Fantastic question. Just a brief reply (and only one mode of several possible replies). Suppose you take away the word "belief" from your question. That we can "hold" or "consider" contradictory thoughts or ideas is no big deal -- after all, whenever you decide which of multiple mutually exclusive beliefs to adopt, you continuously weigh all of them as you work your way to your decision. Having that capacity is all you really need to obtain (say) the specific benefit you mention (pursuing multiple lines of inquiry simultaneously). When does a "thought" become a "belief"? Well that's a super complicated question, particularly when you add in complicating factors such as the ability to believe "subconsciously" or implicitly. On top of that let's throw in some intellectual humility, which might take the form (say) of (always? regularly? occasionally?) being willing to revisit your beliefs, reconsider them, consider new opposing arguments and objections. Plus the fact that we may easily change our minds as...

I presume this phrase refers to the "The one thing I know is that I know nothing" remark attributed to Socrates? Well, one form of paradox occurs when you are simultaneously motivated to endorse a contradiction -- i.e. both accept and reject a given proposition, or assign the truth values of both true and false to it. And that seems applicable in this case. On the one hand what Socrates is asserting is that he knows nothing (after all, if he KNOWS that he knows nothing, then since knowledge usually implies truth, it follows that he knows nothing). But then again on the other hand the very assertion seems to disprove it, since he KNOWS it, and therefore knows not nothing, but something. So he simultaneously seems to be asserting that he knows something and that he does not know something. Now you may not find this particularly paradoxical -- you might be tempted to resolve it directly (by rejecting one of the two propositions). But I suppose it's called a paradox because reasonably good cases can be made...

Perhaps this: true by definition, v. true by means of some correpondence between their meanings and the world. "Bachelors are unmarried" is logically true, ie true by meaning, because that is how we use the definitions involved; it's a matter of convention and meaning that that sentence is true, and thus one doesn't need to go investigate the world whether it's true -- indeed it's not making a claim primarily about the world at all, if it's truth matter is a function of definition. Contrast with "bachelors live longer on average than average man." Ths is NOT merely logically true, true by definition -- we must go do a study to find out fi it's true, and thus to learn something substantive, some fact, about the world.. Logical truths are trivial because we learn from them no new facts about the world, beyond the meanings of the words involved. hope that helps-- ap

This is a great question, which deserves a book-length answer. (And in some possible world, perhaps, I would give such an answer.) For many philosophers the 'logically possible' means something like the 'non-contradictory', which (for many) also yields something like the 'limits of intelligibility.' That is, you may imagine the possibility of a world in which logic does not apply, but that is not a world we can grasp, make sense of, in any way. (I can imagine a 'round square' or a 'married bachelor,' I can say those words, but as soon as I try to make sense of such a thing I pretty much have to give up.) So it's not really apparent that we can even meaningfully entertain the notion you're working with, that there are/might be 'possible worlds' in which logic doesn't apply. In light of that it seems plausible to hold, instead, that by 'possible world' we mean 'logically possible world,' i.e. worlds the description of which does not involve any contradicitons (and worlds in which logic is applicable)....

Well, one CAN do that, since I myself in fact do (and many, many others) .... But of course what you're asking is more like "is it morally permissible to violate one's own principles?", or something like that ... Assuming that one's principles are correct (i.e. that you are right to believe that animals shouldn't be willfully killed etc.), then it seems clear that the answer must be no, because it's not morally permissible to do that which is morally impermissible! But that seems so clear that I wonder if that really is, ultimately, your question. Weakness of will is a well-known (and much discussed phenomenon), and a paradigm case of weakness of will is precisely that where you cannot bring yourself to do that which is right (and so when I succumb, and eat meat, I condemn myself for not being able to live up to my own standards). But you seem to be getting at a much deeper question, which the weakness of will case is merely a simple case of: is there a moral obligation to avoid contradictions, to...

I don't really, but it is one of my biggest pet peeves, from the perspective of one grading students' philosophy papers! ... My guess would be that on many such occasions, the person means something like "valid" -- where "valid" does NOT mean the technical deductive notion but something closer to "true"! (They will often say, "P is not logical," clearly meaning that P is false ...) Occasionally people use it with a defeater: "P seems logical, and yet here's why it's false ..." On such uses they seem to mean "apparently true, even if not really true." Rarely do they use it with anything very close to its basic sense, if not quite "deductively valid" then at least bearing some relationship to arguments and conclusions (where to say "P is logical" would be to say "P is based on some form of argument") .... ap

Rather than offer a response to this excellent question, let me just refer you to a paper whcih essentially raises and discusses the very same problem: Susan Haack's "A Justification of Deduction," from the journal Mind in 1976 (try vol 85, n. 337 I believe). Also, Lewis Carroll (as in "Alice in Wonderland" has a similar, more fun version of it -- "What the Tortoise said to Achilles" -- also in Mind, in 1895 or so ... Check them out! ap

Great set of thoughts, here. But maybe one quick mode of response is to remark that much depends on just what you take the word "cause" to mean. You could take it to mean something like this: "x causes y" = "y if and only if x", as you've suggested. Then, granting that both cases above are cases fulfilling the "if and only if", sure, giving birth would count as a cause of the later death. But now two things. (1) Why should "cause" mean precisely that? Wouldn't it be enough if the x reliably yielded the y, even if things other than x could yield the y too? (i.e. couldn't you drop the 'only if' part, so 'x causes y' would mean 'if x, then, y', even if it might also be true (say) that 'if z, then y'?) Going this route would preserve your intuition that both cases above are cases of causation, but focus on whether your particular definition is the best one. (2) Perhaps more importantly, though, one might examine the 'pragmatics' of causation -- how people actually use the word, different from how...

If by 'logical' you mean 'a decent argument can be constructed of this form' then i would say the answer is yes -- but if you mean 'an absolutely convincing argument ...' then, well, you don't find too many of those anywhere in philosophy -- my favorite version of the kind of argument that Allen Stairs mentions is some version of the fine-tuning argument -- which observes how perfectly fine-tuned features of the universe seem to be, such that they could easily have been otherwise, and yet had they been otherwise then human life (conscious, rational, moral life) would not have been possible -- and goes from there to argue that it is reasonable to think this didn't occur by chance -- a good source on this topic would be any of Paul Davies' recent books ... best, ap

lucky you, with so much time on your hands and with such interesting interests! there are numerous secondary expositions of Godel etc. -- I personally love Douglas Hofstadter's way of explaining it (in Godel Escher Bach and also his more recent Strange Loops) ... but Rebecca Goldstein has a recent book on it (haven't read it, can't speak to its quality) -- http://www.rebeccagoldstein.com/books/incompleteness/index.html good luck ap