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Where is the Logic?

By Harry R. Lloyd

The Art of Logic

Eugenia Cheng, Profile, 2019

Think Again: How to Reason and Argue

Walter Sinnott-Armstrong, Pelican, 2019

On 21 January this year, President Trump backed a bill released in the US Senate asking his government to allocate $5.7 billion towards a US-Mexico border wall. Twenty days earlier, the President had tweeted that ‘MEXICO IS PAYING FOR THE WALL’. For those living through the Trump era, it seems that the political world is mired in contradictions.

Are we living through an era of unparalleled disrespect for logic? Perhaps not. From right-wing fascism to Communist hysteria, the twentieth century has also weathered its fair share of populist political upheavals. Many of these periods were accompanied by a similarly cavalier attitude to logical thinking. For example, Lin Yutang’s popphilosophy volume The Importance of Living – the bestselling nonfiction book in America in 1938 – expresses, in several places, a ‘profound distrust in logic’. This distrust clearly grew out of Lin’s despondency at the failure of appeals to ‘reasonableness’ in galvanising the German population to resist the Nazis. Lin was also influenced by domestic political events. In 1937, President Roosevelt drew attention away from the cynicism of his plan to pack the Supreme Court with New Deal supporters by involving his opponents in logical-legal arguments about the constitutionality of the policy.

Distaste for logic also emerged during India’s independence struggle. Rabindranath Tagore, the great ‘national poet of Bengal’, believed that his compatriots should express their nationalism in strictly emotional terms, eschewing logical and factual arguments. ‘A mind all logic is like a knife all blade; it makes the hand bleed that uses it,’ he once warned. True patriotism, Tagore believed, could not be achieved by way of logic. That the overuse of logic can be self-undermining has some modern parallels. Many still hold onto the myth, for example, that the Remainers lost the 2016 EU referendum because they relied too much on logic and too little on appeals to emotion. In comparison with past incidents, however, the Brexit crisis is, if anything, notable for the fact that a certain residual affection for logic has survived. This is evinced by the deployment of pseudo-logical argument forms to justify the pet projects of the populists. For example: ‘We must respect democracy, therefore we must leave the EU soon.’

Most of us agree that democracy should be respected. But does it follow from this that we must leave the EU soon, regardless of the terms of the divorce? Only if Brexit regardless of the cost is exactly what 52% of the population voted for. Brexiteers who refuse to debate the details of their arguments have a lot in common with the ancient Sophists, who were adept at dressing up their rhetoric to look like incontrovertible logic.

Even if the mystique of logicism survives, the real science of logic doesn’t get much respect right now. Perhaps that’s part of the reason for the present crises on both sides of the Atlantic. To jump straight to the conclusion that lack of respect for logic is responsible for the current mess would itself be a case of illogical thinking – a fallacy of post hoc ergo propter hoc (‘the two things happened together, therefore the one caused the other’). Still, the correlation between declining respect for logic and the West’s political challenges are striking. What’s more, there is a plausible causal mechanism linking these two events. Reasoning from circumstantial facts to a policy decision requires us to engage in argumentation, in the technical sense of a truth-preserving inference from premises to a conclusion.

Those less skilled in logic are more liable to be hoodwinked by fallacious arguments, like the famous Yes Minister ‘politician’s syllogism’: 'We must do something; / This is something: / Therefore, we must do this.' The ‘Politician’s syllogism’ has long been the stock-in-trade of stump speakers like Boris Johnson, whose reputation for dynamism is his chief asset.

In the face of such irrationality, Walter Sinnott-Armstrong and Eugenia Cheng have both come to the conclusion that our society is in need of a lesson on logic. Cheng and Sinnott-Armstrong both view logic as a living discipline, bound up with the warm and tangible facts of political life. In their new books The Art of Logic and Think Again: How to Reason and Argue, they discuss the value of their profession through topics ranging from feminism and safe spaces to nationalism and welfare policy. The novelty of their approach lies in their preoccupation with the structure of the arguments about these issues, rather than the issues themselves.

What exactly is an argument anyway? A logician will tell you that an argument is a set of sentences – one of which is the conclusion, and the others of which are the ‘premises’. An argument is valid when its conclusion must be true if all the premises are true, and a valid argument with true premises is called sound. If one can give a sound argument for some conclusion, then any reasonable listener ought to be persuaded that the conclusion is true.

In order to determine whether an argument is sound, we have to have some method for deciding on the truth of its premises. Introductory logic books are often guilty of assuming that this is unproblematic. So far as most factual premises like ‘the earth is getting warmer’ are concerned, the introductory textbooks are probably right. Scientists can look at reliable data and decide (at some specified level of certainty) whether it is likely that the Earth is warming or not. But when it comes to political arguments, our premises are rarely just about facts. We often have recourse to premises like ‘governments should help people whose living standards are well below average’, and ‘The will of the majority is more important than traditional moral strictures’. These are evaluative rather than factual premises.

The evaluative premises that one subscribes to can often be justified by reference to other, more basic, evaluative judgements. Ask someone why they support liberal political parties, and they might say that it’s because they support the welfare state. Ask them why they support the welfare state, and they might say that governments ought to help people whose living standards are well below average. Ask them why these people are owed help, and they might say that all human beings are equal, and so deserve to be treated as such. But ask them why that’s the case, and you’ll probably get the answer ‘just because’. You’ll have reached a foundation stone of their belief system; what mathematicians would call an axiom.

What can we do when we find that we disagree with someone else’s axioms? This is not a question that Sinnott-Armstrong really deals with. He suggests that argument can help us to understand one another’s axioms, and that even mutual understanding is badly lacking in contemporary public discourse. There are, nonetheless, a couple of tactics available to us to persuade someone to reconsider their axioms. Cheng takes an admirable shot at outlining these tactics.

The strictly logical option is to try to demonstrate to our interlocutor that their axioms are inconsistent. Often people fail to think about the implications of their axioms in all possible situations. In certain situations, our opponent in an argument might have two axioms that contradict each other. She needs to reject or modify at least one of them (unless she is happy living with cognitive dissonance, in which case there really isn’t much use arguing with her). Philosophers refer to this process of axiom revision as the ‘method of reflective equilibrium’, a term coined by the political philosopher John Rawls.

The other way to induce someone to change their evaluative axioms is to make an appeal to their emotions. ‘Do you really feel that way?’. It is to Cheng and Sinnott-Armstrong’s credit that they both explicitly acknowledge the importance of emotions. Cheng even makes emotions the subject of The Art of Logic’s penultimate chapter. The crucial point is that emotions need not be antithetical to logic. In the hands of demagogues, emotions like fear and hatred can be used to ride roughshod over demands for logical reasoning. But there are also quite a few enthusiastic proponents of enlightenment thinking whose love of logic can take on a distinctly emotional tenor. (Think of Steven Pinker or A. C. Grayling.)

Cheng’s sustained focus on our evaluative premises marks an innovation in the introductory literature on logic. Unfortunately, Cheng lets this innovation hijack her approach to the subject. Reading The Art of Logic, one is given the impression that evaluative agreement plus logical validity constitutes a magic bullet that can cure all political ills. Cheng’s own faith in logic is clearly emotionally charged, so it is perhaps unsurprising that Cheng believes we should solve political problems by blending emotionality with mathematical thinking – to declaim otherwise would be an act of performative self-contradiction! But Cheng’s line of thinking is fundamentally flawed, because it neglects the importance of factual disagreements.

Even when two people share the same evaluative axioms, if they ‘do not agree on crucial facts, they are unlikely to agree about what to do in the face of the facts’ (as Sinnott-Armstrong pertinently reminds us). Critically, many of today’s most vehement political disputes are driven by factual rather than evaluative or logical disagreements. In the USA, most voters probably agree that foreign powers should not be allowed to intervene in Presidential elections. But there is vociferous disagreement over whether there actually was significant Russian interference in the 2016 election. In the UK, most voters seem not to understand that ‘frictionless’

exporting by British manufacturers into the EU is by definition incompatible with withdrawing mutual recognition of product-testing regulations. The average UK voter probably already has a pretty good intuitive grasp of the notion of ‘logical inconsistency’, even if he doesn’t call it that. What he lacks is knowledge of which things are logically inconsistent with each other.

Logic popularisers like Cheng and Sinnott-Armstrong are right to judge that factual questions are in principle easier to settle than evaluative ones. But our society may – and often does – lack a collective will to reach agreement on factual disputes. Fans and critics of Trump alike are unmotivated to sift through hundreds of pages of evidence to settle the question of Russian interference; in any case, most of the Mueller report has been redacted by congressional Republicans. Climate changes deniers see little reason to reevaluate the comforting prejudice that blasé fossil fuel use is morally unimpeachable. Popular willingness to better engage with facts will require a shift in political culture. Our society does too little to inculcate public engagement in policy questions, and the civic virtue of intellectual humility is on the wane.

These are not, however, problems that can be addressed by the logicians and their textbooks. It is not just that Cheng pays little attention to factual disagreements. In fact, her discussion of policy questions in The Art of Logic often manages to conceal the relevant factual disagreements, by oversimplifying the questions at issue. Take, for example, her discussion of the dispute between people who think that ‘social services should be expanded to give more help to vulnerable people’, and others who think that ‘social services should be cut to save money and stop encouraging laziness’. Cheng suggests that this disagreement can be ‘clarified’ by thinking about things in terms of false negatives and false positives. ‘A false negative, in this case, is someone who deserves help but doesn’t get it; a false positive would be someone who doesn’t deserve help but does get it’. She then asserts that the following conditional statements are logical truths:

1) If you care more about false negatives than false positives you will believe in expanding social services.

2) If you care more about false positives than false negatives you will believe in reducing social services.

These assertions are actually false, because they ignore the relative incidence of false positives and false negatives under different levels of social services provision (this is, of course, a debate about a matter of fact). Even if you care a lot more about individual false negatives than you do about individual false positives, it may still be perfectly logical for you to be in favour of reducing social services – provided you believe that the number of false positives this would eliminate would far outweigh the number of false negatives that it would create. In any case, Cheng’s analysis abstracts away far too many other aspects of the debate for it to be of any real use to us. Some may argue, for example, that expanding social services will have adverse consequences other than false positives, like creating a pattern of dependency on welfare. Cheng’s decision to brush all this under the carpet is highly irresponsible. It gives the impression that logic simply consists in a set of intimidating verbal tricks – like rephrasing things in terms of ‘false negatives’ and ‘false positives’ – that liberal academics can use to silence the concerns of their opponents. Cheng’s arguments turn out to be just as sophistic as the pseudo-logical arguments of the ERG.

The Art of Logic also drastically oversells the power of Cheng’s sometimes tendentious brand of ‘mathematical thinking’. In a chapter on paradoxes, Cheng observes that many verbal and mathematical operators have a ‘double negative makes a positive’ structure – like the word ‘not’, or the operation of multiplying by minus one. Combining this observation with a hefty (albeit unacknowledged) dash of her own intuition, Cheng jumps to the conclusion that the word ‘tolerant’ also conforms to this structure:

If you’re tolerant of tolerance then that is tolerance.

If you’re intolerant of tolerance that is intolerance.

If you’re tolerant of intolerance that is intolerance.

If you’re intolerant of intolerance that is tolerance.

Although most will agree that intolerance of intolerance is not a blameworthy sort of attitude, there isn’t any reason to think that the concept ‘intolerance of intolerance’ bears any further analysis of the sort proposed by Cheng. Intolerance of intolerance isn’t tolerance. It’s just intolerance of intolerance, and that’s the end of the matter. Yet again, syntactical swashbuckling has trumped philosophical rigour.

Sinnott-Armstrong’s book Think Again is a much more conventional, straightforward introduction to logical argumentation than The Art of Logic, and is therefore probably the better choice for the beginning logician. Sinnott-Armstrong is less optimistic than Cheng about the power of logic: he acknowledges that ‘audiences must be receptive before arguments can accomplish anything’. Sinnott-Armstrong is also always careful to walk the reader through applying the concepts that he mentions, rather than just defining them. Particularly valuable are the lists that he gives of common valid and invalid argument forms, and of common fallacies. Think Again also covers suppressed premises and the induction-versus-deduction distinction, two topics that are conspicuously absent from The Art of Logic.

Although Sinnott-Armstrong is more cautious than Cheng, there are still flashes of brilliance in Think Again. Punctiliously even-handed, when Sinnott-Armstrong makes his views known they tend to evince admirable nuance and insight. SinnottArmstrong reconciles support for safe spaces with a thoughtful defence of the maxim that we should always endeavour to engage with those whose views we deeply disapprove of. The maintenance of safe spaces is

perfectly compatible with the general point that we need to encounter opponents in order to learn from them. Nothing is wrong with using safe spaces at certain times in order to prepare ourselves to encounter opponents at other times – as long as everyone eventually does get out and encounter opposition often enough to understand that opposition.

Passages like this are a shining example of what a logical mind, coupled with a knowledge of the facts, can accomplish.

In sum, then, Sinnott-Armstrong has produced a good guide to logic for those who are unfamiliar with it, but one whose approach to the subject is fairly run of the mill. By contrast, Cheng has produced a book that approaches the subject in novel and unfamiliar ways. Her insights into real-world ‘axioms’ and the interaction between logic and emotion will be of interest even to seasoned logicians. But The Art of Logic is marred by mistakes and overt partisanship. A good exercise in logical thinking for readers as ‘woke’ as Cheng would be to sort through her deductions, deciding which are logically sound and which are not. Tempering the force of some of Cheng’s conclusions makes us realise that logic is not a panacea for political disagreement. Only when logic and emotional introspection are combined with proper knowledge of the world do they have the power to shape our lives for the better, and get us out of this mess.

HARRY R. LLOYD teaches game theory at Lady Margaret Hall, where until recently he studied PPE. He flunked his finals logic exam.

Art by Ellena Murray


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