The Art of Logic

• Contrary to how it might seem, maths isn't about right and wrong, and nor are most arguments. They're about the sense in which something is right and wrong, depending on world views. If people disagree, it's often a result of different points of view stemming from different fundamental beliefs, not that one is right and the other is wrong.

I: The Power of Logic

1. Why Logic?

• The idea of logic is to have clear rules so that conclusions can be dran unambiguously and consistently by different people.
• The rules of scientific discovery involve experiments, evidence and replicability. The rules of mathematical discovery involve logical proof. Mathematical truth is established by constructing logical arguments, and that is all.
• Maths is the study of how logical things work: it's the logical study of how logical things work.
• A powerful aspect of abstraction is that many different situations become the same when you forget some details.
• Mathematics is a framework for finding similarities between different parts of science, and my research field, category theory, is a framework for finding similarities between different parts of maths.
• When we look for similarities between things we often have to discard more and more layers of outer details, until we get to the deep structures that are holding things together.
• Making analogies is the essence of mathematical thinking, where we focus on important features of a situation to clarify it, and to make connections with other situations. In fact mathematics as a whole can be thought of as the theory of analogies.
• In normal language people judge things not only by context but also relative to their own experiences; logical explanations need to be independent of personal experiences.

2. What Logic Is

• Unlike evidence, logic tells us when something has to be true, not by cause and effect, not by probability, not by observation, but by something inherent that will never ever change.
• Logical implication says that "if" one thing is true "then" another must be true, using logic.
• The primary aim of normal language is communication, whereas the primary aim of logical language is to eliminate ambiguity.
• The closer to a purely logical implication we get, the more obvious it should sound.
• Logical conclusions are true all along whether or not a human notices it.
• Long chains of implications often require us to package many connected ideas into a single unity so that we can build on them more easily, like vacuum packing our clothes. What we gain in the process is new insights and deeper understanding.
• Problems of logic:
  • Gaps in the logic
  • Incorrect inferences
  • Handwaving
  • Incorrect logic

3. The Directionality of Logic

• Time and causation flow in one direction only, and so does logic, and we must be careful not to make errors in direction.
• A --> B. The statement that we get by turning the arrow round is called the converse of the original statement.
• "Only if" is a way of expressing the converse of "if" - logic is flowing in the opposite direction.
• Working hard is a necessary but not sufficient condition for doing well. It is not sufficient because you also have to work hard in the right sort of way, and if you think otherwise you are making a converse error.
• When two things are logically equivalent, the implication flows in both directions, so we use this symbol: A <--> B.

4. Opposites and Falsehoods

• People are not very good at dealing with grey areas and nor is logic.
• The law of the excluded middle says that true and not true are the only two options we are going to deal with at the moment. So all types of not true have to be included. All the grey area has to be included in one side or the other so that there is effectively no middle any more.
• Absorbing the grey area into one side or the other is a simplification, but at least not incorrect or contradictory. Whereas denying its existence altogether is where black and white thinking usually goes wrong.
• If A -> B, then the contrapositive statement is B is false -> A is false. This statement is logically equivalent to the original.
• If A -> B then the converse, B -> is logically independent of the original.
• If A -> B, then negating A and B individually A is false -> B is false is the contrapositive of the converse, and so logically equivalent to the converse.
• A scientific law is something that has been determined to be probably true, to within levels of certainty accepted by science.

5. Blame and Responsibility

• Logical connectives (and and or) are the way of sticking together simple logical statements into complex wholes.
• Logic and maths require things to be unambiguous without us having to understand things according to context.
• The aim of intelligent rational humans shouldn’t be simply to be logical, but rather, to be logical in a useful way.

6. Relationships

• In category theory, mathematicians discover the highlighting relationships between things is often much more illuminating than just thinking of things in isolation.
• Category theory tells us it is always important to be clear what context we’re thinking about. 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 (a victim) while others tend to view themselves in a context that makes them overprivileged.
• If we all become more adept at seeing things from both a privileged and a non-privileged point of view, we will achieve greater understanding of disadvantaged people’s struggles but also of the actions, whether malicious or ignorant, that cause bigotry and oppression.

7. How to Be Right

• Refining your scope means being precise about what world of objects you are focusing on.
• If you qualify a statement with “almost all”, “most”, or “some”, together with “in my experience”, you can almost never be wrong.
• Finding truth in someone’s sweeping statement by applying the right qualifiers can lead to greater understanding of what people are trying to say and where disagreements are coming from.
• Vacuous truth or a condition being vacuously satisfied are cheats, such as “all the elephants in the room have two heads”.
• If you are very precise about how you qualify your statements, you can always be right.
• A useful way to be a rational person is to look for the sene in which things are true rather than simply deciding if they are true or false

II: The Limits of Logic

8. Truth and Humans

• This is somehow how proofs work - at some point you just decide to stop filling in gaps any more, because you think that justifying things any further would not help.
• When we’re thinking up a proof we often use gut instinct, bague suspicions, hunches, inklings, we look for things that slightly remind us of other things, we wait for flashes of inspiration. We then try to fill in all those things using logic, but only after we’ve used many not-entirely-logical processes to get out first ideas in place.

9. Paradoxes

• Paradoxes occur when:
  • Logic contradicts itself - We need to take more care over how we set up our logic, our definitions, or the scope of our thinking.
  • Logic contradicts intuition - We shouldn’t necessarily trust our intuition, or we should take some time to understand quite where that intuition is coming from.
• The liar paradox - “I’m lying!” or “Don’ take my advice!”
• Carroll’s paradox warns us that there is always a level of meta-logic controlling our logic, and we can only understand this by keeping those levels separate. If we don’t decide enough is enough, we will never be able to conclude anything.
• Zeno’s most famous paradoxes are to do with motion, distances, and infinitely small things. They are falsidical paradoxes, where a fault of logic has been hidden in the argument, and that is what causes the strange result.
• Hilbert’s hotel paradox is a veridical paradox that challenges our intuition around infinity. It warns us that we can’t just extend our intuition about finite numbers to infinite numbers, because strange things start happening. Those things aren’t wrong, they’re just different.
• Gödel’s paradox - As soon as a statement is allowed to reference itself, strange loops can be caused. He showed that it is possible to make the statement “This statement is unprovable.” using the language or arithmetic, thus showing that any mathematical system that includes arithmetic must be incomplete. This is, again, a veridical paradox.
• Russell’s Paradox - The barber who shaves everyone who doesn’t shave themselves. It led to the careful axiomatisation os set theory so that we have different levels of sets, a bit like how we have different levels of logic.
• We could decide that we are going to be tolerant of people’s ideas, but not necessarily of their meta-ideas. Their intolerance of other people’s ideas would then count as a meta-idea, and we wouldn’t feel required to tolerate it.
• Shared knowledge and meta-knowledge at all levels is an important tool against this sort of manipulation.

10. Where Logic Can’t Help Us

• One source of frustration when learning new languages is that there seem to be huge quantities of rules to remember, and also huge quantities of exceptions. It’s a tricky combination of logic and non-logic.
• We can trace back the etymology of the language we speak now, to see how it came to be the way it is over time, through gradual morphing, borrowing from other languages, and sometimes misunderstanding. But as with logic, at a certain point we get back to a starting point that we can’t explain.
• Words like “cat” are the starting points of language and they must once have come from some sort of free or random association. Not all concepts make sounds that we can’t explain imitate in our naming of them.
• We don’t learn to speak our native language logically, we do it by immersion, by copying, by emotional connections, and by desire.
• One of the reasons that making decisions is hard is that in most cases logic narrows down the possibilities for us, but several plausibly logical choices still remain. Life is very complicated and mush of it is unknowable, with the result that we often end up in situations that logic can’t completely decide for us, and it’s easy to get stuck in indecision.
• A time pressure is one of the things that helps or requires us to override logic, because logic is too slow.
• Sometimes something starts logical and then by repetition we embed it somewhere deeper in our consciousness s to that we can access it more quickly than by logical thought processes.
• Accessing something by feelings is often faster than accessing them by logic.
• Areas where we have insufficient information include sport, weather, economics,
• The prisoner’s dilemma requires you to trust the other to be perfectly logical and for them to do the same for you.
• Climate change is a type of commons dilemma, which focuses more on ongoing situations and different timescales of benefit, whereas the prisoner’s dilemma focuses on the curious combination of logic and suspicion causing a system to collapse.

III: Beyond Logic

11. Axioms

• Axioms in maths are analogous to our personal core beliefs.
• Most of us get our personal beliefs from some combination of our upbringing, society, education, life experience, and gut feeling.

12. Fine Lines and Grey Areas

• The idea of believing all the logical implications of your other beliefs is called “deductive closure - a set of statements is deductively closed if it also contains everything you can deduce from all the statements in the set.

13 Analogies

• When we forget details about situations, many different situations start looking the same. Abstraction is a way of finding what different situations have in common.
• An analogy is a similarity between two different situations, and abstraction takes that similarity very seriously and treats it as a situation in its own right.
• Arguably, all of mathematics comes from finding similarities between different situations, and when we study them as concepts in their own right we have moved on level more abstract than before.
• Abstraction is a way to pivot between different situations that have something in common.
• Usually in normal life we make the analogy without being explicit about what the abstract version is.
• You need to find the right level of abstraction for a situation. This is like shining a light at an appropriate distance so that you can see enough detail but also enough context around the thing you're looking at. In a way, for abstraction, this consists of forgetting as many details as possible while still retaining the truth of what you are trying to study.
• Thinking about something more abstractly appears to take us further away from concrete ideas, it enables us to pivot further away from where we started, and thus encompass more ideas.
• Analogies are like bridges that can take us anywhere - so we'd better be careful what bridge we choose.
• All analogies break down somewhere. That is the whole point of an analogy: it is not the same as the original thing; it is similar in some way but therefore also different in some way.
• The best we can do is explore different levels and find out what levels cause analogies to appear and to break.

14. Equivalence

• Nothing is actually the same as anything except itself.
• In fact, all equivalence and sameness depends on what you take into account and what you ignore. So as maths becomes more advanced it becomes more and more about finding the sense in which things are the same and the sense in which they are different. The more dimensions something has, the more different ways there are in which it could count as the same, and the more subtle it becomes.
• False equivalence and false dichotomies in general lead to false arguments and fabricated division between people who might not really be disagreeing. This can be exploited by politicians, the media, or just people who have more to gain from discord than from unity.

15. Emotions

• Emotions do not lie. They are never false. If you feel something you are definitely feeling it.
• The most powerful place to be is in the middle, where logic and emotions coexist. Not only do they not need to compete with one another, but they can even strengthen each other.
• Fear causes people to override logic, and that is a good thing in emergency situations. But fabricating fear in order to get people to override logic is not a good thing.
• Many arguments come down to tensions between the idea of individuals and the idea of groups. This applies to the idea of individual vs group responsibility, the extent to which everyone should look after themselves or whether there should be group care. It applies to whether people think a group's treatment by society has any bearing on an individual's.

16. Intelligence and Rationality

• Intelligence consists of Reasonable + Powerfully logical + Helpful, which consist Framework + Logic + Techniques + Emotion
• Believing everything that follows logically from your core beliefs corresponds to the logical notion of "deductive closure". The idea that your beliefs should not cause contradictions corresponds to the logical notion of "consistency".
• A key component of "reasonableness" is that there should be some sort of framework for verification and adjustment.
• The openness to changing one's conclusions or axioms in the face of evidence is an important sign of rationality.
• Loyalty means not changing your support over minor issues. Blind support means not changing your support over major issues, or any issues at all.
• It is important to maintain the distinction between statements arrived at via a framework and those without. It is tempting to try and distinguish between facts and falsehoods, but if you follow logic carefully you should find it difficult to say for sure what a fact is.
• The notion of what counts as a reasonable framework is sociological, just as the notion of what counts as a valid mathematical proof turned out to be sociological. One group of people thinks that the scientific method is the most reasonable framework, whereas another group thinks it is a conspiracy. One group thinks that the Bible is the most reasonable framework, and another group thinks it is a piece of fiction
• The most glaring sign of unreasonableness is if someone is absolutely unprepared to change their mind about something.
• Scepticism is an important part of rationality, and loyalty is an important part of humanity, but both become dangerous when taken to extremes.
• I call someone's trust or scepticism blind if they can't justify it to many steps.
• Powerful rationality starts with abstraction and has three main components:
  • Paths made of long chains of logic
  • Packages made of a collection of concepts sturcture into a new compound unit
  • Pivots using levels of abstraction to build bridges to previously disconnected places.
• Abstraction is the discipline of separating out relevant details from irrelevant ones, and finding the principles that are really behind a situation in such a way that we can try to apply logic.
• We follow logic forwards, to comprehend all the consequences of one's thinking, and backwards to construct and understand complex justifications of things. This includes being able to axiomatize a system down to very fundamental beliefs, rather than just believing things because you do.
• Powerful rationality involves understanding the way in which individuals are interrelated, forming the whole system. It involves great facility at moving between different levels of abstraction to make different sorts of pivots, to move between different contexts and see many points of view.
• Life doesn't have to be a zero-sum game. There are abundant examples of situations where people collaborate for the greater good, rather than compete. This is the essence of teamwork and communities, and perhaps the very essence of humanity.
• Intelligence is the quadrant of mutual benefit:
  • Martyr - Benefit others, but not yourself
  • Stupid - Benefit neither others, nor yourself
  • Bandit - Benefit yourself but not others
  • Intelligent - Benefit both others and yourself
• If you are only using logic to defeat someone else's argument and promote your own, that is the intellectual version of being a bandit.
• A good argument has a logical component and an emotional component and they work together.