Aristotelean Logic As Antiquated

[Nicko0301]

I was reading Bertrand Russell's The History of Western Philosophy. In it, Russell intimated, with what seems to be disdain, that Aristotle's Logic, though seminal and admirable, is obsolete; and that one mustn't neglect recent advancements in aforesaid field. This prompted a bit of curiosity in me, for I am trying to learn how to reason properly.

My question is this: What works do you feel are important in regard to Logic? And, incidentally, would you consider Aristoteleanism as antiquated?

Thanks

[TheEgoist]

De Interpretations and Prior Analytics contain the majority of Aristotle's ideas on logic. His logic was commented on highly during the Medieval age by the Scholastics, but you won't find all that much about it after that outside of guys like Venn. Aristotle's work on logic is thick and pretty hard to grasp without rigorous reading.

For modern logic, the obvious answers are Gottlob Frege's "Begriffsschrift" and the collected works of Russell and Whitehead in "Principia Mathematica". Hard as Aristotle is, this stuff is insane. PM is 3 volumes in size, Begriffsschrift's logical notation is...what's the word...insane.

Aristotle's logic is foundational, and as with most disciplines, is essential to the development of logic. However, it is extremely limited in its scope and ability. Predicate logic (Modern logic) first started taking issue with the problem of existential import, that categorical logic's statements of particularity must assign existence to entities such as unicorns in order to form a syllogism about them. For example:

Some Unicorns are Brown

Therefore, Some brown things are unicorns

This must take the assumption that some unicorn actually does exist.

Of course, this was only the start. Frege, Russell and Whitehead set out to make logic an explanation, at least in some aspect, of mathematics. As such, it is said to have much larger scope than categorical logic.

There are a thousand criticisms lobbed at categorical logic. They usually deal with things it fails to be able to express. For example, categorical logic cannot make the inference "Every house is a building. Every owner of a house is an owner of a building"

[Schmarksvillian]

For modern logic, the obvious answers are Gottlob Frege's "Begriffsschrift" and the collected works of Russell and Whitehead in "Principia Mathematica".

Those were early important works, but they have been supplanted by more modern, streamlined treatments. Also, Principia Mathematica especially is not a treatise just in logic but also in mathematics.

Hard as Aristotle is, this stuff is insane. PM is 3 volumes in size, Begriffsschrift's logical notation is...what's the word...insane.

I don't know what you consider "insane" about the works. That Principia Mathematica is long does not entail that it is insane, just as the length of any work does not in itself make the work insane. And Frege's 2-dimensional notation does seem awkward to modern readers, but Frege's contribution is not usually counted as his notation but rather the idea of the predicate calculus itself. That is, we use more modern (and "1-dimensional" notation) to express basically the system (or equivalent variations) that Frege devised. But again, that Frege first devised awkward notation doesnt' entail that his work is "insane", nor is the notation, albeit awkard, itself insane. Edited March 5, 2010 by Schmarksvillian

[TheEgoist]

I just meant that it was insane in its notation. Insane here = difficult. Anyway, he was asking for the most influential books (I took it), not just the most recent. Those were what, from what I've seen, shifted the paradigm completely.

[Schmarksvillian]

I just meant that it was insane in its notation. Insane here = difficult.

Ah, got ya. Fair enough.

Anyway, he was asking for the most influential books (I took it), not just the most recent. Those were what, from what I've seen, shifted the paradigm completely.

Again, fair enough. Frege introduced the predicate calculus (actually, not just first order, which I should have said); then used his higher order calculus to derive mathematics. Too bad his higher order system was inconsistent (the first order portion is okay, I think). Then Whitehead and Russell proposed instead a theory of types from which to derive mathematics. (By common account) they failed to the extent that they desired to derive mathematics purely from logic (since infinity, choice, and reduciblity are not logical axioms). Of course, first order set theory also provided a foundation for mathematics, and it seems to have won out as the most commonly accepted foundation even until now. And there have been other alternatives proposed along with other advances in our knowledge on the subject.

/

As to the first post in this thread, was Russell actually disdainful? Maybe he was (?), but one isn't disdainful for merely pointing out that Aristotelian logic is not adequate for even the quite modest logic that goes into ordinary mathematical reasoning, while Aristotelian logic is subsumed by ordinary logic systems such as first order logic. By the same token, we recognize forms of reasoning (modal, relevance, etc.) for which plain first order predicate logic may not be enough.

Edited March 5, 2010 by Schmarksvillian

[TheEgoist]

Well Godel, to my knowledge anyway, showed that mathematics could not be fully or even substantially derived from logic and axiomatic systems. My knowledge on all that is rather limited though

[Schmarksvillian]

One might make take the view that the incompleteness theorem refutes logicism (logicism being the view (or hope, perhaps) that mathematics may be derived from logic alone), but I don't find it entirely clear that the incompleteness theorem does that (of course, much of this depending on what we mean in this context by 'mathematics', etc.). However, I know of no system that fulfills the logicist goal. Yet, there are neo-logicists who have written about various approaches through somewhat changing some of the terms of success, though I am not familiar enough with the writings to opine on them.

Actually, more commonly, it is taken that the incompleteness theorem closes the door on Hilbert's program, not really that incompleteness shuts down logicism. Again, though I think it is safe to say that it is the common view that incompleteness puts the kibosh on Hilbert's program, there still are people who argue that at least parts of Hilbert's program may be salvaged or even that the crux of Hilbert's program remains undamaged by incompleteness. These are philsophical matters upoon which one must conclude or not for oneself.

Edited March 8, 2010 by Schmarksvillian

[TheEgoist]

Straying from the original topic since no one else seems engaging, do you have any background in this stuff, SV?

[Schmarksvillian]

Many years ago (too many) I did some undergraduate study along with some graduate seminars taken while I was an undergraduate. In the last few years I've gone back to the subject as a hobby. In that context I've compiled a set of systematic, comprehensive, exhaustive notes in which I provide a theorem-by-theorem, definition-by-definition set theoretic derivation of the basics of mathematics, including arithmetic, the construction of the reals, abstract algebra, graphy theory, topology, mathematical logic, and other areas. However, due to limitations of time, at this point the development hasn't gotten past the mere basics (say, the first few chapters of a text in each of the aforementioned subjects). Moreover, I have not myself yet mastered much of even the mid-level material.

In other words, my knowledge and understanding are solid up to a certain modest point, but then falls off quickly when the subject matter gets more into the graduate level.

To make matters worse, lately I find I'm starting to pursue a completely unrelated creative endeavor, so the logic and math are taking a poor back seat. But I want to do at least a little bit each week so that I don't lose complete touch with it.