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Extra info for Stevens' Handbook of Experimental Psychology, Methodology in Experimental Psychology
Axioms extending the ordinal theory of semiorders to the kind of thing analyzed by Wiener (1921) are in Gerlach (1957); unfortunately, to obtain a full interval-scale representation with thresholds involves very complicated axioms. 3 of FM II. The axioms are complicated when stated strictly in terms of the relation of semiorders. Algebraic Theory of Thresholds Probabilistic Theory of Thresholds The work following Wiener on algebraic thresholds was only revived in the 1950s and may be found in Goodman (1951), Halphen (1955), Luce (1956), and Scott and Suppes (1958).
The atoms are ordered in qualitative probability by the relation ∼. We also introduce a restricted conditional probability. If i > j then ai | A j is the conditional event that the change of state will occur on trial i given that it has not occurred on or before trial j. ) The qualitative probability ordering relation is extended to include these special conditional events as well. The two postulated properties, in addition to (i), (ii), and (iii) given above, are these: (iv) Order property: ai ∼ a j iff j ≥ i; (v) Memoryless property: ai+1 | Ai ∼ a1 .
There, such statistical notions as the median and such elementary mathematical notions as that of continuity were assumed without further explanation or deﬁnition. Another ingredient, less important from a formal standpoint but of considerable importance in practice, are the questions of whether notions deﬁned in terms of the primitive concepts should be introduced when formulating the axioms and whether auxiliary mathematical notions are assumed in stating the axioms. The contrasting alternative is to state the axioms strictly in terms of the primitive notions.