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Re: Notation
Posted:
May 27, 2013 1:46 AM


On May 26, 9:31 pm, fom <fomJ...@nyms.net> wrote: > On 5/26/2013 10:01 PM, Ross A. Finlayson wrote: > > > > > Nice of you to introduce Boolos, a 20th centory logician who may well > > have already covered the desirable features of theory with the > > acknowledgment of the limitations of either the purely synthetic or > > analytic, and the necessity of their combination of the total and > > wholistic. > > >https://duckduckgo.com/?q=George+Boolos > > Are you surprised that I actually read moderns? Well, > the description theory is a long list of 20th century > developments and relates to a significant body of > researches ignored by "mathematical logic". Its onset, > however, is significant in the works of Frege and Russell. > > > Then, of note is a reference to Boolos' rehabilitation of Frege's > > Grundgesetze, here in terms of the action in philosophy to have the > > technically analytic result in the utility of the synthetic for > > philosophically logical and logically philosphic foundations for > > mathematics, there's an example of such technical underpinnings as > > might be of note and use to represent a suitably concise form for > > discussion on the general matter: general matters. > > Zalta has done a nice introduction to Frege and the modern > day rehabilitation of his ideas, > > http://plato.stanford.edu/entries/fregelogic/ > > Of course, set theoretically this would involve one with > Quine's New Foundations. That too has a resurgence. I > am interested, but the general mathematical community > dictates consideration of set theories evolving from > Zermelo as primary. > > > > > > > > > > > Far from it from that to be said to accomplish this goal, along the > > lines of the Hilbert program is not to complete mathematics then to > > integrate for completion the theoretical foundations of mathematics in > > as to their completion, yet, that is the statement of what would see > > in the basically analytic and synthetic as to, for example, an > > axiomless system of natural deduction, for inference, induction, and > > reason. This is synthetically from null or void, and analytically: > > that the first principles are final causes, for mathematics as a > > whole. > > > I'm reminded of your earliest posts, fom, and their plainly > > enumerative content: from what simplest principles and observation, > > is thus reason evidenced? In describing equality as tautology and > > identity, where is it that what are otherwise seen as the simplest > > logical primitives are themselves structured, and, how is it so that > > then: this truthpreserving theory encompasses "no paradox" and "all > > and none"? > > Circularity is inherent to mathematical practice. Before solving > a problem, analysis dictates considering a plausible solution and > working backward or both backward and forward. The final presentation, > however, is synthetic and forwarddirected from premises to conclusions. > > The tautologous intepretation of "A is A" as "A=A" predates the logical > atomism required to make sense of the compositional language structure > associated with the modern paradigm of formal logic. But, the history > of mathematics leading to "number systems" as abstract axiomatic > systems with deductive calculi dictate treating identity in relation > to "systems". Thus, identity, diversity, and negation are intimately > bound. > > The notion of "priority" begins with Aristotle and grounds his > epistemological argument for distinguishing between "demonstrative > science" and "dialectical argumentation". Late nineteenth century > and early twentieth century trends organized the natural sciences > into a hierarchy of priority. Logicism is a philosophical statement > of priority in this sense. So, when considering logicism critically, > one is confronted with priority as the fundamental issue. > > However, the semantical notions of Frege and the philosophies that > most influenced the received paradigm rose to prominence at the > expense of Aristotle's epistemological arguments. Analysis of > belief took the form of propositional attitude reports rather than > the rhetorical argumentation of the Aristotelian dialectic. So, > Aristotle's epistemological justifications had been deprecated. > > This minimal influence of Aristotle's "Posterior Analytics" is > actually documented in the link, > > http://plato.stanford.edu/entries/demonstrationmedieval/ > > "In the seventeenth and eighteenth centuries, with the new > Platonisms, the antiAristotelian bias of the new science, > and developing empiricism, the theory of demonstration came > to be ignored in mainstream philosophy, although it persists > as an element of Thomism." > > For my part, I eventually came to justify my use of circular > definition for the introduction of 'morphemic' predicates > in terms of Aristotle and Leibniz' interpretations of Aristotle. > Naturally, "objects" such as denotations of constants and > functions should not be introduced with such syntax. But, I > had to figure out how the use of "undefined language symbols" > came to be introduced. That is found in Bolzano, and, it is > related to notions of definition in relation to syllogistic > logic interpreted extensionally. Leibniz' logic is intensional > as are his explanations of individuation and naming. So > there is a mismatch in directionality. > > What is novel in my construction is that I recognize the > role of both directions in justifying the axioms. A 'framework' > must be established intensionally (impredicatively). But, > the ontological foundation and concreta must be extensional > so that the modeltheoretic utility of the axiom of regularity > applies. > > At the core of this is recognizing the identity/diversity > dichotomy in relation to a system as opposed to the selfidentity > of logical atomism. > > > > > That would be of general interest, particularly in the general > > assignment of an integrated logic with general bibliography, in as to > > the parallel carriage of fact from the various to and from the > > specific and macro and micro and synthetic, and analytic. > > > Basically, logic is reversible in: that the complement: is itself. > > > And the infinite is infinite. > > ... unless you consider a theory with completed > infinities. Then the infinite is the absolute infinite. > > (The singular is to the universal as the individual is > to the infinite  a paraphrase of Kant)
Thank you that's quite remarkable, your presentation is impressive.
Priority monism as extreme: a duallyselfinfraconsistent (or here, plurallyself, but dually) urelement founds complete consistent theory, as of zeroeth order logic.
Not just simple: the simple.
Then with mathematics as prior the mathematical sciences, for science, and logic prior the mathematical logic, for logicism, then philosophy the root of reason, for logic, then as above where the highest (or lowest, as it were, superlative) grounds for concern of the domain of discourse are as to the most fundamental objects, or object in monism (also monadism, monadology for the thing as singular and unique and of things) as foundation: then we see that a regular/wellfounded theory of sets isn't truly fundamental, foundational, in as to where mereology and collection would apply to the object(s) of monism as more fundamental.
That then as clear (that the empty set is structure, ZF has definitions and nonlogical axioms etcetera as to be upon something more fundamentally primitive) does and doesn't then resolve for the consideration of a simple theory of logic for mathematics to said theoretical framework, for reasonable definition of all things. As above in this consideration of the construction of the universe, here as to etymology as clear, our synthetic regular set theory sees extension of the theoretical framework concomitant with a not necessarily regular (as it necessarily wouldn't be a la Russell) fragment or subtheory, itself as well: a theory of sets. Infinity and Regularity, the axioms (and nonlogical or proper axioms) aren't independent the others of ZF, indeed, quantification over the finite sees Russell.
Then, for a superRussellian theory or superCantorian as it were, here really though superZermeloaen (zerMaelien) as it has the regular and the irregular "cumulative hierarchy" for Zermelo's regularity, justifications abound for that for the comprehensive treatment of here: sets. It is to be clear that Infinity and Regularity are restrictions of comprehension, and via Skolem and Levy it is that ZF is as to the countable or back to its own defined constant, via models there are irregular models of ZF and models of ZF are ZF's content (as not wellfoundable), via forcing were it consistent that AC is consistent else trichotomy of cardinals wouldn't be, that these are proscriptions to simplify a realm of discourse, at cost to the realm of discourse.
Then, as to the duallyselfinfraconsistent urelement for zeroeth order logic, I was intrigued with the apocryphal note of Boolos:
"Boolos argued that if one reads the secondorder variables in monadic secondorder logic plurally, then secondorder logic can be interpreted as having no ontological commitment to entities other than those over which the firstorder variables range. The result is plural quantification."
I note that for having declared that zeroeth order is n'th order or logic, in these pointed discussions.
That is as to our "usual" existential and universal quantifiers and how they define each other (or as to how the existential is built from the universal): it is still to the plural on the universal in quantification, for each / for any / for every / for all, that it is too simple to subsume those as the same.
Then, defining the universe of sets and constructible universe of sets: there's no such thing in ZF. So: what theory is it? I ask that and think it's A Theory (the null axiom theory), for what it's worth.
It so proves.
Regards,
Ross Finlayson



