William Stanley Jevons

William Stanley Jevons

Logician and economist · 1835–1882 · Britain

A precise contribution

Jevons investigated how a finite logical problem could be represented by combinations of attributes and solved by eliminating combinations incompatible with the premises. His mechanical reasoning apparatus, commonly called the Logic Piano, made that operation physically inspectable. It was an early, important logical machine, not a general-purpose computer or a machine that discovered arbitrary mathematical truths.

The comparison with present-day satisfiability solving is useful when it specifies a shared operation. It becomes misleading when it claims that every theorem prover or processor descends directly from this one device. The history includes earlier logical devices and multiple subsequent mathematical and engineering traditions. Oxford's exhibition describes both the surviving piano and an earlier pocket logic calculator. Oxford historical exhibition.

Life and work in context

Born in Liverpool on 1 September 1835, Jevons studied at University College London, including mathematics and logic with Augustus De Morgan. He interrupted his studies to work as an assayer in Sydney, returned to Britain in 1859, and received his MA in 1862. He subsequently taught at Owens College, Manchester, and took a chair at University College London in 1876. He drowned near Hastings in August 1882. His scientific interests included measurement, statistics, political economy, logic and scientific inference. These facts do not justify reconstructing his childhood conversations or making a psychological explanation of his theories. Scholarly biography.

Useful landmarks are Pure Logic (1864), The Coal Question (first published in 1865), The Substitution of Similars (1869), The Theory of Political Economy (1871), The Principles of Science (1874), and Studies in Deductive Logic (1880). The machine was developed in the late 1860s; a contemporary report places his Royal Society presentation on 20 January 1870. Distinguishing construction, presentation and publication prevents a misleading single invention date. Jevons study and bibliography, contemporary meeting record.

What the logical alphabet represents

For attributes A and B, an arbitrary object can belong to both classes, A alone, B alone, or neither. In modern binary notation these four possibilities are 11, 10, 01 and 00. A universal premise constrains which kinds of objects may occur. “Every A is B” rules out 10.

It does not identify the classes A and B. The correct modern statements are A ⊆ B or A ∩ B = A. Equality A = B would additionally forbid objects that are B but not A. Consequently, replacing an occurrence of “A” by “B” indiscriminately is not a valid explanation of this inference.

Jevons's historical class algebra and substitution project deserve study in their own notation. The binary tables here are an original modern teaching reconstruction, not a claim that the machine used contemporary propositional syntax. Burris's account is especially useful for avoiding that anachronism. Technical history of the class algebra.

Worked mechanism: eliminate types, then test a conclusion

Assume three classes A, B and C, and these universal premises:

1. Every A is B.
2. No B is C.

Begin with eight possible membership types for an individual. A 1 means membership; a 0 means nonmembership.

A	B	C	Decision
0	0	0	Allowed
0	0	1	Allowed
0	1	0	Allowed
0	1	1	Forbidden by premise 2
1	0	0	Forbidden by premise 1
1	0	1	Forbidden by premise 1
1	1	0	Allowed
1	1	1	Forbidden by premise 2

Every allowed type with A = 1 has C = 0. Therefore the premises entail “No A is C.” But they do not entail “Every B is A,” because 010 remains allowed.

An allowed type is a possibility, not an assertion that an object of that type exists. The premises permit A to be empty. If we separately add “Some A exists,” then at least one object must occupy type 110. This existential requirement must be tracked separately from merely crossing out forbidden types.

The distinction also exposes a common error: elimination is not the closed-world assumption. If both 000 and 001 remain possible, we have not established whether a particular unconstrained object is C. We cannot label every unproved proposition false. A conclusion follows only if it holds in every model satisfying the premises.

What exhaustive representation costs

With n independent binary attributes, a complete alphabet has 2^n types. Four attributes give 16; twenty give 1,048,576. A mechanism that explicitly represents every type encounters exponential growth even if each elimination is simple. This is a limitation of that representation, not a proof that every solver must inspect every assignment on every problem.

Modern solvers can use partial assignments, propagation, learned constraints and shared representations. Comparing them with the piano should begin with invariants: which possibilities have been excluded, why was exclusion sound, and what evidence establishes a conclusion? The useful inheritance is a way to formulate an inspectable problem, not a universal genealogical claim.

Economics and inference without slogans

Jevons also helped develop marginal utility analysis. Marginal utility is a change or derivative of utility with respect to quantity; diminishing marginal utility is an additional behavioral assumption, not part of the definition of a derivative. Equating marginal utility per unit of price requires a specified optimization problem and suitable interior conditions.

His work on coal raises a separate systems question: an improvement in resource use per operation need not reduce total consumption if the number of operations rises enough. The arithmetic is conditional. If energy per job halves and the job count triples, total energy rises by a factor of 1.5. This example is our calculation, not a measured historical result or a universal demand law. Scientific and economic inference require assumptions about observation, behavior and uncertainty beyond deductive elimination. Jevons's economic and scientific projects.

Exercises

1. Start with “Every A is B” and “Every A is C.” Determine the allowed membership types. Does “Some B is C” follow? Reconsider after adding “Some A exists.”
2. An implementation stores one byte for every binary membership type. Calculate its storage for 24 and 30 attributes. Explain why this calculation does not prove that all logically equivalent algorithms require that amount of storage.
3. A service originally uses 2 energy units per job and runs 100 jobs. A redesigned version uses 1 unit per job and runs 250 jobs. Calculate both totals and the percentage change. Identify the missing empirical evidence before using this example to predict a real service's consumption.

Compare your reasoning with the worked solutions

1. Possibility and existence

The first premise forbids 100 and 101; the second forbids 100 and 110. The remaining types are 000, 001, 010, 011 and 111. Type 011 is permitted, but no premise says it occurs. A model containing only a 000 object satisfies both premises while making B ∩ C empty. Thus the existential conclusion does not follow. Adding a witness in A forces that witness into 111, and it therefore witnesses “Some B is C.” The extra premise changes existence, not the elimination rules.

2. Representation cost

2^24 bytes is 16,777,216 bytes, or 16 MiB. 2^30 bytes is 1,073,741,824 bytes, or 1 GiB. These counts describe an explicit array representation. A compact formula such as “Every A is B” can describe a very large set of permitted assignments without storing every row. Compact representation does not automatically make all inference easy, but the cost of one representation is not a lower bound on all algorithms.

3. Conditional rebound calculation

The original total is 2 × 100 = 200 units. The new total is 1 × 250 = 250 units, an increase of 50/200 = 25%. The efficiency improvement is real even though the total rises. A prediction needs evidence about demand, prices, capacity, behavior and the chosen system boundary. It also needs the counterfactual job count without the redesign. The example establishes a possibility under stated numbers, not an observed causal effect.