Consider the following game that costs \$2 to play: You roll a fair, six-sided die. You are awarded a \$6 prize if, and only if, you roll a six; otherwise, you get nothing. Should you play the game? Well, considering the odds, the average payout - or "expected utility" - is (1/6)x(\$6)=\$1, which is *less* than the \$2 cost of playing. Therefore, since over many trials you would lose out, you should not play this game. That line of reasoning sounds OK. But let's say you are given a chance to play only once. What sort of bearing does this "average payout" argument have on this special "one shot" case? If you are in this for a single trial, it is not obviously irrelevant what the trend is "over many trials?"

Good question. My own view is that what happens in the long run is irrelevant to the rationality of betting (or in your case not betting) according to the odds in the single case. I think that it is a basic principle of practical rationality that your choices should be guided by the probabilities and that, surprisingly, there is no further justification for this. A first point. You say 'over many trials you would lose out'. Well, if you are talking about a finite number of trials, that's not guaranteed. It is possible--indeed there will be a positive probability--that in a finite number of trials you will win even if you bet against the odds. All we can say it that the probability of winning over many trials is low. So now we are just back with the original problem. Why is it rational to avoid doing something just because the probability of success is low? Does the situation change if we think about an infinite number of trials? Well, it's not even obvious that you are guaranteed to...

Is an event which has zero probability of occurring but which is nonetheless conceivably possible rightly termed "impossible"? For instance, is it "impossible" that I could be the EXACT same height as another person? I take it that the chance of this is zero in that there are infinitely many heights I could be (6 ft, 6.01 ft, 6.001 ft, 6.0001 ft, etc.) but only one which could match that of a given other person exactly; at the same time, I have no problem at all imagining a world in which I really am exactly as tall as this other.

Probability theorists often consider random trials with infinitely many possible outcomes each with probability zero--for example, the probability that a quantum particle will be at some particular point in space. In such cases, the probability that the result falls within an (infinite) set of such outcomes need not be zero----the probability that the particle is in some region of space, say. I don't see that there is anything paradoxical here. It's true that cases like this violate the 'additivity' assumption that the probability of a disjunction of non-overlapping outcomes is the sum of the probabilities of the individual outcomes. But there's nothing manadatory about this assumption when we are dealing with infinite sets of outcomes, and probability theories covering this kind of case are perfectly consistent. The question asked whether we should use the term 'impossible' for probability zero outcomes like the particle being at some particular point in space. I'd say not, ...

These days, you often hear about criminal trials in which genetic predispositions to violence are invoked as factors mitigating moral culpability. Strictly speaking, though, isn't all our behavior -- good and bad -- dictated by an interaction of our genes and environment? If genes direct us in any case and at all times, does it really make sense to cite genetic determination in the instance of bad acts, as if these were exceptional cases?

You are quite right. There seems no good reason why genetic causes should absolve us from moral responsibility any more that other causes of our behaviour. This is a point that has often been made by Richard Dawkins. If there is a threat to free will and moral responsibility, it is determinism per se, not genetic determinism in particular. Of course, there remains the question of whether determinism does undermine free will and moral responsibility. Compatibilists say no--they say you are free as long as your actions issue from your own conscious choices, even if those choices themselves are determined by your genes and environment. Incompatibilists say yes--if your actions are ultimately determined by causes beyond your control, then you aren't free, even if the determination proceeds via your conscious choices. But, either way, genetic causes have no special status. Compatibilists will say that genetic causes, like other causes, don't undermine your freedom when they influence your...

I have a question about probability (and baseball). Say that a hitter has consistently hit .300 for many years. Now, suppose that he begins a new season in a slump, and hits only .200 for the first half; should we infer that he will hit well above .300 for the second half (and so finish with the year-end .300 average we have reason to expect of him), or would this be an instance of the gambler's fallacy?

If the hitter were just a batting machine that averages .300 in the long run, then it would indeed be an instance of the gambler's fallacy to think he would end up with his normal .300 even though he's .200 halfway through the season (just as it would be a fallacy to suppose a fair coin that has come down heads 5 times in a row is more likely to come down tails than heads over the next 5 throws). But our hitter isn't a batting machine, and one respect in which this may matter is that he may try harder in the second half of the season so as to keep up his record of hitting .300 each season, and this may itself make him score well above .300 for the second half. What we have here is an instance of the 'reference class problem'. Should we consider the hitter's second-half performance as an instance of (a) the class of all his half-season performances (in which case we should expect him to average only .300), or should we consider it as an instance of (b) the class of his-second-half-season...

I was reading a philosophy article, which just talked about "the functions of..." (the subject was law, but that isn't what I'm interested in now). I am not being able to find any introductory philosophy text on "function", not even in Wikipedia (apart from entries on mathematical functions). What is a function?

You can think of a function of X as an effect E that X is designed to produce, or that X is supposed to produce, or that it is the purpose of X to produce. When is X designed to produce/supposed to produce/has the purpose of producing E? The most obvious case is where one or more conscious agents have deliberately brought about X because they desired E and believed that X would cause it. Here is another kind of case. Biological natural selection (or some similar natural process) has led to X because in the past (previous versions of) X caused E, and E is the kind of effect favoured by natural selection. (Polar bears have white fur because in the past that camouflaged them from their prey . . . etc.) Most philosophers of biology would probably say that in this case too the white fur has the function of camouflaging the bears, even though no conscious agent brought this about. Many would be equally happy to say that the white fur is designed to camouflage the bears/supposed to...

Let's say that a virus spread throughout the world and damaged the areas of the brain that are responsible for emotions. The entire population was affected and could no longer experience any emotional reactions, although their reason and intellectual ability was unimpaired. Would morality change if we no longer have any emotional reaction to cheaters, thiefs, inequity, or tragedy? Maybe it's difficult to answer such a hypothetical, but any opinions would be appreciated.

On views of morality that I find plausible, your virus wouldn't stop us judging that certain things (cheating, inequity . . .) are wrong, even though it would probably mean that we were not longer motivated to avoid them. (But on other 'non-cognitivist' views, which tie moral judgements to our motivations, this would mean that we would cease even to judge that those things are wrong.) A loss of emotional reactions is likely to undermine more than just moral motivation. In his book 'Descartes' Error' Anthony Damasio argues that without emotional reactions there would be no effective decision-making of any kind. Damasio describes a patient with severe damage to his prefrontal lobes. This patient could see the pros and cons of alternative courses of action (such as Tuesday versus Wednesday for his next appointment) but would discuss the options interminably without ever reaching a decision. This suggests that emotional reactions to envisaged situations are an essential part of the mechanism...

Why is the Law of Contradiction so important to Philosophers? Can its truth be proven? What are the consequences for philosophy of an answer to the question, what's at stake? Thanks so much for your help. Krystle

Here's what I said to an earlier question asking why contradictions are bad (November 23). As you'll see, my answer implies that it's not just philosophers for whom the avoidance of contradiction is important. It's important for anybody who is aiming at the truth. "A simple answer is that sentences of the form 'A and not-A' cannot be true. So if you're aiming at truth, such sentences should not be endorsed. (And if other claims led you 'A and not-A' by valid reasoning (reasoning that never goes from truths to falsehood) they cannot all be true either, and so similarly shouldn't all be endorsed.) Why can't sentences of the form 'A and not-A' be true? Because of the meanings of 'not' and 'and'. The classical semantic analysis of 'not' is that prefixing it to a sentence ('A') gives you a new sentence ('not-A') that is true in just those circumstances where A is not true. The analogous analysis of 'and' is that placing it between two sentences gives you a new sentence that is true just in...

Is time a logically coherent notion in the way we commonly understand it?

We normally think of the passage of time in terms of a 'moving present'--a point that moves steadily futurewards along the temporal dimension, so to speak, and carries us along from our births to our deaths. However, many philosophers, from McTaggart on, have argued that this idea is incoherent, and that 'now' no more refers to a genuine feature of reality than 'here' does. On their view (the 'B-series view) 'now' is an indexical term that simply refers to whatever time you are at, just as 'here' refers to whichever place you are at. It is doubtful whether the idea of a moving present is strictly incoherent. But, even if it isn't, our best theories of reality may well do without it. Perhaps we can explain everything we want to explain, including our experience of the passage of time, without positing a moving present. Indeed some philosophers argue that, even if you do posit a moving present, it is no help at all in explaining the things we want to explain. There is a huge literature on...

What is the principle underpinning logic's rules aimed at avoiding contradiction? We know that contradiction is "bad", e.g. if a line of argument can be reduced to the statement A & ~A, (or if such an assertion can be extracted from an argument) the argument may be invalid. Where does this principle that contradiction leads to invalidity come from? Is it a "just because" axiom? Is it from overwhelming empirical observation? (I've certainly never seen something that both is and isn't at the same time.) More broadly (if this isn't too much) what is the relationship between basic concepts in epistemology (non-contradiction and cause/effect) and axioms of logic? What metaphysical connections or bindings exist between axioms of logic, epistemology, and objective reality? Thanks

A simple answer is that sentences of the form 'A and not-A' cannot be true. So if you're aiming at truth, such sentences should not be endorsed. (And if other claims led you 'A and not-A' by valid reasoning (reasoning that never goes from truths to falsehood) they cannot all be true either, and so similarly shouldn't all be endorsed.) Why can't sentences of the form 'A and not-A' be true? Because of the meanings of 'not' and 'and'. The classical semantic analysis of 'not' is that prefixing it to a sentence ('A') gives you a new sentence ('not-A') that is true in just those circumstances where A is not true. The analogous analysis of 'and' is that putting it between two sentences gives you a new sentence that is true just in case the original two sentences are both true. Putting these together, it follows that 'A and not-A' cannot be true. (There are other (non-classical) ways of analysing 'not and 'and'; most (though not all) will similarly explain why 'A and not-A' cannot be true.)

What makes a question a "philosophical" question? It's easy enough to understand that scientific questions, for example, don't fit here, but beyond such obvious eliminations I'm at somewhat of a loss to come up with some more rigorous criteria.

Somephilosophers think that philosophy has a special subject matter (theanalysis of concepts?) that distinguishes it from science and otherfirst-order enquiries. I don't find this persuasive (apart fromanything else, it would make philosophy trivial). In my view,philosophy deals with just the same kind of subject matter (nature,morality) as other kinds of enquiry. Even so, we can distinguish two ways in which philosophical questions are distinctive. The first way relates to the generality of the categories philosophy deals with. Wherescientists think about viruses, electrons or stars, and medicalethicists think about abortion and genetic enginerring, philosophersthink aboutspatiotemporal continuants, universals and identity, and about duty andvalue. These latter categories do not relate tospecific topics, but structure all our thinking. The second way inwhich philosophy is special cuts across the first. Not all philosophical issues are of...