Last updated 5 Oct 2007: Originally created in 1999 I thought it was time to make the file more easily readable in web browsers, and add a few new links. Reformatted (reduced width) 14 Dec 2016
We had several email exchanges which prompted me both to extend my thinking about emotional and other affective states, and also to summarise some of the ideas in way that might be useful for others: the journalist certainly understood most of what I was saying, though we had to have several cycles of correspondence to remove some of the misunderstandings (e.g. bringing out the distinction between 'having an emotion' and 'feeling an emotion'. Most people, like the journalist at first, believe mistakenly that the first implies the second.
I have not included the messages from the journalist, but quoted bits of the messages to which I was responding. His name is replaced with "Journalist" throughout.
NOTE: 5 Oct 2007
I can now see no harm in pointing out that the Journalist was Jeremy Webb, writing for New Scientist.
The New Scientist article starts28 March 1998, Magazine issue 2127
"DO birds consider whether the nest they have built is better than the one built in the next tree?" asks Aaron Sloman, professor of artificial intelligence and cognitive science at the University of Birmingham. "I doubt it." If his assessment is correct, it's a safe bet that you won't find a bird that feels pride. Sloman's point is that being proud isn't easy. At the very least you need a sense of "self" and a way to compare yourself with others. So feeling superior takes a higher level of mental complexity than most animals can muster. That doesn't apply to all emotions. The fear that keeps animals out of the path of a predator or a speeding car, for instance, is almost universal across species and needs little or no thought. But it's the complex emotions like pride that fascinate Sloman. He believes that they arise naturally from the information-processing ...
> The quoted bits of the journalist's messages > are all formatted like this.
I'm busy reading through the papers on your website as I write. I I haven't got to pride yet--perhaps it's not there--but I would love to discuss the idea of looking at how one might make a proud machine/agent. If you have addressed this subject, we may be able to use your thoughts as a way into discussing your entire approach to consciousness.
> DDD suggested that we should look at "pride" in terms of the > structures that might be needed to support such a complex emotion. He > thought that you may have given some thought to how pride might be > instilled in a machine/agent.
I haven't thought much about pride, though I have thought about grief, shame, embarrassment, guilt, anger, irritation, regret, excited anticipation, carelessness, and several others.
> I'm busy reading through the papers on your website as I write.That could take you an awfully long time.
I've just added some new ones to the Cognition and Affect project directory. http://www.cs.bham.ac.uk/research/cogaff/
I'll be discussing related topics (robots in love?) at a literary
society meeting I have (foolishly?) agreed to give a talk to on 21st Feb
at the RFH. (Link to event now doesn't work.)
This talk eventually led to a paper on love being published:
> I haven't got to pride yet--perhaps it's not there--but I would love > to discuss the idea of looking at how one might make a proud > machine/agent. If you have addressed this subject, we may be able to > use your thoughts as a way into discussing your entire approach to > consciousness.The approach is based on the assumption that everything depends on the underlying architecture -- to be more precise the information processing architecture.
(This is not necessarily a physical architecture: just as a complex software system can have an architecture which is not a physical architecture, though it is *implemented* in physical mechanisms).
When you know what sort of architecture is implemented in the brain of an animal or machine you can (if you have a good theory) work out the possible states and processes that can be supported by that architecture.
(E.g. think of how the periodic table of the elements, based on a theory of the architecture of matter, generates a systematic collection of types of "elementary" stuff, which when combined with more architectural detail, a theory of valency and chemical bonds, generates an even larger variety of types of chemical compounds, and processes in which compounds get transformed.)
At present we have only fragments of a theory of types of architectures and the processes they can generate. We can make progress by simultaneously working top down trying to "unpack" familiar types of mental states and processes (e.g. using philosophical methods of conceptual analysis, and some kinds of psychological research) and also bottom up trying to work out what sorts of architectures are implemented in brains and other mechanisms and what they can do. (This needs to be related to the evolution of human-like brains: our architecture reflects much of our evolutionary history.)
If we are lucky, the top down and bottom up approaches will meet up in a fruitful way (as I am already beginning to find in discussions with brain scientists).
A few thoughts on pride, provoked by your message.
Pride is a fairly sophisticated and subtle collection of states and processes involving
- long term attitudes (taking a pride in one's work, being proud of being a Brum),
- relatively short term moods (sharing the pride of one's football team, or nation, or family during a day of celebration of some triumph or recognition),
- emotional states (a thrill of pride when your child wins a race, or your team beats another more famous team, or a distinguished person praises your achievement...).
These all involve some kind of high valuation of oneself or something one is associated with or associates oneself with. It's a valuation both in relation to how one values others, and also how one thinks others evaluate one.
In some cases this evaluation can be distorted, e.g. people value their family or their class or their skin colour or their achievements more highly than those of others and wrongly think that that evaluation is justified and should be shared by everyone.
Then there's a more active kind of pride which has to do with reasons for taking decisions or performing actions, e.g. refusing help, rejecting pity, not wanting to show your fear or other weaknesses, or being too proud to admit that....
What all this implies is that having the full range of varieties of pride involves a very sophisticated cognitive and motivational architecture including lots of beliefs about achievements of oneself and others, the qualities of oneself and others, the appreciation or recognition of those qualities and achievements by others, and beliefs about sorts of actions or happenings which can affect that recognition.
Alongside beliefs it also requires desires of various sorts, including a desire to achieve or preserve that recognition (= fear of losing it), desires to take actions which will enhance it, or draw attention to it.
It usually also involves not just the belief that others value one's qualities and achievements but also pleasure in the fact that they do.
These combinations of beliefs and desires (and other things) can interact with all sorts of other information in forming attitudes and and decisions in various more or less subtle ways.
Some of this can happen relatively unselfconsciously.
However, the human architecture also includes various abilities to be aware of internal states and processes and to experience pleasure and pain (things which I don't think have yet been adequately analysed). So some kinds of pride include these additional self-reflective states and processes.
Some emotional states involve partial loss of control of attention and thought processes, e.g. the person who wallows in the pride of his achievement and cannot put it out of his mind in order to get on with his work, or a task of thinking about someone else's needs.
(That presupposes an architecture in which one is sometimes in control of ones attention and thought processes: you can't lose control if you never have it. I don't think that's possible for a rat, or a newborn baby, or the simplest sorts or robots of the future.)
I think all of the kinds of pride (or most of them?) involve some kind of awareness of other agents with values related to ones own.
Without that relation to a social group, one could feel pleased with oneself, relieved at having achieved a difficult and possibly dangerous goal, self-approving in various ways, etc. but I don't know that we'd regard those states as examples of "pride".
However these questions about the boundaries of pre-theoretical concepts are often unanswerable: the concepts are inherently indeterminate (like the concepts of "water" and "air" prior to the development of the atomic theory of matter.)
The upshot of all this in relation to questions such as whether pride could occur in a machine is that it would require a whole lot of other things to be in place in the architecture before there's any chance of pride occurring. The requirements for perception, belief, and desire are much simpler.
E.g. I suspect that many animals have quite complex mental states and processes without being capable of most of the above kinds of pride, if any, e.g. ants, dragonflies, rats, birds. I am not sure about chimps and gorillas: it's very hard to tell what sorts of categories they use in their thinking especially their self-evaluations.
I am fairly sure that a newborn human infant is incapable of pride, whereas within a couple of years the architecture and the conceptual framework has developed sufficiently for certain kinds of pride (a two year old can, in a simple way, be proud of his/her appearance or achievements, but not proud of his nation, for instance!).
Further analysis could perhaps point to kinds of brain damage which would remove the capacity for various kinds of pride by removing some of the kinds of capabilities on which they depend: e.g. the ability to consider how one is evaluated by others.
The architectural requirements for the ability to have or experience pride are closely related to requirements for shame, humiliation, guilt, embarrassment, and other things. Most mental states come in elaborate families and cannot occur in isolation.
Well enough of my rambling. That's just a brief sketch from a "top down" point of view. A lot more would have to be said about the kinds of information processing architectures which could do all that. But that's a very long story, involving a lot of ongoing research with only partial and provisional results.
Is this relevant to what you wanted? Feel free to phone me at home in the morning if you wish.
It occurs to me that if you wish to regard pride as one of the sins, that would rule out reasonable or justifiable pride, such as
Taking pride in one's work, because it is well done.
The cases where pride can be foolish, wicked, unjustified, or counter productive include things like.
Some of these states depend on awareness of one's own vulnerability.
Most involve the ability to think about how others think of oneself, and wanting to be rated highly.
Most/all involve the notion of some kind of social ranking, or ranking in levels of approval by society, colleagues, acquaintances, etc.
And maybe more.
I once, as a child, had a dog: a black and white mongrel called Spotty. I remember an occasion when we were out walking. A much smaller dog came rushing out of the garden of a house we were passing, barking furiously at Spotty, causing Spotty to run away with his tail between his legs. The other dog then stopped and strutted back. I thought the episode extremely funny and stood laughing loudly. Spotty turned, looked at me, then walked off limping, as if he had been hurt. But the other dog had not got near him. A short time later something distracted him and he stopped limping.
I don't really know what was going on, but at the time I thought he had felt humiliated by my laughing at him (a blow to his pride???) and he had tried to regain my sympathy by feigning injury.
Who knows what was actually going on in that brain?
I see the current issue of New Scientist has an article on deception by animals.
Thanks for your message, and also for putting so much effort into this.
> My understanding now is that you do not "plug emotions into your > model" but that emotions arise out of your model--they are emergent > properties.This is NOT true of all emotions, but is true of the (evolutionarily new) sorts of characteristically human emotions that interest me (grief, infatuation, jealousy, excited anticipation, apprehension, embarrassment, shame, etc. etc.).
[[Note: I now call these tertiary emotions. There are also primary and secondary emotions. (A.S.)]]
As you have probably already grasped these all require:
1.a. deliberative capabilities the ability to create and consider and evaluate new possibilities, e.g. for what might happen, for actions, for things to say, etc. 1.b metamanagement capabilities the ability to monitor one's own state, to evaluate it, to (attempt to) change it.
Some of them also require awareness of being part of a social system.
Some of them require acceptance of an external system of values, going beyond one's own personal desires, preferences, etc.
Remember that the word "emotion" is multiply ambiguous (57 varieties?) and is used by different people to refer to different things. Most psychologists who study emotions do not focus on the above. Poets and playwrights and some sorts of therapists do.
I allow that the word "emotion" also refers to evolutionarily older states and reactions, e.g. nausea produced by a horrible smell, paralysing terror, being startled, sexual reactions, and perhaps others which we may share with rats and other animals which are not capable of the sophisticated "new" emotions.
What I say about these *older* emotions is different from what I say about the *newer* human emotions. In particular a subset of the older ones may be pre-programmed genetically via patterns and reactions in the limbic system and perhaps other mechanisms. I.e. they may depend on specific "emotional" mechanisms as opposed to emerging from the interactions of a variety of other mechanisms required for intelligence.
Another important point which we did not discuss is that there's a difference between *having* an emotion (being in an emotional state) and *feeling* the emotion, though most people unthinkingly assume that if you have one you feel one.
That's not so: a person can be angry or infatuated or envious without realising it. It can be very clear to all their colleagues or friends however. This is commonplace in plays and novels because it is commonplace in life.
> So, just as a file server gets into a spin when it is > overloaded and becomes useless to its users, so your nursemaid MIGHT > "feel hassled" by having too many babies to care for and it cannot ^^^^ > plan its next action. (Overload could be equivalent to being hassled.)
Yes, but it could *be* hassled without *feeling* hassled.
That requires the meta-management (self-monitoring) facilities to detect the state. The state could occur and go undetected. In fact part of the emotional education of humans is helping them to detect and classify some of their emotional states, so that they can begin to manage them better.
The simulated nursemaid I showed you was not capable of *feeling* hassled even when it *was* hassled (i.e. when it could not cope with the flood of new motives, when there were too many babies all getting into trouble at once).
(Incidentally the nursery scenario allows more kinds of troubles than I showed you: e.g. in an over-crowded room thuggery can develop! However the implementation of this is still very shallow).
Ian Wright's more sophisticated nursemaid, which I did not show you, could *detect* that it was hassled and react by raising its attention filter threshold.
You could regard that as a primitive form of "feeling hassled" going beyond merely being hassled.
But it had no capacity to find that state unpleasant: it merely reacted by taking pre-programmed action to remedy the state. Experiencing something as pleasant or unpleasant requires more sophisticated mechanisms than those we discussed, and my ideas on that are still very vague.
So, I'd rephrase what you wrote, without the word "feel" as:
> the nursemaid might > "become hassled" by having too many babies to care for, so that > it never completes any planning process because it is constantly > interrupted by being forced to attend to a new goal. > (Overload could be equivalent to being hassled.)
> The class of emergent properties that you have done most work on is > the perturbant emotions--shame, grief, guilt etc--which intrude by > continually generating motivators of a higher insistence than the > agent's attention filter.
Yes, but that's just one facet of what's going on. The other facets include
Some "OLD" emotions may have genetically programmed emotion mechanisms, e.g. freezing in terror. I have mostly ignored those, not because they are uninteresting but because I thought enough had been said about them by other people (e.g. Joe le Doux)
The "NEW" emotions are far more complex and in at least some cases are not direct consequences of explicit emotion mechanisms but rather emerge from interactions of multiple subsystems concerned with motivation, deliberation, attention, evaluation, etc.
What the "new" and "old" emotions have in common is that there's some (possibly partial) loss of control.
(The loss can be dispositional: it doesn't manifest itself unless certain circumstances occur, e.g. being reminded of one's dead child.)
> Then, pride in one's work might be a state that emerges once, say, > the nursemaid has an understanding of what a good job is compared to a > bad job.
There may be different sorts of pride in one's work, accessible to different sorts of animals (or artificial agents) of varying sophistication.
For a non-social animal there may be awareness of the fact that jobs can be done well or badly, one in time to meet some need or too late to meet the need, done with a lot of effort or with little effort, etc.
If those cases can be recognized and discriminated, then on completion of a job which has been done well, in time, with little effort, etc. there may be a period of assessment when the quality of the achievement is recognized. This is one form of taking pride in one's achievements.
(This assessment of a job as well or badly done could be part of an adaptive mechanism feeding back into weights which will affect choices in future activities, for instance. I.e. it's a pre-requisite for certain kinds of learning.)
A further level of sophistication would be the ability to remember the evaluation of *previous* performances and the ability to detect whether things are getting better or worse. An animal which could detect that there is a steady improvement over several tasks may get additional satisfaction from a job well done, over the simple recognition that it was well done. I have no idea which animals other than humans can do this comparison with past performances.
For a social agent an additional factor may be comparison with how *others* perform on similar tasks. I.e. one can evaluate one's performance in relation to how well others are perceived to perform on similar tasks.
Where the performance involves direct competition (competing for a mate or fighting over some food) the recognising achievement of the task itself involves the comparative assessment (I got it, he didn't).
In other cases more sophisticated cognitive abilities are required if one's own performance is to be compared with others. Do birds consider whether the nest they have built is better than the one built in the next tree? Do they consider whether their chicks are bigger, or more numerous, than those in the next nest? I doubt it. (But could be wrong.)
Yet another level of sophistication involves thinking about how *others* evaluate one's performance, and how they rank it. I may be able to tell that I am doing better than another individual without having the conceptual sophistication required to think about whether other people think I am doing better than that individual. For agents which have that conceptual sophistication there may be a recognition that others in the group rank them highly.
This recognition may or may not lead to further internal processes, e.g. a glow of satisfaction. This requires not only detection of the fact that one is ranked high, but also evaluation of that fact as good or bad.
So, depending on its cognitive sophistication, an agent may be able to:
> Would it need the metamanagement layer to say "you should > feel good about yourself", or could it emerge just from the management > layer?You are forcing me to try to design very complex explanatory mechanisms in a short time, and I am very likely to make mistakes!
However I hope my comments above help to show that there are different types of recognition of how well one has done, involving different kinds of cognitive and evaluative capabilities, and in complex cases these interact in intricate ways.
I don't think instructions to oneself to feel good could achieve anything directly. However there are explicit evaluations of states as good or bad, and that is part of what it is to feel good or bad about those states.
(Self evaluation is a complex phenomenon on which I am still not very clear.)
If you are finding some of this hard to understand, it's not your fault, because
(a) the theory is complex and difficult
(b) it is still patchy and unclear in many places
> Then there are the emotions that require a knowledge of self and > status within a social group--such as being too proud to talk to > someone.
That in turn subdivides into various different cases, e.g.
and more besides.
> With the exception of grief, which you go into in one of your > papers, do you have a clear idea of what kind of state an agent would > have to get into to feel any of the other emotions? ^^^^
If we replace the word "feel" above with "have", the question is about what sorts of states are involved in having other emotions. Yes, I have thought about and discussed a number of other cases with colleagues and students.
In general it is unprofitable simply to think about a particular emotion. Rather, in order to understand any particular case you have to think about a range of cases and how they are similar and how they are different.
For example, I might ask a student to analyse and compare these:
X is angry with Y X is irritated with Y X regrets what Y did X wishes Y had not done that X is angry/irritated etc. with himselfor
X regrets doing A X is embarrassed about having done A X is ashamed of having done A X feels guilty about doing Aor
X admires Y's house X would like to have a house like Y's X is envious of Y because of Y's house X is obsessively envious of Yor
X loves his children, wife, family, etc X loves his country X loves his job X loves ice cream X loves skiing X is in love with Y X is infatuated with Yor
X is afraid that Y will happen X is terrified that Y will happen X is apprehensive about Y X is nervous about Y X is cautious because Y may happen ....or
X likes/enjoys looking at Y X finds Y beautiful X finds Y impressive X finds Y awe-inspiring X finds Y sexually attractive X finds Y interesting, unforgettable, ... X cannot help staring at Y (even though Y is horrible) ...
Not all of these are emotions.
I could try to dig up some old notes on anger and related emotions, if you are interested.
[[ Now in
Towards a Grammar of Emotions, originally in New Universities Quarterly, 36,3, pp 230-238, 1982 ]]
> Perturbant emotions and (possibly) hassle both emerge because of > resource limitations. Is it your contention that all emotions are > bound up with resource limitations,
> or will some emerge through other > quirks of your model?quirks???
The "old" primeval emotions come not from mechanisms with resource limits but from rapid global effects of powerful "alarm" mechanisms (e.g. the limbic system). Emotions involving global interrupts are the ones most studied by psychologists and brain scientists, I think.
In humans it looks as if this mechanism can be trained to produce new kinds of reactions and to do so in relation to new kinds of things. (I think a lot of religious indoctrination is based on this.)
So the architecture supports different sorts of states which are called "emotions" in ordinary language. It's only a subset of those states which involve perturbance i.e. actual or potential loss of control of attention. I happen to be interested in those partly because I want to know what it is to be in control of thought processes. (Which I assume chickens and new born babies are not).
I.e. the perturbant emotions are specially interesting to me not as emotions but as something that has to be explained by a theory of the high level control mechanisms in a human mental architecture.
> Of course labelling a state that an agent gets into as an emotion > is one thing. The big question, then, is can an agent feel an emotion? > Does it need sensory actuators to generate pain or malaise when the > agent is feeling sad?When you *feel* sad, angry, hassled, excited about your friend's arrival, infatuated, that's because your self-monitoring mechanisms have detected and classified your state.
In most cases this does not involve sensory actuators and it need not involve pleasure or pain apart from the evaluation that is already there in the state detected.
In some cases sensory actuators are involved. When grief makes you cry you will experience the physical/physiological processes involved in crying. But there is far, far more to grief than crying or any other physical manifestation which may be detected by physical sensors.
> One final point. Have you any idea how the most basic emotions, > such as startle, fright etc, emerge from the workings of the reactive > layer? Or is the reactive layer designed specifically to emit such > emotional signals? (I realise that these emotions are not your primary > interest. I just wondered whether you had thought about them?}
I hope my comments about the old emotions answer that. I don't know details, but I presume the limbic system has various global control capabilities which account for some of the things you are talking about.
There may be other more subtle things like reactive mechanisms which for some reason keep intruding on the deliberative layer. E.g. if there's a rumbling noise in the background it may be of no concern to you and you may be able to shut it out. But under some circumstances you keep finding your attention drawn back to it as something needing explanation. This might go via some sort of anomaly detector which is not part of the global control system (required to make you freeze, flee, change direction, etc. etc.), but still has the ability to attract attention.
I think itches and various bodily desires work like that. As they grow more and more intense they gain in their ability to attract attention (interrupt the deliberative mechanism). I.e. they acquire higher insistence in my terminology. I don't see any need for them to go through any global alarm system (the limbic system) unless states are detected which are potentially life threatening, or whatever.
Whether these should or should not be called emotions is a matter of taste. Some people would and some wouldn't.
I don't care either way: the labels are not important, only what the phenomena are like in detail and how they fit into the explanatory framework provided by the architecture.
I have a suspicion that I have answered at too much length to be of any use to you. Does it help?
Letter written after being sent a draft of the article.
Thanks for letting me see your draft. I am sorry I could not respond earlier: I have had a visitor today.
I have some comments on your questions, and some suggested changes to your draft.
> a) How do you respond to Rodney Brooks' suggestions that evolution > would have created less rigid structures and connections in the mind > than you propose?
I don't think Rodney Brooks has read my proposals.
[[Note added 2005. Since about 2002 Brooks has been withdrawing from the position he took in the 1990s]]
Anyhow there's nothing particularly rigid about my structures as far as I know -- the whole point is to provide increased flexibility. The remarks you quote from him sound as if they were based on his seeing a very short summary.
Most of the main features I postulate correspond only to well documented human abilities (e.g. we definitely do have long term associative memories, we do have motives and desires, e.g. you really do want to write this paper (I think), we definitely do make plans (e.g. you planned in advance to write the draft then send it to me for comments). We can notice things about our thinking (e.g. you have expressed a negative judgement about your thoughts expressed in your final paragraph.)
And of course we have reactive mechanisms, including well known reflexes of many kinds and startles, etc.
So when I talk about evolution having produced such things I am not going far beyond common knowledge, and I am not saying that evolution is *constrained* to produce these things (e.g. it solved the survival problems differently in bees and termites).
Moreover, my papers have been so *vague* about the interactions between the layers that I can't see how they *constrain* evolution.
That's a serious weakness in my work: the vagueness still needs to be removed. Until then, it's just a very schematic and incomplete specification of one region of design space.
Some of Rod's followers make a different point. I am suggesting that animal brains implement a fairly *modular* structure combining a variety of identifiable functionally distinct components.
The alternative view (strongly represented at Sussex University) is that evolution produces an unintelligible mish-mash of relatively simple reactive behaviours, which works nevertheless.
The first thing to say is that nobody knows, and the people who are dogmatic on one side or the other of this issue are just that: dogmatic. We have to keep an open mind.
I say in many papers that my proposed architecture is merely one sort of possibility in a space of designs and we have to understand what that space is like, what alternative possibilities there are and what sorts of trajectories can occur through the space.
Secondly I have suggested that contrary to the Sussex/Brooks view there may be reasons why evolution would favour a modular structure rather than an unintelligible mish-mash when it is creating something with a very large collection of capabilities.
That's because (a) in some cases the un-modular system would require far more storage than something composed of re-usable re-combinable modules; and (b) the ability to do without deliberation, planning, etc. (the functionality of my second layer) would require a far longer period of evolution to create explicit solutions to the need to cope with the same variety of contexts.
This is a very short summary of a more complex argument, which still needs to be spelled out more fully than I have managed so far. The argument may have flaws, but as far as I know it has not even been considered by the defenders of the Sussex/Brooks position (excellent researchers, friends of mine, etc. but sometimes too narrow and dogmatic!!!)
The Sussex position is based partly on their own very interesting evolutionary experiments in which they evolved circuits capable of performing rather simple tasks. The circuits worked but were very hard to understand, and did not seem to have any clearly identifiable modules with well defined functions. They extrapolate from this to the claim that animal brains may be similarly unstructured and non-modular.
But that's a totally unjustified leap from very simple systems to far more complex systems. My arguments for the evolutionary value of modularity depend on the complexity and the variety of capabilities in diverse situations.
> b) If you managed to create a community of agents sophisticated enough > to feel pride in all its forms, do you think they could evolve a > feeling that excessive pride is a bad thing?
I think it is usually not helpful to ask:
Could machines etc. do so and so?
Instead always ask:
How could they do so and so?
The first question is an invitation to undisciplined and premature speculation. The second is a request for reasoned arguments and explanations.
As far as your question is concerned my undisciplined, wild speculative answer is: I don't see why not, since (some) humans did and we are therefore an existence proof of possibility.
As to HOW it happened, that's a far more interesting question, and it is part of a more general question about how values evolve. I don't have any kind of nice short answer to that. I suspect it's another one of those simple looking questions which have a very complicated answer.
> Thanks for your help on this. It's certainly given me a deeper > understanding of AI.
I hope that this sort of work will give people a deeper understanding of what we are!
You've had a hard task, and I am impressed that you've managed to compress so much into such a short space. I think the compression has produced some minor infelicities and inaccuracies so in what follows I've suggested a changes which I hope are helpful. I've also added explanatory comments for you in parentheses, not for inclusion in the text.
I hope it all makes sense.
Best wishes. Aaron
Updated: 5 Jul 2008; 14 Dec 2016 (Minor re-formatting)
Updated: 5 Oct 2007 (mostly re-formatting)