fom
Posts:
1,969
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:UYdnUnFksIIUDrMnZ2dnUVZ_gdnZ2d@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 truthfunctional >> 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 leads to two possibilities. So, the question asks if 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 selfreferentiality > (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 noncircularity. 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 truthfunctionality reflects negation. There is no truthfunction 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 prerequisite for the possibility of any experience of the world. > Along similar lines, counting and distinguishing may very well be > apriory 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 prelogical or premathematical. >
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 about angels or intelligences ('that 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/identityrelative/#1
In my (nonstandard) 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 FregeRussell "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 at this point. I wish to return to your mention of "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 "selfidentity" 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 fourfold system of categories is described according to a sequencing in time. In discussing the schematism of those categories, the fourfold 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 *timeseries*, the *timecontent*, the *timeorder*, and lastly to the *scopeoftime*."
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 sumtotal 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 sumtotal 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 notbeing. 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 'nonmortal' does not enable us to declare that we are thereby representing the object of a mere notbeing; the expression leaves all content unaffected. A transcendental negation, on the other hand, signifies notbeing 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 reiterating the Aristotelian admonition against negating substance. 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.

