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inadequacy of Aristotlean logicSubmitted by frankly1 on Sat, 20071006 22:18
Hi All, I'm new to this forum so I hope this is an appropriate place to bring up this issue. I tried posting the following issue on another forum devoted to Objectivist philosophy, but haven't yet gotten much of an effective response. Just to give you a quick summary of my background, I was once (from about 19681973) 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. My distinct recollection from that time was that both Peikoff and Binswanger claimed that all of deductive logic was fully comprehended under Aristotle's logic. (I gather Peikoff gave a 1974 series of talks on logic  I'd expect that that claim was repeated there, though I don't have access to it, so I can't verify that). In fact, however (and this is my objection), 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 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". You may wish to examine Aristotlean syllogistic logic to verify this point: you will note that every single inference involves a "unary" property, i.e., a property strictly of an individual, such as the property of being human, and no properties that obtain between more than one object, such as x being taller than y. Now I remember quite distinctly Peikoff dismissing essentially all of modern "symbolic" logic (in part because of the socalled "paradoxes of material implication"). He argued 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. Just to reduce my problem to its absolute basics, ask yourself if Aristotlean syllogistic logic manages to capture even the following inference: Premise: There is some person who loves every person. Conclusion: For every person, there is a person who loves him/her. Note that the inference in the opposite direction is not logically valid. Note too, this inference involves a nonunary, relational predicate, 'x loves y'. It is precisely this sort of predicate that Aristotlean syllogisms could not express. Isn't it pretty obvious that this inference is a purely logical one, and that it is logically valid? What then comes of the claim that Aristotlean syllogisms capture all of logic?

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