Induction & Probability

This is Unit Seven of Logic: Induction and Probability. Everything since Unit Three has been about arguments whose conclusions are guaranteed. This unit is about the other kind — the ones that make their conclusions likely, which is what nearly every argument outside a mathematics department actually does. There is one property that separates the two, and it goes deeper than the familiar necessary-versus-probable line. A valid argument stays valid whatever else you learn. An inductive argument's strength moves the moment new information arrives — its form untouched, its force changed. The unit covers sampling, analogy, five classical designs for finding causes, the probability calculus, and inference to the best explanation. And it dwells on two specific errors, because between them they account for a startling amount of harm in medicine, security, and the news. By the end you will be able to do five things. One: say why new evidence changes strength and never touches validity. Two: name the two ways a sample fails, and which one no amount of size will fix. Three: give the four explanations that any correlation admits. Four: compute "at least one" the easy way, and say why rarity outvotes a test's accuracy. Five: rank competing explanations by four criteria, and say why they conflict.

One property, and it is the deepest difference between the two kinds of argument. Deductive validity is settled once and for all by the form. Inductive strength is permanently open to revision by facts that were not part of the argument at all.

The property.

Logic 331 Strength Revisited

Mention that the sample was self-selected and a strong argument weakens on the spot, with nothing about its structure altered. A valid argument is untouchable this way; a strong one never is.

The most common inductive argument in public life: measure a portion, conclude about the whole. It fails in two ways, and only one of them is what people worry about — the other is untreatable by the fix everybody reaches for.

The form.

Logic 332 Enumerative Induction

Some proportion of those measured have the property, so roughly that proportion of everyone does. Everything depends on the measured group resembling the rest, which is the next entry's business.

The two failures.

Logic 333 Sample Size and Sample Bias

A famous 1936 poll gathered over two million replies, drew them from the wrong list, and called the election spectacularly wrong. Size is no protection against a crooked sample — it only makes the error more confident.

Reasoning from resemblance is everywhere, and it has a specific weakness worth naming: the number of similarities is easy to count and almost beside the point. What matters is whether the similarities bear on the conclusion.

The form.

Logic 334 Argument from Analogy

Two things share several properties, one has a further property, so expect the other to have it too. Nothing here is guaranteed — the only real question is how much the shared properties support the leap.

What makes one strong.

Logic 335 What Makes Analogies Strong

Predicting how a car handles, the drivetrain matters and the paint does not. A single relevant match outweighs twenty irrelevant ones — and tallying matches without weighing them is how weak analogies acquire impressive-looking numbers.

Five designs for isolating a cause, worked out in the nineteenth century and still the shape of every controlled experiment. Each makes an assumption, and the assumptions are where they break in the field rather than on the page.

The first.

Logic 336 Mill's Method of Agreement

Five people fall ill, five different meals, one dish in common. The idealisation to watch is "one factor in common" — real cases share dozens, so this design narrows the field rather than settling it.

The second.

Logic 337 Method of Difference

Two situations alike but for a single factor, and the effect appears in only one. This is the controlled experiment in miniature, and its force depends entirely on how genuinely alike everything else was.

The two, combined.

Logic 338 The Joint Method

The factor turns up wherever the effect does and is missing wherever it is not. The strongest of the five designs — and still only a correlation, with the rival explanations two topics from here still standing.

When you cannot remove the factor.

Logic 339 Concomitant Variation

The effect rises and falls in step with the factor. For anything that cannot be switched off — temperature, exposure, dose — this is the only one of the five available.

And the fifth subtracts.

Logic 340 Method of Residues

Take the part of an effect that known causes account for and assign what is left to what remains. Its accuracy is exactly the accuracy of your list of known causes — one you missed comes back as a phantom.

The methods worked on a case, and then the limitation that applies to all five equally. Whatever design produced it, a correlation is compatible with four different stories, and only one of them is the one people announce.

Watch them run.

Logic 341 Applying Mill's Methods

Narrow the candidates by what the affected cases share, eliminate survivors by comparing cases that differ, then corroborate what is left by watching it vary. The methods can only eliminate candidates that made the list — one never considered is never ruled out.

The four explanations.

Logic 342 Correlation vs. Causation

Ice cream sales track drownings, and summer explains both. Any correlation admits coincidence, causation running either way, or something behind both — and the work is eliminating three rivals, not applauding the survivor.

And what "the" cause means.

Logic 343 Necessary and Sufficient Causes

Oxygen is required for a fire and guarantees nothing. Most causes worth discussing are neither — they contribute. Naming one of them "the" cause is a selection made by our interests, not something read off the world.

Before the calculus, a question about what the numbers are. Three quite different things get written as probabilities, and they answer to different evidence — though all three obey the same arithmetic, which is why the distinction is easy to forget.

The three readings.

Logic 344 Three Interpretations of Probability

Counting equally possible outcomes, measuring an observed long-run rate, or stating a calibrated degree of confidence. Nothing in the calculus distinguishes them — the interpretation question is about what the number is claiming.

Four rules for combining probabilities. Each has a condition attached, and the pattern from Unit Five repeats exactly: the arithmetic is easy, and the errors are all about applying a rule where its condition does not hold.

The first.

Logic 345 Conjunction: Independent Events

Two events that do not influence one another: multiply. Independence is the licence, and multiplying dependent events as though they were independent is this calculus's characteristic mistake.

When they do influence each other.

Logic 346 Conditional Probability

Drawing two aces without replacement: the second draw faces a different deck. And note that the chance of one given the other, read in the two directions, gives routinely different numbers — a fact two topics from here is built on.

Combining alternatives.

Logic 347 Disjunction Rules

Two outcomes that cannot both happen: add. Otherwise add and then remove the overlap, because it got counted twice. That subtraction is Unit Four's inclusive reading, now costing arithmetic.

And the most useful trick here.

Logic 348 The Negation Rule

The chance of something not happening is one minus the chance it does. Asking for at least one, compute the chance of none and subtract — done directly it becomes a combinatorial swamp.

Three problems, and the sequence to notice is that the rule gets chosen before any number is written down. Which rule applies depends on two facts about the world, and assuming either one is how the arithmetic ends up impeccable and wrong.

Watch three get worked.

Logic 349 Probability Walkthrough

Ask first whether the events can overlap and whether one changes the other, then pick the rule those answers point to. Independence is a claim about the situation and never a default — treating dependent draws as independent quietly changes the question.

The unit's most consequential idea. When something is rare, a genuinely good test still produces mostly false alarms — and the reason is arithmetic rather than any failing of the test. This is the reasoning error with the largest real cost outside a classroom.

The idea.

Logic 350 Base Rates

How often something occurs before any test is run. Where it is rare enough, rarity overwhelms accuracy and most positives are wrong — and this gets neglected in medicine, in security screening, and in the news.

The formula.

Logic 351 Bayes' Theorem

Updated confidence equals how well the hypothesis predicted the evidence, weighted by how plausible it was beforehand, scaled by how common the evidence is. It is the legitimate way to turn a conditional around — the illegitimate way was named two topics back.

Worked by counting people.

Logic 352 Bayes Worked Example

Take a concrete population and count who lands in each group — that makes the result visible without the algebra. The starting rate does all the work here, and it is the piece people throw away first, reasoning from the test's accuracy alone.

Two ways of reading pattern into randomness. The first is famous. The second is not, and it is worse, because it does not merely mislead — it manufactures causes that were never there, and then rewards the people who act on them.

The famous one.

Logic 353 The Gambler's Fallacy

Five heads in a row, so tails is owed. The coin has no memory and no debts. The near-miss worth knowing: if the process might not be independent at all, a long streak is real evidence of that.

And the worse one.

Logic 354 Regression to the Mean

An extraordinary first season followed by an ordinary second. Extremes are partly luck, and luck does not repeat to order. Criticise after a bad result and improvement follows — which teaches exactly the wrong lesson.

The unit's last form of inference, and it is the one science actually runs on: pick the hypothesis that best accounts for what you see. Four criteria decide "best", they can pull against each other, and the capstone runs all of it against a real headline.

The form.

Logic 355 Abduction

Hoofprints, whinnying, and missing oats settle it without anyone seeing a horse. But this supports rather than certifies — best available is a ranking, and rankings move when a new candidate arrives.

The four criteria.

Logic 356 What Makes an Explanation Best

How much it accounts for, how little it assumes, how well it sits with what is already established, and whether it can be checked. They can conflict — a bolder theory may account for more — so ranking stays a judgment the criteria guide.

The capstone. One headline, taken apart.

Logic 357 Capstone: A Causal Claim in the Wild

Start with the design, then the rival explanations, the sample, and the starting rate hiding behind a relative figure. A relative risk quoted with no starting rate beside it cannot be read at all — which is the form headlines prefer.

Five things you can now do. You can say what separates the two kinds of argument at the deepest level: new evidence moves an inductive argument's strength and cannot touch a valid one. You can name the two ways a sample fails, and you know a crooked sample is not repaired by making it bigger — only made more confident. You can give the four explanations any correlation admits, and you know the work is knocking out three of them rather than announcing the fourth. You can compute "at least one" through its complement, and you know why a rare condition plus a good test still yields mostly false alarms. And you can rank explanations by scope, simplicity, fit with what is known, and testability — while knowing those four can pull against one another.