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The natural aging process brings with it natural changes in sexual function. It doesn't necessarily get worse, but our physiology changes as we get older. The danger is that people come to accept the stereotypes that aging equals a loss of sexual desire and ability. But we are sexual creatures, no matter what our age; and sex is important for our life quality. If you hang up the towel on your sex life, you can risk losing one of the most important parts of your life-and your health.
It helps reduce stress and improves self-esteem. Sex is good for you; it keeps your pipes working! You should continue to have good sex for the same reason you should continue to get good exercise: it's a natural form of taking good care of yourself-and your relationship. If your sex life has been put in the garage, open the door and start your engines-whether you feel like it or not. Once you do, you'll feel tremendously relieved and glad you did. You have a responsibility to make the rest of your life as full it can be. A continuing sex life can actually extend your lifespan.
Desire or libido starts the engine.
If your desire is down, check with your doctor. If a man's desire is down, he should visit his doctor and have his testosterone levels checked. Most men's testosterone levels go down with aging. So the first step is to measure these levels to see if they are abnormal. The FDA estimates that there are over five million men in the U.S. with low testosterone levels, a dangerous medical condition known as hypogonadism. But many of these men are not getting testosterone treatment. But make sure to get a reliable bioavailable testosterone test before attempting testosterone therapy. As men age, they tend to lose their ability to remember at a greater rate than women. The single best predictor of poor memory in middle-aged men is low testosterone. And when men are given it back, their memory can improve. So that is a huge bonus! (And Viagra cannot work properly if low testosterone levels are not corrected.)
It's important to know that inability to achieve an erection can be a sneaky symptom of heart disease. So while you're there, have your heart checked too! And your doctor should ask you if you smoke-since the number one drug that causes impotence is tobacco!
Know that untreated depression can lead to loss of libido and sexual function in men and in women. Ironically, antidepressant medication, while helping with depression, can reduce libido and desire.
Here are a few things to consider to keep things moving in the right direction:
Take it easy, go slow and enjoy every aspect of intimacy. It may take a bit longer to start your engine, but part of being older is the fact that we are more patient and we know what satisfies us. So go slowly and don't judge yourself or each other. Take your time, and enjoy a playful nature, and let it progress naturally into foreplay, and whatever else follows. Sex does not have to equal intercourse. You can still be sexual and sensual, even if intercourse isn't working for you. Let go of judgments, intercourse should not be the goal. If some sexual activity is uncomfortable or doesn't work, try something different-get creative!
Smooch it up! The beginning of a romantic relationship often begins with a kiss. In the converse, the absence of kissing can begin the slow descent toward lack of intimacy or worse. You don't just abruptly stop cuddling or having sex, you first stop kissing. Never let kissing fade in your relationship, or sooner or later, the whole thing can fade. There is nothing that tastes as sweet as a kiss, and nothing creates a more powerful "seal."
Get playful. You're never too old to honor and enjoy each other's bodies. Try sexy lingerie, even kinky costumes-why not? Take a shower or a bubble bath together. Follow that with a slow and tender massage with scented oils and lotions. Don't forget the ambiance of candlelight. Use whatever tools you want to keep that engine purring.
Speaking of tools... Why not visit lingerie or 'adult toy' shops. If that's not your cup of tea, there are sites online that can provide stimulating items of interest, as well as privacy and confidentiality. Remember too, that it's normal for women over 60 to need a little boost in the natural moisture department. There are plenty of safe and effective lubricants that can help bring back the joy of slip and slide, and when applied by your partner, it can become an erotic and welcomed stimulant.
Get creative. Leave sexy notes for your partner to find. Surprise them with little gifts and tokens of your love and intentions. Why not go dancing together or enjoy a flirtatious rendezvous? Take a salsa class or ballroom dance class and stir things up a bit. Sign up for a class on massage techniques. Give each other hand massages, even pedicures!
Take advantage of proper timing. Enjoy intimacy with your partner at any time of the day or night that works for you. It's not uncommon for the 60+ generation to have more energy in the morning and afternoon, so take advantage of that energy and enjoy a morning or an afternoon delight.
Snuggle alert! Take the pressure off. There's nothing more deflating than worrying about performance. Emphasize satisfaction, not performance. Decide together that whatever happens happens, so enjoy the beauty of cuddling before, during, and after sex. Simply holding each other, and enjoying openness and sensuality can be the best part of intimacy-that's what makes it
Get the giggles! There's nothing more attractive than having a bond of humor and laughter with your mate. Play silly games, create your own lingo with private meanings and special words, go ahead and be a tease with each other. Laughter is the best medicine, enjoy the good times!
The Dating Game. Remember when you used to date? Recreate some of that same youthful and spontaneous energy by making a date with each other. Go and do things you've always meant to do, but never found the time to do before. Create fresh, romantic, and fun things to do, and make new memories together!
Communicate. The biggest block to a positive sexual life is our inability to talk about our feelings and desires. Many older people were raised to not talk about sex, so it can be a difficult subject to broach. If you can't talk about it with your partner, find a way to talk about it with somebody, not necessarily a therapist but a friend. Expressing your feelings initially is the best way to open the door to discussing it with your partner. That may be all it takes, a little nudge may open the door to communication.


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Premature Thursday Gaming Thread (Xbone reverses all it's DRM policies)
Well, there we go then, that was a failed experiment before it even took to market!
Also it would appear Steam will soon allow people to share games based on some code found in the latest update.
ah well, for too many people the damage is already done
I've always been a PS girl anyway.Antelope, twentynine, and JaguarPirate this'd this
Still trying to figure out
what filth and depravity "PS" stands for.
Probably too late for a lot of people
That fact that they even had it planned at all has lost them a few customers who wouldn't trust them again regardless of this turnabout, although in reality I suspect that would be a small - if very disproportionally vocal - minority.
Puts an interesting slant on things now though, especially as there were a lot of people more impressed by MS's E3 showing but turned off by the DRM issues.
Now they really need to clarify exactly how the entertainment side of things is going to specifically work in the various regions.
Xbox has looking better exclusives thus far
PS4 has greater indie support (though this means very little to me as I have a PC and these indies will probably for the most part be available on PC) however PS4 is still way cheaper and has the better hardware and hmmmmm.
History has taught us that Sony tend to have more exclusives though
and tend to support the console for longer. The only Xbox exclusive that I find interesting was the Project Sparks one. That one looks really cool, like a modern Populous.
I'd say Project Spark was more Little Big Planet in scope
only way more ambitious
They seem to have the art style nailed down too.
PS4 has got a better RAM spec of course and is 'technically' a superior machine
but I think the standard of line of it being a much better machine is over the top.
I'd expect a lot of multiformat games to largely be produced to a similar spec so I'm not sure how often you'd see a noticeable PS4 improvement.
As for the difference in cost it's probably not as clear as is being made out, it's shipping with what I'd assume is a more expensive controller to produce what with the added trigger rumble and Kinnect which would probably ship at £100+ on it's own. The fact that people don't want Kinnect ad standard is a separate issue, but don't forget just how quickly that first version flew off the shelves so once again there's likely a sizeable number of people who do want it.
Also, what's the gaming history of the most powerful console necessarily being the best/most successful anyway?
Anyone basing a purchase largely on the margins of the spec sheets is clearly a bit of an idiot.
I just said thats why I might buy one
never implied it would win the generation based off of it.
Wasn't aiming that at you!
But have read a lot of comments that basically seemed to have pretty much made their decision based almost solely on what is 'technically superior'.
Oh right
the only reason is the hope that it'll allow full 1080p and 60fps for longer that Xbone. It probably wont though
Eurogamer yesterday
were suggesting a lot of PS4 demo builds are running at 1080p30 rather than 1080p60. Whether that's because it's unfinished code or because that's what's realistically possible at this stage of the console lifecycle is an unknown.
Yeah but if most games this next generation are going to be multiplat anyway
then you might as well side with the better specced console, because it's going to be able to provide more technically impressive exclusives. If we're going by history here, should we really be that skeptical that Sony won't deliver on this? Even if you don't care about graphics, the indie library is impressive already.
I can't for the life of me understand how people can justify spending a significant amount of money on a more expensive lower specced machine from a company that is notorious for promising amazing exclusives and then falling through with it and jumping ship at the first opportunity. What was the last truly great game on 360 that wasn't a Halo?NoNoNoRobots this'd this
Dust: An Elysian Tale (Now on PC)
Limbo (Now on PC)
Bastion (Now on PC)
Limbo's on PS3 now too
Don't get me wrong, I want Microsoft to do well here, some of my favourite games ever were born on the Xbox (Mass Effect is still incredible) but I really feel like they've dropped the ball the last few years. They started off really well and then relaxed and got arrogant.
Oh yeah I was agreeing with you
as only one of those game came out within the past year and none of them are exclusive anymore.Antelope this'd this
Just playing devil's advocate on this really
The spec's a pretty good tiebreaker to use if there's not much else to differentiate but I don't see it as such a gulf that it should be the main deal breaker.
Of course you could also argue that standard Kinnect functionality gives Xbox something more unique from a tech point of view which may offset other issues (and I don't think the PS Eye is really comparable for several reason)
And people do seem to be losing their shit over that mech shooter game.
I think the main difference here is that usually the most expensive machine is the most powerful
not in this case though, that could be a key difference this time around.
Also I've not heard great things about the new Kinect, some people saying it doesn't feel that much better than the current one in terms of latency & reliability.
there is a lot we don't know yet about both machines
Can't say I have much faith in Kinnect but sales do show it's something a lot of people want so is liable to be more of a system seller than a better RAM spec. Similarly a lot of that media integration stuff which gets slated will probably be an attractive feature.
For an awful lot of people MS's pitch of 'you can do voice and motion control as standard and integrate into all you other AV stuff' will be much more appealing than saying 'our machine will play those same games at a slightly better frame rate'.
I guess the camera technology is great
they just need really good software to back it up. And I guess that's why hackers have been able to make more impressive tech demos with Kinect than Microsoft have themselves.
Also I know what you mean podge, I feel like people put too much emphasis on graphics when they don't really have as much a bearing on how fun a game is to play compared with gameplay. It's more about a great art style over just raw graphics or realism anyway I reckon.
I think people forget about performance though
Frame Rate, Load Times, Amount of objects on screen at once, level of complexity that can go into an engine... etc etc etc
Many people fall into the trap of thinking BETTER SPEC = Better graphics, I can assure you that 60 FPS games feel and play a lot better than games running at 30FPS*
*In certain genres obviously it means feck all in a game like The Walking Dead.
So if you got 2 machines with the same line up and one runs it better than that's the machine I'll go for
but that's just me, I'm more tech minded than the average consumer.
- podge this'd this
I disagree
all the major networks that I've seen do a summary of E3 made it pretty clear about the DRM, including the backlash on it
I don't think the general public A) Know what E3 is and watched coverage of it
and B) Even know what DRM is
I don't think the major market share gives a shit and by removing it now they have prevented a lot of potential damage HOWEVER amongst the gaming community it has further engrained mistrust amongst them.... though a couple of exclusive titles and they'll break whatever 'protest' they have going on because gamers are fickle at the best of times.
It was on the 6 o'clock news here, I don't know about the UK though
Even my granddad knew (not representative of the whole population i know, but he knows next to nothing about that kind of thing).
Are pre-orders ever indicative of this kind of thing? Because it was really struggling on pre-orders compared with the PS4, despite the amount of coverage the Xbox had pre-E3 and with having a release date. Also, you have to remember that the Xbox One simply won't be available in most countries at launch, or wasn't even playable, so most of Europe couldn't even buy one at that stage. I'm not sure what the region-lock rules are now though.
They've scrapped Region lock as well
I still don't think it even registers as a thing with most of it's potential customers though. Most people that I'm aware of knew the was a new Xbox coming but that was about the extent of it. They didn't even seem that fussed about the DRM (those who knew) citing that they'll just like the convenience of keeping their friends list from Xbox 360.
Online multiplayer will require PS+ at $5 per month
but you also get free games
I suppose that balances itself out, maybe.
I've begrudged giving Xbox 20 odd quid a year just for the privilege.
I'm looking forward to going back to Sonyville.
Xbox will be giving you free games soon
and the online service on Xbox is far superior to Playstation even to this date BUT these games start with Halo 3 and Assassins Creed 2 which is hardly inspiring.
Well they said 2 free games a month to keep forever
and those are the first 2 games
you can also get fable 3
but dont, its shitStarmixer this'd this
bloody hell
who owns an Xbox and doesn't have Halo 3. smdh
Not sure the xbox live being gods gift and PSN being shit has been true for a long time
i have both consoles and use psn a lot more and have never had any problems with it. Not to mention with PS+ being a requirment i doubt there will be any difference in quality for the next generation, even if you thought there was one now.
PSN does seem to have more periodic maintenance downtime than XBL
while overall there's probably not an awful lot between them it can be hard to shake off perceptions.
Downloading, installing and updating on PS3 for example have always been much slower which appears to be as much a problem as how it's processed as it is the online infrastructure. Should be sorted out next gen by the looks of things but time will tell.
And of course there's the little issue of last years PSN hacking situation, from which their reputation is still no doubt suffering.
does this mean people will finally shut up about it?
I mean, I really like games a lot, but COME ON
You realise it was those people not shutting up that probably helped change this, right?
I reckon people who didn't care about the DRM should still get an Xbox One with DRM and put up with it (I'm not being completely serious here, but come on...)
that's a good point
but it also really ignited the sort of people who use the word fanboy a helluva lot. And I'm getting older by the day; can't be reading that shit without becoming at least somewhat reminiscent of crank.
True, it did seem like an excuse for a lot of people who wouldn't usually care
to come out of the woodwork and be insufferable pricks about it. There's disagreeing with something, then there's being an absolute jerk about it.
To be honest I was happy enough to just think 'that looks a bit shit, I won't buy one'
Rather than spending months moaning about itcolossalhorse and JaguarPirate this'd this
Just to change track a bit
has anyone played Build A World? It looks incredible, very very thinly veiled minecraft, but it looks more impressive and varied. I haven't managed to install it yet but the gameplay videos look really promising
(I think this might be down)
Good PR move
Probably wasn't legal in some non-American bits of the world anyway. Would've been funny to see that dragged through the courts, though.
Talking actual games: PS+ freebies for July (from 3/7/13)
Battlefield 3
Saints Row 3
Payday: The Heist
Ninja Gaiden Sigma Plus
Unit 13
just signed up to PS+ this month
so, so good - Catherine, The Cave, ICO/Shadow of the Colossus, Uncharted 3 ... looking forward to this year of games
Old Gods expansion for Crusader Kings II has turned the game from a 7/10 to a 9.5/10
hooked just like i was with EU3
- justanothersheeldz this'd this
This is good news.
I ploughed lots of hours into it when it first came out but whenever I've tried to get back into recently it just doesn't happen.
I installed Rollercoaster Tycoon 2 onto my old laptop
and it runs like a dream (obviously). Can't wait to build Alton Towers again.
- justanothersheeldz this'd this
Hotline Miami 2: Wrong Number sounds interesting
It sounds like they're overreaching and writing themselves into meta-hell, but I'm very much intrigued. After all the world isn't short of pretty good sequels: I'd rather they swung for the fences and produced a heroic failure any day.
Also: Dark Souls II. I hate that they're teasing this almost a year in advance. My initial reservations have melted away to leave a quivering mass of IWANTITIWANTITNOW! Now my 360 is broken I can't even play the first one to tide me over. Harsh times.
So excited for Hotline Miami 2
Though it's been decided that a better title would have been Hotline Miami 3: Wrong Numbercolossalhorse this'd this
The Last of Us is good isn't it
Finding it really tricky though, I can only have played for an hour or so and I'm already struggling.
My cunning plan:
Wait for a year or two until both consoles have had a chance to build up a games library, and any early bugs to be ironed out, and hopefully drop in price a little.
Then decided which one has more games I like, then get that if I feel like it.JaguarPirate and shadyadie this'd this
No you fool!
You've got to decide NOW!podge, Yesiamaduck, Antelope, colinzealuk, JaguarPirate, and colossalhorse this'd this
currently on that Dishonored DLC where you're playing as Daud
enjoying it; the fact that you play a character with a voiceover makes it feel even more like Thief (which I'm gonna have to get a next-gen console for, I think)
Xbox live
My Live connection just dropped out upon exiting a game and won't reconnect. I've tried to access the suggested XBox.com help site, but Xbox.com appears to be stuffed. Anyone else having this issue? I'm assuming it isn't just me (because my network connection to the internet is how I'm writing this) and that Xbox.com is currently struggling with traffic.
PC Gamers - GOG have a summer sale on now
And today there are loads of great daily deals as it's the longest day.

subset: commoncrawl


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The construction of an outdoor museum is blocking and polluting the river that leads to Phong Nha Cave
The Phong Nha – Ke Bang National Park and the Citadel of the Ho Dynasty of Vietnam are among 37 heritage sites proposed for inscription into the UNESCO World Heritage List as natural and cultural properties.
Addressing the on-going session of the World Heritage Committee in Paris, Director General of the United Nations Educational, Scientific and Cultural Organisation Irina Bokova said that the 35th session will examine the World Heritage List, the state of conservation of 169 World Heritage properties and increasing challenges to the preservation of these sites.
World Heritage is a reminder of all that unite humanity, she said, adding that it is a reminder also of the ties between culture, nature and societies.
World Heritage sites can be tremendous vectors for dialogue, reconciliation, development and knowledge, she noted.
She took the occasion to call on UNESCO States Parties to focus on risk preparedness and long-term management to protect irreplaceable and priceless heritage of humanity.
To date, the World Heritage List numbers 704 cultural, 180 natural and 27 mixed properties in 151 out of its total 187 States Parties.


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- Janice Cain
- 3 years ago
- Views:
1 ALAN HÁJEK WHAT CONDITIONAL PROBABILITY COULD NOT BE ABSTRACT. Kolmogorov s axiomatization of probability includes the familiar ratio formula for conditional probability: (RATIO) P(A B) = P(A B) P(B) (P (B) > 0). Call this the ratio analysis of conditional probability. It has become so entrenched that it is oftenreferredtoasthedefinition of conditional probability. I argue that it is not even an adequate analysis of that concept. I prove what I call the Four Horn theorem, concluding that every probability assignment has uncountably many trouble spots. Trouble spots come in four varieties: assignments of zero togenuine possibilities; assignments ofinfinitesimals to such possibilities; vague assignments to such possibilities; and no assignment whatsoever to such possibilities. Each sort of trouble spot can create serious problems for the ratio analysis. I marshal many examples from scientific and philosophical practice against the ratio analysis. I conclude more positively: we should reverse the traditional direction of analysis. Conditional probability should be taken as the primitive notion, and unconditional probability should be analyzed in terms of it. I d probably be famous now If I wasn t such a good waitress. Jane Siberry, Waitress Over three hundred years ago, Fermat and Pascal began the enterprise of systematizing our pre-mathematical, pre-philosophical concept of probability. Orthodoxy has it that this task was completed in 1933, when Kolmogorov provided his axiomatization. In the process, he systematized another pre-theoretical concept: conditional probability. He identified conditional probability with a certain ratio of unconditional probabilities, according to the formula: (RATIO) P(A B) = P(A B) P(B) (P (B) > 0). Iwillcallthistheratio analysis of conditional probability. 1 It has become so entrenched that we seem to have forgotten that it is an analysis. My main purposes in this paper are to cast doubt on its adequacy, and to argue Synthese 137: , Kluwer Academic Publishers. Printed in the Netherlands.
2 274 ALAN HÁJEK that conditional probability should be taken as the fundamental notion in probability theory. 1. THE RATIO IS AN ANALYSIS, NOT A DEFINITION, OF CONDITIONAL PROBABILITY What I have just said may strike you as absurd. You may say that conditional probability is just a technical term that is defined by this formula. If so, you are in good company. For example, Jackson (1987, p. 12) writes: it is important to take the conditional probability of consequent given antecedent to be defined as the probability of the conjunction of the antecedent and the consequent divided by the probability of the antecedent. (The emphasis is his.) Skyrms (1980, p. 156; 2000, p. 119) and Earman (1992, p. 36) similarly speak of (RATIO) as the definition of conditional probability, just to mention two other prominent authors, among many. It is quite understandable that they speak this way it is part of the orthodoxy, after all. And it only underscores, to my mind, the extent to which the ratio analysis has become dogma. To be sure, if the conditional probability of A given B were a purely technical notion, then Jackson, Skyrms, Earman, etc. would doubtless be right. 2 There is nothing problematic about introducing the symbol into the language of the probability calculus, and defining it in terms of the ratio formula. You can even intone the words the conditional probability of A given B when reading P(A B) out loud, if you insist. If these words are not loaded with philosophical and commonsensical associations, then this is no more contentious than saying the probability of A slash B, and taking that as convenient shorthand for the ratio. But the words are loaded. They are tinged with associations with models of learning, rational belief revision and decision-making, with analyses of confirmation, causal dependence, the semantics of conditionals, and so on. Jackson goes on to deploy intuitions about conditional probabilities in his defense of Adams Thesis that the assertability of the indicative conditional A B goes by the conditional probability of B,given A.For example, he writes: Take a conditional which is highly assertible, say, If unemployment drops sharply, the unions will be pleased ; it will invariably be one whose consequent is highly probable given the antecedent. And, indeed, the probability that the unions will be pleased given unemployment drops sharply is very high (12). Likewise, Skyrms goes on to deploy intuitions about conditional probabilities in his defense of his account of laws as resilient generalizations; and Earman goes on to deploy intuitions about conditional probabilities in his defense of the Bayesian analysis of
3 WHAT CONDITIONAL PROBABILITY COULD NOT BE 275 confirmation. They are clearly appealing to some commonsensical notion (albeit refined by philosophical reflection) rather than to something that is merely stipulatively defined. After all, we had the concept of conditional probability long before we donned our philosopher s or mathematician s caps. In ordinary language, conditional probability statements can be made using locutions of the form it s likely that p,givenq, it s improbable that r,ifs, and variations on these. In The Piano Man, Billy Joel sings, I m sure that I could be a movie star, if I could get out of this place : his probability for becoming a movie star is high, not unconditionally, but conditionally on his getting out of this place. And so forth. Conditional probability is not just a technical notion such as zero-sum game or categorical imperative or Turing computability. Rather, it is a familiar concept, like game, or moral duty, and I daresay more familiar than computability. It is there to be analyzed, if possible. (If it is not possible, then it follows immediately that the ratio does not provide an adequate analysis of it.) Our choice of words in reading P(A B) as the probability of A, given B is not so innocent after all. And far from being a stipulative definition of that concept, I will argue that the ratio formula is not even an adequate analysis of it. 3 In the sequel, I will follow convention in using the notation P(A B) as shorthand for the ratio P(A B)/P(B). However, I want you to resist the further convention of reading it as the probability of A given B. When I want to speak of the conditional probability of A, givenb, while remaining neutral as to how this might be analyzed (if at all), I will simply write P (A, givenb). I will call B the condition. My strategy will be as follows. After some stage-setting in Section 2, I will prove in Section 3 what I call the Four Horn theorem. Its upshot is that every probability assignment has uncountably many trouble spots. Trouble spots come in four varieties: assignments of zero to genuine possibilities; assignments of infinitesimals to such possibilities; vague assignments to such possibilities; and no assignment whatsoever to such possibilities. The next four sections show how each sort of trouble spot can create serious problems for the ratio analysis. Section 4 rehearses, then gives a new gloss on, an old problem: that the analysis is mute whenever the condition has probability zero, and yet conditional probabilities may nevertheless be well defined in such cases. Section 5 discusses problems for infinitesimal assignments to the unconditional probabilities that figure in (RATIO). Section 6 shows that the analysis cannot respect the fact that various conditional probabilities are sharp even when the corresponding unconditional probabilities are vague. Section 7 argues that various conditional probabilities are defined even when the corresponding unconditional
4 276 ALAN HÁJEK probabilities are undefined, and indeed cannot be defined. In Section 8, I canvass many instances from actual scientific and philosophical practice. The dogma has it that the ratio is an adequate analysis of the probability of A,given B, irrespective of the interpretation of probability itself. I give arguments against the ratio analysis for both subjectivist and objectivist construals of probability. If (RATIO) fails as an analysis of conditional probability, how can we explain its undoubted utility? In Section 9, I argue that it provides a constraint on conditional probability: when P(A B) and P(B)are both sharply defined, and P(B) is non-zero, the probability of A given B is constrained to be their ratio. In the cases that I discuss, these conditions are not met, so there is no constraining to be done by the ratio. I conclude more positively in Section 10: we should reverse the traditional direction of analysis. Conditional probability should be taken as the primitive notion, and unconditional probability should be analyzed in terms of it. 2. BACKGROUND There is some philosophical and technical background that I will be assuming in my arguments Subjective/Objective Probability I assume that probability comes in at least two varieties: subjective and objective. I will stay neutral as to how exactly these should be understood; suffice to say that my examples will include putatively paradigm cases of each. The following characterizations, though not fully fleshed-out and in some ways problematic (topics for other occasions), will be more than adequate for my purposes here: 4 Subjective probability: Probability is interpreted as the degree of belief, or credence, of a rational agent. Let the agent be you, for example. There are various proposals for attributing a single probability function P to you. For example, de Finetti (1980) regards P as capturing all information about the bets that you are prepared to enter into: P(X) is the price that you regard as fair for a bet that pays 1 unit if X occurs, and 0 otherwise. According to van Fraassen (1990), P encapsulates all information about the judgments that you make for example, if you judge it as likely as not that Madonna will win a Nobel prize, then P (Madonna will win a Nobel prize) = 1/2. Ramsey (1926), Savage (1954) and Jeffrey (1983b) derive both probabilities and utilities (desirabilities) from rational preferences (given various assumptions about the richness of the preference space, and certain
5 WHAT CONDITIONAL PROBABILITY COULD NOT BE 277 consistency assumptions on the preference relation). In a similar spirit, Lewis (1986a, 1994a) analyses P as the probability function belonging to the utility function/probability function pair that best rationalizes your behavioral dispositions (rationality being given a decision-theoretic gloss). Objective probability: This is often called objective chance. It exists heedless of our beliefs and interests. Long run relative frequency is typically a good guide to determining it. (Some, e.g., Reichenbach 1949, think that, more than being a good guide, such relative frequency should be identified with objective chance; but I part company with them there see my 1997.) Chance also places constraints on rational credence, via something along the lines of Lewis (1980, 1986b) Principal Principle moreonthat in Section 8.5. Some authors (e.g., Popper (1959a), Mellor (1971), Fetzer (1983), and many others see Gillies (2000) for a survey) regard it as a kind of propensity, or disposition, or causal tendency for an outcome to be brought about by an experimental arrangement. For example, according to Popper, a probability p of an outcome of a certain type is a propensity of a repeatable experiment to produce outcomes of that type with limiting relative frequency p. Lewis among others contends that chances evolve over time, the past (relative to any given moment of time) always having a chance of 1. Quantum mechanical probabilities are standardly regarded as objective. Chance values are plausibly determined by the laws of nature. Some authors (e.g., Lewis 1986b, p. 118) regard a deterministic world as one in which all chances are 0 or Vague Probability Let us drop the assumption that a given proposition is assigned a unique, sharp value, and allow instead that it may be assigned a range of values. I will follow van Fraassen s (1990) representation of vague subjective probability (which develops proposals by Levi (1974, 1980) and Jeffrey (1983a), among others). I will then extend it to the case of objective probability. Suppose that your state of opinion does not determine a single probability function, but rather is consistent with a multiplicity of such functions. In that case, your opinion is represented as the set of all these functions. Call this set your representor. Each function in your representor corresponds to a way of precisifying your opinion in a legitimate way. Imagine, for example, that your probability for Diego Maradonna winning an Oscar is not a sharp value; rather, it lies in some interval, say [1/2, 3/4]. Then your representor contains all probability functions that assign a sharp value in the interval [1/2, 3/4] to Maradonna s winning, and that are otherwise com-
6 278 ALAN HÁJEK patible with your state of mind. This might be thought of as a probabilistic analogue of the supervaluational approach to vagueness. More controversially, let me suggest that we remain open to the possibility of vague objective probabilities. For example, perhaps the laws of nature could be vague, and if these laws are indeterministic, then objective probabilities could inherit this vagueness. 5 Be that as it may, I submit that vague objective probabilities, if there are any, could likewise be understood in terms of representors. For example, a chance that is vague over the set S corresponds to a set of sharp chances, taking on each of the values in S Probability Gaps Furthermore, we can make sense of probability gaps propositions that get assigned no probability value whatsoever under both the subjective and objective interpretations. As a point about probability functions in the abstract, this is familiar enough. We begin with a probability space (, F, P ), where is some set of elementary events, or possible worlds, and F is a sigma-algebra of subsets of. P is defined on all the members of F, but on no other propositions. So any subset of that does not belong to F can be thought of as a probability gap as far as P is concerned: a proposition to which P assigns no value. Decision theory recognizes the possibility of probability gaps in its distinction between decisions under risk, and decisions under uncertainty: in the latter case, probabilities are simply not assigned to the relevant states of the world. Statistics recognizes such gaps in its distinction between random and unknown parameters: probabilities are not assigned to values of the latter. And indeed it seems that we must allow for probability gaps on all of the accounts of subjective probability given above. On the betting account, this is clear: suppose that you are currently not prepared to enter into any bet regarding X for example, you have never even considered X. (Of course, you might later become prepared to enter into such bets, having giving X some thought, or upon coercion.) Likewise on the preference-based accounts, for your preferences may not be defined over all propositions. Probability gaps are a little harder to come by on van Fraassen s and Lewis accounts: if you lack determinate judgments concerning X, or if you lack the relevant behavioral dispositions, then it seems that all precisifications of the probability of X are legitimate, so that your probability for X is vague over the entire [0, 1] interval and that seems to be something different from a probability gap. But suppose that you resolutely refuse to assign X a probability, or for some reason cannot assign X a probability. Then any precisification of the probability of X
7 WHAT CONDITIONAL PROBABILITY COULD NOT BE 279 seems to distort your opinion, giving it a value where you choose to remain silent. In that case, your opinion would be better represented as having a gap for X. Perhaps, for example, you regard X as lacking a chance value altogether, and you think that rational opinion concerning X should follow suit. Or perhaps X involves a future action of yours upon which you are deliberating; more on this in Section 8.4. Or perhaps you regard X as non-measurable a set that simply cannot be assigned a probability, consistent with certain natural constraints that one would want to impose. Non-measurable sets notoriously arise in interesting and surprising ways. For instance, consider the case of assigning a uniform probability distribution P across the [0, 1] interval. Certain subsets of the interval simply cannot be assigned any probability whatsoever by P, and so are non-measurable (with respect to it). It is not that they receive zero probability; rather, they are probability gaps. We may prove this, à la the textbook, as follows: By Kolmogorov s axiomatization, P is countably additive: for every infinite sequence of mutually disjoint propositions A 1, A 2,...in its domain, ( ) P A n = n=1 P(A n ). n=1 Assume further that it is uniform in the sense of being translation invariant: for any real number, and any set X, P(X)= P(X+ ),wherex + is the set produced by adding (modulo 1) to each member of X. Consider the equivalence relation on [0, 1] satisfied by x and y iff x y is rational. This partitions [0, 1] into equivalence classes. Let C be a choice set containing exactly one member of each of these classes (here we assume the axiom of choice). For each rational r in [0, 1], let C r be the set you get by adding (modulo 1) r to each member of C. There are denumerably many C r s, and they are mutually exclusive. Their union is [0, 1]. Suppose for reductio that each C r receives some probability value; by the assumption of uniformity every C r receives this same value. Then either this value is 0, in which case P(C r ) = 0, or it is some positive number, in which case P(C r ) =. By countable additivity, we have either 1 = 0 or 1 =. Contradiction. This completes the reductio. We cannot maintain the assumption that each C r receives some probability value but if any one of them doesn t, then they all don t (by the assumption of uniformity). All the C r s are thus non-measurable sets. That is, we have proven the existence of infinitely many probability gaps in the [0, 1] interval a surprising and profound result that I will appeal to later on. If we can
8 280 ALAN HÁJEK interpret P both subjectively or objectively, we have both subjective and objective probability gaps indeed, uncountably many Infinitesimal Probabilities, Countable Additivity, and Regularity Robinson (1966) exploited compactness to show that there are nonstandard models of first order analysis containing positive infinitesimal elements: positive numbers smaller than every positive real. Consider a first order language appropriate for real analysis in particular, for each real number r, there is a name r. The theory of analysis consists of all the sentences of this language that are true according to the standard model. Consider the theory that is the union of the theory of analysis, with all sentences of the form r <y, for each real r. Each finite subset of this theory has a model; hence, by compactness, this theory has a model as well. Any such model (and there are many) is a non-standard model of the real numbers. The denotation of y in such a model is an infinite element, greater than each of the standard reals. Its reciprocal 1/y is a positive infinitesimal element. 6 In what follows I will drop the word positive before infinitesimal : while 0 is strictly speaking an infinitesimal (the only real one, in fact), my interest in infinitesimals will concern the positive ones, and when philosophers speak of infinitesimal probabilities, it s typically the positive ones they have in mind. Kolmogorov s axiomatization stipulates that probability measures are real-valued; hence according to that axiomatization, there can be no infinitesimal probabilities. We may wish to relax the stipulation, and indeed philosophers such as Skyrms (1980) and Lewis (1980) explicitly embrace infinitesimal probabilities. However, the stipulation is not idle. For as I have said, Kolmogorov also required probabilities to be countably additive. But the notion of countable additivity does not even make sense in the context of non-standard analysis, and there is no obvious analogue of it. For the very statement of countable additivity requires the taking of a limit as n tends to infinity of a sequence of partial sums of probabilities, and it turns out that this is not something that can even be said in the language of non-standard analysis (the set of natural numbers, which gets appealed to in the ɛ-n definition of the limit, is not an internal set). Finally, a probability function is said to be regular iff it assigns probability 1 only to logical truths, and 0 only to contradictions. (For our purposes, we might equally define a regular function as one that assigns 1 only to the necessary proposition(s), and 0 only to the impossible proposition(s) any differences that there might be between the two definitions will not matter to us here, and I will use both sentential and propositional locutions.) A number of authors propose versions of the idea that
9 WHAT CONDITIONAL PROBABILITY COULD NOT BE 281 rational credence functions are regular (sometimes calling them strictly coherent ), among them Kemeny (1955), Jeffreys (1961), Carnap (1963), Edwards et al. (1963), Stalnaker (1970), Shimony (1970), and Appiah (1985). Note that an agent who updates by conditionalization upon some item E of evidence ends up with an irregular probability function, for P(E)becomes 1. So the proposal of regularity contradicts another widely endorsed claim about rationality: that updating should take place by conditionalization. For further opposition to the demand for regularity, see de Finetti (1972) and Levi (1978). 7 We turn now to the promised Four Horn theorem. Each of the four sections that follow it (Section 4 7) focuses on a type of trouble spot for the ratio analysis: zero probabilities, infinitesimal probabilities, vague probabilities, and undefined probabilities. The discussion in these sections is largely self-contained, and the reader who is impatient with technicalities (or simply impatient) may want to skip directly to them. What the theorem, coupled with a widely accepted assumption, show is that the trouble spots are inescapable, and that they are, to put it mildly, plentiful. There s trouble, and a lot of it. 3. THE FOUR HORN THEOREM 8 We begin by proving a theorem concerning probability measures (that need not be total functions). As usual, they are real-valued, and sharp (that is, single functions). We then relax these assumptions, generalizing the theorem for probability functions that may also take infinitesimal values, giving a corollary for such functions that are regular, and then a theorem concerning vague probability assignments. This culminates in the Four Horn theorem, which puts the results together. This theorem, combined with an assumption regarding propositions, implies that every probability function has uncountably many trouble spots. These trouble spots, moreover, will be central to my attack on the ratio analysis. PROBABILITY MEASURE THEOREM. Let be an uncountable set. Any (real-valued, sharp) probability measure fails to assign positive probability to uncountably many subsets of. Proof. Let uncountable be given, and let P be a (real-valued, sharp) probability measure. Let D be an uncountable set of disjoint subsets of. (The disjointness will come in handy when we appeal to finite additivity.) For each n =1,2,...consider G n ={S D: P(S) 1/n}.
10 282 ALAN HÁJEK Each set in G 1 has probability at least 1; it thus has at most 1 member. Each set in G 2 has probability at least 1/2; it thus has at most 2 members (by finite additivity of P )... For each i, G i has at most i members (by finite additivity of P ), and is thus a finite set. Therefore, i G i is countable, being a countable union of finite sets. This is the set of members of D that receive positive, non-infinitesimal assignments from P (since for any positive, non-infinitesimal x, there is an n such that x 1/n). This leaves uncountably many members of D, that is, subsets of,thatdonot receive positive, non-infinitesimal assignments from P. Since P is real-valued, it does not assign positive, infinitesimal probabilities; thus, the word non-infinitesimal was redundant in the previous sentence. That is, we have shown that uncountably many subsets of do not receive positive assignments from P. Q.E.D. How can a (real-valued, sharp) probability measure P fail to assign positive, non-infinitesimal probability to a given set? There are two ways: by assigning it 0 probability, or by assigning it no probability whatsoever. Now let us relax the assumption that P is a measure, allowing it to take infinitesimal values; in that case we will still call P a probability function, even though strictly speaking we have gone beyond Kolmogorov s axiomatization. Then there is a third way in which a probability function P can fail to assign positive, non-infinitesimal probability to a given set: by assigning it infinitesimal probability. We have a strengthening of the probability measure theorem: PROBABILITY FUNCTION THEOREM (infinitesimals allowed). Let be an uncountable set. Any (sharp) probability function either assigns 0 to uncountably many subsets of, or assigns infinitesimal probability to uncountably many subsets of, or fails to assign probability to uncountably many subsets of. Proof. The proof is as before, up to and including: This leaves uncountably many members of D, that is, subsets of, that do not receive positive, non-infinitesimal assignments from P. Then we reason as follows. The following union of sets is uncountable: {S D : P(S) = 0} {S D : P(S)is infinitesimal} {S D : P(S)is undefined}. Thus, at least one of these three sets of subsets of is uncountable (for if each of them were countable, their union would be also). Q.E.D.
11 WHAT CONDITIONAL PROBABILITY COULD NOT BE 283 With minimal extra work, we get: REGULARITY THEOREM. Any regular probability function defined on uncountably many mutually disjoint propositions assigns infinitesimal probabilities to uncountably many propositions. Proof. Let P be a regular probability function defined on an uncountable set D of mutually disjoint propositions (and on other sets, too, of course). The proof goes through as before. Then note that since P is regular, it assigns 0 only to the empty set. Thus {S D: P(S) = 0} cannot be uncountable; indeed, it has at most one member. Nor can {S D: P(S)is undefined} be uncountable; indeed, it is empty (by definition of D). This leaves {S D: P(S) is infinitesimal} as the set that must be uncountable. Q.E.D. So far, our theorems have concerned (sharp) single probability functions. Vague probability assignments, on the other hand, are represented by (nonsingleton) sets of such functions, or representors. So now let the term probability assignment subsume both sharp probability functions and vague probability assignments. VAGUE PROBABILITY THEOREM. If a probability assignment defined on an uncountable algebra of propositions is vague at all, then there are uncountably many propositions to which it assigns vague probability. Proof. Let a vague probability assignment V defined on an uncountable algebra of propositions U be given. It corresponds to a representor with at least two probability functions defined on U. Choose two such functions, P 1 and P 2. V assigns vague probability to each proposition on which they differ. Hence it suffices to show that they differ on uncountably many propositions. Since P 1 and P 2 are distinct, there is some proposition D U on which they differ: P 1 (D) = P 2 (D). Thus, P 1 (D c ) = P 2 (D c ),whered c U also. Since U is uncountable, either (i) there are uncountably many propositions in U disjoint from D, or (ii) there are uncountably many propositions in U disjoint from D c. (Any proposition X U that has non-empty intersection with both D and D c can be split into two parts: X D U, which is disjoint from D c,andx D c U, which is disjoint from D.) Without loss of generality, suppose (i)
12 284 ALAN HÁJEK is the case. If P 1 and P 2 differ on uncountably many of these propositions, we are done. So suppose that P 1 and P 2 differ on only countably many of these propositions; they thus agree on uncountably many of them. But for each such proposition A on which they agree, they disagree on D A: P 1 (D A) = P 1 (D) + P 1 (A), since D and A are disjoint, = P 1 (D) + P 2 (A), since P 1 and P 2 agree on A, = P 2 (D) + P 2 (A), since P 1 and P 2 differ on D, = P 2 (D A), since D and A are disjoint. Each of the uncountably many candidates for D A is a proposition for which P 1 and P 2 differ; each of them is thus a proposition to which V assigns vague probability. Q.E.D. Every probability assignment is either sharp or vague. Putting our results together, we have the FOUR HOUR THEOREM. Any probability assignment defined on an uncountable algebra on an uncountable set either 1. assigns zero probability to uncountably many propositions; or 2. assigns infinitesimal probability to uncountably many propositions; or 3. assigns no probability whatsoever to uncountably many propositions; or 4. assigns vague probability to uncountably many propositions (where the or s are inclusive). We now make an assumption about propositions, one that many will regard as not just true, but necessarily so. ASSUMPTION. There are uncountably many mutually disjoint propositions. JUSTIFICATION. For each real number in [0, 1], there corresponds a distinct proposition. For example, imagine throwing an infinitely fine dart at the interval. For each r in the interval, there is a proposition L r to the effect that the dart lands exactly on r. Moreover, these propositions are mutually disjoint. If you prefer, consider instead a particular radium atom; for each time t > 0 there is a proposition that the atom decays exactly at time t, disjoint with all the others. Or if you prefer to think in terms of possible worlds (be they concrete, abstract, or what have you), and propositions as sets of possible worlds: there are uncountably many possible worlds, thus
13 WHAT CONDITIONAL PROBABILITY COULD NOT BE 285 infinitely many mutually disjoint propositions (e.g., the singleton sets of these worlds; cf. Lewis 1986c). We are now in the realm of metaphysics rather than mathematics. Combining our mathematical theorem with our metaphysical assumption, we have the more startling result: any probability assignment whatsoever has uncountably many trouble spots: propositions that it assigns zero probability, infinitesimal probability, vague probability, or no probability at all. If it is defined on an uncountable algebra of propositions of an uncountable set, the mathematics alone does the work. If it is not, the metaphysics chimes in: whatever algebra it is defined on, outside that algebra it is undefined on uncountably many propositions. The following four sections address the four sorts of trouble spots for the ratio analysis that we find in the Four Horn theorem. It will prove convenient to take them slightly out of order: zero probability assignments (Section 4), infinitesimal probability assignments (Section 5), vague probability assignments (Section 6), and no probability assignment whatsoever (Section 7). It is inescapable that any probability assignment lands on at least one of the four horns. Moreover, it is noteworthy that each horn is actually occupied by various interesting probability functions. In each section I will give examples that are both paradigm cases of what it is like to land on the relevant horn and of independent philosophical interest. 4. THE ZERO DENOMINATOR PROBLEM Suppose that we have a probability assignment that lands on the first horn: it assigns zero probability to uncountably many propositions. Then for each such proposition, all probabilities conditional on it are undefined as far as the ratio analysis is concerned. After all, (RATIO) comes with a proviso: the condition must have positive probability. Jackson remarks: The restriction to cases where P(B) > 0 is less severe than may at first appear (11, adapting his notation). His argument for this is essentially that all rational credence functions are regular. Again, I am only using Jackson as a foil, and this opinion is widely voiced. But I believe that on the contrary, the restriction is more severe than may at first appear. The ratio analysis fails to give values to certain conditional probabilities that apparently are well-defined. Let us call this the zero denominator problem. The problem is well known. 9 Indeed, given how many probability textbooks go out of their way to caution the reader that probability 0 does not imply impossible, it is perhaps surprising that more is not made of it, at
14 286 ALAN HÁJEK least in philosophical circles. In any case, let me mention some particularly interesting cases in which it arises some familiar, some less so Some Textbook Examples Continuous random variables give rise to non-trivial events of probability 0. Associated with such a random variable is a probability density function (think, for example, of the bell -shaped normal distribution). The probability of a continuous random variable X, taking a particular value x, equals 0, being the integral from x to x of X s density function. Yet it would seem that various probabilities, conditional on this event, are well-defined, for example: P(X = x, given X = x) = 1. Indeed, here and throughout the paper, we hold this truth to be self-evident: the conditional probability of any (non-empty) proposition, given itself, is 1. I think that this is about as basic a fact about conditional probability as there can be, and I would consider giving it up to be a desperate last resort. Some others are comparably basic; in the case before us they include: P(X = x, given X = x)= 0, P(T, given X = x)= 1, where T is a necessarily true proposition (e.g., everything is self-identical ), P(F, given X = x)= 0, where F is a necessarily false proposition (e.g., not-t). Less trivially, various conditional probability assignments based on judgments of independence are compelling for instance: P (this coin lands heads, given X = x) = 1/2. Further examples are provided by the behavior of relative frequencies of events in independent, identically distributed trials. Various laws of large numbers concern the convergence of such relative frequencies to the corresponding true probabilities. However, they do not guarantee that relative frequencies will so converge; rather, the convergence only occurs with probability 1. This caveat isn t just coyness a fair coin could land heads every toss, forever, although its doing so is an event of probability 0. If you think that this sequence of results cannot happen because it has probability 0, then you should think that no sequence can happen, since each sequence has the same probability. Again, however, various probabilities
15 WHAT CONDITIONAL PROBABILITY COULD NOT BE 287 conditional on such events would seem to be well defined: P(the coin lands heads forever, given the coin lands heads forever) = 1, and so on. We may regard the probabilities in these cases objectively or subjectively. Other cases arise specifically for objective probabilities Chances It would appear that objective probability functions are not regular. For example, it is plausible that the chance function P t at any given time t lands on the first horn of the Four Horn theorem. Lewis, for example, contends that the past is not chancy: any proposition whose truth value is settled at a time earlier than t is assigned either 1 (if it is true), or 0 (if it is false) by P t. And there are uncountably many such propositions that are false. Suppose that a coin landed heads at 12:00, and consider P 12:01 (the coin landed tails, given the coin landed tails). This conditional probability should equal 1, but according to the ratio formula, it is undefined. Similar points can be made using each of the uncountably many false propositions about the past as the condition. Or consider some proposition about the future for example, an eclipse will occur at 15:52, and the conditional probability: P 12:01 (an eclipse will occur at 15:52, given an eclipse will occur at 15:52). This should equal 1, whether or not the condition is false. And according to Lewis among others, if the world is deterministic, then all false propositions have chance 0 at any time. In that case, if the condition is false, this conditional probability is undefined according to the ratio analysis; if the condition is true, change the example, replacing occur with not occur. Thus, the adequacy of the ratio analysis is held hostage to empirical matters: the world had better not be deterministic! Clearly, the ratio analysis cannot have empirical consequences. So if the conditional probability is 1, irrespective of whether the world is deterministic, the analysis is refuted. Similar points can be made using conditional probabilities involving T and F, as before. The proponent of the ratio analysis of conditional chance is thus committed to saying either (1) the chance function (at any given moment of time) is regular; or (2) conditional chances of the form P (X, givenz) are never defined when P(Z) = 0.
16 288 ALAN HÁJEK (1) is implausible; logic and chance surely do not align so neatly. In any case, (1) appears to be an empirical claim, one hardly discoverable by pure conceptual analysis. But (2) is hardly appealing, either, as I have argued. Indeed, conjoined with the claim that the chance function is irregular at some time, it contradicts a truth we hold to be self-evident Subjective Probabilities Still other cases arise more naturally for subjective probabilities. Let P be the credence function of a rational agent. If, contra Kemeny, Jeffreys et al., P is irregular, then it assigns 0 to at least one logically possible proposition. Let Z be such a proposition. I claim that rationality requires the following assignments: P (Z, givenz) =1,P (Z c,givenz) =0,P (T, given Z)=1,andP (F, givenz) = 0. But an advocate of the ratio analysis cannot even grant that rationality permits these assignments. We have already noted that updating by conditionalization leads one to irregularity. Suppose, for example, that you begin by assigning probability 1/2 to a given die landing with an odd face showing up. The die is tossed, and you conditionalize on the evidence that the die landed odd. You then assign probability 0 to the die landing with an even face showing up on that toss. Still, surely rationality permits you to assign P (die landed even, given die landed even) = 1 (and, I would add, even requires it). We have, then, a rational credence function P that assigns a conditional probability that cannot be equated to a corresponding ratio of unconditional probabilities. There are uncountably many propositions on which conditionalization could take place. It seems, then, that there are uncountably many candidates for Z for some rational credence functions. They thus land on the first horn of the Four Horn theorem. But that is just icing on the cake one such candidate is all I need to make my point. We have a dilemma. Either: (1) all rational credence functions are regular; or (2) for all rational credence functions P, conditional probabilities of the form P (X,givenZ) are never defined when P(Z) = 0. As we have seen, various authors opt for (1). They thus commit themselves to saying that updating by conditionalization is always irrational. They also seem to have a novel proof of the non-existence of God (as traditionally conceived) for an omniscient agent who is certain of which world is the actual world is thereby convicted of irrationality! By their lights, rationality requires a certain form of ignorance regarding all contingent propositions, or else false modesty. And even if regularity is such a requirement on rationality, we all fall short of such requirements. But when
17 WHAT CONDITIONAL PROBABILITY COULD NOT BE 289 we (irrationally?) assign 0 to a contingent proposition A, we still seem to have no trouble giving various corresponding conditional probability assignments that are apparently rationally required, and surely rationally permitted for example, P (A, givena) =1. 10 (2) may seem to be more palatable, then. But do we really want to say that rationality does not even permit conditional probabilities such as P (Z, givenz) to be defined? 4.4. Symmetric Distributions Symmetric probability assignments over uncountable sets give rise to particularly acute problems for the ratio analysis, because many (indeed, uncountably many) non-trivial conditional probabilities with probability zero conditions are apparently well-defined, and their values are obvious. Here are some familiar examples. (i) (Due to Borel.) Suppose that we have a uniform probability measure over the Earth s surface (imagine it to be a perfect sphere). What is the probability that a randomly chosen point lies in the western hemisphere (W), given that it lies on the equator (E)? 1/2, surely. But the probability that the point lies on the equator is 0, since the equator has no area. And the problem could be made to go the other way. We could embed the Earth s surface in a much larger space, relative to which it has measure 0. Yet P (the point lies on the earth s surface, given the point lies on the earth s surface) = 1, and still P (the point lies in the western hemisphere, given the point lies on the equator) = 1/2. (ii) Imagine throwing an infinitely fine dart at the [0,1] interval. Suppose that the probability measure for where the dart lands is uniform over the interval the so-called Lebesgue measure. What is the probability that the dart lands on the point 1/4, given that it lands on either 1/4 or 3/4? 1/2, surely. But the probability that the point lands on 1/4 or 3/4 is 0 according to the uniform measure. The probability functions associated with these examples land on the first horn of the Four Horn theorem: they assign probability 0 to uncountably many propositions. In (i), we could have asked a similar question for each of the uncountably many lines of latitude, with the same answer in each case. In (ii), we could have asked a similar question for each of uncountably many pairs of points in [0, 1], with the same answer in each case. In all of these cases, the ratio analysis fails to yield an answer for the conditional probability, yet it is seemingly well-defined in each case. We could interpret the uniform probability here either objectively or subjectively. In (i), there might be some genuinely chancy process that determines the selected point without any bias perhaps a random number generator based on some quantum mechanical event is used to specify a
18 290 ALAN HÁJEK latitude and a longitude. And supposing, as we might, that you believe that this is indeed the selection mechanism, your own subjective probabilities should be correspondingly uniform. Similar points apply to (ii). You might try to recover the desired answers by taking appropriate limits. You might imagine, for example, a sequence of nested strips around the equator, shrinking to the equator in the limit, and define the conditional probability of W, givene, as the limit of the sequence of ratios of the form P(W E)/P(E). You would need to rule out in a non-question begging way strips that give the wrong answer: for example, we could have converging on the equator a sequence of strips, fat in the west, thin in the east, yielding answers greater than 1/2. Of course, the proposal would not rescue finite cases in which the zero denominator problem arises e.g., conditioning on the die landing odd. But more to the point, the proposal is not the ratio analysis, but rather a substantially more complicated analysis in which ratios appear. Note that the assumption of uniformity in the last examples is not essential to producing the zero denominator problem; it is only there to make the answers obvious. Since there are uncountably many propositions to distribute probabilities over, a real-valued probability measure must assign 0 to uncountably many of them, for it does not assign any infinitesimal values. Note also that it is not the answers themselves that matter, but the fact that there are answers. If for some reason you disagree with me on the answers that I have suggested, but agree with me that there are answers, you should agree with me that the ratio analysis is inadequate. You may be tempted to bite the bullet, saying: the conditional probability is undefined in these cases there is no answer, because the ratio analysis gives no answer. If this conflicts with our intuitions, too bad for those intuitions. Our intuitions about cases involving infinities were never that trustworthy, anyway. This is often the right way to respond to a surprising result. Consider the result that the set of even numbers and the set of integers have the same cardinality. This offends the intuitions of many of us. Yet I would not want to run the following superficially parallel argument: Mathematicians have not adequately analyzed our concept of size of set, for they saddle us with an unintuitive result. Rather, I would say that if this result conflicts with our intuitions, too bad for those intuitions. They were never that trustworthy, anyway. And I might say that of the earth s surface and dart examples, too, if that were the end of the story. But it is not.
19 WHAT CONDITIONAL PROBABILITY COULD NOT BE Kolmogorov s Elaboration of the Ratio Formula It turns out that an elaboration of the idea behind the ratio analysis, also due to Kolmogorov, delivers our intuitive answers of 1/2. So our intuitions were apparently quite sound, and in no need of revision. Kolmogorov himself was well aware of the zero denominator problem, and this is what motivated him to elaborate the ratio analysis to handle cases such these. I will briefly describe the Kolmogorov elaboration, though this is not the place to show how it delivers the desired answers. He introduces the concept of probability conditional on the outcome of a random experiment. The experiment is represented by a sub-σ -field F of the total σ -field under consideration. We define the random variable P(A F ), the probability of A conditional on F, which assumes different values according to which propositions in F are established as true on the basis of the experiment. P(A F ) is a random variable (whose existence is guaranteed by the Radon-Nikodym theorem) which satisfies the following condition: (RV) B P(A F )dp = P(A B) for all B F Compare this to the simple transposition of (RATIO): (PRODUCT) P(A B)P(B) = P(A B). It turns out that this gives us a way of coping with the zero denominator problem, while preserving the guiding idea behind the simpler ratio analysis. 11 We see, then, that we have no need to revise the intuitions that tell us that certain conditional probabilities take particular values in examples such as these. It might thus be thought that my attack on the ratio analysis is misdirected, and that my real target should be the full-blown Kolmogorov analysis. In response to this, let me firstly observe that it is clearly the ratio analysis that has the limelight in the philosophical literature, and the full-blown analysis is relatively rarely discussed by philosophers. 12 Indeed, various well-known philosophical tomes on probability do not mention Kolmogorov s elaboration (e.g., Fine (1973), Rosenkrantz (1981), Earman (1992), Howson and Urbach (1993), among many others). But more importantly, I think that the full-blown analysis is also inadequate. There are cases in which the probabilities of conjunctions required by the right-hand side of (RV) are vague or undefined, and yet the corresponding conditional probabilities are defined, as we will see in Section 6 and Section 7.
20 292 ALAN HÁJEK 5. INFINITESIMAL PROBABILITIES 13 You might think that this is a job for non-standard analysis. Recall the regularity theorem: any regular probability function defined on uncountably many mutually disjoint propositions assigns infinitesimal probabilities to uncountably many propositions. Since there are such regular functions, we know that the second horn of the Four Horn theorem is occupied. 14 I will now argue that any function impaled on this horn is in an important sense defective. Infinitesimals do not provide the sanctuary for the ratio analysis that one might hope for. It will be helpful to begin by returning to our examples of the earth s surface, and the [0, 1] interval, so that we can see how infinitesimal assignments might naturally arise. One might claim that using infinitesimal probabilities in the ratio formula will deliver the correct answers in these examples: The probability that the chosen point lies on the equator is not 0, but rather some positive infinitesimal, ɛ say; the probability that it lies in the western hemisphere and on the equator is ɛ/2. Likewise, the probability that the dart lands on 1/4 or 3/4 is some positive infinitesimal, twice the probability that it lands on 1/4. (RATIO) then gives the correct answer of 1/2 in each case. One hears this sort of claim a lot in this line of work, and it has considerable prima facie plausibility. However, I find it wanting, for any putative infinitesimal assignment is ineffable. In order to speak of non-standard numbers such as infinitesimals, one must have a non-standard model. We are guaranteed the existence of such a model, if we assume the axiom of choice; but we are also guaranteed that there are many of them. We cannot pick out a model uniquely, by naming it or pointing to it. Any talk of the non-standard model with which we are working is quite misleading. Infinitesimal probability assignments are thus curiously hard to pin down. They are not like familiar probability assignments such as P (the coin lands heads) = 1/2, or even 1/ 2 there is an ineliminable non-constructive aspect to them. (The appeal to the axiom of choice ought to alert us to this.) I would analogize them more to a gesture of the form P (the coin lands heads) = x, forsomex in [0, 1]. This is slightly unfair, perhaps, for we can pin down certain things about them that they lie strictly between 0 and every positive real number, for instance. Nevertheless, when we are dealing with sets on this tiny scale, where minute differences count for a lot, we want things pinned down very precisely. ɛ is a nice name for an infinitesimal probability, but what is it exactly? So when philosophers gesture at infinitesimal probability assignments, I want them to give me a specific example of one. But this they cannot do; the best they can do is gesture. 15
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text:
SO I had taken up the challenge to experience a potent hallucinogenic while seeking a state of deep relaxation and mediation, in the dark. (As opposed to the typical party hype.)
Collected ingredients, had a quick squiz at the procedure, had everything I needed and went ahead with an acid-base alkaloid extraction using HCI, NaOH, and Ether as the non-polar solvent. Big mistake.
While ether would 'Theoretically' work due to it being a non-polar solvent, there are a few things that make it less than ideal.
For one, VERY low boiling point, using ether on a hot summers day is less than practical, not to mention that HCL and NaOH have an exothermic reaction (it give off a lot of heat)
The ether was boiling as soon as it hit the mix. I basically panicked, thinking that I was going to lose my precious ether befoer I had had time to do the extraction, so I wanted to get things rolling. I put the lid on and shook it up....
I do soarly regret it of course, and I am lucky to have many guardian angels (I obviously need them) because even though I had acid and ether go all over me and my face and eyes ect... I managed to escape any injure except for a lot of ichyness and a small mild burn on my right eyelid.
Do not play with ether, its stupid. Ether is so unbelievably volatile, which would be fine, if it wasn't so explosive, bad for the lungs, and unpredictable.
I truly am lucky that it did not ignite.
Back to the DMT extraction... I had what I think is very low success.
I did manage to crystallise something out of my naptha (I went and drove to the next town and got some naptha after the ether explosion)
But it appeared to be the complete opposite of what I was looking for, crystals sharding inward from around an edge, rather than outward. They were also pretty cubic little bastards.
So I smoked that just to check what it was (hahaha just joking, you totally thought I would, didn't you )
I tried again using a different method. Metho, NaOH, Naptha.
I then tried 'de-fatting', and yet the fats did not separate.
All in all, you can tell, i've had a shit day.
1: Fuck Ether.
2. Measure your acids, filling a beaker with 400ml of acid, means your going to have 1ltr before its back to basic.
3. Make sure that the chemicals they send you are actually what you ordered (I have a scientific regent bottle of sodium Hydoxide, not even react with HCL (WTF!?) intend to get a refund on that bottle.
4. Don't try to be cool, and wing it on the recipes, there could be unforceen forces at play, that you are not considering.
5. "Pride comes before a fall." Now I'm not suggesting that I'm not way cooler than the rest of you, because I obviously am, but maybe I shouldn't rub it in so much
6. Listen to Zoas23 sometimes.
7. USE CORRECT SCIENTIFIC METHOD. Do the calculations. Measure out your chems prior to beginning the experiment. Consider your emergency protocols. Take time to eliminate risks. Use your PPE, or Jesus, whichever you prefer. Write notes, AND the method. Writing out the method gives you more opportunity to eliminate mistakes.
I do have something evaporating now, im not very confident in it, but we'll see what happens....
Until next time kids


text:
USA 3, Australia 2
Kai, Wambach, Lloyd
Coverage of women's soccer in the US. If it's not here...it's not important...
West Virginia scored three unanswered goals to post a 3-1 win over Maryland Friday evening at Ludwig Field in College Park, Md.
Adam's 88th minute goal keeps Pali Blues undefeated.
The U.S. U-23 Women's National Team opened its week-long training camp in Portland, Ore., with a resounding 7-1 victory over the University of Portland on April 18 at Merlo Field in a match played in front of 3,500 fans.
Chris Economides is angry. He's VERY angry, and not just because he's one of the guys being sued for $10.8 million by NBT Bank.
Michelle French scored for Pali Blues
Southern Miss women's soccer coach Gail Macklin has resigned after five years of directing the program.
A former Marshall University women's soccer coach will go to trial on May 23 to face charges of driving under the influence and striking two parked vehicles.
FC Indiana, a two-time Women's Premier Soccer League champion now playing their inaugural season in the United Soccer Leagues W-League, announced the field for their first-ever FC Indiana PDA International Cup tournament, set for April 30 and May 2 at Kuntz Stadium in Indianapolis.
The Blues had a few chances during the first half, including a free kick from Danesha Adams that the UCLA keeper, Ashley Thompson, struggled to save. The team didn't break through until the 49th minute of play as Rosie Tantillo finally beat the UCLA keeper. Tantillo has scored five goals during this preseason campaign.
The Women's Premier Soccer League (WPSL) has returned once again to the South Texas area. The South Select will be based out of Houston, TX and this upcoming Summer 2008 campaign will try to establish itself in the highest level of women's soccer in not only Texas - but the world.
Forwards Megan Mischler and Deana Everrett scored in West Virginia's 2-0 win over Canada's U-17 National Team Tuesday afternoon at Dick Dlesk Soccer Stadium.
The Washington Freedom won their first match of the year in a preseason game on Saturday over the Penn State Nitany Lions at the KICKS Against Breast Cancer Tournament by a score of 2-0.
The U.S. women's soccer team qualified for the Beijing Olympics on Wednesday night by defeating Costa Rica 3-0 in the CONCACAF qualifying tournament semifinals in Juarez, Mexico.
The Pali Blues extended their winning streak to three games as they defeated the U.S. Women's Under-20 National Team, 4-2, at The Home Depot Center.
The University of Tennessee has fined women's soccer coach Angela Kelly and suspended her for one game for using an ineligible athlete in an indoor tournament in February.
USA 3, Mexico 1
California upended 19-time national champion North Carolina, 3-2, Saturday night in spring action at Vista Murrieta High School in front of approximately 3,000 fans.
The Northern Arizona United Cheetahs has been awarded a franchise in the Women's Premier Soccer League (WPSL) for this upcoming Summer 2008 season and will compete in the Southwest Conference. The Cheetahs home field will be at Embry-Riddle Aeronautical University in Prescott, and Northern Arizona is also looking to play one or two games in Flagstaff, AZ.
For one fleeting minute on a sun-baked Sunday afternoon, Mexico and their 22,000 screaming fans believed in miracles.
Danesha Adams, Rosie Tantillo, Mele French and Liz Bogus scored to give W-League newcomers convincing win over collegians.
The Washington Freedom announce Thursday the signing of US Women's National Team player and University of Virginia standout Becky Sauerbrunn. Sauerbrunn joins the Freedom as they set out to defend their 2007 W-League Championship.
On Saturday afternoon, Rosie Tantillo stepped onto McAlister Field. It was something she had done hundreds of times before.