Jamwhite

I am certaintly glad that people do not resort to ad hominems when others disagree with them on this forum. For the honest readers of this forum, those who do not approach mathematics as apart from reality. Here is a basic example of the axiomatic approach to mathematics. I am quoting from Pisaturo -- italics his (TIA Nov 2001): What I wish to call attention to here is what I consider the be the essential, epistemological, problem in the theory of a science based on axioms -- it does so by dropping its connection to reality. The first problem concerns the nature of axioms themselves. As Pisaturo puts it, "[a]xioms must be statements about reality, not statements about concepts" [ibid]. Ayn Rand says, "axiomatic concepts identify explicitly what is merely implicit in the consciousness of an infant or of an animal." [ITOE 2nd ed, "Axiomatic Concepts"]. These "axioms" fail every basic test of axioms in Objectivist Epistemology that I can find. Let us contrast the "axiomatic system" of mathematical foundation with Pisaturo and Marcus's. Under their foundation, mathematics needs only one axiom: 1 = 1. Here is what they mean: To me, this is a much better example of an axiom because it relates numbers to units to existents. Thus, mathematics is linked to epistemology and the relationship between between metaphysics, epistemology, and mathematics has begun to be explored. However, let us assume that the original axioms are correct. Notice how the idea of the philosophy of mathematics fades away? In a fully deductive system, there are axioms (which are taken as unquestionable self-evident propositions) and derived proofs/theoreoms/postulates/etc... In such a system, it is the goal for it to exist without reference to anything outside itself. Number, in the example above, is not linked to units or measurement -- it is a primary. Taken that way, the entire system of mathematics can be self-contained without actually referencing anything outside its own definitions. It is for this reason that I say such systems are floating, and rationalistic. When I talk about how worthless the research of my math professors was. It is exactly to the extent that their research was based on floating abstraction that I consider it worthless. My purpose here is simply to present to people that there has been work done to found mathematics on an inductive basis, and that for myself, I found extremely useful in understanding the meaning and purpose of mathematics. For those who are interested, here are the The Intellectual Activists where these articles were published: "The Foundation of Mathematics" by Pisaturo and Marcus, July 1994. "The Foundation of Mathematics II", Sept 1994 "Undermining Reason: The 20th Century's Assault on the Philosophy of Mathematics, Part 1" by Pisaturo, Oct 2000. "Undermining Reason: The 20th Century's Assault on the Philosophy of Mathematics, Part 2" by Pisaturo, Nov 2001 "Undermining Reason: The 20th Century's Assault on the Philosophy of Mathematics, Conclusion" by Pisaturo, Dec 2001 Also, it was either Pisaturo or Marcus who gave a lecture at the Jefferson school a long time ago about the philosophic corruption of mathematics that was sold by 2nd Renaissence. They no longer carry it, and I cannot find a copy to buy. If anyone knows of a copy, please let me know. I would love to have it.

I have a great deal of business using "postmodern" to discuss mathematics, as well as logical positivism. After your colorful response, I went and looked up my original sources on the history of mathematics. In particular, I wish to point out an article series by Ronald Pisaturo called "Undermining Reason: The 20th Century's Assault on the Philosophy of Mathematics" in the Intellectual Activist issues: October 2000, November 2001, and December 2001. From the November issue: Now I verified the dates on positivism and Comte died in 1857, with the main writing is positivism was around the 1870s - 1890s. The Pisaturo articles critic the different attempts at grounding mathematics much better than I could. He also offers a positive theory of the foundation of mathematics which I referenced before -- here. As Pisaturo points out, advanced mathematics has put men on the moon. The problem is all in the philosophic foundations of mathematics -- or what do we mean by the natural numbers. How do validate addition? The answer given by Pisaturo and Marcus (in TIA July 1994 and Sept 1994) is via an induction on the concepts that give rise to mathematics, or by understanding the conceptual common denominator that seperates "some fish" from "5 fish". Now their article goes into this in depth from pre-mathematics onward. Induction, the process of moving from up the abstraction chain, must precede deduction, the process of expanding the breadth of the chain. The problem, historically, of mathematics is that the mathematicians have failed to explain the process their minds had to go through in order to arrive at something -- the induction. I do not fault the mathematicians for trying to defend math, nor do I fault Plato for trying to defend reason. But in both cases, for similiar reasons, all the evidence presented to me shows that both systems are wrong. My experience in mathematics has been this (for context, I was a math minor in college). The axiomatic system of mathematical foundation turned into the equivolent of "where God is dead, anything is possible". The post-doctorial work of most of my professors was in creating or analyzing new systems of mathematics that existed for no other reason then that they could. If nothing else, and there is alot else, this tells me clearly that something is very wrong is mathematics today. Pisaturo's explinations are consistent exactly with that I saw in college.

It does contain a mathematical axiom (just one). Postmodern philosophy attempts to replace induction with axioms. I have yet to figure out where this idea orginated but I believe that it was logical positivism. This article does not subscribe to that method. The article is of the nature of other philosophic writing such as OPAR and ITOE, it inductively shows the foundation of arithmetic and hints at this foundations relationship to all the higher mathematics that results from arithmetic.

(I lied about sleeping, but not in a malicious way. This just occurred to me while I lay there.) Consider the following sentence: "Stop thief in the name of the king. Stop Thief." There are all kinda of propositions popping out of the sentence now. There is king. His messager is yelling at you. He, through his messanger, orders you to stop. Commands, as in the example, can be conclusions to arguments, as can rhetorical questions. This, to me, is another example of the syllogistic nature of human thought.

This is probably be my last post tonight (bedtime). I do not use the term "cram" to refer to syllogisms. My understanding, partly through introspection, is that syllogisms are the mechanism by which our minds put arguments together. I work on mastering syllogisms because I see them in a number of places such as when I do an outline for an article. Or when I feel like I have something missing in an argument in my mind. Or just when I get to a difficult passage in some work. I also wish to add, before I am off, that I do respect your arguments Hal and will give them much thought in the coming weeks and months as I further my study in logic. Though I still disagree with you, I respect many of the reasons that you defend your position and I understand better why someone would hold your position.

I have no idea what that means. For all x that is alive and functioning there is an ultimate value x and life x? Can you give me a couple 'x' examples?

I seem to not be able to connect. Subject-predicate is the definition of a sentence. Aristotelians study sentences. Grammar is the precursor to logic and logic reinforces good grammar. This is what I mean by the depth of the logic. Having clauses in a sentence is exciting it is a sub-premise (if the writer is good) giving details to the main clause. I do find your example interesting. It reminds me of the most quoted Objectivist quote that I am aware of: (Italics mine) Existence exists is of course a proposition. One that is used in the next sentence of the quote. I still see predicate logic as a weak subset with no room to grow and nothing to do excite math-like exercises with small arguments. (which is what I did for a year in predicate logic classes) My goal, my only goal, is to study actual arguments and understand them, and with the exception of some funny sophists, I doubt any famous philosopher has gapping holes in form. Now, I do not deny that valid form is important. But it is a minor issue is analyzing an argument, and the individual only applies it argument by argument and not, as my predicate logic book seems to indicate, by converting the whole paper into a series of symbols.

Are you proposing that a grammatically correct sentence does not contain a subject and predicate? Do you believe that these are not sentences? The first is a compound sentence, and a contradiction. The subject is two plus two, and the predicate is equals four.

Hal, if you would, I wonder if you could comment on your thoughts about the notion of predicate logic of multiple specialized logics. Much of my why I think the field is completely bankrupt rests on the premise that they use "logic" for whatever they want it to be so that their arguments are "valid". Such as intuitionistic logic which denies truth and seeks "confirmation", or relevance logics which have a special symbol (added to true and false) to indicate contradiction?

How is apply logic to my study. This is my answer to Hal's question about Rand and syllogisms. One thing that is important to point out is that most of Rand's writing is inductive, but normally she has a deductive section in the summary. In a particular hard passage, the first thing that I take apart are the propositions, and then I study their relationship to understand the argument being made. For instance, from "Objectivist Ethics" in Virtue of Selfishness: Here is the propositions that I wrote down: 1) What is the relation between ultimate ends or values and the facts of reality? 2) Living entities exist and function necessitates the existence of values and of an ultimate value. 3) The ultimate value for any given living entity is its own life. 4) [Missing] The conditional nature of your life is a fact of reality. 5) The validation of value judgments is to be achieved by reference to the facts of reality. There is a missing premise here (offered an example of what it might be), but when I take the premises apart on paper, I can chew and agree or disagree with them much better. The syllogistic aspects are automated for me, so I look for the term being distributed (ultimate value, and life in my example) and look to see if it has been or what would be required for it to?

Aristotelian (Categorical) logic extends far past the syllogism (to my knowledge predicate knowledge does not extend past the form). For myself, while I read Rand when I come to a difficult passage, I often pick out the terms and look for their syllogistic relationship to each other. (I will post an example in a bit.) As to Kant, if logic is, as Ayn Rand described, the art of non-contradictory identification, then being able to disprove Kant, or any other philosopher must be part of it if you wish to point out their contradictions.

This is known as a enthymeme, a missing premise. There is no syllogistic problem with this statement. You can easily find the premises missing, such as "humans have heads". This is not an argument, but if you used it as a proposition to an argument, then I do not see what the problem is. Two plus two is four. Four is two times two. So Two plus two is two times two. I am curious how you handle it in predicate logic? This is another example of a enthymeme -- some horses are pink. There is no problem in syllogistic logic with moving from all to some. I do not understand how this is case. For the dogmatic Catholics who had to have each term absolutely exact, there was some funny ways around problems, but to my knowledge all such problems are linguistic and not logical. All sentences are subject-predicate. To be strongly focused on a subject-predicate is to be focused on language -- the tool of reasoning. I want to make it clear that it is for the reason that Aristotle is focused on tool of reasoning that I say his is the only valid system of logic. But your statement confuses me. Assuming that it is true that all objectivists are philosophers (a premise I would reject if you mean philosopher as an occupation), how is that different? In both cases you are talking about an attribute of an existent or set of existents. The "particular interpretation of the symbols" means the content of the argument. So, if I understand you correctly, your argument is that predicate logic is superior because it ignores the primary tool of reasoning (the sentence), or content of the argument, and instead focuses only the symbols involved? If this is your argument, then I fully agree and attempted to make that not only the theme of my argument but an indication of the philosophic corruption of the field. If I may ask a couple of questions: what can you learn about the argument itself from only the form? How do you, using logic only, explain the difference between Rand and any other philosopher? Do you not find that you must concede to anyone who can keep the form right that their argument is valid? Aristotelian (categorical)/Rand's logic (as I read it) using the axioms of philosophy are able to invalidate the content of a bad argument which is the only important thing to me.

Having spent the day reading and studying, in response to many of LauricAcid's comments on the Perfecting Logic thread, I do not feel that I have a full enough context to completely invalidate predicate logic. My approach to the subject is to understand its history and the result, but I am not yet strong enough in many of the key areas necessary, especially early 20th century philosophy (during which predicate logic arose as a systematic logic). I also have found almost no good information on the problems, true or imagined, with Aristotelian logic that led to the development of alternative logic systems. However, my reading does indicate that the primary reason for predicate logic's development was an attempt to make all argument mathematical. Leinbiz, for instance, thought "that disputes of all kinds, not merely mathematical ones, could be settled if the parties translated their dispute into the characteristica [language] and then simply calculated." (http://www.rbjones.com/rbjpub/logic/jrh0103.htm). Other research indicates that theirs was a division between linguistic and mathematical logic during the early 20th century. Since I cannot give a full argument about the nature and results of predicate logic, I have confined myself to attempting to point out the essential differences. I shall include a "good faith" account of what I learned in my graduate logic class in college, and then contrast it with the primary topics that I focus on in logic (Aristotelian and Ayn Rand). My source for predicate logic is Logics by John Nolt. The most basic non-Aristotelian formal logic is known as Boolean logic. Boolean logic is limited in what it can analyze to the operators "and," "or," "not," and "if." Its only form of analysis consists of converting the terms between these operators into symbols, and then using the rules of logic to check the validity of the form. (The rules for Boolean analysis are in almost all cases identical to those of Aristotelian logic). An example of a Boolean argument would look like this: If Socrates is a man, then he is mortal. Socrates is a man. Therefore, Socrates is mortal. If a, then b. a; therefore b. Predicate logic is the next stage of Boolean logic, introducing the concepts of universal and particular propositions. It is therefore, just as in Aristotelian logic, possible to talk about some men, or all men. At this point, predicate logic has achieved a relative parity with the Aristotelian notion of syllogisms. By parity, I mean that both systems achieve relatively the same results from the same argument. However predicate logic has not achieved its goal of being a calculus. Thus, it matures into what it currently is a series of logical systems deducted from mathematical theories: abstract algebra and set theory. Among the problems of predicate logic is its inability to maintain any content from the original argument. (It can only use "and," "or," "not," "if," "all," and "some.") To deal with this problem, new symbols were introducted, such as Leibniz's Alethic and Deontic operators, which represented 'possibility' and 'necessity' respectively (Nolt p308). Another example is Kripke, who added operators for "knowing", "has always been the case that", and "obligation" (Nolt p335). In order to handle these new operators, and to later fully integrate with set theory, predicate (now known as "formal") logicians created a new field called "metalogic". This field was highly axiomatized in place of induction and allowed for the resolution of the problems caused by the new operators with the creation of a series of logical systems. Each system was designed to handle a specific category of problems with a specific set of operators and rules. Examples of such systems are free logics, multivalue logics, supervaluations, infinite values logics, fuzzy logics, and intuitionistic logics (Nolt, chapters 15 and 16). My conclusion from these is that they are all rationalistic and floating. Their intention is to create rules such that their arguments are correct, and they keep the student focused so much on the form that the context is ignored. To date, I have never seen, nor could I imagine any use in, anyone using boolean or predicate logic to analyze Ayn Rand. I offer this: the difference between Rand and Kant is not simply that Kant failed to use logic in the correct formal way. What is the alternative? Aristotelian logic is focused on grammar, not mathematics. Its syllogisms handle all the forms of predicate logic, but can also do things like using different verbs without needing any special symbols. It focuses on the relationship of words to concepts and most importantly, Aristotelian logic integrates all logical fallacies. I mentioned in my original statement that the rules of syllogisms should be automatized. No one can analyze the syllogisms of an argument while at the same time taking in the content. So it is important to learn the rules of distribution so that you can automate your ability to validate the syllogistic aspects of an argument. The best logic book that i have ever seen is An Introduction to Logic by H.W.B. Joseph (http://www.papertig.com/Logic.htm). I will quote a section of the table of contents starting with chapter 2. Of terms, and their principal distinctions Of the categories Of the predicables Of the Rules of Definition and Division: Classification and Dichotomy Of the Intension and Extension of Terms and of their Denotation and Connotation Of the Proposition or Judgment Of the Various Forms of the Judgment Of the Distribution of Term in the Judgment: And of the Opposition of Judgments Of Immediate Inferences Of Syllogisms in General In this dense 600-page book, the topic of syllogisms is not even full discussed until chapter 11 (page 249). Furthermore, the material up to that point directly relates to a student's ability to understand a logical argument and to create one himself. However, the Joseph book is very advanced. Loinel Ruby's Logic: An Introduction (http://www.papertig.com/Logic.htm) is much easier, with lots of good Aristotelian content suitable for more casual students of logic. It, too, helps the reader in learning to apply logic to arguments. In addition to the huge repository of material on logic in the Aristotelian tradition, there are also contributions to logic by Ayn Rand. Concept-formation allows for a more detailed understanding of propositions. Also see the "Objectivity" chapter of OPAR -- for instance, under the heading, "Objectivity as volitional adherence to reality by means of logic". The Ayn Rand Bookstore has a number of further topics, including Peikoff's Introduction to Logic (http://www.aynrandbookstore2.com/store/pro...tem=19&mitem=46). Most exciting to the field of logic are the advancements in induction by Objectivism (http://www.aynrandbookstore2.com/store/pro...tem=34&mitem=46 and http://www.aynrandbookstore2.com/store/pro...tem=44&mitem=46). Induction and deduction are the two methods of integrating your ideas. As one last comment, I want to point out that there is nothing you can study in Boolean logic or predicate logic that does not exist in Aristotelian logic. In most cases, such as verbs, these statements are fully integrated already. I hope that this is helpful in explaining my objections to formal logic as such.

I am preparing a post for a seperate thread for many of your other comments. For this one, here is my original source for what I find is the most rational explination of the basis of mathematics: "Foundation of Mathematics" by Ronald Pisaturo and Glenn D. Marcus in The Intellectual Activist Volume 8, Number 4 July 1994 "Foundation of Mathematics II" by Ronald Pisaturo and Glenn D. Marcus in The Intellectual Activist Volume 8, Number 5 September 1994 Included in these two articles is a detailed inductive approach to mathematics which has long been missing.

If you want to learn about logic, you should read "An Introduction to Logic" by H.W.B Joseph (http://www.papertig.com/Logic.htm) and listen to "Logic Thinking" by Dr. Binswanger. If you are seriously interesting in logic, then the induction material available at the Ayn Rand bookstore is wonderful. (Remember induction preceeds deduction) If, however, you are interested in "Predicate Logic", then you will find yourself caught between Objectivism and that "logic". "Predicate logic" divorses thought from the argument is a way that can only lead to rationalism. It pretends that the form of the argument is what makes an argument logical. Many Objectivists have rejected this claim and their material is also available at the Ayn Rand Bookstore. Logic is a rich field in which the form of the argument is really only a small minor piece that you automate when you first learn it. Definition, categorization, conceptual common denominators -- these are all the more interesting topics in the field. While many fewer Objectivists have written about the errors in abstract algebra, set theory, and the other inverted mathematical fields. You can find some good material on this in old Intellectual Activists (though I do not have them available to reference). The problem with the "abstract algebra" approach to mathematics is that it uses deduction in a rationalistic way by igonoring the inductive process that got them there. Much like the classical education school in the past, abstract algebra arbitrarily looks at certain relationships in mathematics, claims, with no basis, that these are axioms, and then presumes to write completely floating systems on top of this imaginary framework.