frankly1

Aristotle's logic is a restricted subset of logic, as it is now understood. Aristotle's syllogisms did not handle n-adic relations, functional relations or extended operations on sets. There are several modalities that Arisotle did not deal with. Metalogical analysis of logical formalisms was also out of reach for Aristotle, so he had no completeness or incompleteness theorems. Aristotle did not deal with multivalued logics where there is a degree associated with the truth of proposition. These logics are appropriate for situations where information is not complete. And this is just a beginning to the list. All beginnings are difficult. Aristotle formulated logic version 1.0. Ba'al Chatzaf I don't disagree with you, though I myself am unclear just how applicable particular so-called non-classical logics, such as modal logic or multivalued logic, really are to real situations. And for what it's worth, axiom systems expressible in the language of Aristotle's logic, that is, that of unary predicates, actually are complete and even decidable, though he certainly never got to a point that that question was something he could prove, or perhaps even contemplate. I do believe that his system of logic was "complete" in the sense that there were no inferences he could not capture in his syllogistic system, given the restrictions in the type of predicates he employed. But I can only say that your own views here seem to conflict with the Objectivist views I remember pretty distinctly from many years ago. Under those views, Aristotle's logic captured all of logical inference. I'm still wondering if standard Objectivist philosophy nowadays allows that Aristotle's logic was deficient in its ability to capture, say, mathematical deductive inference. Indeed, could it even capture this following inference? Premise: There is some person who loves every person. Conclusion: For every person, there is a person who loves them. Note that the inference in the opposite direction is not logically legitimate. Note too, this inference involves a non-unary, relational predicate, 'x loves y'.

Sorry if I came to present my issue in the wrong context -- I'm new to these parts, and so don't have a feel for what's considered appropriate. Just to give you a quick summary of my background, I was once (from about 1968-1973) pretty seriously involved in studying Objectivism, and attended a number of lectures given by Peikoff, and a course at the New School by Binswanger on Objectivism. But one problem I had with the philosophy at the time was that I couldn't understand how it handled the entire question of logical inference. There are a number of things you say that I take issue with, and which replicate the sort of problems I had with Objectivist thought way back in the early 70s. Most importantly is your assertion that "~deductive~ logic was nailed down by Aristotle's treatment of syllogisms." In fact, Aristotle's syllogisms did NOT capture all of deductive logic by any means. In fact, it is demonstrably incapable of capturing the logic of the simplest kinds of mathematical deductive inference -- the very basis of virtually every theory in the hard sciences. While you're right to point out that Aristotle and the Greek philosophers were perfectly aware of Euclidean geometry, and the rigorous inferences that discipline involved, the logical techniques of Aristotle had no way of capturing those inferences. While Euclidean geometry proceeded from axioms via purely logical inference to prove all of the theorems of the geometry, no philosopher or logician until Frege in the late nineteenth century was able to describe the logic of those inferences. The fundamental thing that Aristotlean logic lacked was an ability to characterize logical inferences when those inferences involved relational predicates or properties. Yet in mathematics, it's virtually impossible to find a subbranch in which relational predicates are not utilized. Even the simple assertion, "for every integer there exists a larger integer" involves necessarily a relational predicate -- "x is larger than y". Now one reason for my original post is that I remember quite distinctly Peikoff deriding essentially all of modern "symbolic" logic (in part because of the so-called "paradoxes of material implication"). He argued, as do you, that Aristotle had captured all of logic. He basically claimed that the modern "symbolic" stuff was based in confusions of various kinds. I have been wondering if Objectivist thought has moved beyond this in the many years since. What I find interesting is that others in response to my post seem to have basically accepted as legitimate many forms of modern logic. Is this now acceptable in Objectivist thought? How about people like Peikoff -- have they come around on this? (I'd be pretty surprised in Peikoff's case, given his general extreme reluctance to accept modifications in Objectivist beliefs, once stated.)

Hello All, Maybe this is a good thread in which to raise my related question about axioms. One basic thing I don't understand about Objectivism is how it deals with logical inferences. My strong impression has been that it holds that all of logic was essentially fully captured by Aristotle's treatment of syllogisms. But how can one do even the simplest kind of logical inference when it comes to such things as, say, Euclidean geometry? How could those logically rigorous arguments possibly be handled by Aristotlean logic? Has anyone holding to Objectivist philosophy tried to address this issue?