What exactly does this say, and what does it leave unsaid?
Level 1
Learn it
6 ideas. Read the hook, pick an answer, see the picture.
The Seven Liberal Arts, Hands-On
The Seven Liberal Arts, Hands-On: Logic, Proof and the Quadrivium
Euler's polynomial, n squared plus n plus 41, gives a prime for every n from 0 to 39. At n = 40 it gives 1681, which is 41 squared. Testing refutes a rule quickly; it cannot prove it for every n.
Pick one
You want a new square with exactly twice the area of an old one. What should its side be?
Doubling the side quadruples the area; one and a half times the side gives 2.25 times. Cut the old square along its diagonal: the square on that diagonal holds four such halves, the old square two. Both wrong options are the slave boy's guesses in Plato's Meno (82e, 83e); Socrates leads him to the diagonal (85a-b).
A camping party once split evenly over a squirrel. A man circles a tree; the squirrel keeps the trunk between them. Does he go round the squirrel? William James's answer: it depends on what you mean by "going round".
Pick one
The same probability puzzle is given two ways: as percentages ("1%") or as counts ("10 out of 1,000"). The two versions are mathematically equivalent. Which version do people solve correctly more often?
Gigerenzer and Hoffrage (1995) found people reason about conditional probabilities much more accurately with natural frequencies. Equivalent numbers, different form: the representation changed how hard the problem was.
The First Three Moves
1ObserveRecord what was seen, not a conclusion
2DefineSay what terms mean, ideally by how you would measure them
3RepresentChoose a form that shows structure, like a timeline or graph
Separate observation from interpretation, define terms, then represent the problem before explaining it.
English Expression Mastery: Words, Sentences, Paragraphs and Speech
One word can invent a cause. 'Although the service is running, users cannot log in' is fair to the facts. Swap in 'because' and the grammar is still fine, but the logic is invented.
Pick one
A large 2018 meta-analysis looked at adult bilingualism and executive function. Once publication bias was corrected, what happened to the apparent bilingual advantage?
Lehtonen et al. (2018) found the apparent cognitive advantages did not hold up once publication bias was corrected. That leaves 'a harder language makes you smarter' unsupported. The finding does not show that learning languages is worthless.
The 25-minute weekday session
1Recall5 minutes retrieving earlier words and moves.
2Study a model5 minutes with one author's prose.
3Write8 minutes on the week's evolving piece.
4Revise5 minutes; keep uncertainty in the text.
5Speak2 minutes explaining the point aloud.
Each weekday session loops through recalling, studying, writing, revising and speaking.
How to Read a Great Book: Levels, Modes and the Completion Standard
Adler and Van Doren's Rule 9: you may not agree with a book, or disagree, until you can honestly say "I understand." So how do you know you understand? This page's test: you have made seven things from it.
Pick one
Adler and Van Doren teach slow, active reading of hard books. What do they have you do before the slow reading?
Inspectional reading: skim the title, preface, contents and index, then read the book through once without stopping to look things up. Slow analytical reading is the next level up.
Four Levels of Reading
4SyntopicalRead several books on one subject and build the conversation
3AnalyticalSlow, active reading: state its arguments, then judge them
2InspectionalSkim systematically to learn what kind of book it is
1ElementaryReading the words; assumed here
↑ higher level
Understand a book before you agree or disagree with it, and build up through the levels.
Proof and Precise Reasoning: From Arguments to Theorems
Euclid's proof that the primes never run out is often presented as a proof by contradiction. His own version is a recipe. Give it any finite list of primes, and it produces a prime not on the list.
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Take 'If it rained, the road is wet'. Its truth table has four rows: rain or not, wet or dry. In how many rows is it false?
Only rain with a dry road makes it false. Without rain it promised nothing, so it counts as true, wet road or dry.
What Counts as Proof
Example
Shows a claim holds once
Can fail at the next case
Computation
Checks particular cases
Refutes; proves only if every case is checked
Diagram or simulation
Suggests a truth
Can hide assumptions
Evidence about a model only
Proof
Each step follows from definitions
Covers every case in the domain
Only a valid proof establishes a universal claim (checking every case of a finite domain is one); the rest suggest or refute.
Choosing a Language to Study for Reading: Latin, Greek, French, German and Beyond
Newton's Principia is in Latin. Kant's Critique of Pure Reason is in German. In translation, every sentence reaches you through a translator's choices you never see.
Pick one
A large meta-analysis asked whether adult bilinguals have better executive function: attention control, switching, working memory. Once publication bias was corrected, what did it find?
Once publication bias was corrected, Lehtonen et al. (2018) found the apparent advantage was not supported. Paap and Greenberg (2013) also failed to find it, so no language ranking follows.
Choose by the authors
Latin
Cicero, Augustine, Aquinas, Newton's Principia
Moderate to demanding; few conversation partners
Ancient Greek
Plato, Aristotle, New Testament
Demanding: new alphabet, verbs, particles
Pick Classical or Koine by target texts
French
Montaigne, Pascal, modern literature
Among the easier for English speakers
German
Kant, Hegel, Nietzsche, Frege
Harder than French
Pick the language of the authors you most want to read, not a supposed intelligence score.
What does grammar's practice, 'parse before judging', ask you to do with a short passage?
Grammar's practice is to work out what every word and clause in a short passage is doing before you decide whether you agree. Its made thing is a parsed passage with an error log. Agreement comes after the parse.
Question 2 of 8
In the definition 'a chair is a seat for one person, with a back', which part names the genus?
The genus names the kind of thing, here a seat. The differentia says what sets a chair apart within that kind: it is for one person and has a back. This is Aristotle's classical form, the genus and then the differentia.
Question 3 of 8
You restate a paragraph and show it to someone who holds its view. They say, 'No, that is not what I think.' What follows?
A fair reconstruction is one the author would accept. The test is whether a person who holds the view could read your summary and say 'yes, that is what I think'. If not, any objection you make is aimed at a stand-in.
Question 4 of 8
Hugh of Saint Victor says the exposition of a text includes three things, the letter, the sense and the inner meaning. What is the letter?
Hugh calls the letter 'the fit arrangement of words' (Didascalicon III.8). The sense is the ready and obvious meaning that the letter presents on the surface, and the inner meaning is the deeper understanding found only through interpretation and commentary.
Question 5 of 8
Someone defines 'chair' as 'anything you can sit on'. What is wrong with this definition?
A definition must admit everything it must and exclude everything it must not. 'Anything you can sit on' admits every chair, but also a tree stump, a wall and the floor, so it is too wide. The Academy asks you to decide whether such a case is a discovery or an error.
Question 6 of 8
Augustine names two causes that prevent what is written from being understood. What are they?
Augustine names 'two causes which prevent what is written from being understood': unknown signs and ambiguous signs (On Christian Doctrine II.10.15). Either kind can be proper or figurative. Natural and conventional is a different sorting, by where a sign's meaning comes from.
Question 7 of 8
Two facts: 'The backup finished at 02:00. The disk filled up at 03:00.' Which sentence claims more than these facts show?
'Because' asserts that one event caused the other, but the two facts give only an order in time, so the connective is not earned. The page's own example, 'Because the service is running, users cannot log in', has fine grammar and invented logic.
Question 8 of 8
A rule book defines 'tool' as 'a hand-held device with a metal head, used for striking'. Which case shows that the definition is too narrow?
A definition is too narrow when it excludes what it must admit. A screwdriver is a tool, but it has no head for striking, so this definition shuts it out. The three hammers all fit the definition, so none of them can show that it is too narrow.