What follows from what, and how well do we know it?
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6 ideas. Read the hook, pick an answer, see the picture.
Proof and Precise Reasoning
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.
Pick one
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.
In 1877 Peirce had a kind word for the ostrich. One that buries its head as danger approaches, he wrote, 'very likely takes the happiest course.' The catch is other people.
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Peirce says inquiry has exactly one job. What is it?
Inquiry starts with the irritation of doubt and stops once belief is firm, true or not. That is why Peirce then asks which method settles belief best.
Four Ways to Settle Belief
Tenacity
Cling to a view, avoid challenges
Other people's views bring doubt back
Authority
An institution fixes doctrine
Other times and places believe differently
A priori
Adopt what seems agreeable to reason
Beliefs follow taste and fashion
Science
Belief caused by something external
Slower, but mistakes can be corrected
Only the method of science rests on something independent of our opinions, so it can correct itself.
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".
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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.
Knowing the ending makes it look more predictable. Baruch Fischhoff reported in 1975 that people told an outcome rated it likelier than people kept in the dark. The fix: write your prediction down first.
Pick one
A table lamp will not light. Four suspects: the bulb, a dead outlet, the lamp's switch or cord, or a wall switch controlling the outlet. Which first test splits them best?
A working lamp in the same outlet splits the outlet suspects from the lamp suspects in one go. A new bulb, or a look at the switch and cord, tests one suspect. Changing everything at once may fix it without telling you why.
From Surprise to Test
1ObserveState the surprise without explaining it
2HypothesizeGenerate at least three different explanations
3DeduceAsk what else should be true if each is right
4PredictWrite down expectations before running the test
5TestPick the observation that separates hypotheses most
6UpdateRecord the result and revise your beliefs
Commit to predictions before you look, and run the test that best separates the explanations.
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).
Edmund Gettier's 1963 paper is three pages long. Its cases, where a justified belief is true by luck, cast doubt on defining knowledge as justified true belief. Over sixty years on, no proposed repair has won general agreement.
Pick one
You need a square with exactly twice the area of a given square. How long should the new square's side be?
Twice the side quadruples the area. One and a half times the side gives 2.25 times. In Plato's Meno the boy tries both, then sees that the square on the diagonal is exactly double.
Six-Text Reading Path
1Peirce 1877How does belief get settled, and which way is best?
2Peirce 1878What does it mean for a concept to be clear?
3CliffordIs it ever right to believe on thin evidence?
4JamesWhen may we believe before the evidence is in?
5Plato's MenoCan inquiry begin from ignorance?
6GettierIs justified true belief enough?
Read in order, then write a milestone essay stating your own answer and its strongest objection.
Read this argument: 'Whales are not fish, because whales are mammals and no mammal is a fish.' Which statement is the conclusion?
The conclusion is the claim the other statements are offered to support. The word 'because' introduces the premises, that whales are mammals and that no mammal is a fish, so here the conclusion comes first: whales are not fish.
Question 2 of 8
Which of these is a definition of 'prime number', in the sense the Academy gives to the word 'definition'?
A definition draws a boundary around a word so that it admits every prime and excludes every non-prime. A list of instances, a single example and a loose description do not do that. The Academy says explicitly that 'a prime is like 7' is not a definition.
Question 3 of 8
Many independent teams, working in different places and trying to break the result, all find that water boils at a lower temperature at high altitude. Which of the Academy's five statuses fits this claim?
Established means the claim has survived repeated measurement and attempts to break it, and could in principle be overturned. Demonstrated is for what follows by valid proof from premises stated in the open.
Question 4 of 8
'The vendor says the fault is in the driver, so the fault is in the driver.' Which added premise would make this inference valid?
A hidden premise is one the argument needs but does not state. Only the claim that the vendor's word is true makes the conclusion follow. The other options make the conclusion more likely at most, so the inference would still be invalid.
Question 5 of 8
In Aristotle's classical form, a definition names a genus and a differentia. In 'a triangle is a closed plane figure bounded by three straight sides', which phrase gives the differentia?
The genus is the kind the thing belongs to, here a closed plane figure. The differentia is what sets a triangle apart within that kind: three straight sides. 'Bounded by straight sides' alone would also fit a square, so it does not set the triangle apart.
Question 6 of 8
A valid proof, starting from stated definitions, shows that the sum of two even whole numbers is even. Which status fits?
A claim that follows by valid proof from premises stated in the open is Demonstrated. It can only fall if a premise falls, so repeated testing is not what earns the label.
Question 7 of 8
An argument is valid, and its conclusion is false. What must follow?
Validity means that if all the premises were true the conclusion would have to be true. A false conclusion therefore means at least one premise is false, so the argument is not sound.
Question 8 of 8
A report says 'the new release is slower.' Which rewrite is an operational definition of 'slower'?
An operational definition fixes a concept by the operations used to measure it: what is measured, where, and what result counts. Only the second option does this. The others restate the feeling or the conclusion.