How to work with this page

This page is the workshop for Lesson 4 of the Logic unit. You work through it alone, in about 45 minutes, with your Lesson 4 handout beside you. It has one job: to make sure that after this lesson you never again treat a pile of confirmations as a proof — and never again treat a failed prediction as if it told you which premise to give up.

Each part is built the same way:

1. Explanation — short, with the lesson’s own examples.
2. Exercise — you type or choose, then press Check. Nobody sees your mistakes. The page counts your first try; that number is for you.

The page has three stages. Stage 1 puts the lesson’s case on trial and asks you to take a seat before you have the tools. Stage 2 gives you the tools: Hume’s fork and the circle, the black swan and falsifiability, the auxiliary-hypothesis move and its counterfeit. Stage 3 sends you back to your verdict with the tools in hand. Your sheet tells you when each stage is due.

Recall — Lesson 3 from memory

Before anything new: eight questions on Lesson 3, without your notes. Guess first: how many of the eight will you get? …… of 8.

Stage 1 — The case first

Before 2008, the Royal Bank of Scotland — like Lehman Brothers and most major banks — measured its risk with Value at Risk (VaR) models built on about twenty years of market data. VaR gives a number of the form “we will not lose more than £X on 99% of trading days”, calculated from the historical distribution of price moves. The models had performed well across thousands of trading days: a large sample, a consistent pattern, predictions repeatedly confirmed. What they had never encountered, because it lay outside their data range, was a correlated collapse of several asset classes at once combined with a shutdown of the short-term funding markets RBS depended on. In October 2007, with market conditions benign for as far back as the data went, RBS led the €71 billion acquisition of ABN AMRO on what the regulator later called “two lever arch folders and a CD” of due diligence. The purchase used up the capital cushion that might have absorbed the 2008 shock; the UK government then injected £45.5 billion for an eventual 84% stake. Nassim Nicholas Taleb had published The Black Swan in 2007, arguing that the very reliability of inductive models in finance made them dangerous, because the events that matter most are the ones outside the distribution of past observations.

The modellers’ argument, laid out:

1. On every trading day in our twenty-year sample, losses stayed inside the distribution the model describes.
2. The sample is large and the pattern has no exceptions.
3. Therefore, tomorrow’s losses will stay inside it too.

Stage 2 · Part A — Hume’s fork and the circle

A1. Two kinds of proposition

Hume divides everything we can assert into two bins, and nothing sits in both:

Induction generalises from what we have observed to what we have not. The generalisation needs a bridge — nature is uniform: the unobserved will resemble the observed. Hume’s fork asks where that bridge stands. Not among relations of ideas: “the next loaf will not nourish” is no contradiction, so no demonstration can secure uniformity. So it is a matter of fact — and matters of fact are supported only by experience. Which experience? The experience that nature has been uniform so far. That is the very inference we were trying to justify.

A2. The circle, and the chicken

Every defence of induction that appeals to its track record has the same shape: induction has worked so far, therefore it will work. The step from “so far” to “will” is an induction. The defence uses the very inference it was supposed to underwrite.

Russell’s chicken makes the point without a syllable of philosophy. Fed every morning without exception, the chicken concludes by induction that it always will be. Then one morning the farmer comes and wrings its neck. Nothing was wrong with the chicken’s reasoning: the sample was large, the pattern exceptionless. Enumerative induction guarantees nothing about the next case, however big the sample. Russell’s own moral, in Problems of Philosophy chapter VI, goes one step further: induction has rational warrant only if it is backed by a separate Principle of Induction, which experience cannot prove and which would have to be accepted a priori.

Part B — The black swan and falsifiability

B1. The asymmetry, and two criteria

Before Europeans reached Australia, every observed swan was white; “all swans are white” was a superbly confirmed generalisation. Then black swans were found. Popper’s point: no number of confirming instances can prove a universal claim, but one counter-instance can disprove it. A universal claim makes an infinite promise; a refutation only has to be found once.

Two criteria were proposed for what makes a claim respectable:

Neither criterion says a claim that fails it is false. The positivists said meaningless; Popper said unscientific. Keep that distinction — it is the most common error on this page.

Part C — The auxiliary hypothesis and the counterfeit

C1. Which premise does a failed prediction condemn?

A theory never faces the world alone. To predict anything it needs helpers — auxiliary hypotheses: the instruments work, nothing unseen interferes, the initial conditions are as measured. The inference looks like this:

1. If Newton’s laws hold and no unseen body perturbs Uranus, then Uranus will follow this orbit.
2. Uranus does not follow this orbit.
3. Therefore: not (Newton’s laws hold and no unseen body perturbs Uranus).

That is modus tollens, and it is valid. But look at what it condemns: the conjunction. Some premise is false — it does not say which. Le Verrier in 1846 chose to keep Newton and drop the auxiliary: he postulated an unseen planet, calculated where it would have to be, and Galle found Neptune there the same year. The auxiliary-hypothesis move — revising a background assumption rather than the core theory — was rational practice, and the lesson says exactly why: the rescue issued a new falsifiable prediction, and the prediction succeeded.

The counterfeit is the ad hoc rescue: a revision that absorbs the failure and predicts nothing new. “The model was right; 2008 just doesn’t count.” Same shape as Le Verrier, without the new planet to look for.

Stage 3 — Back to the case

Your verdict, with the tools

Take out what you wrote at Stage 1. Now rewrite it in three sentences:

1. Your seat (not rational — rational — the wrong game).
2. Your reason — and say which sense of “rational” it runs on: the philosopher’s (immune to Hume’s challenge), the practitioner’s (best available methods), or Taleb’s (relative to the domain). Hold that sense for the whole answer.
3. The price your seat pays. Hume’s seat: name a forecasting method in a tail-risk domain you would call rational — or say what status scientific prediction then has. The practitioner’s seat: name the technical failure precisely enough that your fix would have changed the answer in 2007, not only in hindsight. Taleb’s seat: say what the bank does on Monday morning without the model, and why that is not Hume’s scepticism in a trader’s jacket.

Then check your own paragraph:

• Did you keep one sense of “rational” from the first sentence to the last? Calling the modellers irrational by the philosopher’s standard and prudent by the practitioner’s, in successive sentences, is not nuance — it is the confusion this lesson exists to end.
• Did your reason name the inference (induction from inside the observation window; a rescue with or without a new prediction), not the outcome (the bank collapsed; the £45.5 billion)?
• Did you say the price in your own words, not the sheet’s?

Only now: compare with the three model paragraphs — two good ones, one bad one.

Done?

You can now:

• sort a claim onto Hume’s fork, and say why “nature is uniform” falls between the tines;
• find the circular step in any defence of induction that appeals to its track record;
• say what one black swan does that a thousand white ones cannot;
• keep meaningless, unscientific and false apart;
• say what a failed prediction condemns — some premise — and what a rescue must do to count as science;
• catch “confirmed” being used for “proved” — in your own essays first.

Go back to your sheet.