Yes, if I flip a fair coin 3 times I have a 1 in 2 3 (i.e. 1 in 8, i.e. 12.5%) chance of throwing three heads. How do we get that result? The rule is that if P and Q are independent events, then the chance of (P and Q) = chance of P x chance of Q. Likewise, if P, Q and R are independent events, then the chance of (P and Q and R) = chance of P x chance of Q x chance of R. If each of P, Q, R as a 1 in 2 chance, then the chance of (P and Q and R) is 1 in 2 3 . But, no, if I make 8 trials at throwing three heads I don't have a 100% chance of pulling it off. For the trials are independent events. And the chance of any one trial being successful is still 1 in 8, irrespective of what happened in the previous trials. Likewise, the chance of any one trial being un successful is 7 in 8, irrespective of what happened the previous trials. So the chance of eight trials being unsuccessful is (7/8) 8 , which is about 0.34. So the chance of getting three heads at least once in 8 trials is .66, i.e....

"Ok, ok," says Anne, "fair point. But I guess I'm not really hung up about the visual thing. A booming voice from the sky would do. Or even a few more signs like burning bushes in the Moses story. But something in your face and unmissable. Sure, people say they have evidence of God. But why does this 'evidence' all seem so flakey and disputable? If God really exists and is all-powerful and all that, why doesn't he make his own existence just obvious to us?" Warming to her theme, Anne might continue: "Some people go on and on about the bible being the word of God. But why should I believe that if I don't already believe in God? Some people talk about religious experience. But what people 'experience' seems to depend on what they already believe (after all, Tibetan monks don't have visions of the Virgin Mary do they?). And some people claim to find evidence of God's presence in the everyday world around us. Like in all the disease and suffering? Or like when they make up dodgy creation...

Yes, different notions are indeed at stake here. We need to distinguish physical probabilities from evidential probabilities. Physical probabilities, also known as chances , are what are involved when we say, for example, that An atom of plutonium 238 has a 50/50 chance of decaying within 88 years. Smokers have a greater chance of getting lung cancer than non-smokers. The chance of rolling 1-1 with a particular throw of a pair of fair dice is 1/36. Note, the half-life of a plutonium atom is an objective physical property of it (a property it has independently of our beliefs about it). Likewise the probability of rolling "snake eyes" is a physical property of the chance set-up. And physical chance is related to another kind of physical property, namely the long-run frequency with which certain events turn up in a sufficient number of trials. For example, in the long-run, about 1 throw in 36 will turn up snake eyes. But philosophers argue over the relationship between the chance...

Start with a different case. Take the sentence "Whales are a kind of fish". Once upon a long time ago, that would have been taken to be a truism. And someone who then denied "Whales are a kind of fish" would probably have been suspected of not understanding "whale" (or "fish") -- whales are "by definition" a particular big kind of fish, it would probably have been said, just as bachelors are a particular kind of unmarried man. Philosophers of the time might even have said it was "true by definition", the kind of thing that is true just in virtue of the meanings of the words involved (" analytic " as some later philosophers would put it). Yet nowadays we do routinely deny "Whales are a kind of fish". So if it really was once true just in virtue of the meanings of the words, and now it isn't true, we'd have to conclude that the meaning of the words has changed. But is that right? Have we definitely come to change what we mean by the words "whale" or "fish"? Or is it perhaps that we have...

There are a number of issues raised here. Let's do a bit of separating out. Take the sentence "Verdi died over a hundred years ago." That's true. It isn't made true by something happening now . The event whose occurrence makes that sentence true is something that happened in the past, in the 1901. (And this isn't a case like the founding of Rome by Romulus and Remus, a mythical event. Verdi was a real person. His death is an event that actually took place.) I wonder just what is being said, then, by "there is effectively no actual past". Is it being claimed that really was no such actual event as Verdi's death after all (just as there was no actual event of the founding of Rome by Romulus and Remus)? The past is a blank and all history is a myth? That's absurd. Or is it being claimed that Verdi's death isn't now actual, i.e. isn't now happening. But that is trivial -- no one disputes that! So it is not immediately clear what sensible but interesting view can be expressed by "there is...

The Guinness Book of Records says (or at any rate, used to say) that the world’s longest lasting rainbow was continuously visible over Sheffield for some six hours on 14 March 1994. Here's a picture (I had an office in the Arts Tower at the time which is why I know about it!) Now, did the Book of Records do some research to check that at every single second from 9am to 3pm, someone or other was perceiving the rainbow? I very much doubt it! I guess that the Book of Records supposed that, to establish the record, it was enough to have grounds for thinking that,at any moment in those six hours, someone suitably placed and lookingin the right direction would have seen the rainbow (it was continuously perceiv able , though not necessarily continuously perceived). And the Book of Records is speaking here entirely in accordance with our ordinary ways of talking about rainbows as continuing to exist unperceived. I see a dramatic double rainbow, go to fetch someone to see...

Not surprisingly, philosophers have always had a tendency to wildly overrate philosophizing. Let me strike a cheerfully skeptical note! Just before the "unexamined life" remark, Socrates says "this is the greatest good for a man, to talk every day about virtue and the other things you hear me converse about examining both myself and others". Which is, frankly, absurd. Sure, a few people have a taste for philosophical discussion about virtue (and no doubt it is a good thing that some people are given to think about such things). But it is just daft to suggest that if philosophizing isn't your scene, then you are missing out on "the greatest good", and somehow your life isn't really worth living. Maybe you just prefer to spend time with your friends, or having sex, or going to the opera, or sailing, or hill-walking, or working as a doctor, or bringing up a family, or acting, or gardening, or raising money for Oxfam, or playing string quartets, or doing any of the myriad other things that...

There is a lot of questions here. Let me pick up on just one. Suppose Jill has devoted all her energies to her family, has centred her whole life around them. And suppose her husband, unknown to her, is a serial deceiver, holding her in contempt; one child is a crooked fraudster; the other (again, still all unknown to Jill) is a wastrel and drug-addict. In this sad situation, even if her ignorance is bliss, Jill's life is not going well. The meaning she thinks she finds in her endeavours is in fact an illusion. In this case, we could hardly "salute the meaning" her devotion lends her life, for that meaning just isn't there. Yet it could still, for all that, be the right thing to leave her in ignorance -- there will be cases and cases. What is our relation to Jill? How strong is she? What would befall her if she wakes up and smells the coffee? We can't possibly give a general rule here. But even if we think we should in the particular case leave things be, we wouldn't be "leaving well alone". That...

The short answer is "no". It might take "experience" and "culture" (in a broad sense) to understand some sentence or other representation M . But it certainly doesn't follow from this that we can't know (as "objectively" as you like) whether M is correct. For a stark illustration of the point, take the case where M is a bit of mathematics, e.g. take the claim "There is an infinite number of prime numbers". Tounderstand this statement you have to call on some background experience in elementary mathematics (the use of numbers in counting and so on). We might say that to understand thissentence requires being inducted into a bit of mathematical "culture". But the fact that your understanding of this claim might besaid to rely on your mathematical "culture" in some broad sense,doesn't mean that (once you've got the understanding) you can'tacquire objective knowledge whether the claim is true. In fact,the claim is a theorem, and you can get knowledge of its truth-- objective...

Just one comment, not really on the main thrust of Allen's response, but on his remark "Some people see the death of a fetus -- even a very early-stage fetus -- as the moral equivalent of the death of a full-fledged person such as you or I." I think it is much more accurate to say that some people, when discussing abortion, proclaim that they see the death of a very early-stage fetus (we ought to say "embryo") as the moral equivalent of the death of a full-fledged person. But though some might proclaim that, very few indeed seem actually to believe it. And that is revealed by the fact that very few indeed think of the natural death of an embryo as the moral equivalent of the natural death of a full-fledged person (or indeed, of a neonate). While the natural miscarriage in the very early weeks of a pregnancy may, for some, be a misfortune, very few people regard it as the moral equivalent of the death of a newly born baby (for example, if a woman is rather cheerfully relieved to...