Majority logic I

When I went for an interview at Oxford for my undergrad degree one of the things we had to do was a kind of logic test. It had things like “All men wear hats, some men wear ties” and you were asked what it was possible to conclude from that. Presumably they were looking for something like “Some men wear ties and hats”. One of the questions was “Most men wear ties, most men wear hats”. I said that it wasn’t possible to conclude anything from this. But it was then explained to me that if hat-wearers and tie-wearers are both in a majority, then there must be some who are both: the proper conclusion is that “Some men wear ties and hats”.

“All men” is standing in for the universal quantifier (x) and “Some men” stands for the existential quantifier (Ex). But “Most men” is awkward to put in similar terms without resorting to some sort of logic enriched with predicates relating to numerical proportions. But that sort of “statistical logic” is too complex. The fact that some predicate applies not to all but to a majority of some domain is an easily understood fact that does not need a whole logic of proportions behind it. Let’s make “Most of” a new kind of quantifier: (Mx). What properties does (Mx) have?

• 

This is the property my interviewer at Oxford was exploiting. If most x’s are P’s and most x’s are Q’s, then some x’s must be both.

• 
• 

These two properties simply say that if all men wear hats then most men wear hats and if most men wear hats then some men wear hats.

Do we really need a third quantifier here? Is there some way to express “most of” in terms of universal and existential quantifiers? I’m not sure there is. Here’s a first stab. When you say “(Mx)Px” what you are really saying is:

• 

But that’s no good, because that expression still contains an “Mx”. We are kind of going beyond standard predicate logic, but only a little bit. Perhaps Mx shouldn’t be a quantifier, but a property of predicates? But I don’t think that is a particularly satisfying suggestion. A property of predicates would be a third-order entity, and that seems extravagant given the modest goal of formalising the easily understood concept of a majority.

Soon I’ll post again and discuss the “dual” of (Mx): “Only a few men wear hats” “Hat-wearers are in a minority”.

A note on the symbolism: I wanted to use and but I couldn’t think of a way to “invert” the letter “M” in a similar fashion. It shares the vertical symmetry of “A” but when reflected around a horizontal axis as is, you get a “W”, which is no good if the idea is to come up with a new symbol. So I reverted to old-style quantifiers. Suggestions regarding how to “symbolise” majority are most welcome.

Misplaced outrage?

It’s about time I weighed in on the ongoing “MP expenses scandal”. First, the outrage seems a little disproportionate to the amount of money involved. If you want to be outraged by massive wastes of the taxpayers’ money, be outraged at the bank bailouts, the trident replacement, the ID card scheme, the “super-database” or something in that vein. (If you consider all of the above reasonable and proportionate expenditure, I’m there’s something else shockingly expensive you’d want to add to the list…) Or how about being outraged by the demented profligacy of the European Parliament decamping to Strasbourg for four days a month simply because the French won’t give it up. We* pay for the upkeep of a building in Strasbourg which is empty most of the year. We also pay to have the 600-odd members of parliament shipped down to Strasbourg from Brussels along with various aides advisers and so on. We also put them up in hotels for the few nights they are there. That is madness. The current expenses fallout is small fry, all things consdered.

But what does bug me is the insistence by the MPs that they were acting within the rules. This is a morally bankrupt way to justify actions that are clearly wrong. Regardless of whether it really is the case that the expense claims under scrutiny were acceptable given the current wording and structure of the rules, the claims should not have been made. Whether or not the claims conform to the word of the law, they are emphatically at odds with its spirit. People who cannot make this elementary distinction should not be in government. The illegality or otherwise of the MPs actions is not what is at stake, it is their moral authority that has been undermined and that is more worrying.

On a related note, here is another reason I don’t want David Cameron to be Prime Minister. A quote from the leader of the opposition in today’s Guardian:

We have to acknowledge just how bad this is. The public are really angry and we have to start by saying “Look, this system that we have, that we used, that we operated, that we took part in – it was wrong and we’re sorry about that.”

It’s not the system that is at fault here, Mr. C. It’s the people abusing it. Of course, the system is faulty if it is open to such flagrant abuse and it does need reform, but that is not why people are angry. People are angry because their elected officials, those people who are supposed to be representing their interests are exploiting the system to line their own pockets (or furnish their own homes…) What Cameron is trying to do is deflect anger away from himself and his party and turn peoples’ outrage on the current expenses system. But people aren’t angry with a system that allows abuse. No system is perfect. Why this affair has provoked so much anger is because it shows politicians engaging in what is effectively benefit fraud. These people are not fit to be in charge and that is true of the Conservatives as well as Labour (and possibly all the other parties too, I haven’t read anything about Lib Dem or SNP or whatever…). How are we to accept new laws, new databases, new intrusions of our privacy as being for “the greater good” or “in our best interests” when the people who come up with these schemes are dishonest and fraudulent. Lord Naseby apparently suggested that the public’s confidence in parliament has been so damaged that it should be dissolved and new elections held. But that wouldn’t solve the problem, I don’t think. I’m not sure what could, since it seems the skills needed to get elected are not the same skills as the ones that make someone a good representative, a good decision maker or a good leader.

That said, I’m sure many of the expenses claims have been portrayed in a way so as to make them seem maximally suspicious and with little or no right to reply. At least some of the supposed “dodgy” claims are probably legit. What is worrying is the sheer number of dubious expenses claims that seem to be flying around. Some of them surely have to be dodgy, I suppose…

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* By “we” I mean taxpayers in EU states. Though as a student I probably contribute so little tax money that my outrage is unwarranted, as it’s not really my money they’re wasting…

RIP I.J. Good

I was sad to read that I.J. Good died recently. I only heard about him a couple of weeks ago. Even from the fairly academic work of his I was reading you got a sense that he was a really interesting character with a unique sense of humour. His obituary confirms that impression.

In other news, I’m desperately trying to finish various papers and so on that need doing and I’m not really progressing particularly fast. Not that that is really news. That seems to be my default status. In the coming weeks I have to finish up short papers on the following topics:

• Frege on the definition of numbers
• Why probability is not always the best way to represent uncertainty
• Quantum Mechanics and Structural Realism
• Similarity relations and the theory-world link

Interesting topics all (apart from the first one), but it’s frustrating to have all these little bits and pieces to get done when what I really want to do is get down to a Big Project like my literature review… Although the second and fourth topics at least have some bearing on the topic of the lit rev, so that’s a bonus.

Two upcoming conferences that I will be attending:

• Scientific Realism Revisited
• Graduate Conference: Philosophy of Probability

Exciting stuff, no?

Data as a mass noun

I was told off yesterday for saying something like “Our data is incomplete…” Now, I know that “data” is a plural (as is media). But I thought it legitimate to use data as a mass noun rather than a count noun. This is, apparently, an “institutionalised mistake”. I’m not so sure. If everyone talks that way, doesn’t that make it OK? That said, I get annoyed when people say “less” when they mean “fewer”, so perhaps I shouldn’t be arguing this point… But in the case of “data”, I don’t see anything wrong with using it as a mass noun. Information, it seems to me, is a continuous mass-like thing. Perhaps the point is that data are still discrete units of information or some such… Anyway, I see nothing wrong with “the media” as a singular. I know that media is the plural of medium, but “the media” or maybe “the Media” is a way of referring to all those associated with any of the diverse media as if they were a homogenous mass. The Media is a thing – it is what feeds you entertainment, celebrity and to a lesser extent information. It does this through various media; the medium of television, the medium of radio and so on…

I am, in general, pedantic and fussy about things like this, but data as a mass noun and media as a singular noun don’t seem to bother me. I don’t really know why. Perhaps I should try and be annoyed about them, for the sake of consistency…

Dutch books

I’ve been thinking about the Dutch Book Argument (DBA) recently. I think the constraints on rational betting preference that underpin the force of the argument are unreasonable. At least they are not always reasonable. I think a better way to think of the argument is as a conditional argument: “Given these rational betting preference conditions, this is how rational degrees of belief should be constrained.” Then you can have a whole series of different DBAs with different betting preference conditions leading to different constraints on credence. It would be interesting to see how one would have to constrain betting preferences in order to have you beliefs behave like upper and lower probabilities or Dempster-Shafer belief functions…

I think this isn’t to undermine the force of the DBA, but to reinforce it. With this wider framework we can understand why people often fail to reason probabilistically. We can understand what aspects of rational betting preference are “non-probabilistic”.

That’s not to say that the DBA isn’t without its flaws. Some elements that concern me are:

• Using betting behaviour as a proxy for belief
• Existence of exactly specific numerical credence (and utility)
• Reasoning as calculating expected utilities
• The idealisations involved in discussing “ideal rational agents”: utility maximising, purely self interested, perfect calculating agents…

Practical politics consists in ignoring facts

The title is a quote from Henry Adams.

I despair for modern politics when David Cameron tries to ally himself with “the common man” while at the same time engaging in petty point-scoring about Titian’s age. Thanks Dave. Really constructive politics we’re doing here. And it is faintly worrying in a 1984 kind of sense when someone then tries to edit wikipedia to make almighty Dave look better. And worrying in a kind of Mr. Bean sense when they get it wrong. And worrying in a more serious sense that time was wasted on this exchange when the country is in dire financial straits. Not to mention issues of climate change that only grow more urgent.

That’s not to say I would never vote Conservative (although instinctively I’m probably closer to something like what Labour used to stand for.) I’d never vote for the current Labour government either. I’ve nothing against Mr. Brown. I think he’s serious and possibly even boring, but experienced and probably fairly good at politicky stuff. And I’m in no position to make any stronger judgements of his ability etc. What I object to is Jacqui Smith. (a) she spells her name in a really stupid way. (b) her voice grates on me whenever I hear her interviewed. She looks permanently put-upon and harrassed and sounds it too. (c) she seems to be forging ahead with all sorts of surveillance type policies despite widespread disapproval. This point is particularly galling for me because in theory I would be open to some sort of national ID card scheme if it could be made useful and worthwhile. The current (unpopular) scheme is hamstrung not only by the anti-ID card lobby, but also by worries over security of the data and the cost involved. And the fact that the card as it is wouldn’t be useful. My suggestion is make the card such that people will find it a convenience to have it, rather than force them to carry it. (d) she has twice ignored the advice of groups set up to explore the reclassification of drugs. Why this is annoying is because it speaks of a basic disregard for the scientific facts which, I feel, should be at the basis of policy decisions. The argument given for ignoring the advice is that it “would send the wrong message”. But if the decision was effectively made before the advice was given, why spend money on having these people produce the advice in the first place? Commissioning a report seems to carry with it an implicit duty to pay attention to the recommendations put forward and to make any decision at least partly on the basis of the report.

None of these are particularly well thought out arguments, nor are they based on any careful collating of all the relevant information. But that’s exactly the problem. I would in general be predisposed to go out and vote, but the impression I get of the current crop of candidates is fairly negative. Politicians should be trying to convince me they deserve to be in power. And I am going to be convinced only by a cogent set of principled policies. And I am not going to go out and read party political manifestoes. It is the duty of the politicians to get their message across to me. My impression is that manifesto promises are often reneged. And politics seems to be all about criticising the other guy. (That could be because in terms of ideology or policies, there is no real difference between the two main parties any more.)

This is all a bit of rant really. I’ve probably done nothing more than show how ill informed I am. Never mind, eh?

Prize for the best title for a conference ever: Tickle your catastrophe.

Enumerating structure on sets, again.

The number of partitions on an N-membered set is called its “Bell number“. The number of strict total orders, I’m not sure about that… There is some relevant information here.

Well it’s good to see those problems solved, and I’m glad they weren’t trivial… Here is a more general issue. How many relations are there on a set? How many transitive? And so on.

So, a relation is simply a set of ordered pairs. So there are 2^X possible relations if there are X ordered pairs. For a set with N elements, there are N^2 ordered pairs. Thus 2^(N^2) relations on N elements. To give you an idea of scale, there are 65,536 relations on the 4 element set. But there are only 15 partitions on the 4 element set (not 18 as I said before; I counted each 2+2 partition twice.) So the properties of being transitive, reflexive and symmetrical must seriously cut down the number of relations! The only thing I’ve concluded in answer to this question is that there is an upper bound on the number of possible total relations on N elements: 2^(N^2) – 2^([n^2]/2). This is because the property of being total means that the size of the set of ordered pairs has to be big enough to contain at least one of (a,b) or (b,a) for every a,b. So small sets of pairs are ruled out.

Ennumerating structures on sets

I have a question: for a set of N elements, how many different ways are there of partitioning it? In other words, how many distinct equivalence relations are there on a set? Similarly, how many distinct strict total orderings are there? How many weak partial orderings?

All of these things (equivalence relations, orderings…) are determined by sets of ordered pairs on the set. So is it easier to calculate numerical formulae for each type of structure on the set, or is it easier to demonstrate that, for example there are more weak partial orderings than there are partitions… It’s trivial, for example, to show that there are at least as many weak partial orderings as there are strict total ones, since each strict total order is equivalent to a weak partial order.

There could be quite a lot of partitions. A partition is a subset of the power set. Or an element of the power set of the power set. So the absolute upper bound on partitions for a 4 element set is 2^(2^4) = 65,536. Obviously this is a ridiculously high figure, but I can’t see an easier way to get closer to the actual number. I am no good at combinatorics. But for the 4 element set you could probably just list all the possible partitions. First list all the ways the sets could break down:

• 1,1,1,1
• 2,1,1
• 2,2
• 3,1
• 4

Then list the number of ways of splitting up the 4 elements, A,B,C,D into those sets.

• Only one {1,1,1,1} partition
• Six {2,1,1}
• Six {2,2}
• Four {3,1}
• One {4}

For a total of 18 partitions. So there are more partitions than there are subsets. But for the 2 element set, the power set has 4 elements, whereas there are only two possible partitions. My guess is for bigger sets there will be more partitions than there are subsets.

As for strict total orderings, I think there will be N! of them. Because there are N choices for the maximal element, N-1 choices for the next “biggest” and so on. So the 4 element set has 24 strict total orderings on it. N! > 2^N for N>3. As for weak partial ordering, there’s probably even more of them…

What other kinds of structure can one impose on a set? If one thinks of graph theory in terms of set theoretic structure, what then? In undirected graphs, the edges of a graph are pairs of elements of the set of vertices. I just did a little googling and learned that there is something called “graph ennumeration” which is where one studies the number of graphs with a certain number of vertices that have certain properties. Much like the classification of finite groups, I suppose. So for example there are no asymmetric graphs with less than 5 vertices. (Or possibly 6?)

Anyway, that’s just what I was thinking about this morning, when I should probably have been trying to understand the game theory lecture…

The end of time…

This New Scientist article made me wonder about time. The latest super-clocks are sensitive to differences in height (actually differences in gravitational field) small enough that their accuracy would be affected if they were placed on a slightly lower table, for example. This is madness. Maybe we’ve reached a point where time doesn’t really mean all that much any more. On those kind of scales, at that sort of precision, maybe time just stops being a useful concept. Like the length of a coastline stops being a useful concept when you measure it too accurately. (Because it’s a fractal). Or temperature; if you keep zooming in, you reach a point where the “temperature” of the volume becomes meaningless – if you zoom in enough that the volume contains maybe one atom or even no atoms, then surely temperature is unhelpful. Perhaps the same thing happens to time when we start trying to subdivide it into 10^-18ths of a second or whatever it is they are doing…

Here’s another article on time where some people seem to be reaching the same conclusions… I’m not sure these articles will be available – I’m on the campus network and they might have some sort of subscription to the magazine…

Titanoboa – Weighs a tonne and eats crocodiles

I just read about the Titanoboa. This is a snake that lived soon after the Dinosaurs died out. It was huge. I don’t think that quite does justice to it. It has been estimated that these things weighed upwards of 1,100kg and were 13 metres long. That’s a snake taller than your house weighing as much as your car! This thing was a metre wide! HUGE! The craziest part was that these things probably ate crododiles. Let me say that again; it ATE CROCODILES! You could not make up something like that up.