Moral rules and universal laws


The philosopher Saul Kripke introduced the following paradox. Imagine we a mathematical operation called ‘quadition’. This operation was exactly the same as addition, except after if the sum of the numbers is greater than 57, the result is always 125. If we have only ever added numbers smaller than 57, how do we know that we are following the rules of addition, rather than ‘quadition’?  And if we have, what if we redefine quadition to be arbitrarily high (let’s say a million trillion)? More generally, what does it mean to say that some finite set of data is consistent with a rule if it is also consistent with an infinite number of other rules? You could imagine an infinite variety of quus like operations with the same result as any finite set of summations. And if you vary ‘quadition’ further, for example by saying ‘2 quus 2 = 5 if and only if it is November 15th, 2019, at 15:35, otherwise 2 quus 2 = 4', all correct addition can be made consistent with some form of quadition.

Kripke was primarily posing this question as a way of investigating the idea of rule following in language and the nature of meaning. But I think it has a moral application too. One of the most important concepts in moral philosophy  is the idea of the Kantian categorical imperative: the idea that we must act in such a way that we could will our action to become a universal law. But what counts as a universal law? Any action can be consistent with a universal law if we allow for a purely formal notion of universality. “Tell the truth, unless you are person X, if so, do as you please” is a universal law in the sense that it can always be applied. OK, we might say, but moral universality cannot refer to a rule applying differently to different people. But what counts as different? It cannot mean, for example, that circumstances are never relevant. Otherwise, as the Woody Allen joke goes, we could never go to a restaurant, because if everybody went to the same restaurant at the same time and the same thing, there would be chaos. What about only accepting those rules which do not apply differently to different people regardless of circumstances? But we have problems here too. It would seem very strange to exclude all  things conceivable as personal differences. Otherwise moral rules like ‘give what you can’, ‘help others if you are able to’ would not be acceptable as moral rules. OK, so what about universal laws which other people would be willing to subscribe to? That seems more reasonable. But the question then becomes, what is actually added by describing our behaviours in terms of universal laws? Any behaviour is consistent with a universal law, and what really counts in deciding whether it is acceptable or not is something different.

Attitudes towards freedom of movement

A couple of days ago I spotted a rather surprising result on a YouGov poll: by an enormous margin, respondents said that they would like to maintain reciprocal rights of UK and EU citizens to live and work in each other's countries. Of those polled, 67% said such rights should be maintained, 15% said they shouldn't, and 19% said they did not know. In other words, what is generally believed to be the key motivation for Brexit, ending freedom of movement, is not wanted. These results got rather more attention on Twitter than I anticipated, and a number of people responded that they are merely a reflection of how the question is posed: if the question is framed in terms of 'control' of immigration, or simply in terms of support for "freedom of movement", the results would be quite different. I don't doubt that this is correct, but I wanted to briefly respond to this point, as I think it is an important one.

It is famously true that opinion polls can generate multiple answers to what is essentially the same question dependent on how it is framed. There are two possible responses to this. The first is simply to disregard the idea of popular opinion and see it purely as an ephemeral product of how questions are posed. Sometimes this may well be a reasonable assessment, and I think there is definitely such thing as a healthy skepticism around these issues.

But the second response is to say, OK, so what question is the most useful and informative? This of course depends a lot on what you are using the question for. If you are a political strategist, you might well find it very useful that questions which frame freedom of movement in terms of 'control' yield a negative reaction. But if you genuinely interested in trying to get an honest sense of public opinion as a means of guiding policy, the question has to be: "which way of describing freedom of movement is the most honest reflection of what the policy actually is?" No doubt this is itself contested, and there are occasions where the appropriate way of understanding a policy is genuinely a difficult question. But with freedom of movement, I don't see how framing the policy as a reciprocal right (the right for citizens to live and work in one another's countries) is not just the most simple, accurate, and thorough description. The alternatives either simply omit that reciprocity or present fictitious alternatives (one way rights for UK citizens, or systems of 'control' which imply much more effective and benevolent bureaucracies than actually ever exist).

For what it's worth, I don't think that public opinion on these questions should be the sole guide to policy. Stripping large numbers of people of acquired rights might well be wrong regardless of whether it is popular to do so. But it seems tragic for people to lose these rights on the basis of a popular support that does not even exist.

O Level and GCSE Maths Compared


When people of a certain age talk about GCSEs, the conversation may well quickly turn to the old ‘O Level’ qualification, and, more often than not, how much harder these were than anything 16 year olds are expected to do today. Part of the problem with these conversations is that few people are in a position to make a real comparison. Most people have only ever seen one type of exam, and if it is the O Levels, this was a long time ago. Perhaps for that reason, when I was a teenager (I was a weird teenager) the old exams had a certain mystique. As somebody who got a little too much of their sense of self through exam grades, I couldn’t help but wonder how I would have done at them.  While I’ll never know the answer to that question, thanks to the wonders of the internet, it’s certainly possible to compare the two exams, if you are geeky and obsessive enough to do so. What follows are the results of that endeavour with the old Maths O Level, and an attempt to answer that all important question: how much harder was it really?

The first thing to mention is that the O Levels (at least the exams from the ’50s and ‘60s) feel in many ways like very dated exams. They are densely printed, long, and in terms of structure feel a lot more like university exams. They have two sections, one of short questions and one of longer questions of which you choose a few (GCSE Maths papers don’t involve any choice).



Some of the material is very much from a time when people didn’t have calculators. There are lots of questions which are clearly testing your ability to use logarithmic and trigonometric tables and others which ask you to approximate π as 22/7 (at GCSE you’d either just put it in a calculator and round, or give your answer as a multiple of π, but it's still a nice approximation to use for rough calculations). There are then arithmetic problems which require some numerical manipulation to solve, often better thought of first as algebraic expressions to simplify, such as these ones here:


You can see why there would have been a greater focus on numerical skills before calculators, but I think it’s a bit of a shame this kind of numerical manipulation isn't taught as much anymore, as it can help children with algebra.  Part iii) of this question is also a reminder of another pesky feature of these exams: old, imperial units and pre-decimal currency (I couldn't answer part iii without looking up how many shillings there were in a pound!)  

That said, there is a lot about the exams that hasn’t changed at all. The algebra is pretty much the same, both in terms of content and level of difficulty (if we are comparing the O Level to the Higher Tier GCSE, after Gove’s reforms. The Foundation is dramatically less challenging, and prior to Gove’s reforms the GCSEs were considerably less difficult). This question, for example, taken from a 1962 O Level exam could have come straight out the GCSE were it not for the font: 
 



There are a few algebra questions in the O Levels which do seem a little more inventive in terms of what is asked. Question 4 (ii) below, for example, requires students not to solve for a single unknown but a quotient of two. That said, how difficult this kind of problem is very much depends on whether you have been taught to do it before or seen it for the first time, and I haven’t seen enough papers to know if this was a standard problem.


The section B algebra problems also seem to have required a little more in terms of initiative and personal input to solve. This one here, for example, I could only solve easily by introducing two unknowns of my own (I used n for the initial number of items sold and N for the total) which you could then get rid of by rearranging. 

The GCSE does require algebraic proof and introducing an unknown to solve numerical problems, but not introducing several unknowns to derive a purely algebraic expression which does not contain the variables you’ve introduced. An example of the type of problem tested at GCSE is here, which you solve by forming a quadratic (e.g by setting the number of green pens as x and blue as x+ 3). 



One way in which the O Level was definitely more challenging was the geometry. GCSE geometry questions tend to provide diagrams, the O Levels required you to figure out yourself what they would look like based only on worded descriptions. This question here, for example, could easily be a GCSE question, but there is no way they would ask you to do it without a diagram provided.


The proof questions were stylistically different, but don’t seem all that much more challenging. The main difference is volume: the O Level papers seem to have a lot of geometric proof involved, whereas on the GCSE there will only be a couple of questions. Below, for comparison, are two geometric proof questions, first a 1968 O Level and below that a 2017 GCSE question.





Content wise, the other main difference is that the O Level was in some ways a narrower exam. In the 1950’s and 60’s papers I didn’t find any statistics, data handling or probability questions, and very little on volumes, number properties, primes factor decomposition or surds. Algebra, geometry and doing long calculations with tables and slide rules seemed to be the greater focus. That said, the O Level did require knowing some basic calculus, and apart from the Edexcel IGCSE this isn’t taught until A Level (and even with the IGCSE there is no integration). And one exam I found from 1957 also had a rather charming third paper on the history of mathematics. I don’t know if this was a common part of the curriculum or not, but there is certainly something wonderfully quaint about it.



So, all in all, was the O Level Maths exam more difficult than the GCSE? Compared to the exams I sat, pre Michael Gove, I regret to say the answer is certainly yes. Compared to the exams students sit now, I’m less certain. There are certainly fewer gift questions (even the higher tier GCSE still has a some very easy questions at the start of the exam, the O Level seemed to have fewer, and they were less easy). On balance, I would still conclude that the O Levels were a little more challenging, if nothing else because they required more personal input and initiative to do start the questions (introducing your own variables, working out what a geometry problem looks like based on a description). But they aren’t a million miles away from one another.

Tactical Voting: Some Problems

There was a lot of anger yesterday about an online tactical voting tool launched by the pro-remain group Best for Britain. The tool appeared, judging by the results of the 2017 general election, to be suggesting Labour voters tactically vote Lib Dem in what were Labour/Conservative marginals. If the objective is to maximise seats held by parties in favour of a second referendum, this would obviously be an unfortunate recommendation.

The problem is, deciding how to vote tactically is a trickier business than it may seem at first sight. As Tony Yates shows here, tactical voting requires some sense of fixed, or only marginally changing background of other voters who are not voting tactically. The easiest way of attempting to do this is by looking at previous election general election results. But party allegiance at the moment is unusually fluid, and this is not the only data which could plausibly used to get a sense of how things look in particular constituencies. Best for Britain appeared to have based their calculations not just on the 2017 general election result, but more recent opinion polls and the European Election results.

How much weight to give these different types of data is a difficult question, but the answer may well not be zero. It is also not entirely clear who the tactical voters in these elections are. If, for the sake of illustration, it turned out that Lib Dem voters were much more willing to tactically vote Labour than vice versa, it is possible to imagine a situation in which the correct strategy for maximising remain seats were to recommend a Labour vote in a constituency, even if current predictions showed a greater Lib Dem likely vote share in that constituency than Labour without tactical voting. 

As Phil Syrpis points out, these problems may only really apply to a small number of marginal seats. But in the event of a close election, this could cause trouble. Moreover, disputes of this nature breed distrust between supporters of the various remain parties. This is a problem, as tactical voting is unlikely to work if there is too much hostility. People need to trust their second choice party at least somewhat, and are most likely to think of tactical voting as a useful strategy is they think of themselves as part of something bigger, where others might be returning the favour. 

Unfortunately, disputes over how to model the current state of the electorate are to be expected. This is partly, as may have happened with Best for Britain, due to differing opinions about methodology. But in other instances, these will also be mediated by partisan interests. Party campaigners are very likely to select prediction methodologies which show a more favourable picture for their party. This may be down to psychological biases, or just the role they fulfil as party campaigners. But either way, it is likely to cause problems down the line if and when candidates put out contradictory campaign material about the state of the race in their constituency. 

In a better world, perhaps the various remain parties might have circumvented these difficulties via an electoral pact. But not only there have always been near insurmountable political barriers to such an endeavour, that could also have caused a counter measure between the leave parties (though it may well still be the case that the latter happens). None of this is to suggest that voters should not attempt to vote tactically. We are where we are, and we have to make the best of this situation. But the difficulties may turn out greater than anticipated. 

Votes at 16? Some thoughts on a series of bad arguments.

There's a lot of discussion at the moment about whether the franchise should be extended to 16 year olds. I think that's a reasonable enough proposition. If the franchise is about giving fair representation to different interest groups capable of evaluating what those interests are, including 16 year olds doesn't seem like a bad idea (particularly if we are honest about the reality of how capable everyone else is of making sensible and reasonable decisions about their own interests). There's an element of value judgement here, for sure. But before we get into that, it's worth noting which arguments we should ignore, because this debate certainly seems to throw up a whole series of them.

The first kind is the 'slippery slope' argument: the argument that a policy in accordance with a particular principle can lead down the 'slippery slope' towards the most extreme interpretation of that principle. Think of people arguing that votes at 16 might lead to votes at 10. When I did debating at school, we were more or less taught never to use this argument. I think this is perhaps overdoing it a bit, but it does have a problem. The argument relies almost entirely on a metaphor to be convincing. Slope imagery aside, why should votes at 16 lead to votes at 10? There isn't any obvious reason why this would happen. Given that the rhetorical effect of the argument relies on the conclusion's obvious absurdity, it would seem reasonable to assume that one would in fact not lead to another. For slippery slope arguments to work there has to be some kind of actual mechanism for A leading to B, which unless you are talking about whether to go for a winter's hillside walk involves more than invoking a metaphor.

A variation of that argument is a kind of false reductio ad absurdum. "If you gave 16 year olds the vote, you might as well give it to 10 year olds!" Or, on the other side, "If voting is about political knowledge, which 16 year olds supposedly don't have, why not restrict it to those who can pass a politics quiz/in IQ test?" The answer to those both is twofold. Firstly, there might be other, completely different reasons not to (restricting the franchise based on tests would almost certainly harm the worst off, 10 year olds might be a whole lot less responsible than 16 year olds etc). The second problem is this argument overlooks that we may be talking about a continuous scale of trade offs, where a somewhat arbitrary decision has to be made. Having an opinion about where a line is drawn that is arbitrary at the margin (why, other than convenience, not one day after your 18th birthday?) does not imply that you can't usefully discuss substantial changes, or that doing so implies that you don't think there needs to be any lines in the sand at all. Think about setting the speed limit on a motorway. There are a series of trade offs to be made about fuel efficiency, traffic, safety, people getting to where they want on time etc. You have to set a limit somewhere. In the the UK this means 70 mph. The fact that it is precisely 70 mph is kind of arbitrary- it's a round number. But neither the arbitrariness of the precise number, nor the the fact that the question may involve value judgements, mean that you cannot construct meaningful arguments about a substantial change. There might be legitimate reasons to disagree about a 60 mph speed limit vs an 80 mph limit, with neither position implying an even lower or even higher one, and nobody would argue that either of these positions imply support for no speed limit, or no driving whatsoever.

The third kind of bad argument is the argument by analogy. "If people are responsible enough at 16 for that to be the age of consent, surely they are old enough to vote!” Or on the other side: "If people cannot marry without parental permission at 16, surely they should not be able to vote!” What's wrong with this kind of argument is that its persuasive force relies on the assumption that things are the way they are because there is some kind of decent rationale to them. That, as Hegel said, the actual is the rational. In reality these facts are historical baggage often as questionable as anything else. There is little reason to think they reveal some kind of profound truth as to how things should be, or, in the case that they are justified, that those justifications equally apply, or even that there is any great value in trying to force our attitudes to have some kind of overall coherence.

The final bad argument is that those advocating a particular position on votes at 16 have bad motives. "Labour only want 16 year olds to have the vote because they think that they will vote for them", or "Conservatives only want to deprive 16 year olds of the vote because they won't." Both are probably true, but equally irrelevant. There is no reason why a good policy can't come from bad motives. Indeed, historically the extension of the franchise is a great example of this: much of the parliamentary wrangling over women getting the vote pre-1914 were precisely about whether women would be more likely to vote Liberal or Conservative. The fact some Liberals were well disposed towards extending the franchise because they believed it would help them electorally did not make doing so a bad policy.

Once the bad arguments are left to one side, what we are left with is a more prosaic judgement. Are people old enough to make political judgements at 16? I think yes, based on admittedly limited experience (and frankly thought). Less so than people aged 30? Perhaps, but not so sufficiently so to disqualify them from the vote. Do they have legitimate interests, as stakeholders? Sure. In some ways less than people over the age of 18, but in others more. They certainly have unique interests that deserve representation (think education policy), and a direct, if not greater stake, in large national decisions like Brexit. Could somebody reasonably come to different conclusion? Sure. Contrary to popular debate, this isn't a grand question of principle. It's a value judgement about whether to shift a somewhat arbitrary line in the sand.

The Brexiteers Close Ranks

Something very strange has happened over the last 24 hours. The Brexiteers, both in Parliament and in the press, seem to be closing ranks behind Johnson's deal. Or at least they seem to be doing so within the Conservative Party. This is perplexing, as it seems to involve everything they said they hated about the Withdrawal Act, but now with a separate customs arrangement between the Northern Ireland and the rest of the UK (the so called backstop) becoming the policy of first choice.

How could this happen? There are a number of possible explanations. One is fatigue. The issue has dragged on, and, so the (erroneous) thought goes, this will put an end to it. The second explanation is that the ERG are cutting their losses. Perhaps they realise that no deal isn't a viable policy after all, or isn't one that can be achieved within this Parliament, and this is the best (or to any sane observer, worst) they are going to get.

But perhaps something else is going on. As Chaminda Jayanetti wrote here, 'elite cues' have played an important role in determining which positions are seen as acceptable and which are not by leave and remain supporters. Bizarre obsessions with the WTO, outrage at regulatory alignment etc are conjured into reality by key figures in the debate who have mass public following. Some of these may represent genuine niche obsessions, others may well have been political posturing by the ERG to oust an outsider Prime Minister (May) in favour of one of their own (Johnson), perhaps believing that this in and of itself would lead to a better outcome.

There is a strong tribe mentality to all of this. It is as if the most important question is the one once posed by Thatcher: "is he one of us?". In the case of Johnson, leavers can certainly answer that question affirmatively. It is not just that he led Vote Leave, but that since taking office, he has done everything he can to signify his scorn for all the old enemies: remainers, the judiciary and parliamentary processes. At this point all the cues are there. The most emblematic figure of the leave campaign has declared a new triumph, as has Rees Mogg, the most emblematic of the ERG. There is sufficient ambiguity over the terms of the future relationship that people can still imagine themselves painting the canvass much as they'd like, so long as they don't think too hard about the implications of customs border in the Irish Sea. Perhaps, as some in the ERG seem to be thinking, this could still be a path to a no deal outcome if no agreement is ratified within the transition period. In any case,  they can rest assured that Johnson, someone who is most assuredly 'one of them', has their back. Scrutiny is, after all, not necessary between friends.

A Shared Exceptionalism

Two facts stand out as about contemporary electoral politics. The first is that almost nobody thinks they can be tricked or manipulated. The second is that almost everyone thinks lying politicians are manipulating the other side. These two facts ought to cause some pause for thought and introspection.

This is no more evident in British politics than in discussions of Brexit. It's a common (and I think correct) claim among remain supporters that the Leave campaign used dishonest, psychologically manipulative strategies to a kind of frenzied support for leave. Arguments of this sort are generally met with the response that this claim is patronising and insulting: it assumes people are gullible enough to be manipulated in this way. But at the same time, leave supporters are equally likely to claim that remain support is a driven by the after effects of 'project fear', the supposed attempt by the Remain campaign to scare the electorate into voting to remain in the EU. Let's forget for a moment what we think about the truth of these two claims. The point is that they both presuppose that people are capable of being psychologically manipulated and being drawn to bad arguments. Yet it is quite uncommon to think that this capability extends to our own side, let alone ourselves. This is our shared exceptionalism: the commonly held belief that everyone can be manipulated but ourselves.

One further absurdity of this is how poorly it sits with the rhetoric of democracy. This is as the belief that others are being manipulated by sinister forces breeds suspicion of democratic processes, at least when they give the other side a chance of winning. One conclusion that can be drawn from this is to stress the importance of trying to understand the reasoning and stated motives of those who differ politically on their own terms. There is no doubt some value to that conclusion. But I think another, more honest, and perhaps complimentary one would be to admit that we are all capable of holding false beliefs, giving credence to bad arguments and having motivations other we are aware of and tell ourselves we have.