Foundations of Measurement Ch 3 & 4 April 4, 2011 presented by Tucker Lentz April 4, 2011 presented by Tucker Lentz

Agenda 11:00 Chapter 3: Extensive Measurement 12:40 break 12:50 Chapter 4: Difference Measurement 11:00 Chapter 3: Extensive Measurement 12:40 break 12:50 Chapter 4: Difference Measurement

Ch 3: Extensive Measurement

Closed Extensive Structure closed, i.e., when any two objects can be concatenated Connectivity: for all a, b ∈ A, a ≿ b or b ≾ a (p14) 73 No matter what the difference is between c and d, as long as a strictly exceeds b, there is some integer that when multiplied by the difference between a and b will swamp the difference between c and d. there are no negative nor zero elements

Closed Extensive Structure 74

Formal Proof of Theorem 1 80

is a simply ordered group iff is a simple order is a group If a ≿ b, then a○c ≿ b○c and c○a ≿ c○b. is also Archimedean if (with the identity element e) a ≻ e, then na ≻ b, for some n. Theorem 5 (Holder's Theorem) An Archimedean simply ordered group is isomorphic to a subgroup of, and the isomorphism is unique up to scaling by a positive constant.

LEMMA 1: There is no anomalous pair. (p 77) LEMMA 2: Every element is either positive, null or negative. (p 78) LEMMA 3: A, ≿, ◦ is weakly commutative (p 78) LEMMA 4: The relation ≈ on A x A is an equivalence relation (p 79) Formal Proof of Theorem 1

79

81 Lemma 6 and Theorem 2.5 prove the existence ofa real-valued function ψ on D such that such that for all [a, b], [c, d] ∈ D 80

Formal Proof of Theorem 1 Define ϕ on A as follows: for all a ∈ A, ϕ (a) = ψ([2a, a]). We verify that ϕ has the desired properties.

Informal Proof Sketch

Select any e in A; this will be the unit. For any other a in A, and for any positive integer n, the Archimedean axiom guarantees that there is an integer m for which me ≻ na. Let m n be the least integer for which this is true, namely, m n e ≻ na ≿ (m n - l)e. Thus, m n copies of e, are approximately equal to n copies of a. As we select n larger and larger, the approximation presumably gets closer and closer and, assuming that the limit exists, it is plausible to define 75

When concatenation is not closed B is the subset of A x A that contains the pairs that can be concatenated (82) 84

Extensive structure with no essential maximum The new associativity: if a, b can be concatenated, and a ◦ b can be concatenated with c, then any concatenation of a, b and c is allowed (82) 84

Extensive structure with no essential maximum Commutativity and monotonicity: If a and c can be concatenated, and a strictly exceeds b, then c and b can be concatenated, and the concatenation of a and c must exceed the concatenation of c and b. (83) 84

Extensive structure with no essential maximum Solvability postulate: there is no smallest element A that can be concatenated. (83) 84

Extensive structure with no essential maximum Positivity 84

Extensive structure with no essential maximum Archimedean axiom: In the earlier structure we defined na and then had for any b in A, there is an n such that na ≿ b. However, because of the restrictions on B, we may not arbitrarily concatenate elements in A. So axiom 6 defines a strictly bounded standard sequence, and assumes it is finite. (83-84) 84

Extensive structure with no essential maximum 85

Some Empirical Interpretations New concatenation operation: a * b is the hypotenuse of a right triangle formed by rods a and b This results in a structure A, ≿, * that satisfies the axioms of definition 3, and the resulting ψ is proportional to ϕ 2 88

Some Empirical Interpretations “To most people, the new interpretation seems much more artificial than the original one. In spite of this strong feeling, neither Ellis nor the authors know of any argument for favoring the first interpretation except familiarity, convention, and, perhaps, convenience...” 88

Some Empirical Interpretations 89

89

90 A, ≿, ◦ A, ≿ ’, ‖

Velocity In a Newtonian universe, a closed extensive structure A, ≿, ◦ could represent velocity For a relativistic universe we need to introduce extensive structures with essential maxima

Essential Maxima in Extensive Structures 92

92

92

92 Theorem 7 gives the representation required for relativistic velocities

Non-Additive Representations right-angled concatenation and relativistic velocity are two examples of non-additive representations  a third: Consider a positive extensive structure with a scale ϕ additive over  ; then ψ = exp ϕ is an alternative scale, which is multiplicative over . 100

Non-Additive Representations 100

Conventionality of Representations 102 “...despite its great appeal and universal acceptance, the additive representation is just one of the infinitely many, equally adequate representations that are generated by the family of strictly monotonic increasing functions from the reals onto the positive reals...”

Extensive Measurement in the Social Sciences In many cases, there is no concatenation operation appropriate to the entities of interest in social science However, an empirical concatenation operation is not necessary for fundamental measurement There are cases where it works, e.g., subjective probability (ch 5) and risk (3.14.1)

Risk 125

126

127

127

Limitations of Extensive Measurement While I am certain we won’t have time to discuss it, I highly recommend reading the 2 page discussion of this topic (pages ).

End