Short theoretical article: Shackle’s “A Non-Additive Measure of Uncertainty”

This short theoretical journal article clarifies a central feature of Shackle’s Expectation in Economics: uncertainty cannot be represented by an additive probability calculus when the decision is unique, historically situated, and not repeatable. Shackle’s target is not probability within games of chance, but its illicit extension to “mental states of uncertainty.” He argues that the relevant object is not distributable belief, but a non-additive measure of “potential surprise,” closer to a measure of disbelief or non-acceptability.

For this purpose we need a measure of acceptance, of a hypothesis proposed in answer to some question, which shall be independent of the degrees of acceptance simultaneously accorded to rival hypotheses.

The first non-additivity is therefore psychological and logical: rival hypotheses about an uncertain future need not divide a fixed unit of belief. A person may find several incompatible outcomes perfectly possible; the arrival of a new plausible hypothesis need not reduce the plausibility of those already entertained. This is why Shackle shifts attention from confidence to disbelief: many rival hypotheses can have “nothing known against” them.

Thus while the additive character of numerical probability is fatal to its use for describing mental states of uncertainty (that is, states of acknowledgement of ignorance), we have with potential surprise an absolute release from any such requirement.

The second non-additivity concerns decision procedure. Shackle distinguishes “divisible” experiments, composed of indefinitely repeated uniform trials, from “non-divisible” experiments: unique, crucial, or isolated acts whose outcomes cannot be decomposed into repeatable parts. Frequency-ratio probability belongs only to the former. When applied to the latter, it borrows the appearance of precision from contexts where repetition makes sense.

For a non-divisible, unique experiment it is plain that no frequency-ratio can have any meaning or relevance.

This distinction grounds the article’s main methodological claim. In divisible experiments, frequency ratios can make the aggregate outcome knowable in advance; in real decisions, the agent faces ignorance, not measurable risk.

The outcome of a divisible experiment, if frequency-ratios can be established, can be known in advance. It is a non-divisible experiment that confronts us with true uncertainty, that is, ignorance, about its outcome.

Shackle then contrasts two possible ways of using imagined outcomes in choice. An “integrative” solution would let many non-excluded hypotheses contribute additively to the decision. A “focus-values” solution instead concentrates on the few hypotheses with greatest stimulating force. His objection to integration is that mutually exclusive imagined futures cannot coherently be treated as simultaneous contributors to one sum.

To draw stimulus from the supposition that some one idea is true, and simultaneously to draw stimulus from the supposition that a second idea is true, when this second idea is a denial of the first, is rationally impossible.

The article’s structure thus moves from measure, to experiment, to choice. Shackle allows that special additive constructions may be formally devised, but he regards them as psychologically artificial. The more natural account of expectation is focal: what matters is not the sum of all imagined possibilities but the most compelling gains and losses that dominate the agent’s imagination. This makes the essay relevant as an early critique of expected-value reasoning under genuine uncertainty and as a foundation for Shackle’s broader theory of decision as imaginative appraisal rather than calculation.

But my belief in the focus-values solution, as opposed to the integrative solution, depends really on how I conceive the nature of expectation itself, as an act of creative imagination and not of ‘rational’ calculation; for calculation is impossible when the data are incomplete, and in face of ignorance, rationality is a mere pretence.