I've been seeing and hearing alot of talk recently about logical paradoxes and how they somehow "cut at the very heart of logic". I've read that they "take logic off the pedistal that its wrongfully been placed on for so long." That rational, "linear" thinking is now as optional as a moral stance is optional. I've heard someone say that faith in logic is as blind as faith in God. Then I read and hear people talk as if there were nothing outside of logic to begin with. That it is the epitomy of objectivity. That it is built into our universe more fundamentally than matter and energy is and that our very consciousnesses absolutely depend on it. What are these logical paradoxes and do they or do they not ultimately refute logic itself?
yes, postmodernists dont believe in logic, and there are a fair number of them. I'm reminded of a conversation that I read that the mathematical genius Alan M. Turing once had about the potentiality of logical contradictions and thier implication on the validity of mathematics. Essentialy he and a friend were debating over whether logical paradoxes such as "What I am saying now is a lie" meant that the logic math is built up on is also flawed. Alan just said "We just won't call this multiplication." Basicly he meant that, sure little tricks like that are neat, but they don't interfere at all so many other existing logical structures. These paradoxes are all self contained little loops, thier existance dosnt invalidate logic.
Thank you. I kind of figured that they were more word games than anything else. So the postmodernist distain for logic is misplaced? IS there a reality outside of logic? I've heard the term "arrational" used alot instead of irrational so as not to carry the derogatory connotation. It implies a world of arrationality and world of rationality, coming together to produce this world. On the other hand, I've heard some logicians claim there to be NOTHING outside of logic. That anything "arrational" is really rational but so complicated that its rationality is not immediately recognizable What do you think?
Paradoxes such as Goedel sentences, Turing's halting problem, Russell's paradox etc show that logical structures built on proofs of truth and falsity have their limits. Imo more can be learnt from paradoxes than from proofs. Trouble is it's a vast topic. As Spymoose says paradoxes are 'little loops', inconsistent self-references, but that does not make them inconsequential. Their effects reverberate through logical systems and ultimately undermine them completely.
Please provide an example of what you mean here. Present a paradox of the sort referred to, and explain how it reverberates through a particular logical system to undermine it completely. There is too much in the way of apparently 'empty' claims-- fill them in for us. Butte Montana
I try not to make empty claims. Please Register or Log in to view the hidden image! The obvious answer is take Goedel's incompleteness theorems, which state that unless it is very simple no formal axiomatic system can be complete and consistent simultaneously. The proof of this arises in a roundabout way from the 'Liar's paradox'. It means that to the extent that any system of calculation or reasoning is formally axiomatic it is subject to limitations on what it can prove, and how much we can trust the proofs we make within it. It also means that however extensive such a system is made there will always be true propositions that are not theorems within the system, that are not provably true within the system, but which can be known to be true despite this. In other our knowledge of what is true can extend further than any but the simplest formally logical system of proof. The implications of this are far reaching but it's a dangerous topic for a non-mathematician so I'll leave it there. Penrose has explored them at length, and comes close to making it an argument for God. Hawkings has an essay about Goedel and physics at http://www.damtp.cam.ac.uk/strtst/dirac/hawking/ (Russell's paradox slightly preceeded Goedel's proofs but is concerned with the same problem in set theory.) Thus from Goedel we know that even when our axioms are consistent they give rise to logical contradictions, and axiom based reasoning can never produce certain truth. At the same time we know that there can be such a thing as certain truth, for we can know that Goedel sentences are true even though we can't prove that they are. We know this from the paradoxes that arise within formal systems, or rather the paradoxes are why it is true. I hope that's an answer. I'll stop there before I start getting muddled. Canute
Let me re-word what Canute just said to see if I understand it. If my rewording is at all incorrect please correct me. An axiom is a statement whose truth is a "given" and which all other statements are judged against? The axiom is your referent for truth, and so if the statement does not jive with the axiom, then it is deemed false. This goes to show that all truth is not absolute, but is true only relative to some axiom. Since this is the heart of logic, then such things as the "liar's paradox" cut at the heart of logic, because they further subjectify this concept of "objective truth" which we think we attaining through logic. Is my interpretation correct? Whether it is or isn't, my second question is: What makes people who believe in the infallibility of logic believe in the infallibility of logic?
CETBO I agree with that. But these paradoxes show a little more than that. They show that even if ones axioms are consistent and true (accord with reality) contradictions and undecidable propositions still arise. Unless reality itself is illogical then this shows that true knowledge cannot be derived from axioms. Not just because the axioms are unproven, but because axiomatic systems cannot provide certain truth under any circumstances whatsoever. Truth must be known non-axiomatically, in other words truth must be derived from what you know is true, and not from making initial assumptions. This is true even for a system built on just one axiom (God for instance). Thus although it's possible for us to know truths it is impossible to ever prove them, since doing so would require the use of a formal axiomatic system. That's how it seems to me anyway. Paradoxes can do more than this also. Zeno's race between Achilles and the tortoise suggests, if not actually proves, that spacetime must be unquantised. All this is imho of course. The answer to your question is that generally people don't bother to think about this sort of stuff much these days, having been forced against their will through a mass vocational training system that has no time for the study of anything which won't generate future income tax revenues, and thus promotes the idea that logic and metaphyics are not worth thinking about seriously. I agree with Penrose, who believes that we have hardly begun to appreciate the full implications of Goedel's proofs for our ideas of logic and rationality. By the way, I'm not pretending to be certain about any of this. The more you get into it the more complicated it gets. Canute
Please see http://yinyinwang.bravepages.com for paradoxes discussion. Logics is the reasoning precedures, or rules of reasoning. It is a topic of knowledge, so it is validity depends on the origin, the natural behavior. The current knowledge of logics may not be perfect, at least we should know the conditions or limits of its validity. That dose not deny logics.
