This post is a continuation of the examination of the EU Referendum that I began in “Logic, Language and Leaving”. What follows is not a political or economic argument, but a look at the vote to leave the European Union from the point of view of logic. Even many philosophers find the study of logic somewhat dry, but in a world where a different point of view can so readily be dismissed as ‘fascism’, being as clear as we can about what propositions do and do not mean is a necessary enterprise.

I do not engage in Graylingesque casuistry about the legitimacy (or otherwise) of the referendum here. If you found this article via my Twitter feed, then you will already know that I am in favour of Britain leaving the EU, and have been since the signing of the Maastricht Treaty (1991). This is not an attempt to convince anyone of the rightness of the UK’s decision to leave the EU, but an attempt to examine the logical status of the referendum result.

What did we vote for on 23 June 2016?

Competing and overlapping intentions

The first thing to note is that none of us tends to do something for only one reason. Thus, a particular action might be describable in many different ways, according to the intention(s) of the actor. For example, I may walk to the supermarket because I need to buy some milk, but I may also wish to take some exercise or enjoy the weather (or all three). Indeed, my need to buy a pint of milk may be secondary to my need to stretch my legs.

In a now famous passage from After Virtue, Alasdair MacIntyre uses the example of a man at work in his garden. There is no easy answer, MacIntyre suggests, to the question ‘What is he doing?’

The answers may with equal truth and appropriateness be ‘Digging’, ‘Gardening’ , ‘Taking exercise’, ‘Preparing for winter’ or ‘Pleasing his wife’. Some of these answers will characterise the agent’s intentions, other unintended consequences of his actions, and of these unintended consequences some may be such that the agent is aware of them and others not.

After Virtue, p. 206

Thoroughgoing Aristotelean that he is, MacIntyre concludes:

We cannot, that is to say, characterise behaviour independently of intentions, and we cannot characterise intentions independently of the settings which make those intentions intelligible both to agents themselves and to others.

After Virtue, p. 206

This is true of the behaviour that we call ‘voting’. Many supporters of the Labour Party are convinced that Conservative voters are motivated by a hatred of the poor and a delight in state-sponsored cruelty. But in reality, the precise reasons for voting Conservative must be almost as numerous as voters. Some people, presumably, voted for the local Conservative candidate because they happened enthusiastically to agree with every sentence of the Tory election manifesto. Rather more, I suggest, voted Tory because they agreed with more of that Party’s manifesto than with other parties’ manifestos. Yet more no doubt voted Tory to ‘keep hold of nurse’. I voted Conservative in June 2017 and yet disagreed with almost everything in that dismal document. However, I felt that the alternative — allowing Jeremy Corbyn anywhere near power — would be even more catastrophic than allowing Theresa May and her idiotic ministers to continue in power.

General  elections are always going to involve compromise when it comes to personal principles. A person might be Tory on income tax, Labour on rail nationalisation, Green on fracking and Liberal Democrat on constitutional reform, but in the UK we only have one cross to place (unless you are a student!).

But referendums are different. The Swiss have pretty regular referendums, but they are used to them, and so perhaps the Condorcet Paradox does not apply in the Helvetic Confederation. The 2016 referendum on EU membership in the UK presented, however, a binary choice:


Although the referendum posed a polar question, there might have been many reasons for voting ‘remain’ or ‘leave’. As I am a ‘leaver’, I shall suggest some reasons (sic) that may have motivated other ‘leave’ voters (rightly, or in many cases, wrongly):

  1. Wishing to strengthen national sovereignty
  2. The desire to protect British industry and jobs against foreign competition
  3. To ensure that the UK parliament determines immigration policy
  4. To prevent the UK being drawn into an increasingly federalising EU
  5. To reduce the number of immigrants from EU countries
  6. To end Freedom of Movement between the UK and other European countries
  7. To reduce the burden on the NHS, schools and housing
  8. To bring about a reorientation of UK trade policy towards the Commonwealth
  9. To allow the UK to make her own trade deals
  10. To end the authority of the ECJ
  11. To end the authority of the European Court of Human Rights
  12. To prevent the UK being ‘taken over’ by Germany
  13. Protecting the UK from Islamist soi-disant migrants who cross other borders
  14. Hastening the restoration of pounds and ounces, feet and inches, gallons and pints.
  15. To leave the Single Market
  16. To isolate ourselves as far as possible from those ‘on the continent’
  17. To allow the UK to remove VAT from tampons and other sanitary products
  18. To allow the UK government to embark on a programme of nationalisation
  19. To disentangle the UK from EU sponsored free market capitalism
  20. To get rid of German owned supermarkets (like Aldi) to protect British corner shops

I need hardly say that some of the above are silly and illogical, but every single reason is one that I have seen expressed without irony on Twitter, or one that I actually heard on the doorstep during the EU Referendum campaign. A reason for voting a certain way might be illogical, but it is still a reason (just a bad one). Green Party candidates have repeatedly stood for election on the logically inconsistent policies of increased immigration, more generous state benefits and permanent economic recession, and yet over a million people voted for them in 2015.

There has been some polling regarding people’s reasons for voting to leave the EU, which showed fairly consistent responses across the political spectrum:

Screen Shot 2017-08-30 at 20.23.25

Ultimately, though, the reasons why people voted to leave the EU are at best secondary and at worst irrelevant. Just as the only thing we can be sure that MacIntyre’s gardener is doing is gardening, so the only thing we know for certain is that 17,410,742 people voted, on 23 June 2016, for the proposition that the UK should leave the EU. That is to say, what was written on the ballot paper. Similarly, whilst it might be true to say of MacIntyre’s gardener, ‘He is pleasing his wife’ or ‘He is preparing for winter’, it is certainly true to say, ‘He is gardening’. Thus, though it might be true to say of a particular Leave voter, ‘She wanted to reduce immigration’ or ‘He wanted to leave the single market’, the only thing we can say with certainty is, ‘S/he voted to leave the EU’.

Let us imagine for a moment, that we are looking down a street in every garden of which there is a man or woman engaged in some kind of horticultural activity. Each will have different reasons for doing what he or she is doing, and some will have several reasons. There is only one thing we can say of the street’s residents as a whole — that they are all engaged in some form of gardening.

While it may be tempting, for politically expedient reasons, to suggest that (e.g.) ‘voters told us to reduce immigration’ by voting to leave the EU; this is to draw an invalid conclusion from a fallacious form of reasoning.

It is worth noting in passing that pro-Remain politicians have (for the most part) been more careful in the claims they make for the imagined motives of remainers. The view seems to be that people voted to remain in the EU because it seemed to be the least risky option. No one, apart from a handful of Ultras, has interpreted the remain vote as a call for faster and deeper integration with the EU, or for the UK parliament to have even less sovereignty than it currently enjoys. Wanting the same amount of ‘Europe’ is not the same thing as wanting ‘more Europe’.

Aristotle’s Binary Logical World

In Book IV of the Metaphysics, Aristotle gave a clear, though not original, account of three ‘Laws of Thought’ which he regarded as the fundamental building blocks of logical reasoning.


∀A (A=A)

A thing is what it is. (Theresa May’s insistence that ‘Brexit means Brexit’ is thus logically unproblematic, even if politically unenlightening.) This may sound uncontroversial, or even condescendingly obvious, and when it comes to asserting than an apple is an apple and not an orange, it is. However, a brief consideration of its function with respect to moral judgments shows how important this principle is. Suppose we agree that ‘killing’ is ‘the deliberate ending of (human) life’. There should be little argument over whether execution or murder constituted ‘killing’, but what of ‘assisted dying’? We either have to agree that it, too, is ‘killing’ or find another definition of ‘killing’.

Likewise, the debate about ‘equal marriage’ largely focused on competing definitions of ‘marriage’. If ‘marriage’ is ‘a freely contracted partnership between two human beings over the age of 16 which is recognised by a legally constituted state’ then the Law of Identity would pose no problems for us. But if the term ‘marriage’ and the description ‘lifelong partnership between one male and one female human beings’ are held to be logically equivalent/identical, then same-sex marriage would be problematic (viz. illogical). This is not to say, of course, that someone clinging to such a definition is therefore opposed to same sex partnerships being legally recognised and safeguarded under the law. The debate over marriage is not between dangerous gay libertines who want to undermine marriage on the one hand and reactionary homophobes on the other, but between non-identical definitions of ‘marriage’.

A more contemporary example concerns gender identity. If the definition of ‘male’ entails having an XY sex chromosome (and possibly a penis and the ability to produce sperm), then what are we to say of someone who possesses this (or these) quality (or qualities) and yet identifies as a ‘female’ (which is to say, not a male)? If we change our working definition of ‘male’ to ‘a human being who self-identifies as male’ then no problem, but if we are to move beyond obvious tautologies (apples are apples, marriage is marriage, males are males, Brexit is Brexit) and on to logically useful relations of identity (apples are pomaceous fruit of the genus Malus) then we need to take care to find accurate and meaningful definitions.



Nothing can be and not be, or better: nothing can be both true and not true in the same sense at the same time. Thus ‘Brexit’ is either ‘Brexit’ or it is not — a particular course of action cannot be both leaving the EU and not leaving the EU. This ‘law’ is more clearly understood alongside:

Excluded Middle


Everything must either be or not be, or: any proposition is either true or false. Either something is Brexit or not-Brexit.

Again, these principles sound obvious, but as anyone who has taught philosophy to undergraduates knows, people are very keen to argue the toss, usually because they misunderstand what is being asserted. The first mistake is to confuse the negation of a proposition with its opposite. While negation (¬A) is a logically clear notion, ‘opposites’ are not, and are often culturally determined. We can divide the world into ‘apples’ and ‘not apples’, but not into apples and oranges. Oranges, yachts, gorillas and the concept of justice are not apples. So what is the ‘opposite’ of ‘apple’? An orange? Why not a pomegranate or a runner-bean? In the realm of logic, things are never ‘black and white’, but ‘black or not-black’.

The loganberry, for example, is a hybrid of a blackberry and a raspberry. Is it, then, both a blackberry and not a blackberry? As before, it partly depends on how you define ‘blackberry’. You can come down on one side or another, but not both. When is a door not a door? When it’s ajar (a jar… boom boom). But in the real world of cold, hard logic, it’s either open or it’s not. What is at issue is your definition of closed/not-closed or open/not-open. You might need to express open/closed-ness in terms of degrees!

Is a Labradoodle a Labrador? Or a Poodle? It could be either, or both:

‘A Labradoodle is a Labrador’


‘A Labradoodle is a Poodle’

can both be true, as can:

‘A Labradoodle is neither a Labrador nor a Poodle’

But it cannot be true to assert:

‘A Labradoodle is a Labrador and not a Labrador’

I realise that Aristotle’s categories are all rather binary, so in order to clarify I return to the question of gender identity. There are plenty of people who say that their gender identity is ‘non-binary’. By this, I take them to mean that they regard themselves as neither immutably male nor immutably female. Whatever one thinks of the meaningfulness of this point of view, Aristotle’s logical laws still work (and if they didn’t, we would be unable to have meaningful discussions about anything). Let’s suppose that our gender-queer, non-binary person asserts ‘I am neither male nor female, but both’ or ‘I am male and female’.

The basic laws of logic apply because they are concerned not with the opposite of ‘male’ (?female), but its negation. Therefore:

‘Everyone is either male or not male’


‘Nothing is both male and not male’

Even if xe asserts ‘I am male and not male’ I would wager that xe is not employing the concept of ‘male’ in exactly the same sense in both iterations. For example, I am a ‘liberal’ when it comes to free market economics, but not a ‘liberal’ when it comes to guns or Class A drugs (that is to say, I’m not a ‘proper’ Libertarian). I am clearly using the term ‘liberal’ in equivocal, not univocal, senses (economic and social). If some good is to come of the somewhat febrile debate around gender identity, it may well be the rediscovery of metaphysics. Let us hope so.

EU Membership according to Aristotle

Given ∀A (A=A) [identity], we can of course say that ‘membership of the EU’ is identical to (≡) ‘membership of the EU’, but that does not get us very far, being a word-for-word tautology. What we need to do is find some (other) definition of ‘membership of the EU’ which can go after the triple bar (the logic symbol for ‘identical to’) to give us A≡B.

The other laws of thought (Non-Contradiction and Excluded Middle) require us also to assert that:

There exists no state which is both a member of the EU and not a member of the EU.


Every state is either a member of the EU or, not a member of the EU.

But Aristotle’s Laws of Thought are simply the foundations of logical argument, so let’s consider some problems, or fallacies, in the arguments we might use.

The Fallacy of Affirming the Consequent (FAC)

A logically valid (and sound) argument might take this form:

  1. The UK Prime Minister lives at No. 10 Downing Street.
  2. Theresa May is a UK Prime Minster.
  3. Therefore, Theresa May lives at No. 10 Downing Street.

Symbolically, this argument has the following form:

  1. A→B       (If A, then B)
  2. A             (A is the case)
  3. ∴ B         (therefore B must be the case)

This is an example of ‘affirming the antecedent’. An example of the fallacy of affirming the consequent (FAC) might therefore be:

  1. The UK Prime Minister lives at 10 Downing Street.
  2. Theresa May lives at 10 Downing Street.
  3. Therefore, Theresa May is UK Prime Minister.

This (invalid) argument can often slip through, unnoticed, because (as in this case) the conclusion happens to be true. Theresa May is indeed the UK Prime Minister. However, consider the following (logically identical) argument, and the fallacy becomes clear:

  1. The UK Prime Minister lives at 10 Downing Street.
  2. Philip May lives at 10 Downing Street.
  3. Therefore, Philip May is UK Prime Minister.

For the sake of clarity, here is another example of FAC:

  1. If I have run a marathon, then I will feel tired.
  2. I feel tired.
  3. Therefore I have run a marathon.

Symbolically, the form of this (invalid) argument is:

  1. A→B       (If A, then B)
  2. B             (B is the case)
  3. ∴ A        (therefore A must be the case)

The Fallacy of Denying the Antecedent (FDA)

This is closely related to FAC. For example:

  1. If I have run a marathon, then I will feel tired.
  2. I have not run a marathon.
  3. Therefore I do not feel tired.


  1. A→B      (If A, then B)
  2. ¬A          (A is not the case)
  3. ∴¬B       (therefore B is not the case)

Clearly, there are many reasons why I might feel tired aside from having run a marathon (like having gone clubbing till 4am or had a bracing walk). But, note that the following is valid:

  1. If I am 40 years old, I will be 41 next birthday.
  2. I am not 40 years old.
  3. Therefore I will not be 41 next birthday.

This is valid because the form of the argument is (and this is why symbols often matter):

  1. AB         (If, and only if, A, then B)
  2. ¬A             (A is not the case)
  3. ∴¬B          (therefore B is not the case)

In other words, the form of the first premise is not ‘If A then B’ but: ‘If, and only if, A, then B’. A implies B, but B implies A.

Fallacies in Reasoning over BREXIT

One could write a text book of logical fallacies based entirely on the examples that we heard during the EU Referendum and during the Brexit process since. There have been dozens of them, more with each passing day. However in this post I shall limit myself to examining some formal and propositional fallacies.

We see the fallacy of FAC at work in an argument such as:

  1. All EU member states participate in the Single Market
  2. The UK participates in the Single Market
  3. ∴ The UK is a member state of the EU

And, mirroring this, examples of the FDA:

  1. If a state is member of the EU, then it will participate the Single Market.
  2. Switzerland is not a member of the EU.
  3. ∴ Switzerland does not participate in the Single Market.

The problem with this ‘arguments’, and the reason why logical fallacies can be so hard to spot, is that the conclusions of both happen to be (accidentally) true. Why these arguments are fallacious becomes clear when we use other variables to yield false conclusions:

  1. All EU member states participate in the Single Market
  2. Norway participates in the Single Market
  3. ∴ Norway is a member state of the EU


  1. If a state is member of the EU, then it will be a member of the Single Market.
  2. Iceland is not a member of the EU.
  3. ∴ Iceland does not participate in the Single Market.

Here, the conclusions are obviously false because both Norway and Iceland participate in the Single Market, yet are not members of the EU. It will not do to try to argue that Norway, Iceland and Liechtenstein are ‘effectively’ EU members or somehow both ‘in and out’, because this runs foul of the Laws of Thought, namely non-contradiction and excluded middle:

There exists no state which is both a member of the EU and not a member of the EU.

Every state (in the world) is either a member of the EU or not a member of the EU.

It is plain, then, that where denotes EU Member State and G denotes Single Market Member, the following cannot be sufficient:


or even:


It is not a relationship of identity. Whilst it is the case that for any given F, G is implied, it does not work in the opposite direction. Some instances of (e.g. Norway) are not F (EU members). Making it a bit clearer, this is untrue (where x stands for ‘state’):


(For any state, if that state is a Single Market Member then it is an EU Member.) It is false, because this is true:


(There exists a state [x] such that this state is a Single Market Member [G] and not an EU Member [F]). There exists Norway, QED.

The Red Lines of the Red Queen and Humpty Dumpty

Theresa May began well (or at least logically) when she asserted that ‘Brexit means Brexit’ (B≡B). Not a very enlightening phrase, but an indubitable a priori proposition. Since that first day on the steps of Downing Street, however, we have heard from the Prime Minister, her ministers, and MPs, statements that would not disgrace the illogical worlds of Lewis Carroll’s Red Queen or Humpty Dumpty.

It was the Red Queen, you recall, who claimed that with practice one might ‘believe six impossible things before breakfast’. Humpty Dumpty, of course, claimed that ‘[a word] means just what I choose it to mean —neither more nor less.’ For her part, Alice questioned whether he could make words ‘mean so many different things.’

Government (and opposition) statements about Brexit (and therefore the nature and definition of EU membership) are a mixture of the Red Queen and Humpty Dumpty. The Red Queen-like ‘impossible things’ have tended to be political and economic fantasies of about having the ‘same relationship’ with the EU or ‘full Single Market access’ — and generally they boil down to some variant of ‘having one’s cake and eating it’. This phrase is one of those of which the precise meaning is not clear from the literal sense, because the literal sense is irrational (‘Cheap at half the price’ is another example of this — most things are cheap at half the price). You cannot eat a cake and at the same time retain (have) it (so you might eat it again). If it means anything at all in the Brexit sense, then one assumes it means ‘leaving the EU [having cake] while retaining all the benefits of EU membership [eating cake].’

What interest me more are the Humpty Dumpty sort of statements, whereby a word (in this case ‘Brexit’) means ‘just what I choose it to mean — neither more nor less.’

Though they change from day to day, from statement to statement, some of the common ‘red lines’ that have been laid down include:

  1. The jurisdiction of the ECJ (ending it with respect to the UK)
  2. The Single Market (leaving it)
  3. Freedom of Movement (ending it)
  4. The Customs Union (withdrawing from it)
  5. EU Agencies such as Europol and Erasmus+ (withdrawing from)
  6. Non-EU European Programmes such as Eurocontrol (withdrawing from)

Only the ‘Brextremists’ have seriously suggested withdrawing from the ambit of the EU to such an extent that we would no longer seek to participate in EU agencies and programmes, many of which predated not only Maastricht, but even the Common Market and EEC. But the first four — ECJ, SM, CU and FoM — are bandied about endlessly.

I hope that I have already shown that (2) is not identical with EU membership. It is not the case that (EU Membership) and G (Single Market Membership) have a relationship of identity, for there are states which participate in the Single Market and are not members of the EU (Iceland, for example). One might want to recognise, however, that EU member states participate in the Single Market in one way, and EFTA states participate in another (remember that the Law of Non Contradiction says that nothing can be, and not be, in the same sense at the same time.) By way of analogy, all 802 sitting peers in the UK parliament are members of the House of Lords, yet some sit by virtue of being Anglican clerics, some by virtue of being elected (former hereditaries) and some by virtue of being appointed.

So at one level, it is true that leaving the EU will mean that the UK can no longer be a member of the Single Market in the same sense and will therefore have to ‘leave’ the SM. (To continue with our peerage analogy, Rowan Williams sat as a Lord Spiritual as a Bishop and then Archbishop until 31 December 2012. But on 26 December 2012 he was made a Life Peer and so continued to sit in the Lords, but as a Lord Temporal.) It is logically the case that a state which is not a member of the EU cannot participate in the SM qua EU member. But the examples of Iceland et al show that it is possible to participate in the Single Market qua EFTA/EEA member. Whether that is economically or politically desirable is a question for politicians (and those who elect them) and economists, not philosophers. It might also be possible to participate in the Single Market qua some-other-arrangement, such as a UK-EU bilateral having-and-eating-cake agreement, but this seems unlikely in practice, certainly at the time of writing.

Yet when it comes to ECJ jurisdiction, there are no states outside the EU which are wholly subject to the judgments of the ECJ. What about EFTA/EEA countries, then? Do they not transpose ECJ judgments into their own EFTA-wide and domestic codes? Yes they do (though not automatically), but what is crucial is the word wholly and the proviso of ‘in the same sense and at the same time’ in the case of the Law of Non-Contradiction. Iceland herself, unlike British remainers (who are so keen upon retaining Single Market membership that they trash the example of those states who have managed to secure it from outside the EU), is clear that as an EFTA state, she is subject to the jurisdiction of the EFTA Court, not the ECJ.

Let us suppose for a moment that in order to secure even some temporary transitional relationship with the EU it were necessary to agree that some disputes might have to be arbitrated by the ECJ. That would indeed count as ‘ECJ jurisdiction’ of some form, yet it would clearly not be ECJ jurisdiction in the form that we currently endure it as an EU member state. Again, not in the same sense at the same time.

The question of Freedom of Movement is similar to the issue of the Single Market. There have been attempts to argue the following:

(EU Membership)≡(Single Market Membership)≡(Freedom of Movement)

We already know this fails from the point of view of logic, because the first two terms are not identical. But to humour this line of reasoning for a moment, we can point out exceptions in the third term that weaken it further, namely that Switzerland has (for the moment) FoM without either EU or SM membership, and even if we take only the second two thirds of the proposition:

(Single Market Membership)≡(Freedom of Movement)

then Liechtenstein has had SM membership without complete FoM (via §112 of the EEA Agreement). EU membership entails agreeing to FoM, but agreeing to FoM does not entail EU membership. Moreover, under the EFTA pillar of the SM, the legal basis of FoM is EEA labour mobility, whereas under the ECJ pillar there is a much stronger foundation; namely ‘EU Citizenship’.

The question of the Customs Union is not quite so straightforward. A number of prominent Remain supporters (e.g. Chuka Umunna MP) have tried to suggest that EU non-members (e.g. Turkey) are members of the EU Customs Union. This is not the case. Turkey is in a (bilateral) Customs Union with the EU, but not in the Customs Union. The same is true of non-EU Andorra and San Marino. The only EU non-members which are integral parts of the EU Customs territory are Monaco (through a treaty with EU member France dating back to 1963) and other separate, but not independent, territories of a member state (in this case the Isle of Man, The Channel Islands and the Cyprus military bases of Akrotiri and Dhekelia). Some territories of EU member states do not participate in the Customs Union, for example Gibraltar (UK) and Ceuta and Melilla (Spain).

Whilst I find it hard to see any political or economic advantage in the UK’s seeking to participate in the EU Customs Union after Brexit, that is not to say that it is logically impossible. Mind you, it is logically possible for me to become a serial killer, win the Nobel Prize for Physics, or pole-vault over the Houses of Parliament. None is likely and one is far from desirable, to say the least.

A Plea for Sound Logic

I have only scratched the surface of the logical inconsistencies of much Brexit talk without even touching upon fallacies of alternative disjuncts, illicit minors or majors, undistributed middles or modal fallacies. The villain of the piece is to be found in the very first of the Laws of Thought — Identity:

(EU Membership) ≡ X

and therefore, mutatis mutandis:

Brexit ≡ (¬X)

The challenge, if we are to be as clear as we can be about Brexit and our future relationship with the EU, is to be clear about what X stands for. In reality, X is likely to be more like X, Y and Z, for there are few individual aspects of EU membership that we can say have an exhaustive relationship of identity with membership as a whole. Aside from some very obvious facts of membership — like sending MEPs to Brussels, sending ministers to the Council of the European Union, nominating Commissioners and, thus, being considered by the EU as a member — it is very difficult to give a clear definition of what EU membership entails (and cannot entail).

Drawing ‘red lines’ underneath things like the Single Market is illogical and unhelpful. There is nothing illogical in wishing to withdraw from EU membership, but unless politicians stop employing lazy and nonsensical arguments, then the eventual manner of our leaving risks looking ever more irrational.