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Topic: LOGIC & MATHEMATICS
Replies: 96   Last Post: Jun 6, 2013 5:19 AM

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 fom Posts: 1,968 Registered: 12/4/12
Re: LOGIC & MATHEMATICS
Posted: May 31, 2013 10:50 PM

On 5/31/2013 10:36 AM, Julio Di Egidio wrote:
> "fom" <fomJUNK@nyms.net> wrote in message
> news:UY-dnUnFksIIUDrMnZ2dnUVZ_g-dnZ2d@giganews.com...

>> On 5/30/2013 11:20 AM, Julio Di Egidio wrote:
> <snipped>
>

>> is there a criterion
>> for deciding whether either of the complete connectives should
>> be viewed as the canonical complete connective? That is, is
>> there an asymmetry in the dyslexia we call truth-functional
>> logic?

>
> Why should there be a canonical one?

There need not be.

The idea is this. Given some standard formulation of a
formal language based on recursive definition, the syntax
is peppered with negation symbols. Now, there is an
equivalent representation whereby every negation has
been eliminated through the use of NAND. Similarly,
there is an equivalent representation whereby every
negation has been eliminated through the use of
NOR.

The *question* is whether or not this is a fundamental
symmetry. If not, what structures might indicate a
criterion for choosing one representation over another.

So, "the given" is a canonical form that becomes
ambiguous when the negation is eliminated. The *question*
asks if that ambiguity -- that is, that symmetry -- is
truly essential.

It is a question of demarcation. For other questions,
one begins with a language of formulas and establishes
certain normal forms within that collection of formulas.
Those normal forms demarcate a subcollection of formulas
that serve certain purposes. In my case, eliminability
I can find a reason for preferring one over another.

I have satisfied myself on that point. But one thing
at a time as your questions proceed (or you give up
on me...)

> Isn't indeed self-referentiality
> (circularity) the essential character of the (any) purely logical
> system?

My answer to that is yes.

I have done a great deal of work to understand how modern mathematical
logic has reached the point where its foundations are almost
exclusively focused on non-circularity. So, while you see this
condition as a matter of fact, such a claim in the mathematics
community may get you some metaphorical version of tar and feathers.

My notion of a foundational set theory begins with the sentences,

AxAy( x proper part y <-> ( Az(y proper part z -> x proper part z) /\
Ez(x proper part z /\ -(y proper part z))))

AxAy(x in y <-> (Az(y proper part z -> x in z) /\ Ez(x in z /\ -(y
proper part z))))

The "proper part" relation is recharacterized as extensionally defined
"proper subset" after some additional axioms. But, the point with
respect to your question is that I view these as circular definitions.
This is perfectly consistent with an Aristotelian theory of
demonstration -- but, it is a difficult case for a modern foundation
unless one examines how the modern ontology came about

(Note that I would never define a constant or function with such
a definition. Constants and functions utilize the sign of equality
in their definitions. They are not "truth makers" in the syntactic
sense of relations. I have a specific "language construction
paradigm" in mind here that actually fixes my notion of cardinality for
"truth values for a language". But, that is for another time.)

Note that the syllogistic hierarchy of Aristotelian logic has the
structural organization of parts and whole (species and genera).
Aristotle speaks of individuals, but does not "isolate" them or
"define" them. That seems to begin with Aquinas and passes into
mathematical logic through Leibniz' generalization.

So, those two sentences characterize the basic structural features
of Aristotelian classes and they do so without a class abstraction
operator like comprehension.

With regard to circularity, my answer is somewhat more complicated.
That is, there is a particular Wikipedia post of which I am fond,

http://en.wikipedia.org/wiki/M%C3%BCnchhausen_Trilemma

There seem to be few choices. It seems best to seek a foundation
for mathematics and logic that respects those aspects in a
balanced fashion. The emphasis to avoid circularity at all costs
seems unbalanced to me.

> (And, I am intrigued, why would you call such system dyslexic?)
>

It is probably an inappropriate choice of words. I did not magically
come to my positions. In order to recognize the geometric relations
(correctly or incorrectly) that I have resolved upon, I have probably
scratched out truth tables at least 50,000 times. 'T' and 'F' are
meaningless and symmetric.

The asymmetry in the system occurs with the truth functions
associated with 'p' and 'q' in standard representations. These
are projections. But, they are invariant under De Morgan
conjugations. I have named those functions in a way that a
mathematician might begin a proof:

FIX (the first component as)

LET (the second component be)

with their negations named

FLIP (the first component)

DENY (the second component)

so that the mnemonics reflect an order of construction.

The "dyslexia" to which I refer is the idea that truth-functionality
reflects negation. There is no truth-function that is a fixed
point under negation. If there is any sense to be made of attaching
"logic" to some metaphysical truth or some actuality, it does not
come from simple purport that "T represents truth".

There is an invariance associated with the De Morgan conjugation
and the contraposition involutions. Moreover, with respect to
De Morgan conjugation alone, a given axiom from my 4096 axioms
maps to another axiom. The fundamental invariant is De Morgan
conjugation. This is why I do not consider Boolean algebra or
negation as fundamental.

Note that others have identified a role for De Morgan's laws
with respect to classical logic. The difference in what I
have done is that I have attached this role to the nature
of compositional linguistic structure by combining the
intensional sense from Church with the notion of equational
theories from Birkhoff. Do I recognize it as a "cheap math
trick"? Yes. But, it is the math trick that places a
context on the linguistic analysis that allowed Frege to
pass from subject/predicate forms to the compositional
forms that ground our modern deductive calculi.

>> Well, my ideas are actually motivated by considering
>> set theory. So, arithmetic, for me, is inside of a
>> theory of classes...

>
> Kant says that space and time are given prior to experience, in fact as
> a pre-requisite for the possibility of any experience of the world.
> Along similar lines, counting and distinguishing may very well be
> a-priory faculties, and, at least as far as I can presently see, that's
> the confusion: the use of logic or mathematics to describe what is
> properly pre-logical or pre-mathematical.
>

Kant distinguishes between "general logic" and "transcendental logic".
In his dissertation, John MacFarlane does a good job of comparing the
Kantian notion of "general logic" with the modern paradigm,

http://johnmacfarlane.net/dissertation.pdf

But, what of "transcendental logic"?

Well, as MacFarlane observes, general logic

"... abstracts from all content of the knowledge
of understanding and from all differences in its
objects, ..."

Kant
CPR, A54

Again, just before introducing the contrast with
transcendental logic, Kant writes:

"General logic, as we have shown, abstracts from
all content of knowledge, that is, from all relation
of knowledge to the object and considers only logical
form in the relation of any knowledge to other
knowledge."

Kant
CPR, A55

But, then he goes on to say,

"..., a distinction might likewise be drawn between
pure and empirical thought of objects. In that
case we should have a logic in which we do not
abstract from the entire content of knowledge.
This other logic, which should contain solely the
rules of the pure thought of an object, would exclude
only those modes of knowledge which have empirical
content. It would also treat of the origin of the
modes in which we know objects in so far as that
origin cannot be attributed to the objects. General
logic, on the other hand, has nothing to do with
the origin of knowledge, but only considers representations,
be they originally a priori in ourselves or only
empirically given, according to the laws which the
understanding employs when, in thinking, it relates
them to one another. It deals therefore only with
that form which the understanding is able to
impart to the representations, from whatever source
they may have arisen."

Kant
CPR, A55 - A56

So, before reciting more Kant, how can we see this
manifested in the modern developments?

Recall that Frege introduced a descriptivist theory
of names. The same holds for Russell. Frege's version
was more like negative free logic in that any negative
existential statement took the null class as its
semantical object. Out of dissatisfaction, Russell
devised a description theory without the presupposition
failure forcing the interpretation of negative
existentials. It is Russell's version that reflects
the realist, classical logic. But, Russell's version
treats descriptions (and names) as a type of quantifier
that is true when instantiated and false when not
instantiated.

Compare this idea with what Leibniz writes:

"What St. Thomas affirms on this point
here every individual is a lowest
species') is true of all substances,
provided one takes the specific
difference in the way that geometers
take it with regard to their figures."

Leibniz
Discourse on Metaphysics

The origin of the principle of identity
of indiscernibles is a Leibnizian generalization
of an Aquinian idea. Logicist individuation
in relation to the Aristotelian syllogistic
hierarchy originates with -- or, at least, is
motivated from -- Thomas Aquinas by Leibniz'
account. Leibniz, however, introduces a
geometric aspect to the principle that is
ignored in the modern logicist standard account
of identity,

http://plato.stanford.edu/entries/identity-relative/#1

In my (non-standard) view, this is relevant to
the difference between Cantor and Frege. Cantorian
set theory is fundamentally a "theory of ones" and
should be contrasted with the Frege-Russell "extension
of a concept". Cantor defended himself somewhat, but,
his ideas certainly could not withstand the influence
of "Principia Mathematica" in the twentieth century.

It is not my purpose to pursue the topological aspects
"distinguishing" in relation to logic.

So, let me recall what Leibniz wrote concerning the
nature of denotation with a name,

"All existential propositions, though true,
are not necessary, for they cannot be
proved unless an infinity of propositions
is used, i.e., unless an analysis is
carried to infinity. That is, they can
be proved only from the complete concept
of an individual, which involves infinite
existents. Thus, if I say, "Peter denies",
understanding this of a certain time, then
there is presupposed also the nature of
that time, which also involves all that
exists at that time. If I say "Peter
denies" indefinitely, abstracting from
time, then for this to be true -- whether
he has denied, or is about to deny --
it must nevertheless be proved from the
concept of Peter. But the concept of
Peter is complete, and so involves infinite
things; so one can never arrive at a
perfect proof, but one always approaches
it more and more, so that the difference
is less than any given difference."

Leibniz

I would have to look up exactly where I found
this. It is among his logical papers rather
than his more philosophical works. But, it
relates the notion of individuation to a
system of relations with other existents. It
is not the "self-identity" of the ontological
interpretation of

Ax(x=x)

motivated by logical atomism.

So, what now do we find in Kant with respect to
transcendental logic?

"Logicians are justified in saying that, in the
employment of judgements in syllogisms, singular
judgements can be treated like those that are
universal. [...] The predicate is valid of that
concept, without any such exception, just as if
it were a general concept and had an extension
to the whole of which the predicate applied. If,
on the other hand, we compare a singular with a
universal judgement, merely as knowledge, in
respect of quantity, the singular stands to the
universal as unity to infinity."

Kant
CPR, A71

"In like manner infinite judgements must, in
transcendental logic, be distinguished from
those that are affirmative, although in
general logic they are rightly classed with
them, and do not constitute a separate member
of the division."

Kant
CPR, A71 - A72

What I am trying to convey here is that our views of
set theory as a foundational theory must be compared
with Kant's transcendental logic and not directly to
his statements concerning the relation of mathematics
to sensible intuition. Kant would have objected to
the introduction of completed infinities into mathematics
on the basis of his statements. But, it is unclear how
he would have responded to the developments of which he
could know nothing. The description theories that arose
with the foundational studies of Frege and Russell speak
to a need for a theory of individuation in relation to
the totalities of individuals that may form a logical
system.

The relationship of general logic to arithmetization
is practically obvious in Kant. The transcendental
deduction of his four-fold system of categories is
described according to a sequencing in time. In
discussing the schematism of those categories, the
four-fold system is described as

"The schemata are thus nothing but a priori
determinations of time in accordance with
rules. These rules relate in the order of
the categories to the *time-series*, the
*time-content*, the *time-order*, and lastly
to the *scope-of-time*."

Kant
CPR, A145

But, it is not until much later that Kant addresses
the notions of import to modern foundations. Here
are some of his remarks concerning a "universe" of
predicates,

"Although this idea of the sum-total of all
possibility, in so far as it serves as the
condition of the complete determination of
each and every thing, is itself undetermined
in respect of the predicates which may constitute
it, and is thought by us as being nothing more
than the sum-total of all possible predicates,
we yet find, on closer scrutiny, that this idea,
as a primordial concept, excludes a number of
predicates which as derivative are already given
through other predicates or which are incompatible
with others; and that it does, indeed, define itself
as a concept that is completely determinate a
priori. It thus becomes the concept of an
individual object which is completely determined
through the mere idea, and must therefore be
entitled an ideal of pure reason.

"When we consider all possible predicates,
not merely logically, but transcendentally,
that is, with reference to such content as
can be thought a priori as belonging to
them, we find that through some of them we
represent a being, through others a mere
not-being. Logical negation, which is indicated
simply through the word 'not', does not properly
refer to a concept, but only its relation to
another concept in a judgement, and is therefore
quite insufficient to determine a concept in
respect of content. The expression 'non-mortal'
does not enable us to declare that we are thereby
representing the object of a mere not-being; the
expression leaves all content unaffected. A
transcendental negation, on the other hand, signifies
not-being in itself, and is opposed to transcendental
affirmation, which is something the very concept
of which in itself expresses a being. Transcendental
affirmation is therefore entitled reality, because
through it alone, and so far only as it reaches,
are objects something (things), whereas its opposite,
negation, signifies a mere want, and, so far as it
alone is thought, represents the abrogation of
all thinghood.

"Now no one can think a negation determinately, save
by basing it upon the opposed affirmation. Those
born blind cannot have the least notion of darkness,
since they have none of light. The savage knows
nothing of poverty, since he has no acquaintance with
wealth. The ignorant have no concept of their
ignorance, because they have none of knowledge, etc.
All concepts of negations are thus derivative; it is
the realities which contain the data, and, so to
speak, the material or transcendental content, for the
possibility and complete determination of all things."

Kant
CPR, A573 - A575

Admittedly, Kant is thinking of this scenario in
terms of some sort of actualism whereby the system
of predicates is predicatively grounded by substantive
individuals. And, he is here re-iterating the
But, Aristotle associates substance with individuals,
and, the development leading to the situation in
modern foundational mathematics is a logic that can
consider the possibility of individuation.

The relationship of his statements to individuals
are made clear in his footnote,

"In accordance with this principle, each and every
thing is therefore related to a common correlate,
the sum of all possibilities. If this correlate
(that is, the material for all possible predicates)
should be found in the idea of some one thing, it
would prove an affinity of all possible things,
through the identity of the ground of their
complete determination. Whereas the determinability
of every concept is subordinate to the universality
of the principle of excluded middle, the determination
of a thing is subordinate to the totality or sum of
all possible predicates."

Kant
CPR, A572

I apologize for a long post. What Kant says or does
not say is always an issue because the historical
record has not often represented him fairly. I hope
some of these remarks have helped you to think that
what Kant says in the transcendental aesthetic is not
the only material relevant to modern foundations. The
modern logic is distinguished from the Aristotelian
syllogistic logic by how it implements methods of
individuation. Kant did speak of these things. He
simply did not speak of them in the transcendental
aesthetic.

Thanks.

Date Subject Author
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5/28/13 LudovicoVan
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6/4/13 LudovicoVan
6/4/13 namducnguyen
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6/5/13 fom
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6/3/13 Herman Rubin
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6/4/13 Peter Percival
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6/1/13 LudovicoVan
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6/5/13 Peter Percival
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6/2/13 fom
5/28/13 Zaljohar@gmail.com
5/28/13 Charlie-Boo
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