Measurement theory rests on axioms stated entirely in terms of qualitative, empirically observable relations. These axioms specify the structural conditions that an empirical system must satisfy for a valid numerical representation to exist. Unlike the axioms of pure mathematics, measurement axioms make testable claims about the world — and their violations carry diagnostic information about the psychological attribute under study.
Core Axioms
Archimedean: for any a ≻ b, finitely many copies of the "step" from b to a can exceed any element
Solvability: for any a, b with a ≻ b, there exists c such that a ~ b ∘ c
Weak ordering requires transitivity and completeness of the qualitative relation. Transitivity is the most fundamental: if stimulus A is judged louder than B, and B louder than C, then A must be judged louder than C. Violations of transitivity — which occur in multi-attribute preferences and some perceptual judgments — indicate that the attribute cannot support even ordinal measurement.
The Archimedean axiom rules out infinitely large or infinitely small elements relative to others. In practical terms, it ensures that the attribute can be represented by real numbers rather than requiring exotic number systems. The solvability axiom guarantees that equations like a ~ b ∘ c always have solutions, ensuring the scale has no "gaps."
Domain-Specific Axioms
Beyond these universal axioms, specific measurement structures require additional conditions. Extensive measurement requires monotonicity and positivity of the concatenation operation. Conjoint measurement requires independence and the Thomsen condition. Difference measurement requires axioms about the ordering of intervals. Each set of axioms is tailored to the empirical operations available in a given domain.
The genius of the axiomatic approach is that every axiom is a testable empirical hypothesis. When axioms fail, the failure tells us something important about the psychological attribute: perhaps it is multidimensional, perhaps it involves context effects, or perhaps it requires a more complex representation than initially assumed.