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January 25th, 2007

Logic really is awesome you guys.  And if you don't believe me, there's this cool website I just bumped into called MetaMath.  I'm gonna try out its Java applet, proof-writing thingy.  :)

Even cooler is that the website recommends A Primer for Logic and Proof that my math professors wrote!  I have the copy she gave me right here.  The Hirsts are an interesting couple ...

... and it's a small world, again.

a Critique of Pure Logic, albeit informally


I am curious about people's concerns or criticism of logic, specifically philosophers in this case, so I'll just ask: do you think logic, so to say, "breaks-down? and where does this happen?

I've been reading Russell's A History of Western Philosophy and his criticism of Aristotle and Kant is fairly clear to me.  I'm glad that we graduated from propositional logic.  :)  Carnap and Quine extend the concern to language and how predicate logic cannot satisfy, but this seems to be a criticism that some philosophers do not wish to cross.  Why?  Is there some goal, yet criticizing language via logic leads us astray?

I bumped into an essay, The Plurality of Science, written by the author of my first logic textbook, Patrick Suppes, and he made interesting inroads for a middle ground, albeit in the context of science.  Bascially his essay challenged the concept of unifying theories because scientists are reductionistic.  Science reduces observations to an ever increasing specificity by mimicking mathematicians and their deductive abilities:
From Descartes to Russell, a central theme of modern philosophy has been the setting forth a method by which certainty of knowledge can be achieved.  The repeatedly stated intention has been to find a basis that is, on the one hand, certain and, on the other hand, adequate for the remaining superstructure of knowledge, including science.  ... Views about the unity of science, coupled with views about the reduction of knowledge to an epistemologically certain basis like that of sense data, are often accompanied by an implicit doctrine of completeness.  Such a doctrine is often expressed by assumptions about the uniformity of nature and assumptions about the universe being ultimately totally ordered and consequently fully knowable in character.
Suppes explains precisely one of my concerns: the ideal of logic.  He goes on about Cantor and Goedel but I found Russell's comments on predicate logic more convincing.  Still, I believe Suppes stopped short of underminding logic entirely because he does not advocate divergent theories, despite the title of his essay.  In his conclusions, Suppes alludes to a 'monotone progression' of knowledge over time instead of convergence toward unifying theories or divergence from chaotic ones.

Returning to logic, I apply this conclusion to language.  Do we maintain a goal to verify that logic reduces language to a unifying -- or single -- meaning? and ensure that language does not explode into pluralistic whatever-you-want meanings?  Indeed, I believe this is the goal of some philosophers as reflected by the ideals they advocate.  Russell summarized my observation well when he severed epistemology and metaphysics from ethics and, essentially, the intent of the philosopher:
Philosophers, from Plato to William James, have allowed their opinions as to the constitution of the universe to be influenced by the desire for edification: knowing, as they supposed, what beliefs would make men virtuous, they have invented arguments, often very sophistical, to prove that these beliefs are true. ... And when he assumes, in advance of inquiry, that certain beliefs, whether true or false, are such as to promote good behavior, he is so limiting the scope of philosophical speculation as to make philosophy trivial; the true philosopher is prepared to examine all preconceptions.  When any limits are placed, consciously or unconsciously, upon the pursuit of truth, philosophy becomes paralysed by fear, ..." (834-5)
Of course, words usually fall on deaf ears but I daresay that even irrationality and unreasonableness may be examined even if both must be eventually rejected by Russell's idealized philosopher, because Russell himself is limiting what philosophy ought to examine.  Still, he and I find some kind of firm foundation in logic though I wonder if the intentions of other people are satisfied when they write proofs.  One of the edifices that even Russell contributed to was rationality,  yet I find Russell's idealization of logic less exclusive than Aristotle's, for example.  Predicate logic embraced more linguistic attributes, such as singular versus plural nouns, and set theory -- when we talk about of groups and membership, even in layman's terms.  Propositional logic could not adequately account for how we speak, although I we say that in retrospect.  And our ability to look back on the progress of logic is, I believe, most significant, even if it is bound by a monotone Suppes limit, so to say.

One goal of logic, at least that taught in my logic classes, was to analyze arguments but I don't think the analyses made during Aristotles era would have satisifed me because of the inadequacy of propositions and syllogisms.  Today I find the inadequacy arises from our inability to accurately translate the original meanings conveyed in language into formal or symbolic logic.  The ideals of completeness, soundness, and consistency are secondary concerns although I find that these ideals might actually be the cause of my primary concern: logic's inadequacy to verify unity or indicate chaos.