The Probability of Induction

No calculus can assign a probability to an inductive conclusion. The textbook rule that says otherwise would make all reasoning from experience worthless.

by Charles Sanders Peirce26 passages held

English, the original language, and your language

First published: 1878(aged 39)
Original language: English

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Overview

Alphonse Gratry, an Oratorian priest who taught moral theology at the Sorbonne and was elected to Voltaire's old seat at the Académie française, had written that every true induction is an immediate inspiration from on high. Peirce respects that answer more than the attempts to settle the question by juggling with probabilities or syllogisms: it recognises how deep the difficulty goes, and proposes a cause large enough to meet it. He declines it only on the ground that an explanation ought to say how a thing is done. The difficulty is real. Deduction only unpacks what the premisses already contain; synthetic reasoning, which Peirce here calls induction loosely, is the only kind that adds anything; so a calculus that merely rearranges given frequencies cannot possibly hand an inductive conclusion a probability. The textbooks say otherwise. They offer the rule that a man who has watched the tide rise m times may expect it again with probability (m+1)/(m+2) — a rule on which Quetelet, the Belgian statistician, had built a theory of induction. Set m to zero and it announces that an event nobody knows anything about is an even chance. Follow that through, Peirce argues, and Nature becomes a chaos of independent elements in which no inference from one fact to another holds, and every past regularity was a run of luck now ending. He proposes reversing the question: ask not how a conclusion stands against all possible worlds, but what proportion of inferences drawn this way come out true in this one. Five years later, teaching at Johns Hopkins, he took up probable inference again in Studies in Logic, the 1883 volume he edited with his students.

Key concepts

What is the rule of succession, and what is wrong with it?

The textbook rule that a man who has seen the tide rise "m" times may expect it again with probability ("m"+1)/("m"+2). Quetelet built a theory of induction on it. Peirce's objection is that it manufactures a number out of ignorance: the frequencies given in the premisses cannot rate a conclusion that goes beyond them.

What happens if you set m to zero in that rule?

It announces that an event nobody knows anything whatever about is an even chance. Peirce treats that as the reductio. Follow it out and nature becomes a chaos of independent elements in which no fact bears on any other, every past regularity was a run of luck now ending, and reasoning from experience is worth nothing.

What does Peirce mean by calling his opponents conceptualists?

They take probability to measure a degree of belief — how confident a mind is entitled to be, given what it knows. Peirce calls his own view the materialistic one: probability is a ratio of real frequencies in the world. Nearly every dispute in the paper turns on which of those two senses is being assumed at the time.

Why does he treat a priest's claim of divine inspiration respectfully?

Because Alphonse Gratry at least saw how deep the difficulty goes and proposed a cause large enough to meet it, which the logicians juggling with probabilities or syllogisms had not. Peirce declines the answer on one ground only: an explanation ought to say how the thing is done. Inspiration names the gap rather than crossing it.

Is 'induction' here the same thing Peirce means by it elsewhere?

He uses it loosely, for synthetic reasoning in general — any inference that adds to what the premisses already contain. That is wider than the sense he fixes elsewhere, where induction is one of three forms and generalises from a sample. Reading the paper with the narrow sense in mind makes its central claim look far more sweeping than it is.

Themes of the book

What this book returns to, gathered into themes and ordered by how much of the text each one occupies.

No Calculus Can Rate an Induction

Deduction only unpacks what the premisses already hold, so where would a probability for a conclusion that adds something new come from? The textbook answer, followed through, destroys reasoning from experience.

The rule of succession promises that a man who has watched the tide rise "m" times may expect it again with probability ("m"+1)/("m"+2); set "m" to zero and it announces that total ignorance is an even chance. Follow that out and nature dissolves into independent elements where no fact bears on any other and every past regularity was a run of luck now ending. Peirce reverses the question: not how a conclusion fares across all possible worlds, but how often inferences drawn this way come true in ours.

probability
induction
conceptualism
evidence
statistical inference
inductive logic
generalization
synthetic judgment
sampling
ignorance
frequentism
chance
belief

Enter a dialogue

Within this work

passages held: 26
distinct concepts: 54
by passages held, corpus-wide: #755

What this book thinks together

The wheel takes this book’s leading concepts and shows how often they come up together. The wider the band between two, the more of its passages speak of both. Among them, most often together: “induction” and “probability”.

Concept co-occurrence, strongest first 6 concepts, joined by 13 pairings. 30 pairings reach beyond these concepts and are not drawn.
Concept	Paired concept	Shared passages
induction	probability	11
epistemology	induction	9
epistemology	probability	7
evidence	probability	7
conceptualism	probability	6
belief	probability	5
belief	evidence	4
conceptualism	evidence	3
belief	conceptualism	2
conceptualism	epistemology	2
conceptualism	induction	2
evidence	induction	2
belief	epistemology	1

Works held

The author's work in order, this book marked in place. Your-language title first, the original beneath it where the two differ; side chips show which full texts are held.

1867 On a New List of Categories
The earliest item on this shelf, and the root of everything after it: the three-part scheme of quality, relation and representation reappears in the logic papers of 1877–78 and again in the evolutionary cosmology of the 1890s.
EN
1868 Questions Concerning Certain Faculties Claimed for Man
Where Peirce clears the ground. The four denials reached here — no intuition, no introspection, no thought without signs, no conception of the incognizable — are restated as premisses at the head of the sequel.
EN
1868 Some Consequences of Four Incapacities
The sequel, and where the argument arrives. The preceding article asked whether we have certain faculties; this one takes its four denials as premisses and follows them out to a theory of reality and of the self.
EN
1877 The Fixation of Belief
Written for readers with no philosophy behind them: the first instalment of a six-part course in the logic of science, serialised in a general-interest monthly. Everything the other 1877–78 papers do rests on the account of doubt, belief and inquiry set out here.
EN
1878 Deduction, Induction, and Hypothesis
The paper that ends the series. Its central idea — guessing, treated as an inference with rules of its own rather than as a lack of them — is what Peirce kept working at for the rest of his life.
EN
1878 How to Make Our Ideas Clear
The second instalment of the 1877–78 series, nominally about a dead distinction in the logic textbooks. Two decades after it was printed, the American pragmatists took it as their charter.
EN
1878 The Doctrine of Chances
Third in the series, and the point where the account of inquiry acquires a price. Half of it is arithmetic — card packs, whist hands, insurance offices — and the other half asks whether a creature who never lives to complete a run of trials may reason from probabilities at all.
EN
1878 The Order of Nature
The religious instalment of the 1877–78 series. It reaches forward to the essay Peirce published at the end of his life: both ask what a scientific habit of mind is entitled to say about God, and they give different answers.
EN
1878 The Probability of Induction you are here EN
1891 The Architecture of Theories
A prospectus rather than a treatise, and the first of the five articles of Peirce's metaphysical series in Paul Carus's Monist, 1891–93 — the run in which the cosmology it sketches was built.
EN
1892 Man's Glassy Essence
Three things that do not obviously belong together — the constitution of matter, the chemistry of living tissue, and the question whether a church or a crowd can think — joined by the argument that mind and matter differ only in which side you look from.
EN
1892 The Doctrine of Necessity Examined
It stands second in the series, and it is the one that started a quarrel: Carus attacked it twice in his own journal and Peirce defended it once, so that the argument about chance occupied The Monist for fifteen months.
EN
1892 The Law of Mind
Where tychism acquires its partner. The chance of the April paper needed continuity to work in, and continuity gets its name and its defence here — the idea Peirce came to regard as the one his philosophy could least do without.
EN
1893 Evolutionary Love
Here the cosmology turns into an argument about how people ought to live: chance, necessity and love applied to the history of ideas, and to the ethics of a commercial century. It closed the Monist series.
EN
1893 What is Christian Faith?
Addressed past the theologians to two audiences at once — churchmen keeping their membership narrow, and the scientifically educated, shut out of the churches, who were talking about founding one of their own.
EN
1899 Leibniz Rewritten
The one book review on this shelf, from the years after the Coast Survey when Peirce lived partly by notices and dictionary work. It is the only piece here in which he is weighing someone else's system rather than building his own.
EN
1908 A Neglected Argument for the Reality of God
Thirty years after the logic papers and fifteen after the cosmology, Peirce turns the machinery of hypothesis on the question the 1878 series had left open.
EN

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