Ch 4: Difference Measurement

Ch 4: Difference Measurement

Difference Measurement

• In Ch 3 we saw the kind of representation you can get with a concatenation operation on an ordered set A

• “The question arises whether similarly tight representations ever exist when there is no concatenation operation.”

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Difference Measurement

• Extensive measurement: consider a set of movable rods

• Difference measurement: consider fixed points on a line. Consider a set of intervals between points

• We can construct standard sequences in A with an auxiliary, uncalibrated rod to lay off equal intervals

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Difference Measurement

• Denoting elements of A by a, b, e, d, we denote intervals in A by ab, cd, etc.

• We distinguish between ab and ba.

• Comparison with a set of movable rods generates an ordering on the intervals in A.

• ab cd if some rod does not exceed ≿ab but exceeds or matches cd.

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Axiomatization of Difference Measurement

• Holder (1901) showed how the measurement of intervals between points on a line can be reduced to extensive measurement.

• Standard sequences of equally spaced elements a1, a2, a3, ..., where the intervals a1a2 ∼ a2a3 ∼ ...

• Equivalent intervals are identified with a single element, their equivalence class

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Otto Ludwig Hölder

Positive Difference Structures

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Positive Difference Structures

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Positive Difference Structures

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Interpret A as the set of endpoints of intervals. A* is the set of positive intervals, and is a subset of A x A.

Positive Difference Structures

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Transitivity

Positive Difference Structures

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Axiom 3 guarantees that there are no null intervals. Note it also follows that A* is not reflexive or symmetric.

Positive Difference Structures

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Weak monotonicity: this is needed to guarantee that concatenation of non-adjacent intervals gets the right results

Positive Difference Structures

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Archimedean axiom: ana1 = (n-1)a2a1

Positive Difference Structures

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Archimedean axiom: ana1 = (n-1)a2a1

Positive Difference Structures

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Positive Difference Structures

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Algebraic Difference Structures

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We now allow for negative and null intervals, so we don’t need A*.

Algebraic Difference Structures

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Axioms 2 and 3 of Definition 1 are here replaced by Axiom 2. It is a pretty intuitive axiom

Algebraic Difference Structures

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Axioms 3-5 are correspond to axioms 4-6 of Definition 1

Algebraic Difference Structures

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Cross Modality Difference Structures

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Solvability axiom: The first part says that any element in A i x Ai can be matched with an element in A1 x A1. The second part is just the normal solvability property for A1. But because of the first part, it follows that all the Ai have the solvability property. This is also why the Archimedean axiom is formulated for A1.

Finite, Equally Spaced Difference Structures

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Absolute-Difference Structures

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Axiom 3: Betweenness is well behavedi) If b is between a and c, and if c is between b and d, then c and b are between a and d. ii) If b is between a and c and c is between a and d, then ad exceeds bd

Absolute-Difference Structures

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Weak Monotonicity:If b is between a and c and b’ is between a’ and c’, and ab

a’b’, then bc b’c’ iff ac a’c’∼ ≿ ≿

Absolute-Difference Structures

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Solvability: if ab cd, then there is some d’ that is between a ≿and b such that ad’ cd∼

Absolute-Difference Structures

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Archimedean: ai is between a1 and ai+1 for all i, and successive intervals are non-null. aia1 is strictly bounded.

Absolute-Difference Structures

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