1. I don't think there is any reason to suppose that learning about mathematical logic from Principia to Gödel will be any help at all in understanding what is going on with the Frankfurt School. (The only tenuous connection I can think of is that the logical positivists were influenced by developments in logic, and the Frankfurt School were concerned inter alia to give a critique of positivism. But since neither the authors of Principia nor Gödel were positivists, it would be better to read some of the positivists themselves if you want to know what the Frankfurt School were reacting against). 2. Of course, I think finding out a bit about mathematical logic is fun for its own sake: but it is mathematics and to really understand I'm afraid there is not much for it other than working through some increasingly tough books called the likes of "An Introduction to Logic" followed by "Intermediate Logic" and then "Mathematical Logic". Still, you can get a distant impression of what's going on...

It certainly isn't the case that both theists and atheists just assume that everyone knows what "God" refers to. Reflective theists worry about what "God" refers to. And indeed, at least for some atheists, their problems start exactly here too: they listen to what their local friendly theists are saying about the God they supposedly believe in and they find they just can't make enough sense of it. For such an atheist, it isn't that they well know what kind of thing this God would be if he existed but don't think that there's anything that fits the role. Rather, rightly or wrongly, they think that the stories about the alleged being -- at least those told by believers who try to go beyond crude anthropomorphic myth -- fail to describe a coherent role that anything could fit. Round our neck of the woods these atheists of course mostly hear Judeo-Christian stories to be ultimately baffled by; it is the local believers who, as it were, set the agenda for the local unbelievers. It isn't so much a matter...

On the standard account, given two finitely long lines, even of different lengths, their pointscan indeed be matched up one-to-one, e.g. by the kind of projection theteacher indicated. And the possibility of that kind of one-to-one matchingis just what we mean when we say the two lines "contain the same number ofpoints". What makes this possible, despite the different lengths? In part, the fact that there are an infinite number of points along a finite line (the issue we are dealing with here is one of those initially puzzling matters which arises when we deal with the non-finite: intuitions tutored by finite examples can lead us astray). And there being an infinite number of points along a finite line is related to the fact that the points on a line are dense -- that is to say, between any two points, however close together, there is another point . Now, consider a line with end points. Between the left hand end-point a and any other point on the line there is a further point....

Shame on your school! :-)After a basic logic you can either go deeper (more of the same, but pursued to greater depth), or go wider (look at logics that deal with more than do sentential and predicate logic -- modal logic, for example, which has primitive operators for "necessarily" and "possibly" -- and also look at rivals to classical logic. Going a bit deeper: try David Bostock Intermediate Logic , OUP ; Ian Chiswell & Wilfrid Hodges, Mathematical Logic, OUP (not as advanced as its title might suggest). Going a bit wider: try Rod Girle, Modal Logics and Philosophy , Acumen; Graham Priest, An Introduction to Non-Classical Logic (2nd edn: CUP ). Some of each: John Bell, David DeVidi, Graham Solomon, Logical Options (Broadview Press).

Of course we might expect religions to take issues about sexual life and conduct seriously (though with some due sense of proportion, compared with other matters, like issues of social justice -- and it is the seemingly too prevalent lack of that sense of proportion that prompted my passing remark). What is quite bizarre is the kind of daft obsession that leads the Anglican communion to point of breaking up over the question of gay bishops. And what is simply vile is the kind of lunatic obsession that gets women stoned for adultery.

My maternal grandmother was the youngest but one of a Victorian family of ten; her oldest brothers were about twenty years older than her. It doesn't seem at all morally inappropriate that she should have cared about her nearest siblings much more than those hardly-known distant figures who left home when she was a toddler. And she manifested her favouritism in all kinds of ways: surely nothing morally amiss with that! And no doubt the older children who were still at home had their various favourites among the little ones too -- surely nothing amiss with that either so long as no one got too left out. So I can't see that there is anything wrong per se about having favourite siblings and manifesting that favouritism. Where things get more problematic is when numbers get small: it could indeed, as Sean says, then be wrong to manifest preferences too much. But suppose that (because of a family tragedy) you and cousins were brought up together from young: then surely the same would apply. So...

Of course it isn't true! Just walk down the street with your eyes open ... Most of us just don't make it in the beauty stakes. Most of us are just very ordinary -- not even quirkily striking. Tough, but that's life. Thankfully, beauty isn't everything, and with luck we get by. We non-beautiful people even manage to hook up with other non-beautiful people (or at least, that's how it usually goes, though as the poet Robert Graves remarked, beautiful girls can have strange tastes in men ...), and the world rattles on and gets populated all the same. It would quite be as daft to seriously pretend that we are all beautiful as to pretend that we are all very athletic, all very smart, or indeed all very nice. We're not. And just as telling someone dumb that they are smart does them no favours, telling someone particularly plain and pasty that they are beautiful won't make them feel better (they can see what is in the mirror and now have a deluded or lying friend too). Is there any kind of...

My colleagues raise a number of points, some rather puzzling, which deserve more that there is space for here. But some quick reflections: 1. Love of the good, to take Charles's example, may be a fine and noble thing. But something surely can be fine and noble without being beautiful. In fact, by my reckoning, both Charles and Richard seem to be prepared to stretch "beauty" and "beautiful" in ways I don't find at all natural or helpful (I'm wickedly reminded of the old hippie all-purpose "beautiful, man!" when Richard talks of Ghandi). They both seem to think being "worthy of our deep aesthetic delight" is ipso facto sufficient for being beautiful. Well, in so far as I understand the phrase, I would have thought that the Grosse Fuge, Dostoevsky's Crime and Punishment , Titian's The Flaying of Marsyas , and King Lear are, if anything is, worthy of our deepest aesthetic delight. But it would seem a quite inept response to describe any of those as beautiful . The list could be greatly...

You ask a grad student ... Well, I semi-jest, but a good way of making a start on some new topic is indeed to ask someone what the two or three recent "must read" items are. And reading these will firstly tell you whether you are going to find the topic fun/profitable to pursue, and no doubt the bibliographies at the end of the papers or books will give you lots of pointers for where to go next if your decide you want more. If you haven't someone on hand to advise (or someone suitable at the end of an email, or among your facebook/twitter friends), I'd start with the relevant Stanford Encyclopedia of Philosophy article, if there is one yet. They vary from good to stunningly good, and usually have amazingly good bibliographies too. And to fill the gap between bibliographies (necessarily backwards looking, and usually a few years out of date) and the current state of play, you can look at the Philosopher's Index (a bibliographical database any university library should have access to), and ...

You might well enjoy reading the wonderful Philosophers without Gods (edited by Louise Antony, one of the contributors to this site). Some of the essays in that book might help you see that belief need not be "equivalent to superstition and group-think"; other essays question the "advantages to having a moral framework as one finds in [a] chosen/inherited religious tradition"; and other essays again will help in thinking through a decent secular moral framework.

Isn't that simply a false dichotomy? You need the good questions and the right spirit of enquiry. If you've no clear, well-formulated, questions then you'll just produce an ill-directed ramble. If you aren't driven by curiosity actually to explore the good questions, if you lack the desire to follow the argument wherever it leads, then nothing will come of having raised the questions! But I'd just add that there is nothing at all special about philosophy here. It's the same whether cosmology or molecular biology, history or the study of ancient Athens are your thing. Enquiry needs to be guided by good questions and driven on by the right spirit of 'wonder'. Or at least that's the ideal! Of course, there's lots of routine hack work in science and history, mechanically grinding away in a low grade way. And equally --- though don't spread the word too enthusiastically! -- there's lots of routine hack work in philosophy (regrettable, maybe, but people do need publications for tenure, and...