The Art of Logic in an Illogical World – Eugenia Cheng

Thoughts: This book was an enjoyable read, and I took a lot from it. Cheng argues convincingly that logic and emotions are best used in combination, and lays out ways to leverage the strengths of both. To illustrate her points, Cheng draws on examples from current hot-button issues such as sexism, racism, and social spending. I’d recommend The Art of Logic to anyone who wishes their discussions and disagreements could be more productive.

(The notes below are not a summary of the book, but rather raw notes - whatever I thought, at the time, might be worth remembering.)

Cheng, Eugenia. 2018. The Art of Logic in an Illogical World. Basic Books.

Part I: The Power of Logic

1: Why Logic?

• 3: Cheng argues that in order to understand the world, we have to simplify it (at least temporarily), and that there are two ways to make things simpler: to ignore certain details of it, or to “become cleverer”. She argues that logic involves both these aspects: abstraction is the process of ignoring certain details (of course, it’s important not to abstract away the core, central details), which allows us to focus on the most important details.
  • after abstraction and reasoning, it’s important to consider again those details we initially abstracted away
• 9: Cheng defines mathematics as “the logical study of how logical things work”
• 10-11: different meanings for the word “theory”:
  • in day-to-day life: “a proposed explanation for something”
  • in science: “an explanation that [has been] rigorously tested… and deemed to be statistically highly likely to be correct.”
  • in mathematics: “a set of results that has been proved to be true according to logic.” - doesn’t involve evidence, probability or doubt.
• 14: why is abstraction useful? it shows how many situations are in some way equivalent, which allows us to take reasoning we’ve worked out in one situation and apply it to another (after this, of course, it’s important to note where the situations differ in details that have been abstracted away)

2. What Logic Is

• 26: Many debates about public spending can be usefully thought of as debates about false positives (i.e. someone is helped who didn’t need it) and false negatives (i.e. someone who needs help is not given help). If a person feels that it’s really important to avoid false positives, for example, they are likely to favour reducing public services.
  • j: and really, in almost all situations, neither avoiding false positives nor avoiding false negatives is “more important” - it’s just a matter of getting the balance right
• 28: “If a statement follows from pure logic then it has to be true, automatically. Saying it out loud doesn’t exactly add new information, but it does add new insight.” Simple logical statements, then, often sound quite obvious when used in spoken language. But logical statements gain their power when they are stacked on one another, leading you to conclusions that aren’t immediately obvious
• 32: logical symbol for “a implies b” (equivalent to “if a is true, then b must be true”) is a double-shafted arrow: A ==> B
• 36-37: logical arguments can break down in several ways, which can be broken down into two general categories: problems of knowledge, and problems of logic
  • examples of problems of knowledge:
    • unstated assumptions, or stated assumptions used incorrectly
    • incorrect definitions, or definitions used incorrectly
  • examples of problems of logic:
    • gaps in logic, where too many steps are skipped over
    • incorrect inferences: a logical step is not properly made
    • handwaving: arriving at a conclusion without actually using logic
    • and various logical fallacies

3. The Directionality of Logic

• 45: “if” indicates that logic flows one way, “only if” indicates that it flows both ways
• 46f: Cheng notes that if, for example, you’re trying to capture members of a 5-member illegal gang, you know that all the members of that gang are white men, and you know that there are only 5 white men in the country, it makes logical sense to search for and arrest white men as you come across them. “It might even be reasonable if you know there are exactly ten white men in the country, because if you round up five of them the odds are quite high that you’ll get some of the actual criminals. This approach becomes gradually less like logic and more like racism as the country’s population of white men grows.”
• 49: book mentioned: Tristan Needham’s Visual Complex Analysis, where he notes that coming up with a picture of a concept is often harder, but is almost always more illuminating, than working with equations.
  • 51: Cheng uses a Venn diagram to show that A ==> B can be represented as a Venn Diagram with area A entirely enclosed within area B.
• 56: if two things are logically equivalent, then they’re logically interchangeable. Cheng notes, however, that logically interchangeable is not the same thing as emotionally interchangeable, noting that people may have a very different emotional reaction to “Obamacare” than to the “Affordable Care Act”

4. Opposites and Falsehoods

• 61: it’s important to note that the negation of a statement is broader than the opposite of a statement. Consider the statement “Product X is the best product”. The opposite is “Product X is the worst product”, whereas the negation, “Product X is not the best product”, leaves room for Product X to be just about anywhere in the spectrum from good to bad.
• 63: The Law of the Excluded Middle - when dealing with logical statements that can be true or false, we only consider two options: “true” and “not true”. Everything that is “not true” gets lumped together. If you’re not aware of the law of the excluded middle, it can lead to problems, as you might not realize that there can be a lot of variety/nuance among the “not true” cases.
  • j: this is relevant to some of the discussion around racism recently, e.g. Ibram X Kendi’s formulation (NOT antiracist ==> racist) - I can see the value in looking at policies this way, but it can lead to thinking that a marginally racist policy is just as bad as an extremely racist policy.
• 72: “There is one other possiblilty for truth values [than true and false]: it is possible that the truth value for something cannot be determined.”
  • can be due to a logically contradictory statement (e.g. “this statement is false”), or due to the factuality of a statement is uncertain/unknown (e.g. “the universe is infinite.”)
• 73: an implication that is not true is indicated with an implication arrow with a stroke through it: A =/=> B. This doesn’t imply falsehood/negation, it just means we can’t conclude B if we know A.
• 74: modus ponens: if we know “A implies B” then we can infer B from A, thusly:
  • 1. A is true
  • 2. A implies B
  • 3. Therefore B is true.
  • Conclusion 3. can be false for two reasons: either 1. is not true, or 2. is not true (or both).
• 76: contrapositive statements: (A ==> B) is logically equivalent to (B is false ==> A is false)
  • don’t confuse contrapositive statements with converse statements: (A ==> B) is not logically equivalent to (B ==> A)
• 80: “when you work as a professional statistician, if you do not have the right data to support your hypothesis the correct negation is ‘There is insufficient evidence to support this hypothesis and therefore we need more funding in order to pursue the matter further.’” j: nice

5. Blame and Responsibility

• 85: “exclusive or” vs. “inclusive or”: when someone asks you whether you want tea or coffee, they probably mean “exclusive or”: you can have one or the other, but not both.
  • 85-86: logicians tend to take “inclusive or” as the default “or”, as it creates clearer/simpler logical structures.
  • 89: i.e. (A AND B is false ==> (A is false) OR (B is false)) is a nice neat relationship between AND and OR, but it only works with “inclusive or”.
  • 90: can combine multiple ANDs/ORs: (A and B and C and D) is false ==> A is false or B is false or C is false or D is false (again, only works with “inclusive or”)
• 91: Book mentioned: An Inspector Calls by J.B. Priestley: “A woman has been found dead, and gradually more and more people are found to be implicated in her demise, in different ways, from personal to professional to incidental interactions” - if any of these interactions had not occurred, the woman would not have been found dead (i.e. each of them is partly to blame for the death)
• 99-100: in situations where many factors contributed to some outcome, it’s not particularly useful to ask “who is to blame?” and more useful to ask “who is going to take responsibility for changing it?”

6. Relationships

• 113-114: category theory (where people/objects/ideas etc. can belong to multiple categories at once) lets us know that it’s important to know in what context an argument is being made in. Cheng offers examples surrounding ideas of privelige: “Everyone is privileged relative to some contexts and underprivileged relative to some others. Animosity tends to occur when someone is prone to thinking of themselves in a context that makes them underprivileged… while others tend to view them in a context that makes them overprivileged.”

7. How to Be Right

• 117: you can avoid making sweeping (and thus likely innacurate) statements by refining your scope, making it clear which objects/ideas in specific you’re focusing on.
• 117-118: most statements have some element of truth, and it’s often more productive to look for the ways in which the statement is true than to prove it is false. Cheng offers the example of homeopathic remedies: one person may argue that they’re no better than a placebo, while another notes that when they take one such remedy, they tend to feel better. There’s actually no logical disagreement here, since placebos have been shown to improve subjective wellbeing.
• 118-119: it’s useful to be able to formalize statements like “all X are Y” and “some X are Y”. Usually, these are formally stated like “for all X in {some group}, X is Y” and “There exists an X in {some group} such that X is Y”
• 119: vacuous truth, or a condition being vacuously satisfied: e.g. the truth of statements like “all elephants in this room have 2 heads”. Since it’s logically equivalent to “there exist no elephant in this room that does not have 2 heads”, it’s true (assuming there are no elephants in the room)
• 125: “I believe that a useful way to be a rational person is to look for the sense in which things are true rather than simply deciding if they are true or false. Someone might say something that is untrue in strictly logical terms, but perhaps they were really trying to say something else, perhaps something with strong emotional content that we should listen to….”

Part II: The Limits of Logic

8: Truth and Humans

• 133: mathematics is often not done truly logically: someone will sketch out a proof, simplifying out a lot of smaller sub-steps in the proof. “Disagreements can arise when a mathematician doesn’t see how to fill a gap in, but in that case they ask the person who wrote the proof, and the onus is then on the original author to fill at least some of the gap in until the skeptical mathematician becomes convinced.”
  • j: cf. “no one has ever truly played a game of Magic”
• 140-141: the only equations in mathematics that are strictly true are of the form “x = x” - they’re simple tautologies. 1+10=10+1 is not strictly true, but there is some sense in which it is useful and illuminating to view the two sides as equivalent.
• summary of the chapter: we can more easily convince people of something if we choose useful and illuminating comparisons between different situations that are somehow equivalent. It can be particularly useful to keep these various statements’ emotional content in mind.
  • 146: “in summary, we should look at engaging people’s emotions to convince them of logical arguments, rather than using logical argument alone.”

9: Paradoxes

• 149: Book mentioned: Douglas Hofstadter’s Metamagical Themas
  • contains the sentence “Cette phrase en français est difficile à traduire en anglais” - it’s simple enough to translate the sentence literally as “This sentence in French is difficult to translate into English”, but upon doing so, the sentence no longer makes sense.
• 152: Cheng identifies two kinds of paradoxes: veridical paradoxes and falsidical paradoxes:
  • veridical paradox: nothing wrong with the logic, but we arrive at a logical conclusion that disagrees with our views about the world
  • falsidical paradox: there is a logical fault hidden in the arguments
  • Zeno’s paradoxes are falsidical paradoxes - he didn’t know at the time it’s faulty logic to try to add an infinite number of infinitessimal values in the same way that you add a non-infinite number of rational values.
• 156: Gödel’s paradox states, more-or-less, that “any consistent logical system is doomed to have statements that can neither be proved nor disproved, unless the logical system is rather small and boring…. ‘logical’ means it has been built up from axioms in a precise way, and ‘consistent’ means that it does not contain any contradictions, so that if something is true it can’t also be false”
• 156: book mentioned: Douglas Hofstadter’s I Am a Strange Loop (mostly for its title)
• 161: “Battenberg cake” structures. they have the form:

	0	1
0	0	1
1	1	0

• Battenberg cake structures appear when we add odd or even numbers, or multiply positive or negative integers, as well as other places.
  • 162: Cheng argues that it appears when we talk about tolerance: to be tolerant of intolerance is itself a form of intolerance, while to be intolerant of intolerance is tolerance
    • to resolve this apparent paradox, one can be tolerant/intolerant of many things, but to judge a tolerant/intolerant act itself is a different kind of thing, a “meta-tolerant” act.

10: Where Logic Can’t Help Us

• 173: sometimes, logical reasoning breaks down because we have insufficient data. e.g. in theory, the weather can be predicted based on the current state of the universe. But because we don’t have all the data, weather in practice is chaotic, so sometimes predictions arrived at logically will still be wrong.
• 178: “Perhaps the level of trust in a relationship or a community can be gauged by the extent to which they would be able to cooperate when faced with a prisoner’s dilemma”

Part III: Beyond Logic

11: Axioms

• 183: Axioms are statements that we choose to accept as true in order to reason with them.
  • null system: if axioms cause a contradiction, however, “then the whole system will collapse and become the null system, in which everything is both true and false”. if this happens, logic is no use, so you’ve got to go back and try changing one/some of the axioms

12: Fine Lines and Gray Areas

• 201: fuzzy logic: where statements are not absolutely true and absolutely false, but instead we measure the extent to which a statement is true or false
• 203: the intermediate value theorem: “if you have a continuous function that starts at 0 and goes up to some number a, it must take every value in between”
• 205: Cheng notes that when we only deal with black-and-white logic, people can argue themselves into more and more extreme positions without realizing what’s going on. She argues that it’s important for people to have tools like fuzzy logic, probabilities, tolerance of uncertainty, etc in order to extricate themselves from such positions

13: Analogies

• 212: it’s important to find the appropriate level of abstraction for every analogy. If you don’t abstract enough away, then it’s not a useful analogy - the two things you’re comparing aren’t actually equivalent at that level of abstraction. But if you abstract away crucial details, it doesn’t actually illuminate the situation in question. At the highest level of abstraction, everything is equivalent to everything else
• 214: “Divisive arguments often arise because everyone picks the level of abstraction that best suits their argument and refuses to consider the possibility that other levels could in any way be valid.”
• 217: an argument in favour of compulsory voting: if everyone is legally required to cast a vote, it eliminates the possibility of a person being prevented from voting, even while someone else argues that “they probably didn’t vote because they didn’t want to”. There will be some noise in the signal from people who actually don’t want to vote, but it can be argued that this is outweighed by the the impossibility of voter suppression.
• 220: at one level of abstraction, when a religious leader asks someone to believe the things they say, it’s equivalent to a scientist asking someone to believe the things they say - these days, most experiments are difficult/impossible for the layperson to replicate, and they’re appealing to some body of writing to justify their beliefs
• 222: “Analogies can help us to engage our emotions, if we can find an analogous situation that resonates more closely with us. This is an important way that we can find an emotional connection to back up a logical argument.”
• 228: sorting out the logical extreme based on a set of assumptions can be useful: “The purpose of pushing something to extremes is to show that many (or even most or all) general principles have limits to their scope, and the difficult part is not in establishing the principle but in establishing the scope. It is a key to understanding disagreements, as the source of the disagreement is often exactly where to draw the line, rather than the principle itself.”
• 228-229: Cheng notes that analogies often go wrong because people disagree on the reason why two things are analogous. Consider the statements:
  • A is true because of principle X
  • B is true because of principle X
  • B is true
  • Therefore A is also true
• … it’s clear that this is faulty logic - it’s only if principle X is true that we can conclude that A is true. But often, people don’t explicitly state what they believe principle X to be. This leads to the two people being on different levels of abstraction without realizing it, leading to the perception of the other person being illogical. This is why it’s a good idea to make it clear exactly why you think an analogy is appropriate/illustrative

14: Equivalence

• 236: at some level of abstraction, everything is the same as everything else. Thus, it’s better to ask “in what sense are these the same and in what ways are they different?” than to ask “are these the same or not?”
• 240: one logical fallacy is erroniously equating one statement with another, or claiming two things are logically equivalent. Tends to drive arguments to extremes.
• 245: false dichotomies: believing that A ==> (not B) when that’s not actually the case. situations where it can be true:
  • there exists a possible case where both A and B are true
  • there exists a possible case where both A and B are false
• 250-251: Cheng points out that “black lives matter” and “all lives matter” is clearly a false dichotomy. But usually when someone says “Black lives matter”, they’re using it as shorthand for:
  • A: Black lives matter just as much as other lives
  • B: Black lives are currently being treated as if they do not matter as much.
  • C: We need to do something to correct this injustice
  • and the full argument is not just “A”, but “A and B and C”
• 255: “Mansplaining is not just when men explain things to women in a patronizing way. This is a false equivalence. Mansplanation is in fact when a man explains something to a woman despite the fact that there is strong evidence that the woman already knows it, and a man is ignoring this evidence as part of the systemic societal assumption that men know more things than women, whether or not he is consciously applying that bias in this particular case.”
• 260: False false equivalence: “A claim of ‘false equivalence’ itself needs to be justified. Just saying that something isn’t the same does not mean that it isn’t equivalent in some crucial way.”

15: Emotions

• 263: "Being emotional does not necessarily equate to being irrational: I think that is a false equivalence. This takes the form of a walse dichotomy between
  • A: Using emotions.
  • B: Using logic.
• I think this is the type of false dichotomy where it is possible to do both at the same time."
• 269: People can be manipulated using their emotions into doing illogical things. But this isn’t necessarily always a bad thing, especially when you emotionally manipulate yourself into doing something logical, one that you wouldn’t have been motivated to do simply by looking at the situation’s logic.
  • j: i.e. any number of techniques used to build/break habits
• 272-273: Analogies are a powerful way to leverage emotions to convince someone of something. If you can show that situation A is analogous to situation B, and a person feels very strongly about situation B, it can cause them to want to do something about situation A. “Once they feel things differently, they might be able to see the logic differently.”
• 276: “In the process of writing this book and thinking through these arguments very carefully, stripping away layer upon layer to find further abstractions and logical points of view, I have realized how many of these arguments come down to tensions between the idea of individuals and the idea of groups.”

16: Intelligence and Rationality

• 280: two people can both be logical but still disagree
• 280: “Someone who is leading with their emotions might not be able to articulate what is logical about their thinking, but that doesn’t mean it is actively illogical”
• 280: a person is being illogical only if what they’re doing causes logical contradictions. But it’s important to recognize that contradictions within one’s system of belief are the ones that matter here: with a different system of beliefs or different set of axioms, “one person’s logic might look like idiocy to another person”.
• 280: there are three ways to be illogical
  • "your beliefs cause contradictions, or
  • there are things you believe that you cannot deduce from your fundamental beliefs, or
  • there are logical implications of things you believe that you do not believe."
• 280-281: what is and isn’t a fundamental belief is a gray area: one person “may think that ‘marriage is between a man and a woman’ as a fundamental belief, whereas someone else thinks of it as a constructed belief that needs justifying.”
• 286: Cheng argues that the difference between a conclusion arrived at by speculation vs. the scientific method, or between fake news and true journalism, is whether the conclusion was arrived at by using a reasonable, established framework to verify it.
• 287: “Because the notion of what counts as a reasonable framework is sociological, just like the notion of what counts as a valit mathematical proof turned out to be sociological.”
• 293: useful model of intelligence (at least one definition of intelligence) can be found in Carlo M. Cipolla’s book The Basic Laws of Human Stupidity. 2x2 grid with two axes: whether or not it benefits yourself, and whether or not it benefits others.
  • action doesn’t benefit yourself, doesn’t benefit others: “stupid”
  • action doesn’t benefit yourself, does benefit others: “martyr”
  • action does benefit yourself, doesn’t benefit others: “bandit”
  • action does benefit yourself, does benefit others: “intelligent”
• 294: “Intelligent rationality should involve being able to find the logic in someone else’s emotional response as well as one’s own, rather than just calling emotions wrong.”
• 296: “I think a good argument, at root, is one in which everyone’s main aim is to understand everyone else.”
  • components of a good argument, as identified by Cheng:
    • acknowledges/addresses both logic and emotion
    • when a disagreement arises, the people who are disagreeing try to find the root of the disagreement - it should be either a mistake in logic or a difference in fundamental beliefs
    • use powers of abstraction, and engaging the emotions of both parties, work to build a bridge between the different positions.