There are no changes in the rules of logic between Euclidian and non-Euclidian geometry. They reach different conclusions by applying the same rules of logic to different initial conditions. The fact that both are correct and useful doesn't mean that logic is relative and changeable, it's just an acknowledgment that in some spaces parallel lines never intersect, and in others they do. Since different types of space have different properties, it makes sense that you would use different systems to model them. This is no different than acknowledging that some cars are made by Ford and others are made by GM, and that they have different properties.
Nasor: Perception being influenced by system, I'm not sure how any of this critiques what I've written. Hastein: Good point, but I'm not trying to disprove solipsism there as to show that it is an impractical (less valid) way of thinking. One might have a somewhat more exciting mental life, but the cost would be jumping off a building thinking one could fly. One could say, and this is something of my origional point, that there are no real 'truths' - only more or less useful ways of approaching the universe. It's gratifying to see how much black metal incorperates of classical - especially Romantic - music. I think Bathory's 'Blood Fire Death' album was dedicated to Wagner (not that I'm a huge fan of Wagner) Classical was my first love, along with industrial and 80s synth-pop (is that the word?) like New Order and the Cassandra Complex.
It is true that the same logic is used in Euclidean geometry and non-Euclidean geometry, namely, classical first order predicate calculus. (If you need set theory as well, you need to move to second order logic, but as far as I remember Hilbert's axioms can be stated in pure first order terms, though I might be wrong here.) But the claim in Xev's original post that several logics are conceivable is also true. Here is short list of logics that have an accepted status today and are different from classical logic: many-valued logic (admitting at least three truth-values besides classical True and False; here, the sky is the limit, there are even infinite-valued logics), probability logic (formally a variant of infinite-valued logic), fuzzy logic (created by an engineer to model vague concepts), intuitionistic logic (Heyting's logic, in which the classical law of excluded middle is not accepted), free logic (the domain of discourse can be empty - this is prohibited in classical logic), the huge family of modal logics, i.e., the logics of necessity and possibility - a class of inferences ignored by classical logic (historically the first of this family was C.I. Lewis' "logic of strict implication" in his systems S1 - S5), intensional type-theoretic logic (Richard Montague's logic to model a broad class of inferences made in natural language), various logics to treat counterfactual implications of the form "If it were the case that A, then it would be the case that B" (David Lewis in his "Counterfactuals" examines 26 such logics - their semantics is based on a topological generalization of the semantics of modal logic first given by Kripke in the sixties), default (or non-monotonic) logics (here the monotonicity principle of classical logic is given up), deontic logic (a variant of modal logic dealing with obligations and permissions), the logic of common sense reasoning (initiated by Minsky and subsequently developed by AI people and philosophers), dynamic logic (Pratt's idea of modelling abstract computer programs), temporal logics (Prior's "linear time" systems and the subsequent developments using branching future), and the list could be much much longer than this. (I didn't mention quantum logic because I don't really know it, but surely it is not a classical system either.) The landscape is perplexing and one mode of inference judged valid by one of the above logics may well be judged invalid by the other. The natural question arises: which one of them is the real logic? And this question does remind one of the similar question: which geometry is the real geometry? In the case of logic, the choice of a particular logic depends on what type of inferences you want to capture. But valid inferences are not dreams, so each and every logic in the above list seems to cover something of a logical reality (though this concept is not fashionable nowadays). David Lewis had a long series of arguments against the view that thinking about possibilities and necessities can ever be reduced to our mental and linguistic operations. His "modal realism" may be too far-fetched, but I think the case is very similar here to the case with geometry: according to physics, cosmic space is non-Euclidean, so Euclid's original system does not describe reality adequately. In other words, the real geometry of our world seems to be Riemannian. In the same vein, it might well turn out some day that classical logic does not describe reality adequately but some other logic does, probably one still to be discovered.
Xev Your first post seems true. It's a strange topic. We know that (logically) it is impossible to prove anything about reality from within any axiomatic system (within any possible universe). Nevertheless we can know things to be true. Goedel proved both of these things. If our systems of logic are all flawed then I'd say it does not matter too much. However if our rationality is an illusion then we're dead in the water. We might as well assume that we are rational, or at least capable of being rational. In some doctrines there is a perfectly good reason why we cannot prove anything about reality, this being that reality has no true or false attributes. (Buddhism, Spinoza etc) In these doctrines (belief-systems, affirmations, whatever) our epistemology is entailed by our ontology. This is impossible to prove ex hypothesis in these doctrines, but it seems like a reasonable idea. PS. How do I subscribe to a thread? Since the changes I can't see how to do it. Is it automatic if I post a message?
