The Logic of Chance, 3rd edition An Essay on the Foundations and Province of the Theory of Probability, With Especial Reference to Its Logical Bearings and Its — Key Ideas to Explore

The Puzzle of the Pyramid's π

Venn dissects a notorious case: Piazzi Smyth's claim that the Great Pyramid of Ghizeh deliberately encodes the value of π in its side-to-height ratio. Venn refuses to accept this as proof of design. He methodically unpacks the assumptions hidden in any probability calculation about ancient builders. First, he notes that not every height is equally possible—'if too high the building would be insecure, and if too low it would be ridiculous.' Second, the precision of measurement matters enormously: a guarantee to the hundredth of an inch yields a different order of coincidence than one to an inch. Third, one must estimate the relative frequency with which builders might have aimed at that ratio—an impossible task given 'extreme ignorance of the attainments of the builders.' Venn then offers an alternative: the builder might have drawn a circle with radius equal to the pyramid's height, laid a cord along its circumference, and used that cord to mark the base. This method would produce the exact ratio without any knowledge of π, just as 'a teredo can bore … a hole which displays the geometric properties of a circle' without understanding geometry. The example illustrates Venn's broader point: apparent design can arise from simple, non-mathematical procedures.

Cards, Dice, and the Unreality of Textbook Examples

Venn criticises the standard examples of probability theory—dice and cards—for creating an 'unreality about the whole treatment of the subject.' These examples, he argues, are 'very well adapted to illustrate its rules' but are 'of a special and peculiar character.' When writers have searched for illustrations from 'the practical business of life,' they have 'very generally, but unfortunately, hit upon just the sort of instances which … are among the very worst that could be chosen.' The problem is that real-world situations involve shifting conventions: as new card games become popular, 'new combinations acquire significance,' and the 'current estimate of such combinations' changes. A calculation that ignores these social dynamics is misleading. Venn insists that probability must be grounded in stable, repeatable observations—not in the fleeting judgments of card players or the arbitrary assumptions of theorists.

The Logic of Frequency, Not Belief

Throughout the excerpts, Venn consistently treats probability as a property of series or classes, not of single events. He speaks of 'the relative frequency of the ‘design’ alternative' and of 'the relative frequency with which such builders can be supposed to have aimed at that ratio.' This frequency interpretation is the core of his logical approach. He rejects the notion that probability measures subjective confidence or 'credibility.' Instead, it must be based on observed proportions in long runs. The pyramid example shows how easily a plausible frequency argument can collapse when the reference class is ill-defined. Venn's insistence on clear, empirical grounding—and his refusal to let mathematical elegance override logical rigor—makes this work a landmark in the philosophy of probability. Readers should attend to his careful distinctions between types of evidence and his relentless exposure of hidden assumptions.

Venn's third edition is a demanding but rewarding read. He expects the reader to follow detailed logical arguments and to question every appeal to 'common sense' in probability. The pyramid discussion is a masterclass in identifying unstated premises. Readers new to the philosophy of probability may benefit from reading the preface and the pyramid section first, as they encapsulate Venn's critical method. The work rewards slow, attentive reading—each paragraph builds a case against sloppy reasoning.

Venn’s insistence that probability lives in what actually happens, not in our hopes, stayed with me long after closing the book—like a quiet friend pointing at the dice on the table. It reminded me of hours spent with Philosophy — Key Ideas to Explore, where the same patient honesty about uncertainty seemed to breathe between the lines. Strange how old arguments feel so close.