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.