarily identical if, ond only if,
N(x=x) and N(y=y). Thus, granted that the Morning Star and the
Evening Star are each necessarily self-identical, if the Morning Star
and the Evening Star are identical, they are necessarily identical.
Hence the Morning Star and the Evening Star are necessarily
identical. The same is not true, however, for Benjamin Franklin
and the inventor of bifocals. For although Benjamin Franklin is
necessarily self-identical, the inventor of bifocals is not
necessarily self-identical. This is why Benjamin Franklin
and the inventor of bifocals, although identical are not
necessarily identical.
--John
Witt
"John" <john_...@yahoo.com> wrote in message
news:c37480a7.03110...@posting.google.com...
Hello, John and Mitch
I am not sure that my remarks are welcome here, I have had "enough" of the
remarks from (G. Frege)(II)
: stupid asshole, fucking bitch, and all of that childish rhetoric, I cannot
reply to him at all.
I assume that you guys are more mature than He.
As to John's claim that: (John Correy = John Correy) is necessarily true, I
disagree.
It is not sufficient to say, x=y <-> AF(Fx <-> Fy).
Rather, it is sufficient to say, x=y <->. E!x & E!y &AF( Fx <-> Fy).
After all, (ix: x=John Correy) is (John Correy).
E!(John Correy) is just as doubtful as E!(The poster who claims that
~Ax(x=x)).
We cannot assume that E!(John Correy) any more than we can assume
E!(Vulcan)!
(x=y) -> [](x=y) and E!x -> [](E!x), are consequences of Leibnitz's Law.
There are no contingent existences nor contingent identities.
Exisence and Identity (and Membership) are analytic properties.
Can we assume that: []Exists(George W. Bush)?
I don't think so, do you?
Surely, the existence of, George W. Bush, is contingent.
There cannot be an assumption that all names of purported physical entities
refer!
Santa dosen't work, Vulcan doesn't work, Pegasus dosen't work, etc.
"For although Benjamin Franklin is necessarily self-identical, the inventor
of bifocals is not
necessarily self-identical."
Benjamin Franklin = Benjamin Franklin, is not necessarily true.
(the inventor of bifocals)=(the inventor of bifocals), is also not
necessarily true, imo.
Necessary identity and existence, applies to logical/mathematical objects,
not to emprical objects, imho.
Thanks for your patience,
Witt
mitch
Witt wrote:
>
> Hello, John and Mitch
>
> I am not sure that my remarks are welcome here
That may make two of us.
> Necessary identity and existence, applies to logical/mathematical objects,
> not to emprical objects, imho.
The problem then becomes how one obtains criteria by which to distinguish.
Just so you understand why I pay so much attention to Kant, I will quote a
passage from The Amphiboly of Concepts of Reflection. In part, it is also an
answer to Frege's somewhat naive statement,
"Perhaps Kant uses the word 'object' in a rather
different sense;"
Kant writes:
"Before we leave the Transcendental Analytic we
must add some remarks which, although in themselves
not of special importance, might nevertheless be
regarded as requisite for the completeness of the
system. The supreme concept with which it is customary
to begin a transcendental philosophy is the division
into the possible and the impossible. But, since all
division presupposes a concept to be divided, a still
higher one is required, and this is the concept of an
object in general, taken problematically, without its
having been decided whether it is something or nothing.
As the categories are the only concepts which refer
to objects in general, the distinguishing of an object,
whether it is something or nothing, will proceed
according to the order and under the guidance of
the categories.
"1. To the concepts of all, many, and one there
is opposed the concept which cancels everything,
that is, none. Thus, the object of a concept to
which no assignable intuition whatsoever corresponds
is equal to nothing. That is, it is a concept without
an object (ens rationis), like noumena, which cannot
be reckoned among the possibilities, although they
must not for that reason be declared to be also
impossible; or like certain new fundamental forces,
which though entertained in thought without
self-contradiction are yet also in our thinking
unsupported by any example from experience,
and are therefore not to be counted as possible.
"2.Reality is something; negation is nothing, namely
a concept of the absence of an object, such as
shadow, cold (nihil privativum)
"3. The mere form of intuition, without substance,
is in itself no object, but the merely formal condition
of an object (as appearance), as pure space and
pure time (ens imaginarium). These are indeed
something, as forms of intuition, but are not
themselves objects which are to intuited.
"4. The object of a concept which contradicts
itself is nothing, because the concept is nothing,
is the impossible, e.g., a two-sided rectilinear
figure (nihil negativum)
"The table of this division of the concept of
nothing would therefore have to be drawn up
as follows. (The corresponding division of
something follows directly from it.).
[begin fixed width]
1
[quantity]
Empty concept without object
ens rationis
2 3
[quality] [relation]
Empty object of a concept Empty intuition without object
nihil privativum ens imaginarium
4
[modality]
Empty object without concept
nihil negativum
[end fixed width]
"We see that the ens rationis (1) is distinguished
from the nihil negativum (4) in that the fomer is
not to be counted among the possibilities because
it is a mere fiction (although not self-contradictory),
whereas the latter is opposed to possibility in that
the concept cancels itself. Both, however, are
empty concepts. On the other hand, the nihil
privativum (2) and the ens imaginarium (3) are
empty data for concepts. If light were not
given to the senses, we could not represent
darkness, and if extended beings were not perceived
we could not represent space. Negation and
the mere form of intuition, in the absence of
something real are not objects."
It would seem that Kant is in agreement with your humble opinion. However, he
uses the necessity ascribed to mathematical objects to ground the predictive
coherence observed in the empirical realm as objective knowledge.
Frege used a non-self contradictory concept (ens rationis) as the foundation of
his definition of number. I suspect that John's notion of "not self-identical"
corresponds to a notion of absence of an object (nihil privativum) since
absolute metalinguistic notions of identity permit us to apply the concept to
objects absent from the domain of discourse.
:-)
mitch
mitch
mitch wrote:
>
> Frege used a non-self contradictory concept (ens rationis) as the foundation of
> his definition of number. I suspect that John's notion of "not self-identical"
> corresponds to a notion of absence of an object (nihil privativum) since
> absolute metalinguistic notions of identity permit us to apply the concept to
> objects absent from the domain of discourse.
>
Defined parameters of usage, however, do not permit such application. That is why
it is so easy to "win" arguments on this matter.
But, these questions are important because they go back to presuppositions about
*knowing* or *believing that we know* a domain of discourse. Any reference to
truth-in-the-sense-of-what-really-is-true is an implicit reference to "the class
universe," "the set universe," or "the intended interpretation." These are all
definite descriptions without formal sense.
Perhaps everyone should solve the problem by not using them. :-)
Apparently, a large part of philosophy involves complaints about people not using
words they way that they were "meant" to be used. Damn people. :-)
:-)
mitch
Witt
"mitch" <mit...@rcnNOSPAM.com> wrote in message
news:3FAC0E35...@rcnNOSPAM.com...
>
>
> Witt wrote:
>
> >
> > Hello, John and Mitch
> >
> > I am not sure that my remarks are welcome here
>
> That may make two of us.
>
> > Necessary identity and existence, applies to logical/mathematical
objects,
> > not to emprical objects, imho.
>
> The problem then becomes how one obtains criteria by which to distinguish.
Empirical objects are physical, logical objects are not (physical).
Of course, we distinguish material objects by their materialism, and,
abstract objects by their abstractness.
Physical objects are those which our senses commit us to!
If we can admit physical objects into our universe of understanding
we can explain 'applied mathematics'.
I have not read Kant, yet, so I miss much of what you say.
Witt
mitch
Witt wrote:
>
> Physical objects are those which our senses commit us to!
>
>
Close enough for a start. :-)
:-)
mitch
John
(1) states necessary and sufficient conditions for the necessity of
(material) identity:
(1) AxAy(x=y -> (N(x=x & y=y) <-> N(x=y)))
If identicals x and y are necessarily self-identical, then--and
only then--is their identity a necessary one. Beyond this,
(1) states nothing more: from (1) it neither follows that
John Correy is necessarily self-identical nor that John
Correy is contingently self-identical--or indeed that John
Correy is self-identical at all. (1) does not say which of
the foregoing is the case.
We can sit around and argue about whether you or I are necessarily
self-identical. However, although logicians do argue about such
matters--Who else would bother?--it is not as logicians that they
argue but as metaphysicians, or as pataphysicians, or as what have
you?
So, when I claim that (e.g.) Benjamin Franklin is necessarily
self-identical while the inventor of bifocals is not, my main
warrant for this claim is that if Benjamin Franklin is necessarily
self-identical but the inventor of bifocals is not, then Benjamin
Franklin and the inventor of bifocals are not necessarily identical
(although they are identical). In other words, I take the necessary
self-identity of the former and the contingent self-identity of the
latter to constitute, together with (1), an *explanation* for the
contingent identity of Benjamin Franklin and the inventor of bifocals.
To this you might object that these would be also be contingently
identical if both Benjamin Franklin and the inventor of bifocals were
contingently self-identical. To which I would respond that identities
involving what linguistically oriented analytical philosophers refer
to as "rigid designators", are identities whose terms are necessarily
self-identical; whereas identities involving what such philosophers
refer to as "non-rigid designators", are identities whose terms are
contingently self-identical. Therefore, granted that I take rigid and
non-rigid designation as the linguistic marks of necessary and
contingent self-identity--putting the cart back behind the horse,
rather than approaching the matter bass ackwards as it is
fashionable to do these days--and granted that I take
"Benjamin Franklin" and "John Correy" to be 'rigid' designators
and 'the inventor of bifocals' to be 'non-rigid', I conclude
that the contingent identity of Benjamin Franklin and the inventor
of bifocals has as its basis the contingent self-identity of the
inventor of bifocals, while Benjamin Franklin is necessarily
self-identical.
As to whether physical or mathematical objects are contingently
self-identical or necessarily so, some sort of metaphysical argument
(rather than a logical one) warranting one or the other of these
conclusions would have to be made. I suspect that mathematical
properties are both essential in, and necessary to, mathematical
objects--but this is an intuition and nothing more.
Best regards,
John
PS It won't surprise me if Paul Holbach or G. Frege bring in talk
about 'scope', which I think is only peripherally relevant to
discussions of necessary and contingent identity.
mitch
John wrote:
Good point. That is one of the reasons that I have always known that there is a philosophical element involved with
discussions of identity.
Personally, I still have some doubts about self-identical and not self-identical. But I am now fairly certain that if
one accepts Frege's argument for the definition of number as attaching to an object, one should logically accept this
distinction.
>
> As to whether physical or mathematical objects are contingently
> self-identical or necessarily so, some sort of metaphysical argument
> (rather than a logical one) warranting one or the other of these
> conclusions would have to be made. I suspect that mathematical
> properties are both essential in, and necessary to, mathematical
> objects--but this is an intuition and nothing more.
>
> It won't surprise me if Paul Holbach or G. Frege bring in talk
> about 'scope', which I think is only peripherally relevant to
> discussions of necessary and contingent identity.
Well, when you claim that you can define scope so that self-identity is always implicit, you must deal with a Kantian
possibility--...
"The logical determination of a concept by reason
is based upon a disjunctive syllogism, in which the
major premiss contains a logical division (the division
of the sphere of a universal concept), the minor
premiss limiting this sphere to a certain part, and
the conclusion determining the concept by means
of this part."
--Immanuel Kant
"Critique of Pure Reason" A577/605
...--namely, the non-self-identical.
Of course, the concept is still a fiction in Kantian epistemology. However, it is also the reason for the apparent
complexity of his ideas--he does not trivially assert self-identity as self-evident.
:-)
mitch
Paul Holbach
> c37480a7.03110...@posting.google.com>...
Let me bring in a quote:
"One must distinguish between the claim that identity sentences are
contingent and the claim that the identity relation itself is
contingent. For the relation to be contingent, there need to be things
between which it holds merely contingent. For it to be necessary, it
has to be that if the relation obtains between things, it obtains
between those very things of necessity. [...] One can consistently say
that there are contingent identity sentences, though the relation
itself is necessary. Thus one could say that "The first
Postmaster-General of the US was the inventor of bifocal lenses." is
contingent and is an identity sentence, but that if we consider the
object, x, which is in fact referred to by "the first
Postmaster-General of the US" and the object, y, which is in fact
referred to by "the inventor of bifocal lenses", it is necessary that
x is identical to y."
[Sainsbury, M. (1995). Philosophical Logic. In A. C. Grayling (Ed.),
/Philosophy. A Guide Through the Subject/ (pp. 61-122). Oxford: Oxford
University Press. (p. 93)]
Regards
PH
G. Frege
> > [...] However, although logicians do argue about such matters
> > --Who else would bother?-- it is not as logicians that they
> > argue but as metaphysicians [...]
> >
Indeed. There's a wonderful story by K. Lambert concerning this point:
I know that the subject of ontology in particular and
metaphysics in general, can be very perplexing to some
logicians. A splendid example is the late logician Arthur
Smullyan, famous for his resolution of certain modal paradoxes
via the device of the scope of the modal operator. When asked to
take over a seminar in metaphysics for a colleague who suddenly
died, Smullyan spent the whole summer pondering what he could
possibly talk about. On the first day of the seminar in the
early fall, Smullyan was still perplexed, and sat at the head of
the seminar table hunched over in deep thought. He said
absolutely nothing for 15 minutes. The tension became unbearable
for some of the graduate students, and one of them tried to
leave the room softly. Just as he was about to cross the
threshold, Smullyan, head still lowered, said, "What's the
matter, Arbini? Don't you like metaphysics?"
(Karel Lamber, Free Logics, St. Augustin: Academia Verlag, 1997)
> >
> > It won't surprise me if Paul Holbach or G. Frege bring in talk
> > about 'scope' ...
> >
Right, see story above. ;-)
F.
John
> Well, when you claim that you can define scope so that self-identity >is always implicit, you must deal with a Kantian
> possibility--...
>
> "The logical determination of a concept by reason
> is based upon a disjunctive syllogism, in which the
> major premiss contains a logical division (the division
> of the sphere of a universal concept), the minor
> premiss limiting this sphere to a certain part, and
> the conclusion determining the concept by means
> of this part."
>
> --Immanuel Kant
> "Critique of Pure Reason" A577/605
I'm not sure how this cashes out where the logic of identity is
concerned. One might construe the classical "Ax[x=x <-> Ey(x=y)]"
as characterizing by exclusion the relation between self-identity and
identity-with-something: no self-identical is an identical-with-nothing,
and no identical-with-something is a non-self-identical.
In respect of the foregoing, ~AxEy(x=y) might be the minor premise
and ~Ax(x=x) the conclusion.
>
> ...--namely, the non-self-identical.
>
> Of course, the concept is still a fiction in Kantian epistemology.
> However, it is also the reason for the apparent
> complexity of his ideas--he does not trivially assert
> self-identity as self-evident.
>
--John
John
This sounds so much like what Kripke says, either in
"Identity and Necessity" or in _Naming and Necessity_, that I hope
Sainsbury cited him.
Of course, what Kripsbury says represents the Party Line on
contingent identity: There are *statements* of contingent
identity but no instances of contingent identity itself.
Oops! Before I forget, let me forestall the inevitable bUllrich-ism:
"Duh. You're not saying anything *new* you know.
'AxAy(x=y -> (N(x=x & y=y) <-> N(x=y)))' is a theorem of standard
quantified modal logic with identity. (Giggle)"
--John
Witt
Necessary self-identity is necessary existence.
(1a) AxAy(x=y -> ((N(E!x) & N(E!y)) <-> N(x=y))
If x and y are physical objects
I don't see necessity having sense wrt empirical things.
If p is necessarily true then, it is logically true, ie. true in all times
for which mind exists
(within a specified method of decision).
Facts are not necessary, even though they are true. They happen to be true
in our 'possible' world.
[]E!(John Corry), means, John Correy exists is logically true.
But, surely there are times at which John Correy, or any other proper name
of a physical object,
does not refer to anything.
Is E!(John Correy) true at (2,000,000 BC) or at (2,000,000 AD) ? Surely
not!
Can we deduce that E!(John Correy)?
For 'logical objects' it is true, eg. E!1 is true at all times (for which
minds exist).
But, I don't think that any physical object or physical relation, has
logical necessity, do you?
Your distinction between named objects and described objects needs
justification.
Do you axiomatically grant necessary existence to named objects and possible
existence to described objects?
Why is Ben Franlklin more 'necessary' than the inventor of bi-focals?
Note: Vulcan is the proper name given to a described planet which does not
exist.
Is there a 'necessary attribution' in the granting of proper names?
>
> As to whether physical or mathematical objects are contingently
> self-identical or necessarily so, some sort of metaphysical argument
> (rather than a logical one) warranting one or the other of these
> conclusions would have to be made. I suspect that mathematical
> properties are both essential in, and necessary to, mathematical
> objects--but this is an intuition and nothing more.
Our intuituitive 'metaphysics' entails that physical objects are indeed
contingent,
contingent on presence, within a specified time. There are no timeless
physical objects at all, are there?
Given the existence of mind, all empirical things are time dependent, and,
abstract
things are not.
Witt
mitch
"G. Frege" wrote:
Yes.
As much as you hate "quotes from the master" the passage in the postscript
includes the statement,
"If we omit a restricting condition, we would
seem to extend the scope of the concept that
was previously limited."
In "Foundations of Arithmetic" Frege writes:
"Kant obviously--as a result, no doubt, of defining
them too narrowly--underestimated the value of
analytic judgements, though it seems that he did
have some inkling of the wider sense in which I have
used the term. On the basis of his definition, the
division of judgements into analytic and synthetic
is not exhaustive. What he is thinking of is the
universal affirmative judgement; there we can speak
of a subject concept and ask--as his definition
requires--whether the predicate concept is contained
in it or not. But, how can we do this, if the subject
is an individual object? Or if the judgement is an
existential one?"
I guess Frege either missed or forgot about where Kant wrote:
"The supreme concept with which it is customary to
begin a transcendental philosophy is the division
into the possible and the impossible. But, since all
division presupposes a concept to be divided, a
still higher one is required, and this is the concept of
an object in general, taken problematically, without
its having been decided whether it is something or not."
It was preceded in the same section where Kant wrote:
"Before constructing any objective judgement we
compare concepts to find in them identity (of many
representations under one concept) with a view to
universal judgements, difference with a view to
particular judgements, agreement with a view to
affirmative judgements, opposition with a view to
negative judgements, etc. For this reason, we ought,
it seems to call the above-mentioned concepts,
concepts of comparison. If, however, the question
is not about logical form, but about the content of the
concepts, i.e., whether things are themselves
identical or different, in agreement or in opposition,
etc., then since the things can have a twofold relation
to our faculty of knowledge, namely, to sensibility
and to understanding, it is the place to which they
belong in this regard that determines the mode in
which they belong to one another."
For Kant, identity was a many-to-one relationship of representations to
concepts. But, with regard to the self-identity of conceptual
representations, he writes:
"Identity and Difference.-- If an object is presented
to us on several occasions but always with the same
inner determinations (qualitas et quantitas), then if it
be taken as object of pure understanding, it is always
one and the same, only one thing (numerica identitas),
not many. But, if it is appearance, we are not concerned
to compare concepts; even if there is no difference
whatever as regards the concepts, difference of spatial
position at one and the same time is still adequate ground
for numerical difference of the object, that is, of the
object of the senses."
On the other hand, Kant was answering Leibniz. So, he also writes,
"The Inner and the Outer.-- In an object of the pure
understanding, that only is inward which has no relation
whatsoever (so far as its existence is concerned) to
anything different from itself. It is quite otherwise with
a substantia phaenomenon in space; its inner determinations
are nothing but relations, and it itself is entirely made up
of mere relations. We are acquainted with substance in
space only through forces which are active in this and
that space, either bringing other objects to it (attraction)
or preventing them from penetrating into it (repulsion and
impenetrability) [...]. For this reason, Leibniz, regarding
substance as noumena, took away from them, by the
manner in which he conceived them, whatever might
signify outer relation, including also, therefore, composition,
and so made them all, even the constituents of matter,
simple subjects with powers of representation--in
a word, monads."
Strange how the naive notion of set has to do with composition. Equally
strange is the fact that a logicism based on Leibnizian and Humean
principles failed to ground mathematics. What is the evidence of this
failure? Set theory.
Frege was a geometer. The partition lattice of equivalence relations on a
set is a geometric lattice. So, it is not surprising that he recognized the
geometric relationships associated with equivalences and used that
understanding to formulate a formal equivalence by which to compare the
numerical identity of concepts. Quite honestly, it is a testament to his
genius since our knowledge of these things arise from his researches.
But, why on earth would anyone interested in understanding set theory as a
foundation for mathematics accept logical categories (first-order are
objects, second-order are sets of objects) that negate the very separative
properties Frege employed when distinguishing between concepts and the
extension of concepts in his own footnotes? It is the extension that
constitutes the object. It is not the relation that determines the object:
"A geometrical illustration will make the distinction
clear to intuition. If we represent the concepts (or
their extensions) by figures or areas in a plane, then
the concept defined by a simple list of characteristics
corresponds to the area common to all the areas
representing the defining characterisitics; it is
enclosed by segments of their boundary lines.
With a definition like this, therefore, what we do--
in terms of our illustration-- is to use the lines
already given in a new way for the purpose of
demarcating an area. Nothing essentially new,
however, emerges in the process. But the more
fruitful type of definition is a matter of drawing
boundary lines that were not previously given
at all."
Let me emphasize that once more "the more fruitful type of definition is a
matter of drawing boundary lines" and then ask what branch of mathematics
studies boundaries as they pertain to systems of sets?
The problem with identity is that there is an ambiguous correlation between
identity as a relation having global significance and the self-identity we
ascribe under objectification. In modern logic, presupposition of
self-identity appears to be paramount and the question of identity as a
relation appears to be best handled through scope and metalinguistic
hierarchies. But this lends itself to abuse when it comes to defining scope
with an arbitrary introduction of constants. In
http://plato.stanford.edu/entries/frege/index.html#atom
you will find the statement,
"Frege criticized the mathematical practise of
introducing notation to name (unique) entities
without first proving that there exist (unique)
such entities. He pointed out that such ‘creative
definitions’ were simply unjustified."
So perhaps I can be forgiven for thinking that modern notation and a
philosophy-last-if-at-all attitude has led to nonsensical positions
concerning my favorite avocation--mathematics.
It is true, that we could imagine some sort of mapping from "sets inside of
a model" to "objects outside of a model" to justify this sort of procedure.
Or, for that matter, we could simply take sets inside of a model to serve as
the objects of some target system. The latter idea should lead to nonsense
because of problems with content, while the former does nothing to solve
Frege's criticisms.
Personally, I have no problems with scope. Indeed, here is part of a
correspondence to William Elliot--the first newsgroup participant to accuse
me of being "anti-FOL,"
"Nevertheless, there is a problem with the
construction presented thus far. Namely, a
referent has not yet been assigned to the universal
quantifier. With the predicates now available,
however, it is a simple matter to introduce a
greatest class and a least class."
To the contrary, my problem is that someone taught everyone that scope was
well-understood and could be taken for granted.
It wasn't Kant. It wasn't Frege.
:-)
mitch
-----
p.s.
"All our knowledge falls within the bounds of possible
experience, and just this universal relation to possible
experience consists that transcendental truth which
precedes all empirical truth and makes it possible.
"But it is also evident that although the schemata of
sensibility first realize the categories, thay at the same
time restrict them, that is, limit them to conditions
which lie outside the understanding and are due to
sensibility. The schema is, properly, only the
phenomenon, or sensible concept, of an object
in agreement with the category. If we omit a restricting
condition, we would seem to extend the scope of
the concept that was previously limited. Arguing from
this assumed fact, we conclude that the categories
in their pure significance, apart from all conditions of
sensibility, ought to apply to things in general, as they
are, and not, like the schemata, represent them only
as they appear. They ought, we conclude, to possess
a meaning independent of all schemata, and of much
wider application. Now there certainly does remain
in the pure concepts of understanding, even after
elimination of every sensible condition, a meaning;
but it is purely logical, signifying only the bare unity
of the representations. The pure concepts can find
no object, and so can acquire no meaning which
might yield a concept of some object. [...] The
categories, therefore, without schemata, are merely
functions of the understanding for concepts; and
represent no object. This [objective] meaning they
acquire from sensibility, which realizes the understanding
in the very process of restricting it."
--Immanuel Kant
Paul Holbach
> Necessary self-identity is necessary existence.
Depends on what a theory of modal logic is intended to validate!
(Frankly, I惴 not a fan of the Barcan formulas ...)
In other words, the matter of scope is decisive.
I think the only acceptable EG of '[](a = a)' is as follows:
[](a = a) -> Ex[](x = a)
"If a is necessarily identical with itself, then there is something
such that a is necessarily identical with it."
Here, the box is interpreted *de re*, for the necessity lies in the
relation of identity and not in the possession of existence!
> []E!(John Corry), means, John Correy exists is logically true.
This is the de dicto reading, but one could as well read it de re,
i.e. as
"John Correy necessarily exists ."
In classical logic existential statements of the type 'Ex(x = a)' are
theorems which are considered necessary truths *by mere stipulation*,
for--by definition--non-referring singular terms are not allowed in
it.
But since there are in actual fact many non-referring singular terms
in ordinary language, free logic is preferable to classical logic. In
the former, no existential statement of the type 'E!a' [<-> Ex(x = a)]
is a theorem.
> But, I don't think that any physical object or physical relation, has
> logical necessity, do you?
From
[](John Correy = John Correy)
it merely follows that
Ex[](x = John Correy)
"There is something such that it is necessarily identical with John
Correy."
It does not follow that
"There necessarily exists something such that it is identical with
John Correy."
Once again, one must systematically differentiate between de dicto
necessities (necessary truths) and de re necessities (essential
properties)!
"We are going to take *de re* modality to express the essential
possession of a property by an individual. We use the '[]' to
represent *de re* as well as *de dicto* modality. In our symbolism, we
will continue to regard an occurrence of a modality sign as
representing modality *de dicto* if and only if that formula and only
that formula within the scope of modality contains no free occurrences
of any individual variables. The easiest way to apply this test is as
follows:
Delete all parts of the whole formula except for what appears within
the scope of the modal sign under consideration. If what remains
contains no free occurrences of any individual variable, the modality
is *de dicto*. Correspondingly, a modality is considered *de re* if
and only if the formula within the scope of the modality does contain
at least one free occurrence of an individual variable. Hence, in the
following example, the first (outermost) occurrence of '[]' is
modality *de dicto* and the second occurrence of '[]' is modality *de
re*: [](x)[](Cx -> Hx)
In giving rules for quantified modal logic, we frequently will be
forced to take stands on complex and hotly disputed metaphysical
issues involved in understanding essence and existence, and *de re*
and *de dicto* modality."
[...]
7. An individual has a property essentially just in case that
individual has that property in every world in which that individual
exists.
8. The *de re* understanding of 'necessarily' or '[]' is as
'essentially'."
[Konyndyk, K. (1986). /Introductory Modal Logic/. Notre Dame, IN:
University of Notre Dame Press. (pp. 83 + 91)]
Put that way, one can state the following without there being any
inconsistency whatsoever:
Ax(E!x -> []E!x)
"For every x, if x exists, then x is essentially existent."
That is, as long as x exists it is impossible for it to be
non-existent."
It may appear illogical at first sight, but thereæ„€ nothing wrong in
saying that existence is not an accidental but an essential property
of every contingent thing--of every existing contingent thing!
That existence is an essential property means that it is possible for
nothing to lose the property of existence without ceasing to exist!
There is a crucial difference between
"Every existent necessarily_is existent."
and
"Every existent is necessarily_existent.",
for
"Everything necessarily_has the property of existence."
certainly doesnæ„’ mean the same as
"Everything (necessarily) has the property of necessary_existence."
Itæ„€ a real blemish of modal logic that its formulas are very
ambiguous.
For example, in the case of self-identity it might be useful if we悲
write
"Ax(x []= x)"
instead of
"Ax[](x = x)" (both de re formulations).
But in my opinion Konyndykæ„€ interpretation (see above) is agreeable
already.
Regards
PH
Paul Holbach
> (Necessary self-identity is necessary existence.)
In my previous posting I wrote:
"Put that way, one can state the following without there being any
inconsistency whatsoever:
Ax(E!x -> []E!x)"
Correction:
It now seems to me that
Ax[](E!x -> E!x)
is the adequate formulation of
"Everything is essentially such that if it exists, it exists."
"Nothing can exist without existing."
PH
Paul Holbach
> V1qrb.188420$7B1....@news04.bloor.is.net.cable.rogers.com>...
> Necessary self-identity is necessary existence.
I don´t think so!
While
"Ax[]([](x = x) -> E!x)"
is true,
"Ax([](x = x) ->[]E!x)"
is not.
"Modality *de re* is modality thought of as applying to a thing
('res'), more precisely, as a way a thing possesses a property. [...]
Modality *de dicto* is the modality applied to a statement ('dictum').
It refers to the manner or mode of a statement´s being true."
[Konyndyk, K. (1986). /Introductory Modal Logic/. Notre Dame, IN:
University of Notre Dame Press. (p. 78f)]
"Now we are ready to try to give a properly de re reading of
statements like 'Socrates is essentially rational'. An individual is
said to have a given property essentially if and only if it is not
possible that the individual exist and lack the property.
Alternatively, an individual must have the property in every world in
which it exists. This account [...] has some implications of its own
that should be observed. The first implication is that anything that
exists has existence essentially."
[Konyndyk, K. (1986). /Introductory Modal Logic/. Notre Dame, IN:
University of Notre Dame Press. (p. 89)]
"I can sum up what I just argued by saying that in a modal sentence
the copula is not modally neutral. And isn´t this exactly the way we
read such sentences? Don´t they precisely say that a property is had
in the mode of necessity or in the mode of contingency? We start by
remarking that Socrates is a man and then, when our thoughts turn
modal, we want to know whether this property inheres in Socrates in
the necessary way or the contingent way: *how* is he a man? What we
are interested in is *mode of instantiation*. Modals are modes. To say
that modal words modify the copula is the linguistic counterpart of
the ontological doctrine that modality is a matter of the strength of
the instantiation relation: does the object in question instantiate
the predicated property only accidentally or is this a matter of
logical or metaphysical necessity ($). Thus, according to the copula
modifier theory, we do not work with an ontology of modal properties;
rather, we take the stock of non-modal properties and think of them as
possessed in different modes. If you like, the instantiation relation
is the only thing that gives rise to modal properties--the properties
of being instantiated necessarily or contingently.
($: What of 'a is possibly F', where this can be true even though the
object doesn´t actually have the property? In this case, obviously, we
cannot be saying in what mode the object *has* the property, since it
doesn´t have it. Instead, we are saying that the object possibly
instantiates the property, where again the modal expression modifies
the copula, as in 'Socrates possibly-is a man'.)"
[McGinn, C. (2000). /Logical Properties/. Oxford: Clarendon Press. (p.
77)]
In that sense the statements
"Everything is necessarily self-identical."
and
"Everything is necessarily existent."
mean that
"Everything instantiates the property of self-identity in the
necessary way."
and
"Everything (i.e. every existing thing!) instantiates the property of
existence in the necessary way."
The latter statement is, of course, not equivalent to
"Everything (i.e. every existing thing!) instantiates the property of
necessary existence in the necessary way." !
Regards
PH
Paul Holbach
> For although Benjamin Franklin is
> necessarily self-identical, the inventor of bifocals is not
> necessarily self-identical.
? - The inventor of bifocals existed, and whoever was the inventor of
bifocals was, of course, necessarily self-identical as long as he
existed!
PH
John
> > (material) identity:
> >
> > (1) AxAy(x=y -> (N(x=x & y=y) <-> N(x=y)))
>
> Necessary self-identity is necessary existence.
>
> (1a) AxAy(x=y -> ((N(E!x) & N(E!y)) <-> N(x=y))
>
> If x and y are physical objects
>
> I don't see necessity having sense wrt empirical things.
> If p is necessarily true then, it is logically true, ie. true in all times
> for which mind exists
> (within a specified method of decision).
> Facts are not necessary, even though they are true. They happen to be true
> in our 'possible' world.
>
> []E!(John Corry), means, John Correy exists is logically true.
> But, surely there are times at which John Correy, or any other proper name
> of a physical object,
> does not refer to anything.
> Is E!(John Correy) true at (2,000,000 BC) or at (2,000,000 AD) ? Surely
> not!
> Can we deduce that E!(John Correy)?
There's a difference between "N[Ex(John Correy = x)]", which is false,
and "Ex(N[John Correy = x])"--which is true. Not cashing assertions
out in primitive notation leads to a failure to discriminate such
readings, in the first of which "N" has operated on the sentence,
"Ex(John Correy = x)", to make a sentence; and in the second of
which "N" has operated on an open formula, "John Correy = x",
to yield the open formula "N(John Correy - x)", which then gets
'existentially' quantified.
--One reason: replacing the dotted lines in
"... need not have been ..."
by "the inventor of bi-focals" yields the true sentence, "the inventor
of bi-focals need not have been the inventor of bi-focals", while
replacement with "Benjamin Franklin" yields the false sentence,
"Benjamin Franklin need not have been Benjamin Franklin".
--Another reason: (1) is false but (2) is true.
1) Benjamin Franklin didn't have to be Benjamin Franklin.
2) The inventor of bifocals didn't have to be the inventor of bifocals.
>
> Note: Vulcan is the proper name given to a described planet which does not
> exist.
>
> Is there a 'necessary attribution' in the granting of proper names?
>
> >
> > As to whether physical or mathematical objects are contingently
> > self-identical or necessarily so, some sort of metaphysical argument
> > (rather than a logical one) warranting one or the other of these
> > conclusions would have to be made. I suspect that mathematical
> > properties are both essential in, and necessary to, mathematical
> > objects--but this is an intuition and nothing more.
>
> Our intuituitive 'metaphysics' entails that physical objects are indeed
> contingent,
> contingent on presence, within a specified time. There are no timeless
> physical objects at all, are there?
> Given the existence of mind, all empirical things are time dependent, and,
> abstract
> things are not.
>
> Witt
--John
Paul Holbach
> I'm not sure how this cashes out where the logic of identity is
> concerned. One might construe the classical "Ax[x=x <-> Ey(x=y)]"
> as characterizing by exclusion the relation between self-identity and
> identity-with-something: no self-identical is an identical-with-nothing,
> and no identical-with-something is a non-self-identical.
I悲 really like to learn from you what it is like for something to be
non-self-identical!
Let me try to give an answer myself:
- Two things that are identical with each other are not different from
each other--and vice versa.
- One thing that is identical with itself is not different from
itself, and vice versa
- One thing that is not identical with itself is different from
itself.
So what is it like for something to be different from itself?
What does
"a is different from itself."
mean in the first place?
In what respect could something be different from itself?
Well, it could
a) be numerically different from itself
or
b) be qualitatively different from itself.
ad (a):
I am convinced that the notion of 'numerical self-difference' does not
make any real sense!
Is it logically possible for one thing to be not one thing, i.e. to be
two or many things (of the same kind), or no thing at all?! - I donæ„’
think so!
I venture to assert that anybody employing the notion of numerical
self-difference doesnæ„’ really know what s/heæ„€ talking about!
And if 'numerical self-difference' is unintelligible, then 'numerical
non-self-identity' is unintelligible as well!
One thing must be one thing (of the same kind)!!!
In other words, everything is necessarily numerically self-identical!
ad (b):
If something is qualitatively self-different, it possesses some
properties not possessed by it and, hence, must be a self-inconsistent
object:
a /= a -> EP(Pa & ~Pa)
Is it possible for there to be any real objects that are
self-inconsistent?
Well, dialetheists such as Graham Priest believe it is--but I donæ„’!
Summary:
We have seen that the only intelligible meaning of
"non-self-identical" is "qualitatively
non-self-identical"/"qualitatively self-different", i.e. "possessing
some properties not possessed by it"!
Both the notion of numerical non-self-identity and the related one of
qualitative non-self-identity are *inherently* contradictory!!!
In my opinion that is the decisive reason why it is absolutely
impossible for there to be anything non-self-identical!!!
Regards
PH
P.S.:
To avoid a misunderstanding, in *pure* logic there may be consistent
formalistic theories of non-self-identity, but there cannot be any
ontological field of application for those!
mitch
Paul Holbach wrote:
>
> If something is qualitatively self-different, it possesses some
> properties not possessed by it and, hence, must be a self-inconsistent
> object:
>
In
http://ontology.buffalo.edu/smith/articles/mereotopology.htm
you will find a description of a universe that is a proper part of itself.
He calls it counterintuitive. I view it with about the same sense that you
suggest above.
I believe it to be a failure to understand the relationships between mereology
(part), set theory (element), and definite description (equals).
It is not a matter of seeing whose is right and who is wrong. It is not a
matter of this alternative foundation is better than that alternative
foundation. It is a matter of seeing how the identity puzzles are resolving
themselves.
Of course, everyone is so busy comparing deductive calculi and formal systems,
they seem to have forgotten that these things arose from philosophical
considerations that were being debated prior to the introduction of such
calculi.
To the best of my knowledge, the only person out here quoting large passages
from philosophical texts is me.
Perhaps Mr. Greene is right. Perhaps not.
:-)
mitch
Paul Holbach
> > "Witt" <oori...@yahoo.com> wrote in message news:<
> > V1qrb.188420$7B1....@news04.bloor.is.net.cable.rogers.com>...
> > Why is Ben Franlklin more 'necessary' than the inventor of bi-focals?
> --One reason: replacing the dotted lines in
> "... need not have been ..."
> by "the inventor of bi-focals" yields the true sentence, "the inventor
> of bi-focals need not have been the inventor of bi-focals", while
> replacement with "Benjamin Franklin" yields the false sentence,
> "Benjamin Franklin need not have been Benjamin Franklin".
>
> --Another reason: (1) is false but (2) is true.
>
> 1) Benjamin Franklin didn't have to be Benjamin Franklin.
>
> 2) The inventor of bifocals didn't have to be the inventor of bifocals.
You´re mistaken!
It is true that Benjamin Franklin didn´t have to be the inventor of
bifocals, and it is equally true that the inventor of bifocals didn´t
have to be Benjamin Franklin.
It is also true that the person actually referred to by "the inventor
of bifocals" might have not invented the bifocals; and if that were
the case, we certainly wouldn´t nowadays refer to *that* particular
person by "the inventor of bifocals". But all this does in no way
imply that whoever is the one we refer to by "the inventor of
bifocals" is not necessarily self-identical! Even if new historical
research showed surprisingly that the particular person we actually
refer to by "the inventor of bifocals" didn´t actually invent the
bifocals, that person we would then erroneously refer to by "the
inventor of bifocals" would nevertheless remain necessarily
self-identical!
The inventor of bifocals would be necessarily self-identical, even if
that particular person had not invented the bifocals. If that were the
case, then "the inventor of bifocals" would be a pseudo-description
functioning just like any arbitrary proper noun!
We can successfully identify particular objects even by means of
pseudo-descriptions.
"ixFx = ixFx"
is necessarily true, while
"FixFx" is contingently true.
Your basic mistake is to suppose that the contingency of 'FixFx'
implies the contingency of 'ixFx = ixFx'--but this is simply not the
case!
The way we linguistically refer to things doesn´t affect their being
necessary self-identical at all!
For example, we could decide to henceforth refer to Bill Clinton by
"the inventor of bifocals", knowing that not Clinton but Franklin
actually invented the bifocals (the latter might then be re-baptized
"the real inventor of bifocals" ;-)). But that wouldn´t turn
"[](the inventor of bifocals = the inventor of bifocals)"
into a false statement!
Your notion of 'contingent identity' is based on a misconstruction!
Regards
PH
George Greene
: In "Foundations of Arithmetic" Frege writes:
:
: "Kant obviously--as a result, no doubt, of defining
: them too narrowly--underestimated the value of
: analytic judgements, though it seems that he did
: have some inkling of the wider sense in which I have
: used the term.
In other words, I was right and you were wrong when I insisted
that Frege disagreed with Kant on a whole bunch of things,
ESPECIALLY about whether math was analytic or synthetic.
: On the basis of his definition, the
: division of judgements into analytic and synthetic
: is not exhaustive. What he is thinking of is the
: universal affirmative judgement; there we can speak
: of a subject concept and ask--as his definition
: requires--whether the predicate concept is contained
: in it or not. But, how can we do this, if the subject
: is an individual object? Or if the judgement is an
: existential one?"
:
:
:
: I guess Frege either missed or forgot about where Kant wrote:
:
: "The supreme concept with which it is customary to
: begin a transcendental philosophy is the division
: into the possible and the impossible. But, since all
: division presupposes a concept to be divided, a
: still higher one is required, and this is the concept of
: an object in general, taken problematically, without
: its having been decided whether it is something or not."
Kant is just full of shit as usual, here.
The kinds of things that CAN be "possible" or "impossble"
are themselves NOT *OBJECTS*. "An object in general",
no matter HOW hard you generalize it, STILL is NOT general
ENOUGH to ALSO be (not even POSSIBLY) A *PROPOSITION*, which IS
the kind of thing that can be "possible" or impossible.
: It was preceded in the same section where Kant wrote:
:
: "Before constructing any objective judgement we
: compare concepts to find in them identity (of many
: representations under one concept) with a view to
: universal judgements, difference with a view to
: particular judgements, agreement with a view to
: affirmative judgements, opposition with a view to
: negative judgements, etc. For this reason, we ought,
: it seems to call the above-mentioned concepts,
: concepts of comparison. If, however, the question
: is not about logical form, but about the content of the
: concepts, i.e., whether things are themselves
: identical or different, in agreement or in opposition,
: etc., then since the things
What things?
: can have a twofold relation
: to our faculty of knowledge, namely, to sensibility
: and to understanding,
Not exactly.
Some "things" can be both sensed and understood, some "things"
can be neither, and some can be one but not the other.
And in any case "thing" and "object" are BOTH entirely too
BROAD as terms here. You really do need different WORDS for
these 4 kinds of things.
: it is the place to which they
: belong in this regard that determines the mode in
: which they belong to one another."
But again, the German "belong to" of 1870 does not look
anything like what "belonging to" looks like in English in 2000.
: For Kant, identity was a many-to-one relationship of representations to
: concepts.
No, it wasn't.
Can't you just QUIT it with these lame paraphrases?
If you've quoted all the rest of this, why can't you quote Kant
defining identity that way?
: But, with regard to the self-identity of conceptual
: representations,
WTF is a "conceptual representation"??
In everything you have quoted so far,
"concepts" and "representations" are OPPOSED!
You talked above of a relationship BETWEEN *many*
"representations" UNDER *one* "concept"! Representations
are NOT "conceptual"! They are UNDER "concepts"!
he writes:
: "Identity and Difference.-- If an object is presented
: to us on several occasions but always with the same
: inner determinations (qualitas et quantitas), then if it
: be taken as object of pure understanding, it is always
: one and the same, only one thing (numerica identitas),
: not many. But, if it is appearance, we are not concerned
: to compare concepts; even if there is no difference
: whatever as regards the concepts, difference of spatial
: position at one and the same time is still adequate ground
: for numerical difference of the object, that is, of the
: object of the senses."
Well, of course, but that is ONE kind of identity versus TWO
different KINDS of THINGS -- one "sensed" and the other "understood";
one, if not concrete, then at least mentally/sensibly (as opposed
to physically) analogous to concrete, and the other abstract.
: On the other hand, Kant was answering Leibniz. So, he also writes,
:
: "The Inner and the Outer.-- In an object of the pure
: understanding, that only is inward which has no relation
: whatsoever (so far as its existence is concerned) to
: anything different from itself. It is quite otherwise with
: a substantia phaenomenon in space; its inner determinations
: are nothing but relations, and it itself is entirely made up
: of mere relations. We are acquainted with substance in
: space only through forces which are active in this and
: that space, either bringing other objects to it (attraction)
: or preventing them from penetrating into it (repulsion and
: impenetrability) [...]. For this reason, Leibniz, regarding
: substance as noumena, took away from them, by the
: manner in which he conceived them, whatever might
: signify outer relation, including also, therefore, composition,
: and so made them all, even the constituents of matter,
: simple subjects with powers of representation--in
: a word, monads."
You have absolutely no idea what you are talking about here.
Again, WHY?
Paul Holbach
> Of course, everyone is so busy comparing deductive calculi
> and formal systems,
> they seem to have forgotten that these things arose from philosophical
> considerations that were being debated prior to the introduction of such
> calculi.
Yes indeed. - And the philosophical debates continue even *after* the
successful construction of diverse calculi, each of which is provably
consistent.
"No calculus can decide a philosophical problem."
[Wittgenstein, L.. /The Big Typescript/ (Wiener Ausgabe, Bd. 11/
Viennese Edition, vol. 11). Vienna: Springer. (p. 362)]
"Philosophers have been infatuated with the quantifier. Understandably
so, since logicians showed the power and elegance of the predicate
calculus. And it is always tempting to want to put shiny new tools to
use. If we translate some idiom of natural language into quantifier
form we feel we know how it works; we feel we have tamed it."
[McGinn, C. (2000). /Logical Properties/. Oxford: Clarendon Press. (p.
69)]
Regards
PH
George Greene
: > Frege used a non-self contradictory concept (ens rationis) as the foundation of
: > his definition of number.
Frege's defintion of number was irrelevant bullshit.
That is all anyone needs to know about it.
: But, these questions are important because they go back to presuppositions about
: *knowing* or *believing that we know* a domain of discourse. Any reference to
: truth-in-the-sense-of-what-really-is-true is an implicit reference to "the class
: universe," "the set universe," or "the intended interpretation." These are all
: definite descriptions without formal sense.
That is just bullshit. You have no idea what "formal" means if you think THAT
to be the case. Most of the time, since we are talking about abstractions,
"the intended interpretation" IS ITSELF A FORMAL object. The natural numbers,
for example, are a formal object.
--
---
"It's difficult ... you need to be united to have any
strength, but internal issues have to be addressed."
--- E. Ray Lewis, on liberalism in America
Witt
"Paul Holbach" <paulholba...@freenet.de> wrote in message
news:881c8779.03111...@posting.google.com...
Not so, for Russell's description theory.
1. G(ix:Fx) <-> Ey(Ax(x=y <-> Fx) & Gy), see PM *14.1
2. E!(ix:Fx) <-> Ey(Ax(x=y <-> Fx), see *14.2
3. E!(ix:Fx) <-> F(ix:Fx)
4. E!(ix:Fx) <-> (ix:Fx)=(ix:Fx)
5. E!(ix:Fx) <-> EG(G(ix:Fx))
6. F(ix:Fx) <-> (ix:Fx)=(ix:Fx)
7. [](F(ix:Fx)) <-> []((ix:Fx)=(ix:Fx))
By: 6, |-p -> |-[]p, and, [](p <-> q) -> ([]p <-> []q).
8. []E!(ix:Fx) <-> []((ix:Fx)=(ix:Fx)).
Witt
G. Frege
wrote:
>
> Frege's definition of number was irrelevant bullshit.
>
SOMETIMES [?] you are really talking like a raving lunatic, George. (*sigh*)
Now it's rather enlightening to contrast your brain fart with the following
statements:
"The question 'What is a number?' is one which has been
often asked, but has only been correctly answered in our own
time. The answer was given by Frege in 1884, in his /Grundlagen
der Arithmetik/. Although this book is quite short, not difficult,
and of the very highest importance, it attracted almost no
attention, and the definition of number which it contains re-
mained practically unknown until it was rediscovered by the
present author in 1901."
(Bertrand Russell, Introduction to Mathematical Philosophy,
CHAPTER II: Definition of Number.)
"[...] it turns out that the notion of number
itself and likewise all other arithmetical concepts are definable
within the field of logic. It is, indeed, easy to establish the mean-
ing of the symbols designating individual natural numbers, such as
"0", "1", "2", and so on. The number 1, for instance, can be
defined as the number of elements of a class which consists of
exactly one element. (A definition of this kind seems to be in-
correct and contains apparently a vicious circle, since the word
"one", which is about to be defined, occurs in the definiens; but
actually no error is committed because the phrase "the class con-
sists of exactly one element" is considered as a whole and its
meaning has been defined previously.) Nor is it hard to define the
general concept of a natural number: a natural number is the
cardinal number of a finite class. We are, further, in a position
to define all operations on natural numbers , and to extend the
concept of number by the introduction of fractions, negative and
irrational numbers, without, at any place having to go beyond
the limits of logic. Furthermore, it is possible to prove all the
theorems of arithmetic on the basis of laws of logic alone (with the
qualification that the system of logical laws must first be enriched
by the inclusion of a statement which is intuitively less evident
than the others, namely the so-called AXIOM OF INFINITY, which
states that there are infinitely may different things.) This
entire construction is very abstract, it cannot easily be popularized
and does not fit into the framework of an elementary presentation
of arithmetic; in this book we also do not attempt to adapt our-
selves to this conception and treat numbers as individuals
and not as properties of classes of classes. But the mere fact that
it has been possible to develop the whole of arithmetic, including
the disciplines erected upon it -algebra, analysis, and so on-, as
a part of pure logic, constitutes one of the grandest achievements
of recent logical investigations.*)
*) The fundamental ideas in this field are due to FREGE; he developed
them for the first time in his interesting book: /Die Grundlagen der
Arithmetik/ (Breslau 1884). FREGE'S ideas found their systematical
and exhaustive realization in WHITEHEAD and RUSSELL's /Principia
Mathematica/.
(Alfred Tarski, Introduction to Logic)
"One often sees it stated that Frege's work was worthless because
of the inconsistency pointed out by Russell. In fact this is far
from the truth and one must view Frege as the person who made one
of the most important contributions to the foundations of
mathematics that has ever been made."
(J.J. O'Connor and E.F. Robertson, The MacTutor History of Mathematics
archive)
---------------------------
PLEASE, George, try to think next time BEFORE you start to write, ok? Probably
reading some literature would help? :-o
Beaney, M., 1997, The Frege Reader, Oxford: Blackwell
Bell, J. L., 1995, 'Type-Reducing Correspondences and Well-Orderings: Frege's
and Zermelo's Construction Re-examined', Journal of Symbolic Logic, 60: 209-221.
Bell, J. L., 1999, 'Frege's Theorem in a Constructive Setting', Journal of
Symbolic Logic, 64/2: 486-488
Bell, J.L., 1994, 'Fregean Extensions of First-Order Theories', Mathematical
Logic Quarterly, 40: 27-30; reprinted in Demopoulos 1995, 432-437.
Boolos, G., 1985, 'Reading the Begriffsschrift', Mind, 94: 331 - 344; reprinted
in Boolos (1998): 155-170. [Page references are to the reprint.]
Boolos, G., 1986, 'Saving Frege From Contradiction', in Proceedings of the
Aristotelian Society, 87 (1986/1987): 137 - 151; reprinted in Boolos (1998):
171-182. [Page references are to the original.]
Boolos, G., 1987, 'The Consistency of Frege's Foundations of Arithmetic', in On
Being and Saying, J. J. Thomson (ed.), Cambridge, MA: MIT Press, pp. 3-20;
reprinted in Boolos (1998): 183-201. [Page references are to the original.]
Boolos, G., 1990, 'The Standard of Equality of Numbers', in Meaning and Method:
Essays in Honor of Hilary Putnam, G. Boolos (ed.), Cambridge: Cambridge
University Press, pp. 261-277; reprinted in Boolos (1998): 202-219. [Page
references are to the original.]
Boolos, G., 1993, 'Whence the Contradiction?', in Aristotelian Society
Supplementary Volume, 67: 213-233; reprinted in Boolos (1998): 220-236.
Boolos, G., 1994, 'The Advantages of Honest Toil Over Theft', in Mathematics and
Mind, Alexander George (ed.), Oxford: Oxford University Press, 27-44; reprinted
in Boolos (1998): 255 - 274.
Boolos, G., 1997, 'Is Hume's Principle Analytic?', in Heck (1997a), 245-262;
reprinted in Boolos (1998): 301-314. [Page references are to the reprint.]
Boolos, G., 1998, Logic, Logic, and Logic, J. Burgess and R. Jeffrey (eds.),
Cambridge, MA: Harvard University Press.
Burgess, J., 1984, 'Review of Wright (1983)', The Philosophical Review, 93:
638-40.
Burgess, J., 1998, 'On a Consistent Subsystem of Frege's Grundgesetze', Notre
Dame Journal of Formal Logic, 39: 274-278
Demopoulos, W., (ed.), 1995, Frege's Philosophy of Mathematics, Cambridge:
Harvard University Press.
Demopoulos, W., forthcoming, 'Gottlob Frege', The Garland Encyclopedia of
Philosophy of Science, M. Hallett (ed.), Garland Press
Demopoulos, W., 1998, 'The Philosophical Basis of Our Knowledge of Number',
Nous, 32: 481-503
Dummett, M., 1991, Frege: Philosophy of Mathematics, Cambridge: Harvard
University Press.
Dummett, M., 1997, 'Neo-Fregeans: In Bad Company?', in Schirn (1997)
Field, H., 1984, 'Critical Notice of Crispin Wright: Frege's Conception of
Numbers as Objects', Canadian Journal of Philosophy, 14: 637-632; reprinted
under the title 'Platonism for Cheap? Crispin Wright on Frege's Context
Principle' in H. Field, Realism, Mathematics, and Modality, Oxford: Blackwell,
1989, pp. 147-170
Frege, G., 1980, Philosophical and Mathematical Correspondence, G. Gabriel, H.
Hermes, F. Kambartel, C. Thiel, and A. Veraart (eds. of the German edition),
abridged from the German edition by Brian McGuinness, translated by Hans Kaal,
Chicago: University of Chicago Press
Frege, G., 1974, The Foundations of Arithmetic, J. L. Austin (trans.), Oxford:
Basil Blackwell
Frege, G., 1967, The Basic Laws of Arithmetic, M. Furth (trans.), Berkeley:
University of California
Furth, M., 1967, 'Editor's Introduction', in G. Frege, The Basic Laws of
Arithmetic, M. Furth (translator and editor), Berkeley: University of California
Press, pp. v-lvii
Goldfarb, W., 2001, 'First-Order Frege Theory is Undecidable', Journal of
Philosophical Logic, 30: 613-616.
Hale, B., 1994, 'Dummett's Critique of Wright's Attempt to Resuscitate Frege',
Philosophia Mathematica, (Series III), 2: 122-147.
Hazen, A., 1985, 'Review of Crispin Wright's Frege's Conception of Numbers as
Objects', Australasian Journal of Philosophy, 63/2 (June): 251-254
Heck, R., 1996, 'The Consistency of Predicative Fragments of Frege's
Grundgesetze der Arithmetik, History and Philosophy of Logic, 17: 209-220
Heck, R., (ed.), 1997a, Language, Thought, and Logic: Essays in Honour of
Michael Dummett, Oxford: Oxford University Press.
Heck, R., 1997b, 'The Julius Caesar Objection', in Heck (1997a), 273-308
Heck, R., 1993, 'The Development of Arithmetic in Frege's Grundgesetze der
Arithmetik', Journal of Symbolic Logic, 58/2 (June): 579-600; reprinted in
Demopoulos (1995).
Heck, R., 1996, 'The Consistency of Predicative Fragments of Frege's
Grundgesetze der Arithmetik', History and Philosophy of Logic, 17: 209-220
Parsons, C., 1965, 'Frege's Theory of Number', Philosophy in America, M. Black
(ed.), Ithaca: Cornell University Press, pp. 180-203; reprinted with Postscript
in Demopoulos (1995), pp. 182-210
Parsons, T., 1987, 'The Consistency of the First-Order Portion of Frege's
Logical System', Notre Dame Journal of Formal Logic, 28: 161-68.
Pelletier, F.J., 2001, "Did Frege Believe Frege's Principle", Journal of Logic,
Language, and Information, 10/1: 87-114
Resnik, M., 1980, Frege and the Philosophy of Mathematics, Ithaca: Cornell
University Press
Rosen, G., 1993, 'The Refutation of Nominalism(?)', Philosophical Topics, 21/2:
149-186
Schirn, M., (ed.), 1997, Philosophy of Mathematics Today, Oxford: Oxford
University Press.
Schroeder-Heister, P., 1987, 'A model-theoretic reconstruction of Frege's
Permutation Argument', Notre Dame Journal of Formal Logic, 28/1: 69-79
Sullivan, P. and Potter, M., 1997, 'Hale on Caesar', Philosophia Mathematica,
(Series III), 5: 135-152.
Tabata, H., 2000, 'Frege's Theorem and His Logicism', History and Philosophy of
Logic, 21/4: 265-295
van Heijenoort, J., 1967, ed., From Frege to Gödel: A Sourcebook in Mathematical
Logic, Cambridge: Harvard University Press
Wehmeier, K., 1999, 'Consistent Fragments of Grundgesetze and the Existence of
Non-Logical Objects', Synthese, 121: 309-328
Whitehead, A. N. and Russell, B., 1912, Principia Mathematica Vol. II,
Cambridge: Cambridge University Press.
Wright, C., 1983, Frege's Conception of Numbers as Objects, Aberdeen: Aberdeen
University Press.
Wright, C., 1997a, 'Response to Dummett', in Schirn (1997)
Wright, C., 1997b, 'On the Philosophical Significance of Frege's Theorem', in
Heck (1997), 201-244
Zalta, E., 1999, 'Natural Numbers and Natural Cardinals as Abstract Objects: A
Partial Reconstruction of Frege's Grundgesetze in Object Theory', Journal of
Philosophical Logic, 28/6 (1999): 619-660
-----------
F.
mitch
"G. Frege" wrote:
> On 10 Nov 2003 16:40:46 -0500, George Greene <gre...@greeneg-cs.cs.unc.edu>
> wrote:
>
> >
> > Frege's definition of number was irrelevant bullshit.
> >
>
> SOMETIMES [?] you are really talking like a raving lunatic, George. (*sigh*)
>
Well, George is not too big on any sort of historical reference.
By the way, I figured out part of my difficulty. This should be the last time I
post any Kant in a reply to your posts. In any case, you did ask about
comparisons....
-----
In "Critique of Pure Reason," Kant presented
his categories. The category of modality was given by the following entries,
1. Possibility -- Impossibility
2. Existence -- Non-existence
3. Necessity -- Contingency
as derived from the respective judgement types
1. Problematic
2. Assertoric
3. Apodeictic
As I noted in news:<3FAC0E35...@rcnNOSPAM.com>, Kant included a "trivial"
discussion concerning the
completeness of his system based on the consideration of the concept of an object
taken problematically (without regard to existence and non-existence). This
discussion is based on the necessary relation of self-identity as expressed by
'a=a'. In turn, it is clear that the objective of this discussion is to establish
'sense' for the concepts _something_ and _nothing_ as they relate to the problematic
modal alternatives, possibility and impossibility.
It should be noted that Frege uses a similar device in his definition of number.
However, rather than consider the concept of object taken problematically, he
considers it assertorically--that is, with respect to existence and non-existence.
Do you see what is going on here?
Both of them use the necessity of 'a=a.' However, Kant uses it to ground
possibility whereas Frege uses it to ground existence. They are distinct concepts.
-----
Frege may be self-evident for you. He is not for me--especially since there is a
lot in "Foundations of Arithmetic" that is an answer to Kant. Anyway, as you agreed
before, it is a difficult matter finding philosophical logic starting from
mathematics. It would have been a lot easier if I had been sitting in a course on
Frege instead of Kant. :-)
:-)
mitch
G. Frege
wrote:
> >
> > In "Foundations of Arithmetic" Frege writes:
> >
> > "Kant obviously -as a result, no doubt, of defining
> > them too narrowly- underestimated the value of
> > analytic judgements, though it seems that he did
> > have some inkling of the wider sense in which I have
> > used the term.
> >
> In other words, I was right and you were wrong when I insisted
> that Frege disagreed with Kant on a whole bunch of things,
> ESPECIALLY about whether math was analytic or synthetic.
>
Sure, since...
F.
G. Frege
Holbach) wrote:
>
> "Philosophers have been infatuated with the quantifier. Understandably
> so, since logicians showed the power and elegance of the predicate
> calculus. And it is always tempting to want to put shiny new tools to
> use. If we translate some idiom of natural language into quantifier
> form we feel we know how it works; we feel we have tamed it."
>
> (McGinn, Logical Properties)
>
Indeed.
F.
ZZBunker
> Paul Holbach wrote:
>
> >
> > If something is qualitatively self-different, it possesses some
> > properties not possessed by it and, hence, must be a self-inconsistent
> > object:
> >
>
> In
>
> http://ontology.buffalo.edu/smith/articles/mereotopology.htm
>
> you will find a description of a universe that is a proper part of itself.
>
> He calls it counterintuitive. I view it with about the same sense that you
> suggest above.
>
> I believe it to be a failure to understand the relationships between mereology
> (part), set theory (element), and definite description (equals).
>
> It is not a matter of seeing whose is right and who is wrong. It is not a
> matter of this alternative foundation is better than that alternative
> foundation. It is a matter of seeing how the identity puzzles are resolving
> themselves.
>
> Of course, everyone is so busy comparing deductive calculi and formal systems,
> they seem to have forgotten that these things arose from philosophical
> considerations that were being debated prior to the introduction of such
> calculi.
Well, not everybody's doing that. Given that the philisophers
who invented deductive calculi were Lawyers, science has
bequeathed the whole issue to chemists and their mathema-lawyers.
Everybody with regular computers is working on robots.
Paul Holbach
> > "Paul Holbach" <paulholba...@freenet.de> wrote in message
> > news:881c8779.03111...@posting.google.com...
> > "ixFx = ixFx"
> >
> > is necessarily true, while
> >
> > "FixFx" is contingently true.
> Not so, for Russell's description theory.
>
> 1. G(ix:Fx) <-> Ey(Ax(x=y <-> Fx) & Gy), see PM *14.1
>
> 2. E!(ix:Fx) <-> Ey(Ax(x=y <-> Fx), see *14.2
>
> 3. E!(ix:Fx) <-> F(ix:Fx)
> 4. E!(ix:Fx) <-> (ix:Fx)=(ix:Fx)
> 5. E!(ix:Fx) <-> EG(G(ix:Fx))
> 6. F(ix:Fx) <-> (ix:Fx)=(ix:Fx)
>
> 7. [](F(ix:Fx)) <-> []((ix:Fx)=(ix:Fx))
> By: 6, |-p -> |-[]p, and, [](p <-> q) -> ([]p <-> []q).
>
> 8. []E!(ix:Fx) <-> []((ix:Fx)=(ix:Fx)).
You´re not wrong insofar as there really seems to be a problem for the
NFL-logician, for whom
(1) ixFx = ixFx -> E!ixFx
is impeccable.
If the normal rules of modal logic (particularly the rule of
necessitation and the rule of box distribution), are applied to (1),
we indeed get the following result:
[](ixFx = ixFx) -> []E!ixFx
But once again, we´re being confronted with the notorious ambiguity of
modal formulas.
Read de dicto we have:
"If 'ixFx = ixFx' is necessarily true, then 'E!ixFx' is necessarily
true."
The point is that in negative free logic "ixFx = ixFx" is not a
necessary truth, since it is possibly false (in case ~E!ixFx). There
is a possible world in which "ixFx = ixFx" is false because ixFx does
not exist in that world; and if there is a possible world in which
ixFx doesn´t exist, "E!ixFx" cannot be a necessary truth either, since
necessary truth is defined as truth in all worlds!
Even though "(ixFx = ixFx) -> E!ixFx" is a necessary NFL-truth, both
"ixFx = ixFx" and "E!ixFx" are no necessary NFL-truths!
But, luckily, that circumstance does not render the following
implication untrue:
[](ixFx = ixFx -> E!ixFx) -> ([](ixFx = ixFx) -> []E!ixFx)
The antecedent is true but neither the antecedent nor the consequent
of the consequent are true in NFL!
So we have
1 -> (0 -> 0)
1 -> 1
1 !
By the way, the rule of necessitation can also be unproblematically
applied to the following NFL-theorem:
Ax(x = x -> E!x)
[]Ax(x = x -> E!x)
[](Ax(x = x) -> AxE!x)
[]Ax(x = x) -> []AxE!x
Since in NFL both Ax(x = x) and AxE!x are axioms, everything´s fine
here!
What if we interpreted (1) *de re*?
In NFL
E!x <-> Ey(y = x)
holds (by definition), and so we can write
(1*) ixFx = ixFx -> Ey(y = ixFx)
and
[](ixFx = ixFx) -> Ey[](y = ixFx)
"If ixFx is necessarily self-identical, then there is something such
that it is necessarily identical with ixFx."
(It doesn´t mean "... there necessarily is something such that
..."!!!)
This de re interpretation is impeccable, for now both the antecedent
and the consequent are true!
Regards
PH
George Greene
: On 10 Nov 2003 16:40:46 -0500, George Greene <gre...@greeneg-cs.cs.unc.edu>
: wrote:
:
: >
: > Frege's definition of number was irrelevant bullshit.
: >
:
: SOMETIMES [?] you are really talking like a raving lunatic, George. (*sigh*)
Takes one to know one, ASSHOLE.
: Now it's rather enlightening to contrast your brain fart with the following
: statements:
YOU are the one brain-farting here, and TONS AND TONS OF philosophers
which you HAVE NOT bothered to read since they wrote AFTER 1953
HAVE PROVEN that.
: "The question 'What is a number?' is one which has been
: often asked, but has only been correctly answered in our own
: time. The answer was given by Frege in 1884, in his /Grundlagen
: der Arithmetik/. Although this book is quite short, not difficult,
: and of the very highest importance, it attracted almost no
: attention, and the definition of number which it contains re-
: mained practically unknown until it was rediscovered by the
: present author in 1901."
:
: (Bertrand Russell, Introduction to Mathematical Philosophy,
: CHAPTER II: Definition of Number.)
If you're quoting from 1901, you should of course expect to lose
to plenty of people who wrote later.
: "[...] it turns out that the notion of number
: itself and likewise all other arithmetical concepts are definable
: within the field of logic. It is, indeed, easy to establish the mean-
: ing of the symbols designating individual natural numbers, such as
: "0", "1", "2", and so on. The number 1, for instance, can be
: defined as the number of elements of a class which consists of
: exactly one element.
What IS that? "The number of elements of a class"?
And how is that not circular, given that 1 simply re-occurs on the
right as "one"?
: (A definition of this kind seems to be in-
: correct and contains apparently a vicious circle, since the word
: "one",
DUH.
: which is about to be defined, occurs in the definiens; but
: actually no error is committed because the phrase "the class con-
: sists of exactly one element" is considered as a whole and its
: meaning has been defined previously.)
Whoop-de-shit. The fact that you can make 1-to-1 correspondences among
an equivalence class of sets does NOT get you a definition of number.
You need a definition of 1 BEFORE you can have 1-to-1 correspondences.
: Nor is it hard to define the
: general concept of a natural number: a natural number is the
: cardinal number of a finite class.
But WHAT IS a cardinal number?
: We are, further, in a position
: to define all operations on natural numbers , and to extend the
: concept of number by the introduction of fractions, negative and
: irrational numbers, without, at any place having to go beyond
: the limits of logic.
Of course, but all that can be done with ANY OF MANY *DIFFERENT*
reductions of "number" to some PARTICULAR formalism, NONE of which
is ACTUALLY a number (since enshrining any one of them as such would
imply that ALL the others were NOT numbers).
: Furthermore, it is possible to prove all the
: theorems of arithmetic on the basis of laws of logic alone (with the
: qualification that the system of logical laws must first be enriched
: by the inclusion of a statement which is intuitively less evident
: than the others, namely the so-called AXIOM OF INFINITY,
OF COURSE, but again, ALL THIS IS IRRELEVANT because you can do it with
MANY DIFFERENT representations of numbers! Just because Frege's formalism
offered ONE such formalist characterization (AND EVEN AS A FORMALIST CHARACTERIZATION,
*IT* *ATE* *SHIT* -- the class of "all" sets with 3 elements DOES NOT EVEN EXIST
in ZFC!) does NOT mean he gave any coherent argument for why those PARTICULAR
numerals deserved to be "numbers" any more than any others that do the job
equally well.
: which
: states that there are infinitely may different things.) This
: entire construction is very abstract, it cannot easily be popularized
: and does not fit into the framework of an elementary presentation
: of arithmetic;
But here is the most telling refutation of YOUR failed point, FF:
: in this book
now, this is Russell talking:
: we also do not attempt to adapt ourselves to this conception, and treat numbers as individuals
^^^^^^^^^^^/\/\/\^^^^^^^^
: and not as properties of classes of classes.
In other words, this definition sucks so bad that Russell personally is not
even going to use it. Nice person to cite in SUPPORT of the definition,
DIPSHIT.
: But the mere fact that
: it has been possible to develop the whole of arithmetic, including
: the disciplines erected upon it -algebra, analysis, and so on-, as
: a part of pure logic, constitutes one of the grandest achievements
: of recent logical investigations.*)
Indeed it does, but it has NOT A DAMN THING TO DO with Frege's definition
of number AS a definition of number: the whole insight that this achievement
provides is basically that there IS NO SUCH THING as a "number": LOTS OF *DIFFERENT*
things will ALL serve EQUALLY well in the role!
: *) The fundamental ideas in this field are due to FREGE; he developed
: them for the first time in his interesting book: /Die Grundlagen der
: Arithmetik/ (Breslau 1884). FREGE'S ideas found their systematical
: and exhaustive realization in WHITEHEAD and RUSSELL's /Principia
: Mathematica/.
:
: (Alfred Tarski, Introduction to Logic)
Of course THE FUNDAMENTAL IDEAS are due to Frege, but that is NOT saying
that Frege had a coherent "definition" of "number"!
: "One often sees it stated that Frege's work was worthless because
: of the inconsistency pointed out by Russell. In fact this is far
: from the truth and one must view Frege as the person who made one
: of the most important contributions to the foundations of
: mathematics that has ever been made."
A perspective with which I entirely agree, so your decision to quote it here
in opposition to me is all the more stupid.
: (J.J. O'Connor and E.F. Robertson, The MacTutor History of Mathematics
: archive)
:
:
:
: PLEASE, George, try to think next time BEFORE you start to write, ok? Probably
: reading some literature would help? :-o
Please kiss my hairy light-brown ass, you ignorant fuck.
I HAVE READ OR TAKEN COURSES WITH MORE OF THESE BOOKS *THAN* *YOU* *HAVE*, ASSHOLE,
and IF YOU had read any of their relevant portions, YOU WOULD KNOW that nobody
respects Frege's "definition of number" AS that!
Just for a few of them
: Boolos, G., 1990, 'The Standard of Equality of Numbers', in Meaning and Method:
: Essays in Honor of Hilary Putnam, G. Boolos (ed.), Cambridge: Cambridge
: University Press, pp. 261-277; reprinted in Boolos (1998): 202-219. [Page
: references are to the original.]
You will certainly NOT find in THAT one any defense of Frege's definition of
number; LOTS of alternatives will be given.
: Boolos, G., 1997, 'Is Hume's Principle Analytic?', in Heck (1997a), 245-262;
: reprinted in Boolos (1998): 301-314. [Page references are to the reprint.]
That is about Frege's 2nd-order proof of Peano Arithmetic FROM Hume's principle,
and again, it does not restrict the definition of "number"; that proof goes
through under ANY framework in which sets between which there is a 1-1 correspondence
must have the same number of elements, IRrespective of what a NUMBER MIGHT BE!
: Boolos, G., 1998, Logic, Logic, and Logic, J. Burgess and R. Jeffrey (eds.),
: Cambridge, MA: Harvard University Press.
:
: Burgess, J., 1984, 'Review of Wright (1983)', The Philosophical Review, 93:
: 638-40.
:
: Burgess, J., 1998, 'On a Consistent Subsystem of Frege's Grundgesetze', Notre
: Dame Journal of Formal Logic, 39: 274-278
Again, you bring in literature about saving Frege's theorem from Russell's
paradox as though it could somehow be relevant to THIS issue, WHICH IT CAN'T.
: Demopoulos, W., 1998, 'The Philosophical Basis of Our Knowledge of Number',
: Nous, 32: 481-503
Regardless of what the philosophical basis of our knowledge might be, the ISSUE
is, WHAT IS a number? What is THE DEFINITION of number? Is Frege's original
definition correct? This article does NOT purport that it is!
You can lead a horse to water but you can't make him THINK.
: Resnik, M., 1980, Frege and the Philosophy of Mathematics, Ithaca: Cornell
: University Press
This one was rather cute; I took Phil.of.Math. FROM Resnik so I think
I can assure you that he does NOT believe that Frege's definition of number
"is the correct" definition of number.
: van Heijenoort, J., 1967, ed., From Frege to Gödel: A Sourcebook in Mathematical
: Logic, Cambridge: Harvard University Press
I took a whole course on the paradoxes based on this book and
you can certainly believe that nobody in it but Frege and Russell
accepts THAT definition of "number". The IMPORTANT stuff in any case is
what came AFTER Godel.
: Whitehead, A. N. and Russell, B., 1912, Principia Mathematica Vol. II,
: Cambridge: Cambridge University Press.
Again, irrelevant bullshit for the subject at hand.
: Wright, C., 1983, Frege's Conception of Numbers as Objects, Aberdeen: Aberdeen
: University Press.
Well, Wright could be wrong. Precisely as you (or Russell) just pointed out,
in the original Fregean treatment, there is TENSION between viewing ANYthing
as BOTH an "object" AND "a property of classes of classes".
: Wright, C., 1997a, 'Response to Dummett', in Schirn (1997)
:
: Wright, C., 1997b, 'On the Philosophical Significance of Frege's Theorem', in
: Heck (1997), 201-244
:
: Zalta, E., 1999, 'Natural Numbers and Natural Cardinals as Abstract Objects: A
: Partial Reconstruction of Frege's Grundgesetze in Object Theory', Journal of
: Philosophical Logic, 28/6 (1999): 619-660
But the WHOLE point of Zalta is that HE is doing numbers IN OBJECT THEORY,
AND NOT as properties of classes of classes! The fact that he is REconstructing
the theory does NOT mean he is endorsing the original conception of number!
HERE is some literature for YOU that IS relevant to the point under discussion:
"What Numbers Cannot Be", by Paul Benacerraf.
READ IT AND GET BACK TO US.
George Greene
: "G. Frege" wrote:
:
: > On 10 Nov 2003 16:40:46 -0500, George Greene <gre...@greeneg-cs.cs.unc.edu>
: > wrote:
: >
: > >
: > > Frege's definition of number was irrelevant bullshit.
: > >
: >
: > SOMETIMES [?] you are really talking like a raving lunatic, George. (*sigh*)
: >
:
: Well, George is not too big on any sort of historical reference.
That is not the point. The point is that FF IS THE ONE TALKING
LIKE A RAVING LUNATIC. Anybody who is going to quote something
from 1901 as authoratitive when other people have written
against it for a century is raving like a lunatic.
Anybody who is going to post references to dozens of articles
he hasn't even read in support of a thesis, WHEN MOST OF THEM
REFUTE HIS THESIS, is talking like a raving lunatic. Anybody
who thinks that later philosophers who built on Frege's work
and praised him for seminal insights (which certainly includes
most of us in this field) all must therefore necessarily
agree that he had a correct conception of what a number was
is talking like a raving lunatic. It is a lot easier to
collect bunches of historical records and pride yourself on
your collection THAN it is to actually engage the philosophical
issues under discussion, unless you are philosophical by temperament.
G. Frege
<gre...@greeneg-cs.cs.unc.edu> wrote:
Sorry, George, you got that wrong, I guess. The following is NOT by
Russell but by TARSKI (1964).
>
> Whoop-de-shit.
>
I'm sure that's the right (i.e. appropriate) way to argue with TARSKI.
:-)
>
> this is Russell talking...
>
No. TARSKI.
The fundamental ideas in this field are due to FREGE; he
developed them for the first time in his interesting book:
/Die Grundlagen der Arithmetik/ (Breslau 1884). FREGE's ideas
found their systematical and exhaustive realization in
WHITEHEAD and RUSSELL's /Principia Mathematica/.
(Alfred Tarski, Introduction to Logic, 1964)
>
> But WHAT IS a cardinal number?
>
If you would have read "Grundgesetze der Arithmetic" you probably would
know. :-)
"For additional reading we suggest a much older book in the
same direction [...] written originally in German, but now
available in English translation:
G. Frege. The Foundations of Arithmetic. A Logico-
mathematical Enquiry into the Concept of Number.
(English and German text paged in duplicate.)
Philosophical Library, Inc. New York, 1950, xii +
119 pp.
This book [...] was published first in 1884 and is now
regarded as a classic. It is exceedingly well written
and contains a stimulating presentation of basic ideas
which underlie the development of arithmetic as part of
the theory of classes."
(Alfred Tarski, Introduction to Logic, 1964)
Russell again:
"The question 'What is a number?' is one which has been
often asked, but has only been correctly answered in our own
time. The answer was given by Frege in 1884, in his
/Grundlagen der Arithmetik/. [...] this book is quite short,
not difficult, and of the very highest importance [...].
(Bertrand Russell, Introduction to Mathematical Philosophy,
CHAPTER II: Definition of Number.)
F.
G. Frege
>
> Well, George is not too big on any sort of historical reference.
>
Indeed. :-)
F.
mitch
George Greene wrote:
> mitch <mit...@rcnNOSPAM.com> writes:
>
> : "G. Frege" wrote:
> :
> : > On 10 Nov 2003 16:40:46 -0500, George Greene <gre...@greeneg-cs.cs.unc.edu>
> : > wrote:
> : >
> : > >
> : > > Frege's definition of number was irrelevant bullshit.
> : > >
> : >
> : > SOMETIMES [?] you are really talking like a raving lunatic, George. (*sigh*)
> : >
> :
> : Well, George is not too big on any sort of historical reference.
>
> That is not the point. The point is that FF IS THE ONE TALKING
> LIKE A RAVING LUNATIC. Anybody who is going to quote something
> from 1901 as authoratitive when other people have written
> against it for a century is raving like a lunatic.
Who has written against Frege's definition?
In the context of certain set theories, one can specifically define a cardinal number
as "the least ordinal such that..." because the ordinal sequence is well ordered.
But, in a general context of classes, the notion of a cardinal number corresponds
precisely with what Frege wrote.
Remember what I had tried to explain to you before? Language invariance???
Hmm... Since you keep demanding that everyone define everything, I have a question.
Could you please grace us with five or ten lines of whatever it is you take to be
first principles? Given your case history, we would be expecting it in FOL or FOL=.
Hmm... Perhaps you could include the first few iterations of metaprinciples as well
so we have some sense of how your infinite regress starts.
Thanks.
:-)
mitch
G. Frege
wrote:
>
> Frege's definition of number was irrelevant bullshit.
>
??? Huh?
You are really talking like a raving lunatic, George.
IF it WERE "irrelevant bullshit", PLEEEZE try to explain the following
statements by RUSSELL (1920, BTW 10 years AFTER the first publication of
PM), TARSKI (1964), J.J. O'Connor and E.F. Robertson (2002), and Edward
N. Zalta (2002).
"The question 'What is a number?' is one which has been
often asked, but has only been correctly answered in our own
time. The answer was given by Frege in 1884, in his /Grundlagen
der Arithmetik/. [...] this book is quite short, not difficult,
and of the very highest importance [...]"
(Bertrand Russell, CHAPTER II: Definition of Number, Introduction to
Mathematical Philosophy, 2nd ed. 1920)
"[...] it turns out that the notion of number
itself and likewise all other arithmetical concepts are definable
within the field of logic. It is, indeed, easy to establish the mean-
ing of the symbols designating individual natural numbers, such as
"0", "1", "2", and so on. The number 1, for instance, can be
defined as the number of elements of a class which consists of
exactly one element. (A definition of this kind seems to be in-
correct and contains apparently a vicious circle, since the word
"one", which is about to be defined, occurs in the definiens; but
actually no error is committed because the phrase "the class con-
sists of exactly one element" is considered as a whole and its
meaning has been defined previously.) Nor is it hard to define the
general concept of a natural number: a natural number is the
cardinal number of a finite class. We are, further, in a position
to define all operations on natural numbers , and to extend the
concept of number by the introduction of fractions, negative and
irrational numbers, without, at any place having to go beyond
the limits of logic. Furthermore, it is possible to prove all the
theorems of arithmetic on the basis of laws of logic alone (with the
qualification that the system of logical laws must first be enriched
by the inclusion of a statement which is intuitively less evident
than the others, namely the so-called AXIOM OF INFINITY, which
states that there are infinitely may different things.) This
entire construction is very abstract, it cannot easily be popularized
and does not fit into the framework of an elementary presentation
of arithmetic; in this book we also do not attempt to adapt our-
selves to this conception and treat numbers as individuals
and not as properties of classes of classes. But the mere fact that
it has been possible to develop the whole of arithmetic, including
the disciplines erected upon it -algebra, analysis, and so on-, as
a part of pure logic, constitutes one of the grandest achievements
of recent logical investigations.*)
*) The fundamental ideas in this field are due to FREGE; he developed
them for the first time in his interesting book: /Die Grundlagen der
Arithmetik/ (Breslau 1884). FREGE'S ideas found their systematical
and exhaustive realization in WHITEHEAD and RUSSELL's /Principia
Mathematica/."
(Alfred Tarski, Introduction to Logic, 3rd ed. 1964)
BTW, it's this book, which J. L. Kelly is referring to in his General
Topology (1955) as "Tarski's excellent exposition".
"For additional reading we suggest a much older book in the
same direction [...] written originally in German, but now
available in English translation:
G. Frege. The Foundations of Arithmetic. A Logico-
mathematical Enquiry into the Concept of Number.
(English and German text paged in duplicate.)
Philosophical Library, Inc. New York, 1950, xii +
119 pp.
This book [...] was published first in 1884 and is now
regarded as a classic. It is exceedingly well written
and contains a stimulating presentation of basic ideas
which underlie the development of arithmetic as part of
the theory of classes."
(Alfred Tarski, Introduction to Logic, 1964)
"It is reasonable to ask what prompted Frege to produce the
revolutionary /Begriffsschrift/. He wanted to have a precise way of
stating results and of proving them, for he realised the difficulties
of using ordinary language which was necessarily imprecise and
ambiguous. He stated in the Preface to the work that he wanted to prove
the basic truths of arithmetic "by means of pure logic". This aim makes
Frege the first to fully develop the main thesis of logicism, that
mathematics is reducible to logic. However, we should note that he only
applied the thesis to number theory and real analysis. His next major
work /Die Grundlagen der Arithmetik/ (The Foundations of Arithmetic),
published in 1884, was written to achieve the aim that he had clearly
set out in the Preface to the earlier work and present an axiomatic
theory of arithmetic.
After setting his agenda at the start of the /Grundlagen/, Frege looked
at the contributions made by previous mathematicians to two fundamental
questions:
What are numbers? What is the nature of arithmetical truth?
In fact he demolishes all earlier attempts to answer these questions
with brilliant clarity. Perhaps it will come as a surprise to readers
of this article to learn that all attempts to define "number" before
Frege contained logical errors. Indeed this is precisely what he
showed, for these earlier definitions had confused the idea of "number"
with that of "plurality". The plurality "two" refers to a collection of
two objects, for example two chairs, two pencils, two houses, etc. The
number "two" is, however, the class of all instances of the "plurality
two" and so is a "plurality of pluralities" and the logical error which
had been made in not recognising this meant that before Frege's
/Grundlagen/ nobody had managed to give a logically correct definition
of "number". Frege then went on to give his own definitions of the
basic concepts of arithmetic based purely on logic, and from these he
deduced, again using pure logic, the basic laws of arithmetic. Dummett
writes:
The work is fascinating even for those quite uninterested in the
philosophy of mathematics, since in the course of it many ideas
are presented which are of significance for the whole of
philosophy.
One often sees it stated that Frege's work was worthless because
of the inconsistency pointed out by Russell. In fact this is far
from the truth and one must view Frege as the person who made one
of the most important contributions to the foundations of
mathematics that has ever been made. In fact in many ways Russell
is correct when he wrote in his History of Western Philosophy:-
In spite of the epoch-making nature of [Frege's] discoveries,
he remained wholly without recognition until I drew attention
to him in 1903.
Frege's influence in the short term came through the work of Peano,
Wittgenstein, Husserl, Carnap and Russell. In the longer term, however,
Frege has become a major influence on the development of philosophical
logic and the man who seems to have been largely ignored by his
contemporaries has been avidly read by many in the second half of the
twentieth century, particularly after his works were translated into
English.
(J.J. O'Connor and E.F. Robertson, The MacTutor History of Mathematics
archive.
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Frege.html)
"In his seminal work /Die Grundlagen der Arithmetik/, Frege successfully
defined the notion of a 'cardinal number' in terms of the primitive
notion of an /extension/ or /set/. [...]
Using this definition, Frege derived many important theorems of number
theory. It was recently shown by R. Heck [1993] that, despite the
logical inconsistency in the system of his /Grundgesetze/, Frege
validly derived the Dedekind/Peano Axioms for number theory from a
powerful and consistent principle now known as Hume's Principle."
(Edward N. Zalta, Gottlob Frege, Stanford Encyclopedia of Philosophy
http://plato.stanford.edu/entries/frege)
George, you are an idiot, you know.
http://www.albinoblacksheep.com/flash/you.html
F.
G. Frege
>
> Who has written against Frege's definition?
>
Certainly not Ed Zalta:
"In his seminal work /Die Grundlagen der Arithmetik/, Frege successfully
defined the notion of a 'cardinal number' in terms of the primitive
notion of an /extension/ or /set/. [...]
Using this definition, Frege derived many important theorems of number
theory. It was recently shown by R. Heck [1993] that, despite the
logical inconsistency in the system of his /Grundgesetze/, Frege
validly derived the Dedekind/Peano Axioms for number theory from a
powerful and consistent principle now known as Hume's Principle."
(Edward N. Zalta, Gottlob Frege, Stanford Encyclopedia of Philosophy,
2002. http://plato.stanford.edu/entries/frege)
... or Bertie:
"The question 'What is a number?' is one which has been
often asked, but has only been correctly answered in our own
time. The answer was given by Frege in 1884, in his /Grundlagen
der Arithmetik/. [...] this book is quite short, not difficult,
and of the very highest importance [...]"
(Bertrand Russell, CHAPTER II: Definition of Number, Introduction to
Mathematical Philosophy, 2nd ed. 1920)
...or O'Connor and Robertson:
"[Frege's] next major work /Die Grundlagen der Arithmetik/ (The
Foundations of Arithmetic), published in 1884, was written to
achieve the aim that he had clearly set out in the Preface to
[Begriffsschrift] and present an axiomatic theory of arithmetic.
After setting his agenda at the start of the /Grundlagen/, Frege
looked at the contributions made by previous mathematicians to
two fundamental questions:
What are numbers? What is the nature of arithmetical truth?
In fact he demolishes all earlier attempts to answer these questions
with brilliant clarity. Perhaps it will come as a surprise to readers
of this article to learn that all attempts to define "number" before
Frege contained logical errors. Indeed this is precisely what he
showed, for these earlier definitions had confused the idea of
"number" with that of "plurality". The plurality "two" refers to a
collection of two objects, for example two chairs, two pencils, two
houses, etc. The number "two" is, however, the class of all instances
of the "plurality two" and so is a "plurality of pluralities" and the
logical error which had been made in not recognising this meant that
before Frege's /Grundlagen/ nobody had managed to give a logically
correct definition of "number". [...]
(J.J. O'Connor and E.F. Robertson, The MacTutor History of Mathematics
archive, 2002.
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Frege.html)
Actually, it's not clear to me how all this statements of rather competent
people fit together with G. Greene's claim:
"Frege's definition of number was irrelevant bullshit."
F.
Actually, we are told:
"For additional reading we suggest a much older book in the
same direction [...] written originally in German, but now
available in English translation:
G. Frege. The Foundations of Arithmetic. A Logico-
mathematical Enquiry into the Concept of Number.
(English and German text paged in duplicate.)
Philosophical Library, Inc. New York, 1950, xii +
119 pp.
This book [...] was published first in 1884 and is now
regarded as a classic. It is exceedingly well written
and contains a stimulating presentation of basic ideas
which underlie the development of arithmetic as part of
the theory of classes."
(Alfred Tarski, Introduction to Logic, 3rd ed. 1964)
"Exceedingly well written irrelevant bullshit containing a stimulating
presentation of basic ideas which underlie the development of arithmetic
as part of the theory of classes"? Sounds queer, can't help.
George Greene
: >
: > But WHAT IS a cardinal number?
: >
G. Frege <no_...@aol.com> writes:
:
: If you would have read "Grundgesetze der Arithmetic" you probably would
: know. :-)
No, you wouldn't.
Frege certainly didn't.
: "For additional reading we suggest a much older book in the
: same direction [...] written originally in German, but now
: available in English translation:
:
: G. Frege. The Foundations of Arithmetic. A Logico-
: mathematical Enquiry into the Concept of Number.
: (English and German text paged in duplicate.)
: Philosophical Library, Inc. New York, 1950, xii +
: 119 pp.
:
: This book [...] was published first in 1884 and is now
: regarded as a classic. It is exceedingly well written
: and contains a stimulating presentation of basic ideas
: which underlie the development of arithmetic as part of
: the theory of classes."
:
: (Alfred Tarski, Introduction to Logic, 1964)
Indeed, and if you had any skill at philosophy OR logic, you would KNOW
that NONE of that implies that this treatment gave a CORRECT
definition of "number".
: Russell again:
:
:
: "The question 'What is a number?' is one which has been
: often asked, but has only been correctly answered in our own
: time. The answer was given by Frege in 1884, in his
: /Grundlagen der Arithmetik/. [...] this book is quite short,
: not difficult, and of the very highest importance [...].
This, I repeat, is RUSSELL, NOT Tarski, who SHARED Frege's ERROR
of trying to define a number as the class of sets having that number
of elements. In modern set theories, that class does not exist, and that
is just one of many reasons why NOBODY, ESPECIALLY NOT TARSKI,
SUPPORTS *that* WRONG definition of "number".
George Greene
: On 10 Nov 2003 16:40:46 -0500, George Greene <gre...@greeneg-cs.cs.unc.edu>
: wrote:
:
: >
: > Frege's definition of number was irrelevant bullshit.
: >
: ??? Huh?
:
: You are really talking like a raving lunatic, George.
:
: IF it WERE "irrelevant bullshit", PLEEEZE try to explain the following
: statements by RUSSELL (1920, BTW 10 years AFTER the first publication of
: PM), TARSKI (1964), J.J. O'Connor and E.F. Robertson (2002), and Edward
: N. Zalta (2002).
:
:
: "The question 'What is a number?' is one which has been
: often asked, but has only been correctly answered in our own
: time. The answer was given by Frege in 1884, in his /Grundlagen
: der Arithmetik/. [...] this book is quite short, not difficult,
: and of the very highest importance [...]"
:
: (Bertrand Russell, CHAPTER II: Definition of Number, Introduction to
: Mathematical Philosophy, 2nd ed. 1920)
That is 1920 and it is irrelevant bullshit.
It is NOT supported by later treatments.
: "[...] it turns out that the notion of number
: itself and likewise all other arithmetical concepts are definable
: within the field of logic.
This, too, is irrelevant bullshit, as you would know if you read
Benacerraf '64. It is called the multiple reduction problem.
"The notion of number" is NOT "definable". Rather, it is REDUCIBLE
to certain formal entities in MULTIPLE ways, and the fact that there
are MORE than one of them PRECLUDES any one of them (and, therefore, ALL
of them) FROM BEING "the" definition of number. The word "notion" is used
FOR A REASON.
: It is, indeed, easy to establish the mean-
: ing of the symbols designating individual natural numbers, such as
: "0", "1", "2", and so on.
No, actually, in first-order logic, it is completely impossible, as was
shown by Godel.
: The number 1, for instance, can be
: defined as the number of elements of a class which consists of
: exactly one element.
That is circular.
: (A definition of this kind seems to be in-
: correct and contains apparently a vicious circle, since the word
: "one", which is about to be defined, occurs in the definiens; but
: actually no error is committed because the phrase "the class con-
: sists of exactly one element" is considered as a whole and its
: meaning has been defined previously.)
This alleges about classes that a "number" can be associated with them
but it STILL does not say what a number is.
: Nor is it hard to define the
: general concept of a natural number: a natural number is the
: cardinal number of a finite class.
But, again, in FOL, you cannot define "finite" either.
: We are, further, in a position
: to define all operations on natural numbers , and to extend the
: concept of number by the introduction of fractions, negative and
: irrational numbers, without, at any place having to go beyond
: the limits of logic.
This is obviously true, but it is done in the context of choosing some
particular reduction of the numbers to some particular sets. And the entities
that Frege and Russell talked about ARE NOT suitable or correct candidates,
and the treatments that Tarski uses in this book do NOT embrace the Fregean
or Russellian entities, WHICH YOU WOULD KNOW IF YOU HAD READ THE BOOK,
with an eye to engaging the philosophical contents thereof, INSTEAD OF
TRYING TO WIN A PUBLIC PISSING CONTEST.
: Furthermore, it is possible to prove all the theorems of arithmetic on the
: basis of laws of logic alone
This is circular, almost to the point of being false, in light of
Godelian incompleteness. Theorems are BY DEFINITION ONLY the things
that can be proved. You DON'T KNOW that something is a theorem of arithmetic
until AFTER you prove it. The class of things that are known to be true in true
first-order arithmetic must in fact include things that are NOT provable.
: (with the
: qualification that the system of logical laws must first be enriched
: by the inclusion of a statement which is intuitively less evident
: than the others, namely the so-called AXIOM OF INFINITY, which
: states that there are infinitely may different things.) This
: entire construction is very abstract, it cannot easily be popularized
: and does not fit into the framework of an elementary presentation
: of arithmetic;
Nor into first-order logic AT ALL, once you arithmetize provability.
: in this book we also do not attempt to adapt our-
: selves to this conception and treat numbers as individuals
: and not as properties of classes of classes.
In other words, Tarski, being a modern treatment, defines numbers AS SOMETHING
ELSE, NOT as Frege and Russell defined them. The prosecution rests.
: But the mere fact that
: it has been possible to develop the whole of arithmetic, including
: the disciplines erected upon it -algebra, analysis, and so on-, as
: a part of pure logic, constitutes one of the grandest achievements
: of recent logical investigations.*)
:
: *) The fundamental ideas in this field are due to FREGE; he developed
: them for the first time in his interesting book: /Die Grundlagen der
: Arithmetik/ (Breslau 1884). FREGE'S ideas found their systematical
: and exhaustive realization in WHITEHEAD and RUSSELL's /Principia
: Mathematica/."
:
: (Alfred Tarski, Introduction to Logic, 3rd ed. 1964)
That's true, but it does NOT imply that Tarski endorses or supports
that framework; he is just acknowledging what he built upon.
If he had said that the fundamental ideas in automotive engineering
were due to Daimler and Benz, would you conclude that he was advocating
driving 1903 model cars on the autobahn in preference to 2003 models?
Thghat is what you are doing here.
mitch
George Greene wrote:
> : >
> : > But WHAT IS a cardinal number?
> : >
>
> G. Frege <no_...@aol.com> writes:
> :
> : If you would have read "Grundgesetze der Arithmetic" you probably would
> : know. :-)
>
> In modern set theories, that class does not exist, and that
> is just one of many reasons why NOBODY, ESPECIALLY NOT TARSKI,
> SUPPORTS *that* WRONG definition of "number".
But, you are confusing the ontology of number with the ontology of
collections.
That the concept of number has a representation in set theory does not imply
that that representation is ontological with respect to the notion of number.
:-)
mitch
George Greene
Yo, Franz, you need to start putting the authors at the top,
not the bottom, if you are going to keep posting these long quotes.
G. Frege <no_...@aol.com> writes:
(quoting : (J.J. O'Connor and E.F. Robertson, The MacTutor History of Mathematics
: archive.)
: "It is reasonable to ask what prompted Frege to produce the
: revolutionary /Begriffsschrift/. He wanted to have a precise way of
: stating results and of proving them, for he realised the difficulties
: of using ordinary language which was necessarily imprecise and
: ambiguous. He stated in the Preface to the work that he wanted to prove
: the basic truths of arithmetic "by means of pure logic". This aim makes
: Frege the first to fully develop the main thesis of logicism, that
: mathematics is reducible to logic. However, we should note that he only
: applied the thesis to number theory and real analysis. His next major
: work /Die Grundlagen der Arithmetik/ (The Foundations of Arithmetic),
: published in 1884, was written to achieve the aim that he had clearly
: set out in the Preface to the earlier work and present an axiomatic
: theory of arithmetic.
:
: After setting his agenda at the start of the /Grundlagen/, Frege looked
: at the contributions made by previous mathematicians to two fundamental
: questions:
:
: What are numbers? What is the nature of arithmetical truth?
:
: In fact he demolishes all earlier attempts to answer these questions
: with brilliant clarity.
Of course. But his own attempt gets demolished pretty quickly as well.
: Perhaps it will come as a surprise to readers
: of this article to learn that all attempts to define "number" before
: Frege contained logical errors. Indeed this is precisely what he
: showed, for these earlier definitions had confused the idea of "number"
: with that of "plurality". The plurality "two" refers to a collection of
: two objects, for example two chairs, two pencils, two houses, etc. The
: number "two" is, however, the class of all instances of the "plurality
: two" and so is a "plurality of pluralities"
THAT is Frege's definition of number, and the PUREST proof of the FACT that
it is NOT supported generally in the modern context is that THIS CLASS DOES NOT
EXIST in the standard modern set theory (ZFC). Indeed, EVEN in class theories where
you CAN have a class of all SETS with two elements, the intent of this definition
is frustrated because you cannot subsume a pair of 2 CLASSES under that rubric.
: and the logical error which
: had been made in not recognising this meant that before Frege's
: /Grundlagen/ nobody had managed to give a logically correct definition
: of "number". Frege then went on to give his own definitions of the
: basic concepts of arithmetic based purely on logic, and from these he
: deduced, again using pure logic, the basic laws of arithmetic.
All well and good. And he did it WITHOUT a coherent definition of "number".
Which turns out to be possible BECAUSE it fundamentally DOES NOT MATTER
how you "define" "number", as long as you pick SOME reduction that fits into
the axiomatic framework appropriately.
: Dummett writes:
:
: The work is fascinating even for those quite uninterested in the
: philosophy of mathematics, since in the course of it many ideas
: are presented which are of significance for the whole of
: philosophy.
NOBODY AROUND HERE WAS CONTESTING the GENERAL importance of the work!
The QUESTION was about ITS DEFINITION OF NUMBER, WHICH HAS BEEN COMPLETELY
REJECTED in modern treatments!
: One often sees it stated that Frege's work was worthless because
: of the inconsistency pointed out by Russell.
That may have been often in 1964 but the modern perspective is entirely
different.
: In fact this is far from the truth
I know that and you deserve to be strung up for insinuating that I don't.
: and one must view Frege as the person who made one
: of the most important contributions to the foundations of
: mathematics that has ever been made.
All of which we already knew, but calling a "number" a class of all classes of the
same "number" (yeah, there is a REASON why that looks circular) WAS A BEGINNING,
NOT a correct ending.
: In fact in many ways Russell
: is correct when he wrote in his History of Western Philosophy:-
:
: In spite of the epoch-making nature of [Frege's] discoveries,
: he remained wholly without recognition until I drew attention
: to him in 1903.
:
: Frege's influence in the short term came through the work of Peano,
: Wittgenstein, Husserl, Carnap and Russell. In the longer term, however,
: Frege has become a major influence on the development of philosophical
: logic and the man who seems to have been largely ignored by his
: contemporaries has been avidly read by many in the second half of the
: twentieth century, particularly after his works were translated into
: English.
Right, we all get it now.
And we also get better candidates for how to represent numbers.
:
: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Frege.html)
:
:
: "In his seminal work /Die Grundlagen der Arithmetik/, Frege successfully
: defined the notion of a 'cardinal number' in terms of the primitive
: notion of an /extension/ or /set/. [...]
no, actually, he didn't; his notion of extension WAS undermined by Russell's paradox.
: Using this definition, Frege derived many important theorems of number
: theory.
Yes, THAT, he DID do, because, as I've said before, it DOESN'T MATTER WHAT,
specifically, you choose as your numbers, for THOSE derivations.
: It was recently shown by R. Heck [1993] that, despite the
: logical inconsistency in the system of his /Grundgesetze/, Frege
: validly derived the Dedekind/Peano Axioms for number theory from a
: powerful and consistent principle now known as Hume's Principle."
Right. That is "Frege's Theorem" and I have certainly known about THAT
longer than you have.
: George, you are an idiot, you know.
Neither that NOR Frege's & Russell's original definition of number is
supported by ANYthing you have posted.
George Greene
FF on what a number is:
: > : If you would have read "Grundgesetze der Arithmetic" you probably would
: > : know. :-)
After noting that the original Frege, followed by Russell,
thought of a number as a class of equipollent classes, I dismissed
that definition with
: > In modern set theories, that class does not exist, and that
: > is just one of many reasons why NOBODY, ESPECIALLY NOT TARSKI,
: > SUPPORTS *that* WRONG definition of "number".
mitch <mit...@rcnNOSPAM.com> writes:
: But, you are confusing the ontology of number with the ontology of
: collections.
DIPSHIT, *I* am the one who has taken courses in this, and YOU are not!
YOU MAY NOT address me in this tone of voice about this issue!
: That the concept of number has a representation in set theory does not imply
: that that representation is ontological with respect to the notion of number.
THAT IS *MY* POINT, FUCKFACE!
WHO THE FUCK DO YOU THINK YOU ARE, trying to *EXPLAIN*
*THAT* to *ME* ??
"number", to this day, REMAINS at the level of "notion"
PRECISELY BECAUSE NOBODY has yet given a philosophically
persuasive ontology. There are MANY DIFFERENT POSSIBLE
representations of numbers in set theories. There CANNOT
exist logical grounds for ensrhining any ONE of them
as " *the* ontological representative" in preference to the others.
But conventionally we can adopt whatever conventions we
choose. Getting from these multiple reductions DOWN to
"true platonic" "numbers" is something that STILL had
not been achieved A CENTURY AFTER Frege, LET ALONE by
Russell and Frege themselves (which is what FF was
idiotically and ignorantly quoting about).
G. Frege
<gre...@greeneg-cs.cs.unc.edu> wrote:
> >
> > But WHAT IS a cardinal number?
> >
>
> If you would have read "[Grundlagen] der Arithmetic" you probably would
> know. :-)
>
> No, you wouldn't.
> Frege certainly didn't.
>
> Indeed, and if you had any skill at philosophy OR logic, you would KNOW
> that NONE of that implies that this treatment gave a CORRECT definition
> of "number".
>
???
"In his seminal work /Die Grundlagen der Arithmetik/, Frege
successfully defined the notion of a 'cardinal number' [...]"
(Edward N. Zalta)
Too difficult for you to understand? :-o
F.
G. Frege
<gre...@greeneg-cs.cs.unc.edu> wrote:
"In his seminal work /Die Grundlagen der Arithmetik/, Frege
successfully defined the notion of a 'cardinal number' in terms
of the primitive notion of an /extension/ or /set/. [...]
(Edward N. Zalta)
>
> no, actually, he didn't
>
OH PLEEEZE shut up, idiot! :-(
>
> his notion of extension WAS undermined by Russell's paradox.
>
??? Huh?
"Using this definition, Frege derived many important theorems
of number theory. It was recently shown by R. Heck [1993]
that, despite the logical inconsistency in the system of his
/Grundgesetze/, Frege validly derived the Dedekind/Peano Axioms
for number theory from a powerful and consistent principle now
known as Hume's Principle."
(Edward N. Zalta)
BTW, shouldn't RUSSELL himself better KNOW about that than a brainless
asshole completely full of shit?
"The question 'What is a number?' is one which has been
often asked, but has only been correctly answered in our own
time. The answer was given by Frege in 1884, in his /Grundlagen
der Arithmetik/. [...] this book is quite short, not difficult,
and of the very highest importance [...]"
(Bertrand Russell, Introduction to Mathematical Philosophy,
2nd ed. 1920)
F.
P.S.
Ok, enough is enough. No more replies to "George Greene".
mitch
"G. Frege" wrote:
<snip>
> ...or O'Connor and Robertson:
>
> "[Frege's] next major work /Die Grundlagen der Arithmetik/ (The
> Foundations of Arithmetic), published in 1884, was written to
> achieve the aim that he had clearly set out in the Preface to
> [Begriffsschrift] and present an axiomatic theory of arithmetic.
>
> After setting his agenda at the start of the /Grundlagen/, Frege
> looked at the contributions made by previous mathematicians to
> two fundamental questions:
>
> What are numbers? What is the nature of arithmetical truth?
>
> In fact he demolishes all earlier attempts to answer these questions
> with brilliant clarity. Perhaps it will come as a surprise to readers
> of this article to learn that all attempts to define "number" before
> Frege contained logical errors. Indeed this is precisely what he
> showed, for these earlier definitions had confused the idea of
> "number" with that of "plurality". The plurality "two" refers to a
> collection of two objects, for example two chairs, two pencils, two
> houses, etc. The number "two" is, however, the class of all instances
> of the "plurality two" and so is a "plurality of pluralities" and the
> logical error which had been made in not recognising this meant that
> before Frege's /Grundlagen/ nobody had managed to give a logically
> correct definition of "number". [...]
>
> (J.J. O'Connor and E.F. Robertson, The MacTutor History of Mathematics
> archive, 2002.
Thanks for this quote, F.
When I read it, it helped me clear up some ideas.
Arguably, Kant made a significant contribution to the "ontology of object" by
formulating an epistemic framework. But, of course, it was not an ontology.
Part of a quote that George posted reads,
"In order to make epistemic logic possible, idealizations
were made concerning the reasoning capacities of the
agents, and modal systems were proposed to describe such
idealized agents."
Of course, Kant had to introduce *two* faculties of knowledge--namely, understanding
and sensibility--so that one faculty could serve as an idealization of reasoning
capacity and the other as a source of content. In addition, his "transcendental
object" is problematically investigated rather than assertorically introduced. With
respect to his own categories, he is treating objectification in terms of
possibility rather than existence. So, he did what George's quote suggests would
need to be done.
One of my problems with all of this is a poor grasp of philosophical terminology.
When I was reading the internet page on Frege in the Stanford Encyclopedia, it made
reference to his *ontology.* Of course, he is establishing an ontology of number.
Because this is an ontological assertion, it is existential and necessarily violates
Kant's delineation of topics derived from epistemic considerations.
I am reading German philosophers now with the hope of sorting some of this out.
Today, I ran across the following statement from Hegel,
"Thought, as Understanding, sticks to fixity of
characters and their distinctness from one another:
every such limited abstract it treats as having a
subsistence and being of its own."
Since Hegel does take the time to provide a fairly detailed discussion of Kant's
Critical Philosophy (with respect to ontological and metaphysical consequences), I
am more than happy to take this as a respectable source as to how the notion of the
identity predicate is reasonably taken to be a purely logical symbol. But, as the
exposition continues, the identity puzzles are seen to have a long history,
"In his more strictly scientific dialogues Plato
employs the dialectical method to show the
finitude of all hard and fast terms of understanding.
Thus in the Parmenides he deduces the many from
the one, and shows nevertheless that the many
cannot but define itself as the one. In this grand
style did Plato treat Dialectic."
Of course, I should also note that Hegel is treating logic as having three
moments--the Abstract, the Dialectical, and the Speculative. The first quote is
from the discussion of the Abstract moment.
As for the ubiquity of the one-to-many/many-to-one deduction he notes, consider this
simple definition in the introductory material from "General Relativity" by Wald,
"Basically a manifold is a set made up of pieces
that 'look like' open subsets of R^n such that
these pieces can be 'sewn together' smoothly. More
precisely, an n-dimensional, C^oo, real manifold M
is a set together with a collection of subsets {O_n}
satisfying the following properties:
"(1) Each (p e M) lies in at least one O_n, i.e., the {O_n} cover M.
"..."
So, a "one" and a "many" are taken together to compose each "piece." The
"individuals" are "sewn together" by virtue of presumed one-to-one, onto mappings
that agree where the "extensions" of each "piece" overlap.
(Sorry about all of the scare quotes. The precise formulation would just be ugly.)
Anyway, what I am getting at is that one must consider how to get from a
"problematic" treatment of an object to an "ontology of object." Clearly, Frege
advanced that by establishing methodologies incorporating a sensible "ontology of
number as object."
That is where the scope issues become so important.
Whatever categoreal errors I made when I was considering diagrammatic
representations of well-founded sets, I did realize that I was using the quantifiers
intentionally and compensated for that once I had sentences describing my
predication. Intentional use of a universal quantifier can support "parts" that are
not "pieces." But, the existential considerations applied to individuals do not.
Of course, that is one of the reasons why Frege was so strict in characterizing his
definition of an object in terms of concepts.
Clearly, then, if I want an ontology of object, I have to respect the scope of
quantification as understood with respect to existential import. In turn, that
brings me back to Hegel's "fixity of character" regardless of dialectical or
speculative deductions.
This still leaves the identity puzzles in question.
With respect to "parts" and "pieces" we can certainly investigate something along
the lines of David Lewis' "Parts of Classes." But, it seems to lead right back to
Plato's dialectical treatment.
Counterpart theory, on the other hand, sounds much more fruitful. Existential
import and fixity of character establish the need for considering possible worlds.
The modal structure is a natural development of Kant's epistemic topics. I know
Zalta has worked on a paper relating possible world theory and situation theory.
And, Seligman's paper on perspectives is reminiscent of numerous aspects discussed
in Kant. Of particular note, it involves objectification with respect to predictive
coherence of information relative to a perspectival domain. While integration of
these ideas are beyond my abilities, they seem to be have the useful features that
interrelate.
The question that remains, then, would be one of plentitude. That is, how does one
accommodate all the combinations of counterpart ensembles. Once again, my knowledge
of the structural set theory derived from Aczel's work is limited. But, if I am
correctly interpreting the bits and pieces relating to it of which I am aware, the
category of tree algebras (associated work directed to situation theory) would
essentially correspond to what one could deduce from a given perspectival domain. I
do have a paper that claims a supercompact cardinal to be a reasonable support for
simultaneously representing foundation and anti-foundation.
You wrote something before about not knowing my goals. All I can say is that
eighteen years ago I was in a set theory class and inadvertently stumbled on the
identity puzzles. I didn't know that they existed. I sat down and tried to
formalize my intuitions so that I could talk about them. Obviously, I did that
wrong. So, all that remains is to try and rationalize what I do know in the best
way that I can.
I may actually have a start on that now. But, as George has observed, I have lots
of bits and pieces that appear disconnected. The possibility of a coherent
explanation is, however, better than a paradox or a puzzle.
Thanks again.
:-)
mitch
G. Frege
<gre...@greeneg-cs.cs.unc.edu> wrote:
>
> ...in FOL, you cannot define "finite" either.
>
??? FOL??? Did ***anyone*** claim that Frege's definition happened in
FOL???
Oh, hell, George, you are just FULL OF SHIT, that's all.
F.
G. Frege
<gre...@greeneg-cs.cs.unc.edu> wrote:
>
> After noting that the original Frege, followed by Russell,
> thought of a number as a class of equipollent classes, I...
>
Complete bullshit.
F.
G. Frege
>
> Thanks for this quote, F.
>
I'm glad you like(d) it.
>
> Arguably, Kant made a significant contribution to the "ontology of object"
> by formulating an epistemic framework.
>
Indeed. Arguably, KANT was a great philosopher, who presented lots of
VALUABLE ideas. ( We should not forget that FREGE formulated (some of)
his ideas while referring to (some of) KANTs thoughts concerning this
topic(s). That's indeed worth to keep in mind. )
>
> Part of a quote that George posted reads,
>
Oh no! :-( [...]
>
> ...as George has observed, I have lots of bits and pieces
> that appear disconnected.
>
Actually, I would second that... :-)
>
> The possibility of [creating] a coherent explanation is
>
> 0 [even in your case].
I'm sure.
F.
G. Frege
Oh, BTW, George, did I already mention that
http://www.albinoblacksheep.com/flash/you.html
F.
mitch
George Greene wrote:
> FF on what a number is:
> : > : If you would have read "Grundgesetze der Arithmetic" you probably would
> : > : know. :-)
>
> After noting that the original Frege, followed by Russell,
> thought of a number as a class of equipollent classes, I dismissed
> that definition with
>
> : > In modern set theories, that class does not exist, and that
> : > is just one of many reasons why NOBODY, ESPECIALLY NOT TARSKI,
> : > SUPPORTS *that* WRONG definition of "number".
>
> mitch <mit...@rcnNOSPAM.com> writes:
> : But, you are confusing the ontology of number with the ontology of
> : collections.
>
> DIPSHIT, *I* am the one who has taken courses in this, and YOU are not!
> YOU MAY NOT address me in this tone of voice about this issue!
>
I read "Foundations of Arithmetic"
Frege does address himself to Kant's Critical Philosophy
I read "Critique of Pure Reason"
But--and you are quite correct about this--I have not read "Logic, Logic, Logic."
So, I do suffer from the fact that no one has told me how to think about what I have
not read.
>
> : That the concept of number has a representation in set theory does not imply
> : that that representation is ontological with respect to the notion of number.
>
> THAT IS *MY* POINT, FUCKFACE!
> WHO THE FUCK DO YOU THINK YOU ARE, trying to *EXPLAIN*
> *THAT* to *ME* ??
>
mitch
>
> "number", to this day, REMAINS at the level of "notion"
> PRECISELY BECAUSE NOBODY has yet given a philosophically
> persuasive ontology.
And, exactly why should someone be persuaded to take the symbol '=' as a logical
symbol again?
The fact of the matter is that philosophy has not been able to stablize the meanings
of the natural language words it uses over time. I will happily plead ignorance of
correct usage of these terms in situations where I see these terms being used with
consistent meaning in rational conversations.
Discussions with you barely meet the criterion of rationality, and, I would hate to
repeat the terms being used with consistent meaning.
If you wish to explain the hierarchy you are using to locate "notion," I would
gladly consider what you are saying. In the meantime, I will point out that Kant
took the use of definitions as starting points to be a mathematical methodology and
the deduction of definitions as finishing points to be a philosophic methodoolgy.
Given the nature of dynamical truth, I don't think I will last to the end of the
universe to get a definition of number with the expectation you are now portraying.
Be a logician or a philosopher. When you sit on the fence you sound like a soprano.
> There are MANY DIFFERENT POSSIBLE
> representations of numbers in set theories.
Right.
The attempt to ground mathematics under logicistic precepts failed. That does not
preclude interpreting Frege's definitions relative to developments in the German
philosophic tradition from the time of Kant. And, that is not to say that Kant did
not have predecessors who influenced him (I look forward to Leibniz, Wolff, and
Hume).
> There CANNOT
> exist logical grounds for ensrhining any ONE of them
> as " *the* ontological representative" in preference to the others.
Good point.
Who has been quoting philosophers around here?
Do you understand why you do not get a definite ontological representative?
>
> But conventionally we can adopt whatever conventions we
> choose.
safl;kjs lkjdfoaij noasdfo[awj fnosdnfowhnfowif sdfonnhw
Did you get that? Unfortunately, it really was composed as gibberish. But, it need
not have been.
> Getting from these multiple reductions DOWN to
> "true platonic" "numbers" is something that STILL had
> not been achieved A CENTURY AFTER Frege, LET ALONE by
> Russell and Frege themselves (which is what FF was
> idiotically and ignorantly quoting about).
Ahem... Why don't you define "true platonic." Then we'll talk.
I mean... DEFINE "true platonic."
:-)
mitch
G. Frege
> >
> > In modern set theories...
> >
M. Randall Holmes, talking about NFU set theory:
"[...] The natural numbers, and cardinal numbers in general,
are defined using Frege's definition."
George Greene:
"Frege's definition of number was irrelevant bullshit."
It's ALWAYS enlightening to contrast George's claims with REASONABLE
statements. (Obviously the definition of natural numbers in NFU are
just based on "irrelevant bullshit". Good to know. :-)
F.
G. Frege
>
> Frege's defintion of number was irrelevant bullshit.
>
Sure...
M. Randall Holmes, talking about NFU set theory:
"[...] The natural numbers, and cardinal numbers in general,
are defined using Frege's definition."
(This means that the definition of natural numbers in NFU is just based
on "irrelevant bullshit". Thanx for that fascinating insight, George.
You should tell Holmes about that! :-)
http://math.boisestate.edu/~holmes/
F.
G. Frege
>
> M. Randall Holmes, talking about NFU set theory:
>
>
> "[...] The natural numbers, and cardinal numbers in general,
> are defined using Frege's definition."
>
>
Indeed:
"In NF or NFU, one defines cardinals as equivalence classes of
sets under equinumerousness (in particular, Frege's definition
of natural number succeeds!) and ordinals as equivalence
classes of well-orderings under similarity."
(M. Randall Holmes)
Conclusion: George Greene's claims are just irrelevant bullshit.
F.
G. Frege
> >
> > In modern set theories, that class does not exist...
> >
"In NF or NFU, one defines cardinals as equivalence classes of
sets under equinumerousness (in particular, Frege's definition
of natural number succeeds!) and ordinals as equivalence
classes of well-orderings under similarity."
(M. Randall Holmes, author of "Naive Set Theory with A Universal
Set", 1998)
Conclusion: George Greene's claims [in this thread] are just irrelevant
brainless bullshit. Very often he simply doesn't know ANYTHING about a
certain topic he is pretending to talk about.
>
> Discussions with [him] barely meet the criterion of rationality
>
Indeed.
F.
G. Frege
"Frege's defintion of number was irrelevant bullshit.
That is all anyone needs to know about it."
(George Greene)
------------------------------------------------
"The question 'What is a number?' is one which has been
often asked, but has only been correctly answered in our own
time. The answer was given by Frege in 1884, in his /Grundlagen
der Arithmetik/."
(Bertrand Russell)
"In his seminal work /Die Grundlagen der Arithmetik/, Frege
successfully defined the notion of a 'cardinal number' [...]"
(Edward N. Zalta)
"[In NF and NFU] the natural numbers, and cardinal numbers in
general, are defined using Frege's definition."
(M. Randall Holmes)
George obviously has gone mad. (And it's getting worse!)
F.
G. Frege
>
> I read "Foundations of Arithmetic"
>
> But [...] I have not read [G. Boolos'] "Logic, Logic, [and] Logic."
>
The following articles are dealing with Frege's work(s):
Boolos, G., 1985, 'Reading the Begriffsschrift', Mind, 94: 331 - 344;
reprinted in Boolos (1998): 155-170.
Boolos, G., 1986, 'Saving Frege From Contradiction', in Proceedings of
the Aristotelian Society, 87 (1986/1987): 137 - 151; reprinted in Boolos
(1998): 171-182.
Boolos, G., 1987, 'The Consistency of Frege's Foundations of
Arithmetic', in On Being and Saying, J. J. Thomson (ed.), Cambridge, MA:
MIT Press, pp. 3-20; reprinted in Boolos (1998): 183-201.
Boolos, G., 1990, 'The Standard of Equality of Numbers', in Meaning and
Method: Essays in Honor of Hilary Putnam, G. Boolos (ed.), Cambridge:
Cambridge University Press, pp. 261-277; reprinted in Boolos (1998):
202-219.
Boolos, G., 1993, 'Whence the Contradiction?', in Aristotelian Society
Supplementary Volume, 67: 213-233; reprinted in Boolos (1998): 220-236.
Boolos, G., 1994, 'The Advantages of Honest Toil Over Theft', in
Mathematics and Mind, Alexander George (ed.), Oxford: Oxford University
Press, 27-44; reprinted in Boolos (1998): 255 - 274.
Boolos, G., 1997, 'Is Hume's Principle Analytic?', in Heck (1997a),
245-262; reprinted in Boolos (1998): 301-314.
--------------------------
Boolos, G., 1998, Logic, Logic, and Logic, J. Burgess and R. Jeffrey
(eds.), Cambridge, MA: Harvard University Press.
F.
mitch
George Greene wrote:
>
> All of which we already knew, but calling a "number" a class of all classes of the
> same "number" (yeah, there is a REASON why that looks circular) WAS A BEGINNING,
> NOT a correct ending.
>
George,
Go read the books.
Frege spoke in terms of objects and concepts. He took the extension of a concept to be an
object for the purpose of defining a number.
He assumed identity for objects in a pre-determined sense so that he could define a similarity
relation 'equals' between extensions of concepts to serve as a definition of numerical identity
between the objects he called numbers.
Classical first-order logic has nothing to do with this.
Frege's analysis of the problem involves natural language and the use of definite articles.
His definition constitutes singular reference. If you read about the related identity puzzles
that arise from his considerations of natural language, you will find that the questions
resolve around the "informativeness" of identity statements--not Frege's definition of number.
As presentations like Langholm's or Fitting's make clear, the combination of truth and
information may also be considered via Kleene truth, partial models, and, possibly,
non-compositional truth definitions.
I, for one, am happy that you know so much about contstructive interpretations of first-order
logic. Moreover, I think it is amazing how you believe that all perspectives on logic must be
reduced to the paradigm you understand and the presentations with which you agree. And, lest I
forget, I always appreciate your reminders that contemporary opinions are necessarily the
outcome of manifest destiny (Quite honestly, it may be the most exemplary conformity with
Freudian statism that I have ever seen.).
But, sometimes square pegs are square pegs, round holes are round holes, and a cigar is just a
cigar.
:-)
mitch
Witt
news:881c8779.03111...@posting.google.com...
> > "Witt" <oori...@yahoo.com> wrote in message news:<
> > IIVrb.16233$Rah1...@twister01.bloor.is.net.cable.rogers.com>...
> > > "Paul Holbach" <paulholba...@freenet.de> wrote in message
> > > news:881c8779.03111...@posting.google.com...
>
>
> > > "ixFx = ixFx"
> > >
> > > is necessarily true, while
> > >
> > > "FixFx" is contingently true.
>
>
> > Not so, for Russell's description theory.
> >
> > 1. G(ix:Fx) <-> Ey(Ax(x=y <-> Fx) & Gy), see PM *14.1
> >
> > 2. E!(ix:Fx) <-> Ey(Ax(x=y <-> Fx)), see *14.2
> >
> > 3. E!(ix:Fx) <-> F(ix:Fx)
> > 4. E!(ix:Fx) <-> (ix:Fx)=(ix:Fx)
> > 5. E!(ix:Fx) <-> EG(G(ix:Fx))
> > 6. F(ix:Fx) <-> (ix:Fx)=(ix:Fx)
> >
> > 7. [](F(ix:Fx)) <-> []((ix:Fx)=(ix:Fx))
> > By: 6, |-p -> |-[]p, and, [](p <-> q) -> ([]p <-> []q).
> >
> > 8. []E!(ix:Fx) <-> []((ix:Fx)=(ix:Fx)).
>
>
> You´re not wrong insofar as there really seems to be a problem for the
> NFL-logician, for whom
>
> (1) ixFx = ixFx -> E!ixFx
>
> is impeccable.
>
> If the normal rules of modal logic (particularly the rule of
> necessitation and the rule of box distribution), are applied to (1),
> we indeed get the following result:
>
> [](ixFx = ixFx) -> []E!ixFx
>
> But once again, we´re being confronted with the notorious ambiguity of
> modal formulas.
>
> Read de dicto we have:
>
> "If 'ixFx = ixFx' is necessarily true, then 'E!ixFx' is necessarily
> true."
By convention, imo, [](ixFx = ixFx) -> [](E!(ixFx)) is not ambigious.
It is read: If 'ixFx = ixFx' is necessarily true, then 'E!(ixFx)' is
necessarily true.
(ixFx) []= (ixFx) -> ([]E!)(ixFx), is also not ambiguious, it is read
If 'ixFx is necessarily equal to ixFx' then 'ixFx is necessarily existent'.
The 'de dicto' and 'de re' distinctions are needed and made specific for
expressions
containing described objects, independent of modal predications.
D1. f(G(ixFx)) defined f(Ey(Ax(x=y <-> Fx) & Gy)) {de dicto}
D2. (fG)(ixFx) defined Ey(Ax(x=y <-> Fx) & f(Gy)) {de re}
This distinction is required even for non-modal truth functions.
For example: ~(G(ixFx)) <-> ~(Ey(Ax(x=y <-> Fx) & Gy)), {de dicto}
and (~G)(ixFx) <-> Ey(Ax(x=y <-> Fx) & ~(Gy)), {de re}.
E!x -> {f(Gx) <-> (fG)x}
E!(ixFx) -> {f(G(ixFx) <-> (fG)(ixFx)}
>
> The point is that in negative free logic "ixFx = ixFx" is not a
> necessary truth, since it is possibly false (in case ~E!ixFx). There
> is a possible world in which "ixFx = ixFx" is false because ixFx does
> not exist in that world; and if there is a possible world in which
> ixFx doesn´t exist, "E!ixFx" cannot be a necessary truth either, since
> necessary truth is defined as truth in all worlds!
>
> Even though "(ixFx = ixFx) -> E!ixFx" is a necessary NFL-truth, both
> "ixFx = ixFx" and "E!ixFx" are no necessary NFL-truths!
Certainly, eg. (the present king of France)=(the present king of France) is
false and,
E!(the present king of France) is also false.
It seems to me that this version of NFL is simply description logic,
contained within classical logic.
ixFx = ixFx, is invalid. i.e. ~[](ixFx = ixFx).
E!x <-> ixFx = ixFx, is valid. i.e. [](E!x <-> ixFx = ixFx).
>
> But, luckily, that circumstance does not render the following
> implication untrue:
>
> [](ixFx = ixFx -> E!ixFx) -> ([](ixFx = ixFx) -> []E!ixFx)
>
> The antecedent is true but neither the antecedent nor the consequent
> of the consequent are true in NFL!
>
> So we have
>
> 1 -> (0 -> 0)
> 1 -> 1
> 1 !
>
> By the way, the rule of necessitation can also be unproblematically
> applied to the following NFL-theorem:
>
> Ax(x = x -> E!x)
> []Ax(x = x -> E!x)
> [](Ax(x = x) -> AxE!x)
> []Ax(x = x) -> []AxE!x
>
> Since in NFL both Ax(x = x) and AxE!x are axioms, everything´s fine
> here!
These are theorems of classical modal logic, aren't they?
>
>
> What if we interpreted (1) *de re*?
>
> In NFL
>
> E!x <-> Ey(y = x)
>
> holds (by definition), and so we can write
>
> (1*) ixFx = ixFx -> Ey(y = ixFx)
(1*) ixFx = ixFx -> Ey(y = ixFx), is valid in description logic but,
it is not the case that (ixFx) is a value of the individual variable unless
E!(ixFx). i.e. it is not an instance of x=x -> Ey(y=x).
|- Ey(y = ixFx) <-> Ey(Ax(x=y <-> Fx)), because, y = ixFx <-> Ax(x=y <->
Fx).
|- E!(ixFx) <-> Ey(y = ixFx)
> and
>
> [](ixFx = ixFx) -> Ey[](y = ixFx)
I don't think so.
[](Ey(y = ixFx)) -> Ey([](y = ixFx)), is invalid.
[](E!(ixFx)) -> Ey([](y = ixFx)), is invalid.
[](E![ixFx]) -> Ey([](y = ixFx)), is valid.
[ixFx] defined (iy: [](Ax(x=y <-> Fx))).
Witt
>
> "If ixFx is necessarily self-identical, then there is something such
> that it is necessarily identical with ixFx."
>
> (It doesn´t mean "... there necessarily is something such that
> ..."!!!)
>
> This de re interpretation is impeccable, for now both the antecedent
> and the consequent are true!
>
> Regards
> PH
G. Frege
Hi mitch!
Good post. Actually, one of the most comprehensible posts by you I've
read so far! :-)
>
> Frege spoke in terms of objects and concepts. He took the extension of a concept to be an
> object for the purpose of defining a number.
>
Right.
>
> Classical first-order logic has nothing to do with this.
>
Indeed. Actually, we need some sort of _2nd order_ logic to restate
Frege's system... See:
http://plato.stanford.edu/entries/frege-logic/
>
> Frege's analysis of the problem involves natural language
> and the use of definite articles.
>
...in /Grundlagen der Arithmetic/ only.
Later (in /Grundgesetze der Arithmetik/) he stated his considerations in
an extended version of the language and the system originally developed
in his /Begriffsschrift/.
>
> I, for one, am happy that you know so much about constructive interpretations of
> first-order logic.
>
Right. The only PROBLEM: the interpretation(s) he very often DOESN'T
know are just the /most common/ (i.e. generally shared) ones... :-)
>
> Moreover, I think it is amazing how you believe that all perspectives on logic
> must be reduced to the paradigm you understand and the presentations with
> which you agree.
>
Indeed. Want to see an example? :-)
"[...] Nowadays the LOGIC (if it is FOL=) comes with 2 more
INFERENCE RULES defining '=' and there are NO "axioms" for
'='!"
(George Greene)
Incredible uninformed nonsense. How the hell does such a guy even DARE
to write here?!
Obviously he NEVER learned ANYTHING beyond the stuff he was told in the
very classes he attended... :-(
>
> Go read the books.
>
Indeed, this probably is a rather good idea. If he WOULD do that, he
possibly would read things like:
"Adequate axioms of identity are 'x = x' and all instances
of the schema 'x = y -> (phi(x) -> phi(y))' of /substitutivity
of identity/. From these, by quantification theory, all laws
of identity follow."
(W. V. O. Quine)
Just for the benefit of all interested readers:
"2 Characterizing identity
The standard way of considering the concept of identity in mathematics
is linked to a tradition which goes back at least to Leibniz. Frege took
possession of Leibniz's dictum /Eadem sunt quorum unum potest substitui
alteri salva veritate/ to motivate his 'definition' of identity
(equality) in /Die Grundlagen der Arithmetik/ (Frege 1884, §65) but, as
remarked by A. Church, he made a confusion between use and mention, for
what is to be substituted are not the things themselves, but their
names: "/things/ are identical if the /name/ of one can be substituted
for that of the other without loss of truth" (Church 1956, p. 300).
As Church also notes, the confusion between use and mention was
corrected nine years later in the Vol. 1 of Frege's /Grundgesetze der
Arithmetik/ (Frege 1893) but, there, instead of a definition, what we
find is an axiom, written by Church as phi(x = y) -> phi((F)[F(x) ->
F(y)]). Previously in the /Begriffsschrift/ (Frege 1879), Frege used
axioms which can be written as (for any a and b):
(F1) a = a
(F2) a = b -> (F(a) -> F(b))
[...] The axioms (F1) and (F2) are essentially those used in first order
languages when identity is taken as an element of the underlying logic
but, for expressing the general case (involving formulas in general),
F(a) must be read as a formula whatever, while F(b) comes from F(a) from
the substitution of some free occurrences of a by b, provided that b is
free for a in F(a), where a and b are terms whatever; see (Mendelson
1997, p. 95).
[...] let us recall that F1 expresses the intuitive idea that 'every
object is identical to itself', and is known as the Reflexive Law of
Identity, while F2 is the Substitutivity Law [of Identity]."
(D. Krause, A. M. N. Coelho, Identity, Indiscernibility, and
Philosophical Claims, 2002 or 3)
George again:
"it bears [...] stressing that the back-arrow in the
conditional of the 2nd law is absolutely necessary."
(George Greene)
Huh?
"A7: x = x
A8: (x = y -> (alpha -> beta))"
Stefan Bilaniuk, A Problem Course in Mathematical Logic
(Version 1.6, 2003)
":
MA5 a = a
MA6 a = b -> (A -> A(a//b))
: "
K. LAMBERT, Free Logics, 1997.
etc. etc.
>
> [...] I always appreciate your reminders that contemporary opinions are necessarily the
> outcome of manifest destiny (Quite honestly, it may be the most exemplary conformity with
> Freudian statism that I have ever seen.).
>
Indeed. :-)
>
> But, sometimes square pegs are square pegs, round holes are round holes, and a cigar is just a
> cigar.
>
Indeed...
"Substitutivity of identity is x = y -> (phi(x) <-> phi(y))."
(George Greene)
Actually...
"...the law of the substitutivity of identity says that,
for any objects x and y, if x is identical to y, then
if x has a certain property F, so does y."
(Saul Kripke, Identity and Necessity, 1971)
(Compare with quotes by REASONABLE people above.)
F.
G. Frege
In NFL (negative free logic)...
>
> Certainly, eg. (the present king of France)=(the present king of France) is
> false and, E!(the present king of France) is also false.
>
Let's get that straight:
Since E!(the present king of France) is false, (the present king
of France) = (the present king of France) is also false.
This follows from the "rule of denotation" in NFL (which is resembling a
principle that was originally formulated by Russell):
A(a)
------ // if A(a) is atomic.
E!a
Hence we have immediately from that:
~E!a |- ~A(a) // if A(a) is atomic.
For A(x) being 'x = x' we have:
~E!a |- ~(x = x).
>
> It seems to me that *this version of NFL* is simply [Russelian]
> description logic, contained within classical logic.
>
Sure, one might see it this way... (though it's not completely correct,
since FL is not 'identical' with classical logic).
F.
mitch
"G. Frege" wrote:
> On Wed, 12 Nov 2003 23:40:17 -0600, mitch <mit...@rcnNOSPAM.com> wrote:
>
>
> >
> > Frege's analysis of the problem involves natural language
> > and the use of definite articles.
> >
> ...in /Grundlagen der Arithmetic/ only.
>
> Later (in /Grundgesetze der Arithmetik/) he stated his considerations in
> an extended version of the language and the system originally developed
> in his /Begriffsschrift/.
>
A work *I* still have to read.
But, I would still emphasize the importance of the intepretation of definite articles. For all of
the benefit and refinement that deductive calculi bring, the significant advance arises from
investigating the structure of language patterns and how people are using them.
:-)
mitch
G. Frege
>
> But, I would still emphasize the importance of the interpretation of
> definite articles.
>
Sure this IS an interesting/important topic! (Hence especially Free
Logic is engaged with it...)
For example, see the contents of Karel Lambert's, Free Logic - Selected
Essays, CUP, 2002.
Introduction
1. Russell's version of the theory of definite descriptions;
2. Existential import, E! and the;
3. On the reduction of two paradoxes and the significance thereof;
4. The Hilbert-Bernays theory of definite descriptions;
5. Foundations of the hierarchy of positive free definite description
theories;
6. Predication and extensionality;
7. Nonextensionality;
8. The philosophical foundations of free logic;
9. Logical truth and microphysics.
The first article ("Russell's version of the theory of definite
descriptions") can be read online:
http://assets.cambridge.org/0521818168/sample/0521818168WS.pdf
For a starter you might also read Russell's seminal paper:
Bertrand Russell, On Denoting, 1905
http://cscs.umich.edu/~crshalizi/Russell/denoting/
"By a 'denoting phrase' I mean a phrase such as any one of the
following: a man, some man, any man, every man, all men, the present
King of England, the presenting King of France, the center of mass of
the solar system at the first instant of the twentieth century, the
revolution of the earth round the sun, the revolution of the sun round
the earth. Thus a phrase is denoting solely in virtue of its form. We
may distinguish three cases: (1) A phrase may be denoting, and yet not
denote anything; e.g., 'the present King of France'. (2) A phrase may
denote one definite object; e.g., 'the present King of England' denotes
a certain man. (3) A phrase may denote ambiguously; e.g. 'a man' denotes
not many men, but an ambiguous man. The interpretation of such phrases
is a matter of considerably difficulty; indeed, it is very hard to frame
any theory not susceptible of formal refutation. All the difficulties
with which I am acquainted are met, so far as I can discover, by the
theory which I am about to explain. [...]"
>
> For all of the benefit and refinement that deductive calculi bring, the
> significant advance arises from investigating the structure of language
> patterns and how people are using them.
>
Right. Completely agree with you.
F.
mitch
"G. Frege" wrote:
> On Wed, 12 Nov 2003 23:40:17 -0600, mitch <mit...@rcnNOSPAM.com> wrote:
>
> Hi mitch!
>
> Good post. Actually, one of the most comprehensible posts by you I've
> read so far! :-)
>
That is only because you were familiar with the material I was reporting. :-)
I honestly believe you would be surprised in the long haul. Last year my brother was concerned
about me and stopped in to check how I was doing. The visit turned into a six--maybe eight--hour
discussion of how I see mathematics, philosophy, and logic. He certainly could not offer
validations. But, he concluded that the patterns of presentation that I was bringing to his
attention were reported accurately and coherently explained.
One way of looking at what I had been doing might be expressed by an observation made by Husserl.
He expresses some respect for Bolzano on the matter of logic, although he does not abide by the
conclusion Bolzano reaches,
"Excellent thoughts towards the circumspection of
our discipline are to be found in Bolzano's
Wissenschaftslehre, but rather in his preliminary
critical searchings than in the definition he himself
espouses. This last sounds oddly enough: the
theory of science (or logic) is 'the science which
shows us how to present the sciences in convenient
textbooks.' "
The foundationalism of the Vienna Circle (responding to a century of rapid and innovative scientific
discovery) changed how we present mathematics. Does that change the relationship of mathematics and
logic as presupposed by the inventor and early adopters of deductive calculi? With respect to
Russell and his excitement over finding Frege's definition of number, one can find a sober warning
(not directed to anyone in particular, of course) in Husserl,
"Only if one is ignorant of the modern science of
mathematics, particularly of formal mathematics,
and measures it by the standards of Euclid or
Adam Riese, can one remain stuck in the common
prejudice that the essence of mathematics lies in
number and quantity."
I certainly did not expect to be reading so much philosophy this year. But, it seems necessary just
to understand why I do not see things in set theory and logic in quite the same way as others. To
be honest, for the first time in my life I am taking the historical record of how facts became facts
to be extremely important. I can certainly accept a set theory that is completely irrelevant to the
philosophical considerations that comprise a foundational perspective. But, as commonly presented,
the facts of set theory do not satisfy my intuitions concerning a foundation.
Given the personal history of my endeavor along these lines, it probably resolves to the
informativeness of identity statements.
:-)
mitch
George Greene
: On 12 Nov 2003 17:44:11 -0500, George Greene
THAT is NOT a refutation, ASSHOLE.
And I cannot be saying complete bullshit if I am just
saying what *I* did (which I was), AS OPPOSED to what Frege did or didn't do.
What is complete bullshit is your contention that Frege gave a
CORRECT definition of "number".
George Greene
: And, exactly why should someone be persuaded to take the symbol '=' as a logical
: symbol again?
NOBODY NEEDS to be persuaded.
BY DEFINITION, ALL *LOGICAL* symbols are STIPULATED to mean
what they mean. That is what MAKES them logical!
There is no more reason to take = as a logical symbol than there
is to take v or ^ or ~ as one! Those are just the ones we take,
BY DEFINITION, BY CONVENTION, when we CHOOSE to use THAT logic
(obviously, if you want, you can choose both other symbols AND other logics).
Classical first-order predicate logic with equality is not the
only logic in the world and nobody can force you to use it in preference
to any other if you don't want to, but it just happens to be the
default. More to the point, it happens to be adequate to whole
bunches and bunches of things mathematical, and your claims that
it somehow isn't have been completely incoherent.
: The fact of the matter is that philosophy has not been able to stablize the meanings
: of the natural language words it uses over time.
It has never tried to. By definition, the NATURAL language meanings
are stable THERE -- IN natural language: natural language NOT as-used-
by-philosophers, but as used by EVERYbody, in nature, naturally. They
are stable (on any given day) in the general public DICTIONARY.
"In Philosophy" OR IN ANY OTHER Intellectual Field Of Inquiry, BY CONTRAST,
we CREATE OUR OWN theories, we found a WHOLE SEPARATE SEMANTIC WORLD,
where things basically mean what we say they mean. Of course, to even say THAT,
we still have to use natural meta-language in a reasonable way. But that does
not connote any failure to "stabilize". Natural language is INHERENTLY fuzzy;
things shade via "family resemblances" from one place to another. You need to
actually READ some Wittgenstein instead of just quoting it, if you didn't know THAT.
In any case, we are talking about LOGIC and the "meanings" (to the extent
that that term can have any meaning at all, in an abstract logical context)
of the terms occurring in the logical languages are WHATEVER WE SAY they are,
BY DEFINITION, and they ARE stable. Or not; my point is, they are conventional.
We could adopt different conventions or different spellings tomorrow withOUT
changing the paradigm. But in any case, you gotta remember that
INside the logical baby-pool, NOBODY GIVES A SHIT about natural language.
It is just something you resort to clarify what needs clarifying.
: I will happily plead ignorance of
: correct usage of these terms in situations where I see these terms being used with
: consistent meaning in rational conversations.
:
: Discussions with you barely meet the criterion of rationality,
YOUR mama sucks cocks in hell TOO, buddy.
: and, I would hate to repeat the terms being used with consistent meaning.
You have not spotted any inconsistency of meaning in my use of natural
language terms, or in any modern logician's use of them either. You are
just throwing bricks out of frustrated ignorance. ALL you have to do to
prove YOU can think rationally (your sheer projection in imputing that
failure to others instead of yourself is just breathtaking) is QUOTE SOMEBODY
USING some terms with meanings varying in an unexplained or mistaken way.
YOU CAN'T BECAUSE WE DON'T.
: If you wish to explain the hierarchy you are using to locate "notion," I would
: gladly consider what you are saying. In the meantime, I will point out that Kant
: took
Idiot, the word "notion" came from THEM not from me!
And FUCK Kant. Kant DIDN'T KNOW SHIT
and there is NOTHING you can point out from Kant that CAN be relevant!
Modern treatments that praise him praise him for POINTING us in the
directions we have NOW traveled toward, NOT for where he ALREADY WAS!
: of the use of definitions as starting points to be a mathematical methodology and
: the deduction of definitions as finishing points to be a philosophic methodoolgy.
Well, fuck that.
It's just irrelevant.
: Given the nature of dynamical truth, I don't think I will last to the end of the
: universe to get a definition of number with the expectation you are now portraying.
Nobody's asking that.
If you think YOU know what a number is, then state your definition.
But if you think Frege knew, then you're just wrong.
: Be a logician or a philosopher. When you sit on the fence you sound like a soprano.
I don't know what that means.
Nobody has any choice but to sit on the fence, if the evidence available
to them doesn't decide the question. It is certainly the case that no more
evidence is available to you than is to Benacerraf, Putnam, Boolos, Barwise,
Mates, Zalta, et al. And it is certainly the case that there is/was more
evidence available to them than was available to Kant.
: > There are MANY DIFFERENT POSSIBLE
: > representations of numbers in set theories.
:
: Right.
:
: The attempt to ground mathematics under logicistic precepts failed.
No, it didn't.
The fact that it succeeds in multiple ways does not turn success into failure.
Didn't you read FF's quotes of Tarski? Every last mathematical notion YOU
know of CAN be defined under "logicistic precepts". Even the proof that FOL
is inadequate to capture the standard model of PA was ITSELF PERFORMED under
"logical precepts".
: That does not preclude interpreting Frege's definitions
: relative to developments in the German philosophic tradition
: from the time of Kant.
OF course it doesn't! That cuts the OPPOSITE way! At the time of Kant,
THERE WERE NO logicistic precepts. Given that under the logicistic precepts
that Frege himself brought into being, FREGE'S DEFINITION OF NUMBER IS INCOHERENT,
any salvaging of Frege's definitions would HAVE to be in terms of "developments
in the German philosophic tradition at the time of Kant". But who gives a fuck
about THAT? THAT is all just OLD BULLSHIT! THAT is ETHER! THAT is PHLOGISTON!
: And, that is not to say that Kant did
: not have predecessors who influenced him (I look forward to Leibniz, Wolff, and
: Hume).
No, dipshit, you look BACKWARD, and that is your PROBLEM.
: > There CANNOT
: > exist logical grounds for ensrhining any ONE of them
: > as " *the* ontological representative" in preference to the others.
:
: Good point.
:
: Who has been quoting philosophers around here?
:
: Do you understand why you do not get a definite ontological representative?
Do I care? Once you have two candidates, it is obvious that none of them can be
definite. Do I understand why ZFC has more than one set of size n? Yes, I do
understand why, but that is not any sort of philosophical point. The whole point is
that n, whatever it is, must in some sense relate to EVERYthing with n parts, and
trying to relate to EVERYthing at once is just inherently impossible in this
framework; the framework is inherently built in levels. But that is NOT a
BAD thing.
: > But conventionally we can adopt whatever conventions we
: > choose.
:
: safl;kjs lkjdfoaij noasdfo[awj fnosdnfowhnfowif sdfonnhw
:
: Did you get that? Unfortunately, it really was composed as gibberish. But, it need
: not have been.
I said WE can adopt whatever conventions WE choose.
I did not say that MITCH can adopt whatever conventions MITCH chooses.
A community of 1 is hardly big enough to have a convention.
But if you would begin by stating the adoption of your conventions,
by holding your constitutional convention, IN a language that was
ALREADY shared by a larger community, THEN, THEREAFTER, the larger
community could all adopt and share your proposed newer conventions.
: > Getting from these multiple reductions DOWN to
: > "true platonic" "numbers" is something that STILL had
: > not been achieved A CENTURY AFTER Frege, LET ALONE by
: > Russell and Frege themselves (which is what FF was
: > idiotically and ignorantly quoting about).
:
: Ahem... Why don't you define "true platonic." Then we'll talk.
:
: I mean... DEFINE "true platonic."
DEFINE "number".
Once you do that, you will have infinitely many examples
of "true platonic" things before your own eyes, so you will
know what they are. I will give you a smaller number of
examples in a minute.
"DEFINE number".
This is the same challenge I put to Frege and Russell,
and FF's claims that their response was defensible
are excellent grounds for ejecting him from the community.
The fact that he quoted bunches of references (most
outrageously Tarski'64) that EXPLICITLY REJECT the
Frege/Russell definition, IN SUPPORT of his claim that
it was defensible, really ought to get me a public apology
at a bare minimum.
But getting back to your request for a definition,
"true platonic" is IN QUOTES for a reason.
Do you know what I *meant* by "true platonic"?
If so, there is nothing to discuss. If you don't,
then the uncharitable reply would be "Just Google
platonic and platonism and LEARN THE VOCABULARY, DUMBASS",
but more charitably, I can say that I meant "purely abstract";
e.g., the letter 'a' -- as distinguished from any particular
physical 'a' or any particular mental perception of an 'a' or
idea about an 'a'.
But before I define "true platonic", I must simply ask,
"do you think numbers exist"? If so, then you know them
to be "truly" abstract and Platonic in the sense described
above, but beyond that, can you answer the question, "OK, if
you believe in numbers, well, tell me, WHAT ARE numbers?"
They are not any of the symbolic representations OF them
in various theories (those are numerals) but the point we
HAVE achieved is that pretty much anything that can be done
with "true" numbers (whatever they are) can also be done with
these symbols.
George Greene
: mitch <mit...@rcnNOSPAM.com> writes:
: : And, exactly why should someone be persuaded to take the symbol '=' as a logical
: : symbol again?
NOBODY NEEDS to be persuaded.
As a wheel we spent a long time re-inventing,
you can do first-order set theory EQUALLY well in
classical FOPL with *without* = as a logical symbol.
ALL you have to do is re-cast the axiom of extensionality as
Axy[ Az[zex<->zey] <-> Az[xez<->yez] ] . Once you've done that,
^^^^^^^^^^^^^
this part ^^^ will wind up re-occurring so often that you'll want
to abbreviate it. You can abbreviate it by any defined binary predicate
you choose. It doesn't have to be =. But since it DOES HAPPEN to be
an EQUIVALENCE relation (it is provably reflexive, symmetric, and transitive),
and since it is clearly the most IMPORTANT equivalence relation in your
theory (occurring as it does again in the axiom schema of replacement),
YOU MIGHT AS WELL choose =.
George Greene
: "In his seminal work /Die Grundlagen der Arithmetik/, Frege
: successfully defined the notion of a 'cardinal number' [...]"
:
: (Edward N. Zalta)
G. Frege <no_...@aol.com> writes:
:
: Too difficult for you to understand? :-o
No, the problem is that it is too difficult for YOU to understand
what Zalta means by "successful". Frege's definition is NOT successful
by any modern criterion. I already referred you to Benacerraf.
Frege's definition was succesful to the extent that it avoided
the psychologistic errors of previous attempts and that it was
adequate as a foundation for his proof of Peano's Axioms
from Hume's principle. That it IS NOT any kind of definition
of a cardinal number is most loudly proven by the facts that
1) precisely as you quoted, from Tarski 1964, modern writers
DON'T USE that definition -- precisely as YOU quoted Tarski
saying, WE treat numbers as individuals AND NOT as properties of
classes of classes;
2) in the canonical modern set theory, the classes that Frege
and Russell posited as cardinal numbers DO *NOT* EXIST, and
3) as Benacerraf 1964 (which Zalta CERTAINLY knows about) and
whole hosts of other people, including Resnik, all understand,
there simply is NO "definition of number" worthy of the name in
the USUAL natural-language senses of "definition" and "number",
NOT *EVEN* NOW, 100 years later, LET ALONE back then.
If all you are going to do is quote individual lines out
of context on a newsgroup, as opposed to actually engaging
the issues, you are never going to be any kind of philosopher,
and far worse, not much of a historian either.
George Greene
G. Frege <no_...@aol.com> writes:
: ??? FOL??? Did ***anyone*** claim that Frege's definition happened in
: FOL???
I certainly didn't.
:
: Oh, hell, George, you are just FULL OF SHIT, that's all.
So quote something I said that's false.
The whole reason the work in which the definition-of-number
we have been debating MATTERS is BECAUSE FOL *was being invented*
during it. If you want to take frameworks, Frege's definition didn't
happen in ANY kind of L because his theory of extensions was
collapsed by Russell's paradox. The modern work on saving
the baby from that bathwater has to do with the fact that
it DOESN'T MATTER how you define "numbers". What's relevant
is a successor operation. There are MANY ways of doing successors.
The theorems are about how successors relate, IRrespective of
how you reduce/represent it. The proof of the Peano axioms from
Hume's principle is 2nd-order, but the definition itself is not
really at any particular order in modern terms. My point was simply
that "finite" is NOT a primitive concept in this arena -- it ITSELF
stands every bit as much in NEED of a definition as "number".
George Greene
: BTW, shouldn't RUSSELL himself better KNOW about that than a brainless
: asshole completely full of shit?
No, he shouldn't.
If he did, we would all now be doing set theory and logic
in the language of Principia Mathematica instead of in the
far better paradigms and notations we have now.
: "The question 'What is a number?' is one which has been
: often asked, but has only been correctly answered in our own
: time.
That is just a damn lie, and the issue is NOT whether Russell knows
better than *I* do, BUT RATHER, whether he knows better than
Paul Benacerraf and Hilary Putnam do.
: The answer was given by Frege in 1884, in his /Grundlagen
: der Arithmetik/.
No, it wasn't.
: (Bertrand Russell, Introduction to Mathematical Philosophy,
: 2nd ed. 1920)
Who was right about a lot of things, but wrong about this.
[...] this book is quite short, not difficult,
: and of the very highest importance [...]"
He was certainly right about THAT.
: P.S.
: Ok, enough is enough. No more replies to "George Greene".
Famous last words.
You don't have to reply to me.
You have to reply to Benacerraf.
George Greene
:
: >
: > All of which we already knew, but calling a "number" a class of all classes of the
: > same "number" (yeah, there is a REASON why that looks circular) WAS A BEGINNING,
: > NOT a correct ending.
mitch <mit...@rcnNOSPAM.com> writes:
: George,
:
: Go read the books.
Why on earth should I do that if YOU know how to explain them?
: Frege spoke in terms of objects and concepts.
Just as classical first-order logic speaks in terms of terms and predicates.
: He took the extension of a concept to be an
: object for the purpose of defining a number.
Just as any first-order set theory can and does.
: He assumed identity for objects in a pre-determined sense
Just as classical FOPL can between syntactically identical terms
(except that should be "term" singular, since if they are syntactically
identical, they are one term, not two, but the one might be bound
as the value of two different variables)
: so that he could define a similarity relation 'equals'
Well, YOU can call it "equals", but what it IS is "one-
to-one correspondence", and I fully expect that it gets EXPLAINED
AS that in the book.
: between extensions of concepts to serve as a definition of numerical identity
: between the objects he called numbers.
:
: Classical first-order logic has nothing to do with this.
Aside from the fact that this WAS THE BEGINNING of classical first-order logic,
right, it has nothing to do with it.
: Frege's analysis of the problem involves natural language and the use of definite articles.
So what? That doesn't mean he is not inventing first-order machinery.
: His definition constitutes singular reference. If you read about the related identity puzzles
: that arise from his considerations of natural language,
I HAVE, DUMBASS>
: you will find that the questions
: resolve around the "informativeness" of identity
: statements--not Frege's definition of number.
Which means that I took courses on those questions as an undergrad MORE THAN
20 YEARS AGO, and that if you need insight into them, you should be
ASKING ME AND NOT TELLING ME.
: I, for one, am happy that you know so much about contstructive interpretations of first-order
: logic.
I have never seen anything that I would respect as a "constructive interpretation".
In this context, "constructive" has a very specific meaning. Please be careful
not to step on our reserved words.
: Moreover, I think it is amazing how you believe
FUCK you, bitch.
I think it is amazing how you can post complete and utter lies about me
and then claim that *I* believe them.
: that all perspectives on logic must be reduced to
I belive no such thing.
: the paradigm you understand
I understand more than 1 paradigm, dumbass, which is more than you can say for
yourself. You were so busy trying to win an argument that you couldn't
even see FOL when you were present at the creation.
: and the presentations with which you agree.
I know that there are lots of different presentations.
In particular, I knew, WHICH YOU DIDN'T, that 1st-order ZFC
could be presented in BOTH of classical FOPL WITH equality AND without.
: And, lest I forget, I always appreciate your reminders that contemporary opinions are necessarily the
: outcome of manifest destiny
I have never reminded anybody of any such thing.
You NEED to read T.S.Kuhn's "The Structure of Scientific Revolutions".
: (Quite honestly, it may be the most exemplary conformity with
: Freudian statism that I have ever seen.).
No, it isn't, and at least I don't lie ABOUT the people I'm arguing with.
At least I quote them and rebut what they actually said instead of total
bullshit that I made up about them.
George Greene
: >
: > Classical first-order logic has nothing to do with this.
: >
G. Frege <no_...@aol.com> writes:
: Indeed. Actually, we need some sort of _2nd order_ logic to restate
: Frege's system...
OF COURSE.
BUT
You CANNOT DO 2ND order logic WITHOUT SUBSUMING
1st-order IN THE PROCESS!
See:
:
: http://plato.stanford.edu/entries/frege-logic/
:
: >
: > Frege's analysis of the problem involves natural language
: > and the use of definite articles.
: >
: ...in /Grundlagen der Arithmetic/ only.
:
: Later (in /Grundgesetze der Arithmetik/) he stated his considerations in
: an extended version of the language and the system originally developed
: in his /Begriffsschrift/.
:
: >
: > I, for one, am happy that you know so much about constructive interpretations of
: > first-order logic.
: >
: Right. The only PROBLEM: the interpretation(s) he very often DOESN'T
: know are just the /most common/ (i.e. generally shared) ones... :-)
That is just bullshit, Franz. None of the arguments we have been having
involve "interpretations of" first-order logic. They have been about
what is common in PARENTHESIZATION and other entirely trivial
matters of style.
: > Moreover, I think it is amazing how you believe that all perspectives on logic
: > must be reduced to the paradigm you understand and the presentations with
: > which you agree.
: >
: Indeed. Want to see an example? :-)
:
: "[...] Nowadays the LOGIC (if it is FOL=) comes with 2 more
: INFERENCE RULES defining '=' and there are NO "axioms" for
: '='!"
:
: (George Greene)
:
: Incredible uninformed nonsense.
It is not; it is the simple truth.
If anybody chooses to present FOL= another way, there is
nothing they can do about the fact that their way is equivalent
to that way; it wouldn't be FOL= if it wasn't.
: How the hell does such a guy even DARE
: to write here?!
That is NOT a refutation! A refutation is a posting
of SOME OTHER MODERN TREATMENT that chooses not to have THIS
x=y |- phi(x)->phi(y)
as an inference rule!
: Obviously he NEVER learned ANYTHING beyond the stuff he was told in the
: very classes he attended... :-(
:
: >
: > Go read the books.
: >
:
: Indeed, this probably is a rather good idea. If he WOULD do that, he
: possibly would read things like:
:
: "Adequate axioms of identity are 'x = x' and all instances
: of the schema 'x = y -> (phi(x) -> phi(y))'
That's WHAT I SAID, So I can hardly be accused of disagreeing with it
or not knowing it!
: of /substitutivity of identity/.
And I know that people often call the 2nd axiom-schema substitutivity of identity,
but the back-arrow IS semantically IMPORTANT for that, EVEN if it does
not need to be asserted!
: From these, by quantification theory, all laws
: of identity follow."
:
: (W. V. O. Quine)
I SAID THAT, so I again fail to see why I am being accused of disagreeing with it.
: Just for the benefit of all interested readers:
This is quoted completely OUT OF CONTEXT, ASSHOLE. I did NOT SAY
that it was necessar for any PROOF! I SAID it was SEMANTICALLY
necessary because of the NATURAL-language meaning of "substitutivity"!
: "A7: x = x
: A8: (x = y -> (alpha -> beta))"
:
: Stefan Bilaniuk, A Problem Course in Mathematical Logic
: (Version 1.6, 2003)
This confirms ME, NOT YOU, fool: alpha and beta make this
an AXIOM SCHEMA AND NOT an axiom, DESPITE the fact it is presented
with an A in front of it.
But none of this is the point.
THE POINT is that IF we're going to be in a room where
WHICH LOGIC WE'RE EVEN USING can VARY, then it becomes MORE IMPORTANT
to SEPARATE What's part of the LOGIC *from* what's part of the axiomatic
theories being defined and reasoned with IN that logic. IF you are
doing FOL= then it is just BETTER to CALL whatever is governing =
an inference rule, ESPECIALLY when it ALREADY canNOT BE a pure
axiom, when it HAS to be an axiom SCHEMA ANYhow.
: MA5 a = a
: MA6 a = b -> (A -> A(a//b))
: : "
:
: K. LAMBERT, Free Logics, 1997.
This is COMPLETELY irrelevant: classical FOPL *is*not* a free logic.
: etc. etc.
: Indeed...
:
: "Substitutivity of identity is x = y -> (phi(x) <-> phi(y))."
:
: (George Greene)
:
: Actually...
:
: "...the law of the substitutivity of identity says that,
: for any objects x and y, if x is identical to y, then
: if x has a certain property F, so does y."
:
: (Saul Kripke, Identity and Necessity, 1971)
I'm right. Kripke is right BECAUSE you can prove the back-arrow
BY relying on reflexivity.
George Greene
: But, I would still emphasize the importance of the intepretation of definite articles. For all of
: the benefit and refinement that deductive calculi bring, the significant advance arises from
: investigating the structure of language patterns and how people are using them.
"The" significant advance??
The signficant advance that allowed theoreticians to START thinking in FOL
and STOP thinking in terms of natural language is the MOST significant of ALL
these advances, and while investigating how people use natural language
was necessary to bring that about, once it happened, natural language
got almost irrelevant VERY quickly. To the extent that people still
engage in natural-language behaviors that FOL cannot capture, there is
more interesting work to be done in figuring out how to define
better logics that CAN capture it, but the IMPORTANT divide was
crossed LONG ago.
George Greene
: >
: > Frege's defintion of number was irrelevant bullshit.
: >
:
: Sure...
G. Frege <no_...@aol.com> writes:
: M. Randall Holmes, talking about NFU set theory:
:
:
: "[...] The natural numbers, and cardinal numbers in general,
: are defined using Frege's definition."
:
:
: (This means that the definition of natural numbers in NFU is just based
: on "irrelevant bullshit". Thanx for that fascinating insight, George.
: You should tell Holmes about that! :-)
:
: http://math.boisestate.edu/~holmes/
Oh, shut up.
NFU is over 50 years after Frege, and by THAT time, YES,
you can fit his definition into a coherent framework DESIGNED BY QUINE.
But it is NOT like FREGE knew what would be involved in making that
definition supportable. And it is NOT like NFU has supplanted ZFC
or NBG everywhere either. All set theories are NOT created equally-used.
Just because you found one somewhere where this definition could be
salvaged does NOT mean it was a good definition. If it did and it were,
then it, AS OPPOSED to the von Neumann ordinals, would be the COMMONLY
USED definition.
mitch
George Greene wrote:
> I took courses on those questions as an undergrad MORE THAN
> 20 YEARS AGO, and that if you need insight into them, you should be
> ASKING ME AND NOT TELLING ME.
>
Sure George.
There is no reasonable person who knows me that does not know that I am a good listener.
You are just like every other jackass who knows too much and revels in their own self-aggrandization. I
started out talking about the identity predicate. You determined that I did not know what I was talking
about.
Did you offer any insight from your vast store of knowledge? Any references justifying the veracity of your
opposition?
No. Instead you decided to engage in the vulgar, thuggish behavior that characterizes the self-support of
your personal opinion of yourself.
Even so, you still would not have dissuaded me from investigating matters for myself. That is the worst
insult to a person of your character. It is beyond you to let others draw their own conclusions via the
same dignified process (that at least some of) your own instructors probably afforded to you.
You're a player, George. You want to be quoted and believed. And, you have no hesitation sticking your
nose where it can get brown if it will advance your career. If this were a period of time prior to Dalton,
you would be persecuting anyone promoting atomism because it was an "old" theory. You probably would have
bought a lottery ticket to be the man who burned Bruno before that.
Damn, you are an ignorant jerk.
:-)
mitch
George Greene
: On Thu, 13 Nov 2003 02:01:05 +0100, G. Frege <no_...@aol.com> wrote:
: > M. Randall Holmes, talking about NFU set theory:
: sets under equinumerousness (in particular, Frege's definition
: of natural number succeeds!) and ordinals as equivalence
: classes of well-orderings under similarity."
:
: (M. Randall Holmes)
:
:
: Conclusion: George Greene's claims are just irrelevant bullshit.
Dipshit:
Are you even REMOTELY acquainted with the MEANING OF A FUCKING *EXCLAMATION POINT*???
????????????????????????????????????????????????????????????????????????
WHY does the fact that Frege's definition succeeds under this set theory
MERIT an exclamation point?? Why is it A NEWS FLASH?????????
WHY is it a surprise??? Because that definition FAILS in most
OTHER set theories, THAT'S why.
It is a surprise BECAUSE IT TOOK an intellectual effort as HEROIC
as Quine's invention of NF *TO RESCUE* a definition that WAS,
ORIGINALLY, every bit as much irrelevant bullshit as Russell's
Principia Mathematica and ramified type theory.
There are two general approaches here, in all this.
There is a large class of different sets of size n, for any n.
n itself is independent of the individuals in the set.
To define n, one needs to somehow maintain some sort of
relationship to all these sets while NOT caring about their
particular members. The two approaches are, either, to
try to collect all the sets into some 1 larger thing and
call THAT n, or to pick 1 of the sets as a distinguished
representative of the class and call THAT n. Frege and
Russell originally tried the former. IT DIDN'T WORK.
It is just HARDER to try to have classes big enough
to accomplish that. No matter what happens, in fact,
they are NEVER big enough. How, for example, in NF
or NBG or any other theory that represents numbers as
CLASSES of SETS, are you ever going to be able to
express that a collection of 3 *classes* is of size 3?
You necessarily fail the generality of the concept.
There are GOOD REASONS WHY the other tack, the tack that
succeeded first, REALLY IS preferable. I don't prefer it
JUST because the fact that it succeeded first means that
it got taught and entrenched first and that therefore
I learned it first and was thereafter bigoted against
the other way. UNLIKE YOU, I ACTUALLY HAVE STUDIED
NF IN A COURSE, not just quoted random bits of it out
of context in order to win a pissing contest on a newsgroup.
Doing things with bigger and bigger collections (i.e. trying to
represent "3" by something with an uncountable infinity of
things INside it, AS OPPOSED to by something with *3* things
inside it) runs into at least 3 difficulties: 1) the bigger&bigger
collections are never bigger ENOUGH; you always have to stop
arbitrarily at some point; in class theories, you will stop
at the line between sets and classes; 2) as the number of years
that elapsed between the time Frege came up with his definition and
the time somebody came up with a useable set theory that could
tolerate it proves, collections that are too big can get you
in trouble; and 3) Occam's razor: WHY would you want to
represent ANYthing as small as 3 with an INFINITE set, when
a set of size *3* will DO? That's purely a question of
conceptual elegance.
Now, if you think NF is really elegant and people ought
to start teaching it instead of ZF, well, that's your
opinion. But it is NOT the COMMON opinion.
mitch
George Greene wrote:
> IF you are
> doing FOL= then it is just BETTER to CALL whatever is governing =
> an inference rule, ESPECIALLY when it ALREADY canNOT BE a pure
> axiom, when it HAS to be an axiom SCHEMA ANYhow.
What is governing the symbol '=' is the existential import of the intepretion given to
the quantification.
That cannot be expressed with an inference rule.
:-)
mitch
George Greene
:
: >
: > If something is qualitatively self-different, it possesses some
: > properties not possessed by it and, hence, must be a self-inconsistent
: > object:
: >
mitch <mit...@rcnNOSPAM.com> writes: in
: http://ontology.buffalo.edu/smith/articles/mereotopology.htm
:
: you will find a description of a universe that is a proper part of itself.
:
: He calls it counterintuitive. I view it with about the same sense that you
: suggest above.
What language is this description in?
Is the thing's being a proper part of itself consistent, or inconsistent,
with its being equal to itself, or identical with itself, or non-distinct
from itself?
: I believe it to be a failure to understand the relationships between mereology
: (part), set theory (element), and definite description (equals).
How can you say that there is ANY "failure" involed here AT ALL, if the
description of the universe is both syntactically grammatical AND logically
consistent?
: It is not a matter of seeing whose is right and who is wrong.
As a general rule, a consistent set of first-order axioms CANNOT POSSIBLY
be "wrong". It can at worst be inadequate to its intended task.
: It is not a
: matter of this alternative foundation is better than that alternative
: foundation. It is a matter of seeing how the identity puzzles are resolving
: themselves.
You still haven't stated the identity puzzles.
If you are appealing to sophomoric natural language stuff then
the resolutions are pretty trivial.
: Of course, everyone is so busy comparing deductive calculi and formal systems,
: they seem to have forgotten that these things arose from philosophical
: considerations that were being debated prior to the introduction of such
: calculi.
And good riddance.
You, conversely, seem to have forgotten that people couldn't even HAVE
coherent discourse about most of the questions until AFTER achieving
formal language. The neologism was HELPFUL.
:
: Perhaps Mr. Greene is right. Perhaps not.
Never say that out of context. Right ABOUT WHAT?
WHAT DID I SAY???????????????????????????????????
I repeat, the ONE thing I am CERTAINLY right about is that
you HAVE to QUOTE the people you are disagreeing with!
IN context!
George Greene
: One of my problems with all of this is a poor grasp of philosophical terminology.
Not really.
: When I was reading the internet page on Frege in the Stanford Encyclopedia, it made
: reference to his *ontology.* Of course, he is establishing an ontology of number.
: Because this is an ontological assertion, it is existential and necessarily violates
: Kant's delineation of topics derived from epistemic considerations.
:
: I am reading German philosophers now with the hope of sorting some of this out.
It will not help.
More old confused treatments is the LAST thing you need.
George Greene
: Indeed. Arguably, KANT was a great philosopher, who presented lots of
: VALUABLE ideas. ( We should not forget that FREGE formulated (some of)
: his ideas while referring to (some of) KANTs thoughts concerning this
: topic(s).
None of that was ever under debate and I am frankly outraged
that this is being asserted against me.
Everybody values the original contributions of everybody to the
field. But Columbus was WRONG about where India was,
and Frege(&Russell who followed him) was WRONG about what
numbers were. And Kant was WRONG about math being synthetic.
NONE of that is to DISPARAGE any of their GREAT intellectual
achievements! But it does start to bear stressing IF people
are going to say stupid things about modern contexts AND THEN
try to justify them by fallacious appeals to authorities so
old that their perspectives HAVE been superseded.
Thomas Bushnell, BSG
> : It is not a matter of seeing whose is right and who is wrong.
>
> As a general rule, a consistent set of first-order axioms CANNOT POSSIBLY
> be "wrong". It can at worst be inadequate to its intended task.
Well, yes, but it is normal to call an inadequate set of axioms
"wrong", since generally one expects an axiom set to conform to some
intuitive notion or other, and that can certainly be done incorrectly.
George Greene
This is just bullshit, mitch, I'm sorry.
PLEASE be clear about this: This IS a matter of DEFINITION:
I AM NOT stating an OPINION here.
A *logic*, BY DEFINITION, is about what strings FOLLOW FROM what
other strings. In FOL=, BY DEFINITION, x=y |- phi(x)->phi(y)
is an inference rule, AND EVERYTHING that happens around = follows
from that rule and reflexivity. EVERYTHING. BY DEFINITION.
NO EXISTENTIAL INTERPRETATION IS EVER given to the quantification
beyond the requirement that there be something in the domain.
The quantifiers of FOL always quantify OVER AND ONLY OVER
THE TERMS OF THE FIRST-ORDER LANGUAGE. Speaking of "existential
import" around those terms is just nonsensical since there
would be no strings to look at if the terms didn't exist.
This is not META-linguistic in any sense. This is
the DEFINITION of the paradigm.
The existence of textbooks in 2002 that want to call
their directives about equality axioms or axiom-schemata
as opposed to inference rules is for purposes of this
exposition simply irrelevant. It is entirely a nit whether,
when you add equality to classical FOPL without equality,
you do it by adding some axioms or by adding some inference
rules. My decision to speak in terms of adding inference
rules was designed to stress that FOPL with equality is
A DIFFERENT logic.
George Greene
:
: > : It is not a matter of seeing whose is right and who is wrong.
: >
: > As a general rule, a consistent set of first-order axioms CANNOT POSSIBLY
: > be "wrong". It can at worst be inadequate to its intended task.
tb+u...@becket.net (Thomas Bushnell, BSG) writes:
: Well, yes, but it is normal to call an inadequate set of axioms
: "wrong",
No, it isn't.
: since generally one expects an axiom set to conform to some
: intuitive notion or other,
Of course, but conformance is a matter of DEGREE, whereas right/wrong
is a lot more bipolar.
: and that can certainly be done incorrectly.
Sure, but the incorrectness is often likely to produce inconsistency.
If what you have instead is consistent inadequacy, then that may
be due to something OTHER than "incorrectness", such as incompleteness.
While it is true that you can always formalize incorrectly, the more
interesting question is whether you can formalize *correctly*.
Beyond a certain fairly low level of ambition, you can't.
Most notoriously, in the case of first-order arithmetic vs. first-
order logic, you can't.
Thus, the usual first-order approximation, despite its inadequacy,
is usually considered RIGHT rather than wrong, for the simple
reason that the second-order version of precisely THAT axiomatization
IS right (and no first-order axiomatization CAN be right).
And my point is that this is not some strange exceptional case --
it was the FOUNDATIONAL case. It was the ORIGINAL case. It was
the first and MOST important case. OK, I guess that does make
it exceptional after all.
mitch
George Greene wrote:
To the contrary George.
Frege jumped on Kant's use of 0 and 1 as definite references.
George Greene said that everything comes down to 0's and 1's.
How does mitch understand 0 and 1....
Well-foundedness is based on succession. Well-formedness is based on nested (palindromic)
symmetries
-----
0 0 = 0
1 1/2 = 0.50000
2 1/2surd(3) = 0.28868
3 1/4surd(2) = 0.17678
4 1/8 = 0.12500
5 1/8surd(2) = 0.08839
6 1/8surd(3) = 0.07217
7 1/16 = 0.06250
8 1/16 = 0.06250
9 1/16surd(2) = 0.04419
10 1/16surd(3) = 0.03608
11 1/18surd(3) = 0.03208
12 1/27 = 0.03704
13 1/18surd(3) = 0.03208
14 1/16surd(3) = 0.03608
15 1/16surd(2) = 0.04419
16 1/16 = 0.06250
17 1/16 = 0.06250
18 1/8surd(3) = 0.07217
19 1/8surd(2) = 0.08839
20 1/8 = 0.12500
21 1/4surd(2) = 0.17678
22 1/2surd(3) = 0.28868
23 1/2 = 0.50000
24 1 = 1
------
Of course, since everyone is so damn sure they know how the universe works, it is a little
hard to get anyone to look at patterns in data.
There are 24 combinations of the rows (T,T), (T,F), (F,T), (F,F).
That the 24 "counterparts" for a truth table might have a fixed, canonical aspect
presenting itself somewhere within the mathematical literature is, of course, ludicrous.
How does one try to get anyone to think differently? Figure out what they believe and
fight like hell to find a way to make them think differently. You can't get to the
Leibnizian ideal of a unified mathematics and logic unless you have some sense of
self-supported semantics.
:-)
mitch
p.s.
The asymmetry of the extrema is an artifact of the particular list. There are other
numbers for the same structure where the palindromes are totally reflective. The fixity of
0=0 and 1=1 at the endpoints may be compared with a 2-monotonic switching function which
must have two arguments fixed. There is a canonical order for the signatures of
2-monotonic switching functions that is independent of any ordering associated with the
underlying referents.
George Greene
: There is no reasonable person who knows me that does not know that I am a good listener.
They have not tried to talk to you about THESE questions.
: You are just like every other jackass who knows too much and revels in their own self-aggrandization.
No, that's YOU, coming in here with a whole bunch of axioms and a whole bunch of pronouncements
about Flum, Zeigler, Halmos, Langholm, and how you know so much more than we do.
: I started out talking about the identity predicate.
No, you didn't. You started out talking about topological model theory,
and you continued by attempting to disparage set theory's use of the
identity predicate, without knowing what you were talking about.
: You determined that I did not know what I was talking about.
You didn't. In particular, you didn't know that FOL comes in
two flavors, both with equality and without, and that it was
a straightforward matter and a known result to define set theory
WITHOUT an identity predicate.
: Did you offer any insight from your vast store of knowledge?
Yes. I (with needed help from FF) proved to you that anything you might
have to say about equality in set theory was bullshit, since there IS NO
equality in set theory, if you phrase it in that dialect.
: Any references justifying the veracity of your opposition?
References ARE NOT HOW ANYBODY justifies ANYthing, DUMBASS!
People justify things BY PROVING them!
: No.
Yes.
I proved to you that you never had and still don't have any grounds for
saying that there is anything "wrong" about the way EITHER
of FOL *or* set theory treats identity.
: Instead you decided to engage in the vulgar, thuggish behavior that characterizes the self-support of
: your personal opinion of yourself.
Dipshit:
peppering one's conversation with curses does NOT constitute thuggish behavior in
this context. Thuggish behavior is quoting people out of context or not at all,
lying about or distorting their positions, and refusing to answer simple questions
about your statements.
: Even so, you still would not have dissuaded me from
: investigating matters for myself. That is the worst insult to a
: person of your character. It is beyond you to let others draw
: their own conclusions via the same dignified process (that at
: least some of) your own instructors probably afforded to you.
That is not the issue.
MY process ALWAYS involved people beginning with clear definitions
of their terms.
: You're a player, George. You want to be quoted and believed.
Really, I could NOT care less. I want you to quote me
so that you will be reacting to what I actually said and not
to some lame-ass paraphrase, and to make sure, IF you mis-
interpret what I said, that readers generally can SEE that you did.
And as for "believed", all the things I have asked you to believe
are in textbooks too numerous to count, despite FF's skill at hunting
down some that differ in some stylistic detail. It is THEM, not me,
whom you need to believe, just as, on the subject of the irrelevance of
Frege and Russell's old definition of "number", it is von Neumann and
Tarski (who, UNlike Frege and Russell, promulgated a SUCCESSFUL definition),
and later Benacerraf (who stressed the UNworthiness of *any* *particular*
abstract numeral as number), whom FF needs to believe -- NOT ME.
: And, you have no hesitation sticking your
: nose where it can get brown if it will advance your career.
That is A COMPLETE DAMN LIE, you LYING asshole (notwithstanding that
my nose is ALWAYS brown by definition). EVERYthing I am posting here
takes time AWAY from advancing my career.
: If this were a period of time prior to Dalton,
: you would be persecuting anyone promoting atomism
: because it was an "old" theory.
Don't be ridiculous. People were conducting relevant experiments then.
Old theories had been refuted. I can't help it if you have decided to start
so old that YOU AND FF are the ones burning all the people who came after
Russell and improved on him, and then when you get caught, you pull one
person who cleaned up their mess (e.g. Quine/NFU) and then claim that
that shows it was never dirty after all.
: You probably would have
: bought a lottery ticket to be the man who burned Bruno before that.
:
: Damn, you are an ignorant jerk.
Yeah, but I know that there is nothing wrong with identity
in ZFC or in FOL, which is more than you know.
mitch
George Greene wrote:
> mitch <mit...@rcnNOSPAM.com> writes:
>
> : George Greene wrote:
> :
> : > IF you are
> : > doing FOL= then it is just BETTER to CALL whatever is governing =
> : > an inference rule, ESPECIALLY when it ALREADY canNOT BE a pure
> : > axiom, when it HAS to be an axiom SCHEMA ANYhow.
> :
> : What is governing the symbol '=' is the existential import of the intepretion given to
> : the quantification.
> :
> : That cannot be expressed with an inference rule.
>
> This is just bullshit, mitch, I'm sorry.
Actually, it is not. I was going to go on to say that I gladly accept a totally formal
deductive calculus. But, I figured you would just complain about something else.
But, from a purely formal perspective, what would any of this have to do with mathematics or
language then? Should everyone be impressed because you constructed an encoding to do the
same thing that can be done without?
You don't get to claim foundationalism and then expect everyone to accept senselessness,
George. In the end, language has to mean something. If it doesn't, then we should not have
a Council of Economic Advisors, a Federal Reserve Board, an Environmental Protection Agency,
a Food and Drug Administration, etc.
What I mean is that I am more than happy to let you hold the seive or milk the he-goat. But,
if you are going to claim relevance (and please do not try to tell me that the people earning
credentials are going to deny their own relevance), then the abstract manipulations have to
attain meaning somehow.
You cannot define meaning. But, you can use abstraction to obfuscate.
:-)
mitch
George Greene
: George Greene wrote:
:
: > mitch <mit...@rcnNOSPAM.com> writes:
: > : One of my problems with all of this is a poor grasp of philosophical terminology.
: >
: > Not really.
: >
: > : When I was reading the internet page on Frege in the Stanford Encyclopedia, it made
: > : reference to his *ontology.* Of course, he is establishing an ontology of number.
: > : Because this is an ontological assertion, it is existential and necessarily violates
: > : Kant's delineation of topics derived from epistemic considerations.
: > :
: > : I am reading German philosophers now with the hope of sorting some of this out.
: >
: > It will not help.
: > More old confused treatments is the LAST thing you need.
:
: To the contrary George.
:
: Frege jumped on Kant's use of 0 and 1 as definite references.
SO WHAT?? JEEzus! That is NOT helpful!
: George Greene said that everything comes down to 0's and 1's.
I repeat, QUOTE ME OR SHUT THE FUCK UP.
I DID NOT say that.
: How does mitch understand 0 and 1....
It doesn't matter how.
: Well-foundedness is based on succession.
Well-formedness is based on nested (palindromic) symmetries
Another stupid lame over-generalization.
There is more than one kind of well-formedness in the world.
: -----
You really need to watch "A Beautiful Mind". Or read the book if that is
more to your taste. Looking for patterns in huge datasets like this is
a known kind of mental illness. Pat yourself on the back for sharing
it with known geniuses.
: There are 24 combinations of the rows (T,T), (T,F), (F,T), (F,F).
:
: That the 24 "counterparts" for a truth table
There are NOT 24 counterparts for a truth table.
A truth table has a THIRD T/F: the RESULT of which truth-
value unary-peg you pit in each of these 4 binary-holes.
There are 16 different ways of doing THAT for each table (or table-
counterpart), and after you consider these 24*16 alternatives, a lot
of them are identical EXCEPT for order. Each of the ACTUAL truth
tables (of which there 16, NOT 24) occurs in 24 different orders.
WHY should ANY of those orders matter?
Why should ANYbody want to collect
all 24 of them? EVERY finite set of size n has n! different orders.
SO WHAT? Are you going to require us to blow up all our representations
by a factor of (n-1)! ?
IF we are thinking about it as a set AND NOT a list, IF we are INTENTIONALLY
throwing away the order, then why do we respect mitch for saying that the
non-order has to be represented by giving equal weight to all the orders
(the same old benighted Frege/Russell error in defining "number"), when that
is not even possible anyhow (you had to privilege one of the 24 orders
just to present the 4 things to us). might have a fixed, canonical aspect
: presenting itself somewhere within the mathematical literature is, of course, ludicrous.
indeed.
: How does one try to get anyone to think differently?
Physician,heal thyself.
: Figure out what they believe and
: fight like hell to find a way to make them think differently.
No, FIRST, you have to make sure that you're RIGHT, that it is THEY
AND NOT YOU who NEED to think differently.
: You can't get to the Leibnizian ideal of a unified mathematics and logic
WHY would anyone want to do THAT?
Logic is one thing, but how do you defend your contention that
math is another? If it really is, then doesn't that by definition MEAN
they can't be unified? If they can, then how do you show that the existing
paradigm DOESN'T unify them?
: unless you have some sense of self-supported semantics.
"Some sense of self-supported semantics" is in fact more than
just arguably already had by the current paradigm, since you
can interpret terms as themselves, and indeed, interpreting
them as anything else violates Occam's Razor.
: p.s.
:
: The asymmetry of the extrema is an artifact of the particular list. There are other
: numbers for the same structure where the palindromes are totally reflective. The fixity of
: 0=0 and 1=1 at the endpoints may be compared with a 2-monotonic switching function which
: must have two arguments fixed. There is a canonical order for the signatures of
: 2-monotonic switching functions that is independent of any ordering associated with the
: underlying referents.
This is just loud proof that everything you said about ME being a player who
wanted to be quoted and knew too much and was self-aggrandizing ACTUALLY is
more applicable to yourself. mitch smith, lead projectionist of this newsgroup.
mitch
George Greene wrote:
> mitch <mit...@rcnNOSPAM.com> writes:
> : There is no reasonable person who knows me that does not know that I am a good listener.
>
> They have not tried to talk to you about THESE questions.
Wrong. They chose to ignore what I was doing in order to talk about these things because they were a lot
like you.
>
>
> : You are just like every other jackass who knows too much and revels in their own self-aggrandization.
>
> No, that's YOU, coming in here with a whole bunch of axioms and a whole bunch of pronouncements
> about Flum, Zeigler, Halmos, Langholm, and how you know so much more than we do.
>
No.
At no time did I ever start out by saying that I was trying to revolutionize anything. I tried to argue for
a specific context in which to consider alternatives.
You responded to your own expectations and your own motivations.
>
> : I started out talking about the identity predicate.
>
> No, you didn't. You started out talking about topological model theory,
> and you continued by attempting to disparage set theory's use of the
> identity predicate, without knowing what you were talking about.
>
No.
If you ever looked at my axiom set, you would find that I introduced a formal equivalence.
It was in isolating the component subformulas from one another in a manner by which they became logically
equivalent that I began considering alternatives to extensionality. Because I was using diagrams the natural
choice was the one leading to the questions involving relative identity.
>
> : You determined that I did not know what I was talking about.
>
> You didn't.
Sorry ace.
From the contents page of "Symbolic Logic" by Thomas,
Part 1 TRUTH-FUNCTIONAL LOGIC
Part 2 QUANTIFICATIONAL LOGIC
Part 3 QUANTIFICATIONAL LOGIC WITH IDENTITY
As I recall, I was nineteen when I used this book. It is still on my shelf. I like its presentation of
deduction.
> In particular, you didn't know that FOL comes in
> two flavors, both with equality and without, and that it was
> a straightforward matter and a known result to define set theory
> WITHOUT an identity predicate.
>
No. Thomas Jech wrote a definitive text on set theory. He bases his presentation on first-order logic with
identity.
Now, I was considering the identity predicate. Do not be such a cad as to think that I was not considering
about the significance of how set theory can be understood relative to either logic. In fact, one of the
questions I kept asking myself was why Jech chose to use identity when he could just define it. This very
question, in fact, is one of the reasons set theory actually can lead one to the identity puzzles. But, to
get there, one must weaken the treatment of quantification over a domain to a problematic status. Given
Russell's paradox, it is not a difficult thing to do in set theory.
And, by the way, at no time did I not say that identity in set theory was unclear for set models where
quantification is clearly being made with respect to a fixed collection.
>
> : Did you offer any insight from your vast store of knowledge?
>
> Yes. I (with needed help from FF) proved to you that anything you might
> have to say about equality in set theory was bullshit, since there IS NO
> equality in set theory, if you phrase it in that dialect.
>
No. There is a difference between proving things and swearing a lot.
>
> : Any references justifying the veracity of your opposition?
>
> References ARE NOT HOW ANYBODY justifies ANYthing, DUMBASS!
> People justify things BY PROVING them!
>
No. References direct people to the very bodies of knowledge whose interconnectedness led to the
idealizations that made your formal systems possible in the first place. Otherwise, you are little more than
a dog barking.
>
> : No.
>
> Yes.
> I proved to you that you never had and still don't have any grounds for
> saying that there is anything "wrong" about the way EITHER
> of FOL *or* set theory treats identity.
>
You proved nothing. You swore a lot.
Occasionally, you did have insightful remarks that I appreciated.
>
> : Instead you decided to engage in the vulgar, thuggish behavior that characterizes the self-support of
> : your personal opinion of yourself.
>
> Dipshit:
> peppering one's conversation with curses does NOT constitute thuggish behavior in
> this context.
Peppering one's conversation with curses is always vulgar.
Always.
> Thuggish behavior is quoting people out of context or not at all,
> lying about or distorting their positions, and refusing to answer simple questions
> about your statements.
>
Your positions could be better understood if they weren't primarily rants.
>
> : Even so, you still would not have dissuaded me from
> : investigating matters for myself. That is the worst insult to a
> : person of your character. It is beyond you to let others draw
> : their own conclusions via the same dignified process (that at
> : least some of) your own instructors probably afforded to you.
>
> That is not the issue.
> MY process ALWAYS involved people beginning with clear definitions
> of their terms.
>
That process works where there are no questions. It works for computers better than people.
In any case, I did start with definitions. I restricted the context to an exploratory status. And, I found
evidence in philosophical logic that supported the basic sense behind my ideas.
You did not like the fact that I began with a circular definition. Whatever else you did after that was for
purposes that had little to do with my remarks. And, my responses reflected as much emotional reaction as
anything else.
But, unfortunately, there seems to be little that can be done to rectify matters.
>
> : You're a player, George. You want to be quoted and believed.
>
> Really, I could NOT care less. I want you to quote me
> so that you will be reacting to what I actually said and not
> to some lame-ass paraphrase, and to make sure, IF you mis-
> interpret what I said, that readers generally can SEE that you did.
>
> And as for "believed", all the things I have asked you to believe
> are in textbooks too numerous to count, despite FF's skill at hunting
> down some that differ in some stylistic detail. It is THEM, not me,
> whom you need to believe, just as, on the subject of the irrelevance of
> Frege and Russell's old definition of "number", it is von Neumann and
> Tarski (who, UNlike Frege and Russell, promulgated a SUCCESSFUL definition),
> and later Benacerraf (who stressed the UNworthiness of *any* *particular*
> abstract numeral as number), whom FF needs to believe -- NOT ME.
>
Right George. I do have respect for modern authors as well. But, Occam's razor is not about complexity.
How long does one watch disk thrashing before one acts to fix the problem?
>
> : And, you have no hesitation sticking your
> : nose where it can get brown if it will advance your career.
>
> That is A COMPLETE DAMN LIE, you LYING asshole (notwithstanding that
> my nose is ALWAYS brown by definition). EVERYthing I am posting here
> takes time AWAY from advancing my career.
>
:-) That was not how it was intended.
>
> : If this were a period of time prior to Dalton,
> : you would be persecuting anyone promoting atomism
> : because it was an "old" theory.
>
> Don't be ridiculous. People were conducting relevant experiments then.
> Old theories had been refuted. I can't help it if you have decided to start
> so old that YOU AND FF are the ones burning all the people who came after
> Russell and improved on him, and then when you get caught, you pull one
> person who cleaned up their mess (e.g. Quine/NFU) and then claim that
> that shows it was never dirty after all.
>
> : You probably would have
> : bought a lottery ticket to be the man who burned Bruno before that.
> :
> : Damn, you are an ignorant jerk.
>
> Yeah, but I know that there is nothing wrong with identity
> in ZFC or in FOL, which is more than you know.
>
Right.
"...The propositions thus arising have been stated
as universal Laws of Thought. Thus the first of
them, the maxim of Identity, reads: Everything is
identical with itself, A=A: and, negatively, A cannot
be at the same time be A and not A. This maxim,
instead of being a true law of thought, is nothing
but the law of abstract understanding. The propositional
form itself contradicts it: for a proposition always
promises a distinction between subject and predicate;"
Hegel discerned three moments in logic--the Abstract, the Dialectical, and the Speculative. The part
referring to subject and predicate clearly is obfuscated when one maps
A is B
to
A=B
The mathematical community never signed off on Frege's logic and Russell's program. How can you say that
mathematicians consistently use the symbol '=' for substitutivity alone when, in fact, their discipline
evolved within natural language? It is an idealization that cannot be justified.
I have never had any argument concerning the abstract symbol manipulations. It is when we start saying that
it has relevance as anything but abstract symbol manipulation.
:-)
mitch
mitch
George Greene wrote:
Well, I will agree that it would go into the category of "pronouncement" that seems to irritate
you so much.
It is a terse summary of about forty pages of threshold logic and a guess about how to interpret
the endpoints of the list.
As for the other post. Some days you just piss me off. Sorry about it.
:-)
mitch
G. Frege
> >
> > Good post. Actually, one of the most comprehensible posts by you I've
> > read so far! :-)
> >
>
> That is only because you were familiar with the material I was reporting. :-)
>
No, that's not the only reason.
>
> [...] he concluded that the patterns of presentation that I was bringing to
> his attention were reported accurately and coherently explained.
>
Oh, I don't doubt you ABILITY to present accurately and coherently
explanations.
>
> I certainly did not expect to be reading so much philosophy this year.
> But, it seems necessary just to understand why I do not see things in
> set theory and logic in quite the same way as others.
>
Hmmm... Hmmm... I see.
>
> To be honest, for the first time in my life I am taking the historical
> record of how facts became facts to be extremely important.
>
:-)
>
> I can certainly accept a set theory that is completely irrelevant to the
> philosophical considerations that comprise a foundational perspective.
> But, as commonly presented, the facts of set theory do not satisfy my
> intuitions concerning a foundation.
>
I see. [ Same with me. ]
F.
George Greene
: George Greene wrote:
:
: > mitch <mit...@rcnNOSPAM.com> writes:
: >
: > : George Greene wrote:
: > :
: > : > IF you are
: > : > doing FOL= then it is just BETTER to CALL whatever is governing =
: > : > an inference rule, ESPECIALLY when it ALREADY canNOT BE a pure
: > : > axiom, when it HAS to be an axiom SCHEMA ANYhow.
: > :
: > : What is governing the symbol '=' is the existential import of the intepretion given to
: > : the quantification.
: > :
: > : That cannot be expressed with an inference rule.
: >
: > This is just bullshit, mitch, I'm sorry.
:
: Actually, it is not.
Actually, you are simply entirely too ignorant to have an opinion
one way or the other. This is an arena where YOU lack the vocabulary.
In the first place, for the 40-eleventh time, beyond the existence of
the terms themselves, NO existential import IS given to ANYthing
by quantification in first-order logic! So the question of whether it
can or can't be expressed with an inference rule is moot anyhow.
: I was going to go on to say that I gladly accept a totally formal
: deductive calculus. But, I figured you would just complain about something else.
:
: But, from a purely formal perspective, what would any of this have to do with mathematics or
: language then?
You tell me!
Actual mathematicians do do actual PROOFS using deductive calculi, or
at least using them for certain PARTS of the task. The argument they
feel compelled to present is an argument that these things COULD be carried
out in deductive calculi, in principle. It's intellectual hygiene.
It's a sanity check.
: Should everyone be impressed because you constructed an encoding to do the
: same thing that can be done without?
But the whole point is that it COULDN'T be done without, any more than
people could've done calculus if they'd had to stick to roman numerals!
The BETTER encoding was HELPFUL!
: You don't get to claim foundationalism and then expect everyone to accept senselessness,
: George.
Of course you do.
At the bottom, it HAS to be senseless if it's foundational, because IF IT WEREN'T,
you'd be in infininite regress ("explain me the sense of your foundation, please").
: In the end, language has to mean something.
Of course it does, but in what, exactly, does meaning inhere?
The point is, it doesn't inhere, for certain KINDS of things,
in "pointing at" something physical. In order for 1+1=2 to "mean", does
1 have to exist? Does 2? If so, then WHAT IS 1? WHAT IS 2?
YOU CANNOT ANSWER THESE QUESTIONS.
FF's contention that Frege & Russell had answered them a hundred years ago
is laughable, but his mis-interpretation of Zalta and (far worse) Tarski
as ALLEGING that Frege had answered them PROPERLY is more like cause for
crying.
: If it doesn't, then we should not have
: a Council of Economic Advisors, a Federal Reserve Board, an Environmental Protection Agency,
: a Food and Drug Administration, etc.
That's just unrelated. All that is CONCRETE. THE ONLY
reason why people worry that logic-about-math might not refer
is BECAUSE what it refers to IS ABSTRACT. In what sense does ANY pure abstraction
EXIST AT ALL? What is "the definition of" the letter 'a'?
: What I mean is that I am more than happy to let you hold the seive or milk the he-goat.
THANK YOU. IT'S ABOUT *FUCKING* TIME.
: But, if you are going to claim relevance (and please do not try to
: tell me that the people earning credentials are going to deny their
: own relevance),
Shit. I'm a grad student. *I'M* earning a credential.
I'll deny anybody's relevance I want to, INCLUDING mine if necessary.
But my credential is NOT IN philosophy and anybody who is writing a
dissertation on something new is necessarily unsure what it's ultimate
relevance will be.
: then the abstract manipulations have to attain meaning somehow.
They do, that's easy. They just don't attain in the way YOU expect
them to.
: You cannot define meaning.
It would be circular if you could.
: But, you can use abstraction to obfuscate.
But I don't. That's what YOU do when you allege the relevance
of something as complicated as topological model theory and
unate switching theory to something as trivial as identity.
George Greene
: On Thu, 13 Nov 2003 10:50:50 -0600, mitch <mit...@rcnNOSPAM.com> wrote:
: > I certainly did not expect to be reading so much philosophy this year.
: > But, it seems necessary just to understand why I do not see things in
: > set theory and logic in quite the same way as others.
No, really, it isn't. What is necessary in order for you to understand
why you do not see things in set theory and logic in the same way as
others is FOR YOU TO READ SOME SET THEORY AND LOGIC. The reason our
understandings are different is because WE KNOW SOME SET THEORY AND
LOGIC AND YOU DON'T. All this crap you keep spewing about "existential
import" WHEN THERE ISN'T ANY (beyond that the domain be non-empty)
is loud proof of that.
: > To be honest, for the first time in my life I am taking the historical
: > record of how facts became facts to be extremely important.
It is important in its own right, but it is not important to your
understanding unless you know in advance that we made a wrong turn
somewhere. In any case, when your study of how the facts became
facts FINALLY gets past 1870 to 1920, maybe you will FINALLY see how
Russell and Principia Mathematica and his ramified type theory with an
axiom of reducibility and the whole idea of representing something
as small as a number by something as illegitimately huge as
the-class-of-all-classes-with-that-number-of-elements ALL GOT
FLUSHED EN MASSE AS THE BULLSHIT THAT it THEY WERE, why the world at large
coalesced (the LATER invention of NF *not*withstanding, despite
the robotic citations of such idiots as Franz Fritsche) around a set
theory in which such classes could not even exist, and where what
went up a type-hierarchy of levels was the LANGUAGES AND NOT the sets.
: > I can certainly accept a set theory that is completely irrelevant to the
: > philosophical considerations that comprise a foundational perspective.
It can't possibly be "irrelevant"! It has to be the PRODUCT of, the RESULT of,
the philosophical considerations that "comprise" a foundational perspective.
More to the point, given that the FOM list ALREADY EXISTS, who the heck ARE YOU,
as opposed to THEY, to even DEFINE "the philosophical considerations that
comprise a foundational perspective"???? IF *THEY* say that first-order ZFC
is relevant to those considerations and that perspective (indeed, if its
evolution and adoption-as-the-default RESULTED from those considerations),
then WHO THE HECK ARE YOU to say otherwise?? Not that they do in fact say
that. But my point is simply that YOU OWE some EXPLANATION as to how & why
1st-order ZFC as an existing proposed foundation is inadequate.
And you don't have to invent it on your own.
Lots of people have lots of problems with it and there are a great
many subfields of extension. The obvious one is of course simply around
any kind if big/large/TOTAL universes. They now talk about "large cardinal
axioms".
: > But, as commonly presented, the facts of set theory do not satisfy my
: > intuitions concerning a foundation.
Talking about "the facts of" any LOGICAL THEORY is mis-terminology in any
case, but WHY CAN'T YOU JUST STATE, IN ONE paragraph, WHAT IT IS about
first-order ZFC that, in YOUR opinion, unfits it to be a foundation?
George Greene
: > : There is a canonical order for the signatures of
: > : 2-monotonic switching functions that is independent of any
: > : ordering associated with the underlying referents.
It is only canonical among some particular cluster of researchers.
And what exactly do you mean by "signature". Don't tell me that I
lack the vocabulary. Over HERE, "signature" just HAPPENS to mean something
DIFFERENT. What does it mean over THERE?
mitch
"G. Frege" wrote:
When I started this I simply decided that mathematics was, in fact, natural
language. I thought about it like an operating system--a language form reflecting
the basic invariants of whatever permits us to have grammar. So, I viewed it as a
sublanguage in the sense that it needed to be a language in its own right. For a
long time I referred to it in informal discussions as "the minimal portable
communication grammar."
Naturally, one of the questions is whether the concept is vacuous.
I took two metaphysical principles as axiomatic.
1. The solipsist's dilemma is valid.
2. I believe I exist within a community of similarly situated individuals.
As far as thought experiments go, I think it is pretty strict. It is not going as
well as I would like. :-)
:-)
mitch
mitch
George Greene wrote:
> mitch <mit...@rcnNOSPAM.com> writes:
> : > : There is a canonical order for the signatures of
> : > : 2-monotonic switching functions that is independent of any
> : > : ordering associated with the underlying referents.
>
> It is only canonical among some particular cluster of researchers.
> And what exactly do you mean by "signature". Don't tell me that I
> lack the vocabulary. Over HERE, "signature" just HAPPENS to mean something
> DIFFERENT. What does it mean over THERE?
Sorry about that. I should have said parameter list, perhaps. I think the
term "function signature" still applies, however.
:-)
mitch
G. Frege
> >
> > But WHAT IS a cardinal number?
> >
It's certainly not an asshole full of shit;
i.e. it's not a George Greene.
Probably the following might help:
"The Natural Numbers
In his seminal work /Die Grundlagen der Arithmetik/, Frege successfully
defined the notion of a 'cardinal number' in terms of the primitive
notion of an /extension/ or /set/. The insight behind the definition is
that a statement of cardinal number such as 'There are n F-things'
predicates a higher-order concept of the concept F, namely, that it is a
concept under which n things fall. Frege simply defines the (cardinal)
number of the concept F (i.e., the number of Fs) as the extension of the
concept /being a concept equinumerous to F/. On this definition, the
number of planets is identified as the extension of the concept /being a
concept equinumerous to the concept of being a planet/. In other words,
the number of planets is an extension (or set) which contains all those
concepts which, like the concept /being a planet/, are exemplified by
nine objects.
Frege defined the concept of /natural number/ by defining, for every
relation xRy, the general concept 'x is an ancestor of y in the
R-series' (this new relation is called 'the ancestral of the relation
R'). The ancestral of a relation R was first defined in Frege's
/Begriffsschrift/. The intuitive idea is easily grasped if we consider
the relation /x is the father of y/. Suppose that a is the father of b,
that b is the father of c, and that c is the father of d. Then Frege's
definition of 'x is an ancestor of y in the fatherhood-series' ensured
that a is an ancestor of b, c, and d, that b is an ancestor of c and d,
and that c is an ancestor of d.
More generally, if given a series of facts of the form aRb, bRc, cRd,
and so on, Frege showed how to define the relation /x is an ancestor of
y in the R-series/ (this is the ancestral of the relation R). To exploit
this definition in the case of natural numbers, Frege had to define both
the relation /x precedes y/ and the ancestral of this relation, namely,
/x is an ancestor of y in the predecessor-series/. He first defined the
relational concept /x precedes y/ as follows:
/x precedes y/ iff there is a concept F and an object z such
that:
* z falls under F,
* y is the (cardinal) number of the concept F, and
* x is the (cardinal) number of the concept /object other
than z falling under F/
In the notation of the second-order predicate calculus, Frege's
definition becomes:
Precedes(x,y) =df EFEz(Fz & y=#F & x=#[\uFu & u!=z])
To see the intuitive idea behind this definition, consider how the
definition is satisfied in the case of the number 1 preceding the number
2: there is a concept F (e.g., let F=the concept /being an author of
Principia Mathematica/) and an object z (e.g. let z=Alfred North
Whitehead) such that:
* Whitehead falls under the concept /being an author of
Principia Mathematica/,
* 2 is the (cardinal) number of the concept /being an author of
Principia Mathematica/, and
* 1 is the (cardinal) number of the concept /object other than
Whitehead which falls under the concept being an author of
Principia Mathematica/
Note that the last conjunct is true because there is exactly 1 object
(namely, Bertrand Russell) which falls under the concept /object other
than Whitehead which falls under the concept being an author of
Principia Mathematica/.
Given this definition of /precedes/, Frege then defined the ancestral of
this relation, namely, /x is an ancestor of y in the predecessor-
series/. So, for example, if 10 precedes 11 and 11 precedes 12, it
follows that 10 is an ancestor of 12 in the predecessor-series. Note,
however, that although 10 is an ancestor of 12, 10 does not precede 12,
for the notion of /precedes/ is that of /strictly/ precedes. Note also
that by defining the ancestral of the precedence relation, Frege had in
effect defined x < y.
Frege then defined the number 0 as the (cardinal) number of the concept
/being an object not identical with itself/. The idea here is that
nothing fails to be self-identical, so nothing falls under this concept.
The number 0 is therefore identified with the extension of all concepts
which fail to be exemplified.
Finally, Frege defined:
/x is a natural number/ iff either x=0 or 0 is an ancestor of x
in the predecessor series
In other words, a natural number is any member of the predecessor series
beginning with 0.
Using this definition, Frege derived many important theorems of number
theory. It was recently shown by R. Heck [1993] that, despite the
logical inconsistency in the system of his /Grundgesetze/, Frege validly
derived the Dedekind/Peano Axioms for number theory from a powerful and
consistent principle now known as Hume's Principle ("The number of Fs is
equal to the number of Gs if and only if there is a one-to-one
correspondence between the Fs and the Gs"). Although Frege used his
inconsistent axiom Basic Law V to establish Hume's Principle, once
Hume's Principle was established, Frege validly derived the
Dedekind/Peano axioms without making any further essential appeals to
Basic Law V. Following the lead of George Boolos, philosophers today
call derivation of the Dedekind/Peano Axioms from Hume's Principle
'Frege's Theorem'. The proof of Frege's Theorem was a tour de force
which involved some of the most beautiful, subtle, and complex logical
reasoning that had ever been devised. For a comprehensive introduction
to the logic of Frege's Theorem, see the entry Frege's logic, theorem,
and foundations for arithmetic."
(Ed Zalta)
Source:
http://plato.stanford.edu/entries/frege/
"Irrelevant bullshit", right? Sure... Oh hell, go away and die, George.
- O mean, the only thing that is irrelevant here is obviously YOU.
George Greene
: George Greene wrote:
:
: > mitch <mit...@rcnNOSPAM.com> writes:
: > : There is no reasonable person who knows
: > : me that does not know that I am a good listener.
Well, how many REASONABLE people do you know??
I mean, if you know 12 people and only 2 of them
think you are a good listener, then all I can do is feel
sorry for you that you dismiss 10 out of 12 people you know
as unreasonable.
: > They have not tried to talk to you about THESE questions.
: Wrong. They chose to ignore what I was doing
What exactly were you doing? What were they ignoring?
It looks to me like what you are doing IS talking about these
questions.
: in order to talk about these things
Again, this seems self-contradictory. If what you were doing
WAS talking about these things, then their talking back to you
about them could NOT be "ignor[ing] what [you were] doing".
: because they were a lot like you.
No, they're not a lot like me. They think you're a good listener,
but I KNOW you are a SHITTY listener. You came in here with an
agenda, as yet unchanged, to TALK, NOT listen. You had some axioms
related to mereology and order and BY GOLLY, YOU WERE going to talk
until WE listened! You were SO bad at listening and so intent
on talking that you couldn't even be bothered to answer SIMPLE
SHORT questions about your presentation. You are SO BAD at listening
that you will just change the subject randomly and start talking
about people's character, bosses, brown-nosing, database-update
security, and 69 other things just to avoid engaging in a logical
argument that you will lose.
: > : You are just like every other jackass who knows too much and revels in their own self-aggrandization.
: >
: > No, that's YOU, coming in here with a whole bunch of axioms and a whole bunch of pronouncements
: > about Flum, Zeigler, Halmos, Langholm, and how you know so much more than we do.
: >
:
: No.
:
: At no time did I ever start out by saying that I was trying to revolutionize anything.
That does NOT matter. If you say "there's something wrong with the way standard
set theory and logic treat identity", then you don't HAVE to say "I am trying to revolutionize
foundations via an alternative take on equality". What you said to START with INHERENTLY
MEANS that and inherently IS revolutionary! So SPARE us these LYING disclaimers
about how you weren't revolutionary!
: I tried to argue for a specific context in which to consider alternatives.
It is revolutionary to even suggest that first-order ZFC is going to
be inadequate as a context. It's like ASCII: you can ENCODE ANYthing
in it. Exccept, of course, semantic generalizations over ALL of it.
But even when you have to make the semantic ascent, you can still
use informal natural language about an object-language that remains
first-order. IF there was EVER any NEED for an alternative then
you OWED an EXPLANATION WHY!
: You responded to your own expectations and your own motivations.
That is a truism. Everybody does that all the time by definition.
: >
: > : I started out talking about the identity predicate.
: >
: > No, you didn't. You started out talking about topological model theory,
: > and you continued by attempting to disparage set theory's use of the
: > identity predicate, without knowing what you were talking about.
: >
:
: No.
:
: If you ever looked at my axiom set, you would find that I introduced a formal equivalence.
You introduced a distinctness predicate and completely clouded the issue.
Far worse, you never said whether you were in FOL *with* equality or *without*.
That kind of MATTERS if you are going to START by "talking about the identity
predicate". And what the heck does "a formal equivalence" have to do with
"the identity predicate" anyhow? There are lots of equivalence relations in
the world, as well as in your axiom-set, as well as in ZFC. Until you've said
WHICH of them is REALLY identity, you haven't said anything. More to the
point, howe is your "introduction of a formal equivalence"
any sort of refutation of my claim that you did NOT start out talking
about the identity predicate? You DID start out talking about Flum, Zeigler, and
topological model theory. Nothing you can say about your axiom-set is a refutation
of THAT. You COULD'VE talked about your axiom-set WITHOUT disparaging people
for not knowing topological model theory and WITHOUT even mentioning topological
model theory. You could've EVEN convinced people that maybe you had a clue
what you were talking about by posting the definition of a topology.
But you did none of that.
George Greene
: It was in isolating the component subformulas from one another
: in a manner by which they became logically equivalent that I
: began considering alternatives to extensionality.
You don't even know what you MEAN by "alternatives to extensionality".
Did you mean that you were somehow going to stay in a set theory but
have sets a=b in circumstances OTHER than their having the same elements?
If this is what you meant, then, AGAIN, were you going to do this in FOL
*with* equality or *without*? In the normal version, you get the same
theory either way, but if you are going to change, then whether you
change in the framework of the = axiom-schemata or twiddle some OTHER
knob might MATTER.
: Because I was using diagrams the natural
: choice was the one leading to the questions involving relative identity.
"Relative identity"?? What the FUCK is THAT?
THAT is a term entirely lacking from the vocabulary of the standard paradigm;
in both of FOL-with-equality AND ZFC, there is ONE flavor of identity.
: > : You determined that I did not know what I was talking about.
You don't.
: Sorry ace.
:
: From the contents page of "Symbolic Logic" by Thomas,
:
: Part 1 TRUTH-FUNCTIONAL LOGIC
:
: Part 2 QUANTIFICATIONAL LOGIC
:
: Part 3 QUANTIFICATIONAL LOGIC WITH IDENTITY
:
: As I recall, I was nineteen when I used this book.
"Used" is one thing.
"Got an A in a course where it was the primary textbook"
IS ENTIRELY another.
: It is still on my shelf. I like its presentation of
: deduction.
Yet you still did not say, IN YOUR treatment, whether you were in
FOL *with* identity or *without*. AND STILL HAVEN'T SAID TO THIS DAY,
DUMBASS. So having the book on your shelf obviously hasn't DONE YOU
ANY GOOD, has it. MAYBE YOU SHOULD *READ* it.
: > In particular, you didn't know that FOL comes in
: > two flavors, both with equality and without, and that it was
: > a straightforward matter and a known result to define set theory
: > WITHOUT an identity predicate.
: >
:
: No.
YES, DUMBASS, IT IS a known result that you can do first-
order ZFC in FOL withOUT identity, and FF even QUOTED FRAENKEL
DOING it for you!
: Thomas Jech wrote a definitive text on set theory. He bases
: his presentation on first-order logic with identity.
SO WHAT? That does NOT mean that flavors without identity are unavailable!
And Jech wasn't (mostly) DEFINING new things in any case! THAT is a
TEXT! It is a CAPSTONE! THAT is a magnum opus! THAT is a comprehensive
coverage of everybody's everything! It wouldn't surprise me abit
if it even HAD a section on "how to do it in FOL without equality"!
: Now, I was considering the identity predicate. Do not be such a
: cad as to think that I was not considering about the
: significance of how set theory can be understood relative to
: either logic.
I am not being a cad and I do NOT CARE what you were "considering about"!
YOU PRESENTED an axiom-set! It had SEVERAL flavors of equality in it!
Yet YOU NEVER ANSWERED THE QUESTION about WHICH LOGIC you AND your axiom-
set were using! And when you accused set theory of handling identity improperly,
you NEVER SAID WHICH version mis-handled it!
: In fact, one of the questions I kept asking myself was
: why Jech chose to use identity when he could just define it.
SO YOU ALLEGE. NOW.
AFTER the fact. The reason why this allegation is not credible is that
the first thing *I* told you when you alleged that set theory handled
identity wrongly was "in set theory, identity is a DEFINED predicate; it
doesn't need to exist at all; since it is NOT THERE IN THE FIRST PLACE,
it can't POSSIBLY be there wrongly." Your reaction to this was NOT
agreement! It was NOT "I thought the same thing when I was reading
Jech". It was "you don't have the vocabulary" for topological model
theory and unate switching theory.