Introduction to measurement theory

Table of Contents

1. Introduction to the course
2. Relations and properties of relations
- 2.1. Definition and some properties
- 2.2. A zoo of relations
3. Fundamental measurement
- 3.1. Representation theorem
- 3.2. Homomorphisms and scale types
4. Representation of complete preorders
- 4.1. Arithmetic and infinite sets
- 4.2. The ≥ relation
5. When indifference is not transitive

1. Introduction to the course

This course is based on Fred Roberts’ book, Measurement Theory. Most of the exercices come from his book.

1.1. What

Measurement: represent aspects of a concrete system abstractly, using numbers, vectors…
Measurement theory: represent relations among objects using relations among numbers
Representability theorems
Measurement also includes metrology
Measurement theory ≠ Measure theory (which does not start from supposedly observable relations)

1.2. Examples

Measure pollution
Organize society so as to maximize global utility (The Methods of Ethics by Henry Sidgwick)
Distribute resources fairly (A Theory of Justice by John Rawls, Natural Justice by Ken Binmore)
Reduce inequality
Satisfy citizen’s preferences

1.3. Motivation

Introduce rigorous reasoning in decision making
Use mathematical language for precision
Mathematics only takes care of a small part! (Also: political science, sociology, psychology, economics…)
A rigorous introduction to some fundamental constructs of mathematics

1.4. Organization

S1, S2 (, S3) : Relations
S3, S4 : Fundamental measurement
S5 : exam
S6 : A simple representation theorem
S7 : Involving some intransitivity
S8 : exam

Each lecture: about 1h theory, 1h practice

Exams $e$ : average of both exams (equal weight)
Participation $p$ : each lecture, 0, + (significant intervention) or ++ (clever or committed intervention)
Notes in Asciidoc format through GitHub $n$ (talk to me first)
Final grade: weighted average of $e$ (9 points), $max (e, p)$ (3 points), $max (e, n)$ (up to 8 points)

Take notes

1.5. Primitive notions

Set, $\emptyset$
$\in$
$=$
Logic operators: $\neg$ , $\lor$ , $\land$ , $\Rightarrow$ , $\Leftrightarrow$ , $\exists$ , $\forall$
$\notin$
Produce other sets: ${x \in A | \dots}$ , $\cup$ , $\cap$ , $℘$ , $\times$
Also: primitive elements
Ordered pair, sets thereof, domain $𝒟$ , image $I$ , field $ℱ$

References

Refer to the delightful book of Halmos: Naive set theory.
For more context, read Logicomix.

2. Relations and properties of relations

2.1. Definition and some properties

Relation from $A$ to $A$ : subset of $A \times A$ (homogeneous relation, our focus)
Also called Binary relation on $A$
Relations between objects, persons, …
Notation: $a R b$ means $(a, b) \in R$

Directed graph

2.1.1. Properties of relations

In this document, properties of a relation $R$ are expressed with implicit qualifiers: a non bound letter $x$ is implicitly prefixed with $\forall x \in ℱ (R)$

2.1.2. Transitivity

Transitive relation: $a R b R c \Rightarrow a R c$

Transire, to pass to the other side, convey an action
Relates a to c through b

2.1.3. Reflexivity

$a R a$

2.1.4. Start exercices

See Exercices S1. TODO: up to and including exercice 3.

2.1.5. Irreflexivity

$\neg (a R a)$

A relation that is not reflexive is not necessarily irreflexive (similarly for other properties).

2.1.6. Preorder

Transitive and reflexive
(Also called: partial preorder, quasi order)

Here: mostly transitive and reflexive relations

Hasse diagram: $a R b, a \neq b$ , corresponds to a line descending from a to b.

2.1.7. Symmetry

$a R b \Rightarrow b R a$

2.1.8. Asymmetry

$a R b \Rightarrow \neg (b R a)$

2.1.9. Antisymmetry

$a \neq b \land a R b \Rightarrow \neg (b R a)$

2.1.10. Exercices

See Exercices S1.

2.2. A zoo of relations

2.2.1. Equivalence

An equivalence relation is a transitive (, reflexive) and symmetric relation.

Each element in an equivalence relation has an associated equivalence class. The set of equivalence classes, called the quotient by $R$ , is denoted by $ℱ (R) / R$ . It partitions (disjointly covers) $ℱ (R)$ .

Symmetric part of a relation $R$ : $s y m (R) = {(a, b) \in R | (b, a) \in R}$
The symmetric part of a preorder $R$ is an equivalence relation
Intuitively speaking, a preorder defines equivalence classes (namely, its quotient by $s y m (R)$ ) and orders them (possibly partially)
That ordering is a relation on its quotient by $s y m (R)$ and is called the reduction of a preorder $R$
Formally, the reduction of $R$ is defined as ${(a^{⋆}, b^{⋆}) \in (ℱ (R) / s y m (R)) ² | \exists a \in a^{⋆}, b \in b^{⋆} | a R b}$ .

2.2.2. Side-uniqueness, ontoness

Right-unique: $a R b \land a R c \Rightarrow b = c$
A right-unique relation is a function from $𝒟 (R)$ to $I (R)$ ; we can write $R (a)$ to denote the single $b \in I (R)$ such that $a R b$ .
Onto $Y$ (right-total over $Y$ ): $Y = I (R)$
Left-unique (injective): $b R a \land c R a \Rightarrow b = c$

2.2.3. Weak completeness

$a \neq b \Rightarrow a R b \lor b R a$

2.2.4. Order

Transitive, reflexive, antisymmetric
(Also called: partial order)

Intuitively: a (possibly partial) ranking without ties

2.2.5. Complete preorder

Transitive, reflexive, weakly complete
(Often called: weak order)
Intuitively: a ranking with ties
The symmetric part of a complete preorder is an equivalence relation
A complete preorder defines equivalence classes and orders them completely

2.2.6. Complete order

Transitive, reflexive, weakly complete, antisymmetric
(Also called: simple order, linear order, total order)

Intuitively: a ranking without ties

The reduction of a complete preorder $R$ is a complete order on $ℱ (R) / s y m (R)$ .

2.2.7. Generalisation to binary operations

A relation from $X$ to $Y$ is a subset of $X \times Y$ . It is non homogeneous when $X \neq Y$ .
A binary operation $α$ on $A$ is a right-unique relation from $A \times A$ to $A$ whose domain is $(A \times A)$ .
It can be viewed as a function from $A \times A$ to $A$ ; we can write $a α b$ to denote the single $c \in I (α)$ such that $(a, b) α c$ .
Examples: $+$ , $\times$ on $ℕ$ .

2.2.8. Note about terminology

For many authors (excluding Halmos but including Roberts), the set on which $R$ is defined is exogenous, thus a relation is a pair $(A, R)$ with $R \subseteq A ²$ (hence $ℱ (R) \subseteq A$ ). This allows for the possibility that $ℱ (R) \neq A$ . Weak completeness is then defined as $\forall a \neq b \in A : a R b ν b R a$ . Similarly, other definitions (such as reflexivity) then differ from those given here. In this document, we assume $A$ is chosen equal to $ℱ (R)$ , in which case the definitions coincide.

2.2.9. Exercices

See Exercices S2.

3. Fundamental measurement

We want to assign numbers to reflect some properties of some systems.
Given relation $R$ “looks shorter than” on $A = {a, b, \dots}$ , can we assign numbers $f (a)$ so that $f (a) < f (b) \Leftrightarrow a R b$ ?
Similarly for relations “preferred to”, “day with better air quality”.
We might also want to reflect operations such as “combining”: consider “is lighter”, with A including combined objects; can we then assign numbers $f (.)$ so that when $a$ and $b$ combined are lighter than $c$ , $f (a) + f (b) < f (c)$ ; or so that when $c$ denotes the combination of $a$ and $b$ , $f (a) + f (b) = f (c)$ ?
Similarly for relation “preferred to” on sets of objects.
Relation $R$ on $A$ corresponds to relation $T$ on $ℝ$ through function $f$ from $A$ to $ℝ$ : $a R b \Leftrightarrow f (a) T f (b)$ .
(If $T$ is restricted to the image of $f$ , it is determined uniquely by $f$ and $R$ , in other words, $R$ never corresponds to two relations $T 1 \neq T 2$ through a single function $f$ when $ℱ (T 1) = ℱ (T 2) = I (f)$ . Proof: if R corresponds to T1 and T2 through f with ℱ(T1) = ℱ(T2) = 𝓘(f), then x T1 y iff a R b, for any a ∈ f-1(x), b ∈ f-1(y), iff x * T2 y thus T1 = T2.)
Operation $⊙$ on $A$ corresponds to operation $α$ on $ℝ$ through function $f$ from $A$ to $ℝ$ : $f (a ⊙ b) = f (a) α f (b)$ .
$R$ is homomorphic to $T$ iff it corresponds to $T$ through some function $f$ .
$(R, ⊙)$ is homomorphic to $(T, α)$ iff $R$ corresponds to $T$ and $⊙$ to $α$ through the same function $f$ .
More generally, $(R, {⊙_{i}})$ is homomorphic to $(T, {α_{i}})$ iff $R$ corresponds to $T$ through some function $f$ and each $⊙_{i}$ corresponds to $α_{i}$ through $f$ .
The tuple $(f, (T, {α_{i}}))$ is called a measurement scale for $(R, {⊙_{i}})$ .

3.1. Representation theorem

A theorem of the form: under such conditions on $(R, {⊙_{i}})$ , the system is homomorphic to $(T, {α_{i}})$ .
Constructive proof: gives a procedure to build a scale $f$ .
Intuitively: transitivity of $R$ is required for homomorphism to $\geq$ .

Uniqueness: to determine properties of the numbers that transfer to our observations.

3.2. Homomorphisms and scale types

Number of persons VS height of a person
Ratio of weight VS ratio of t°

Given $(R, {⊙_{i}})$ corresponding to $(T, {α_{i}})$ through $f$ , admissible transformation $φ$ from $f (A)$ to $ℝ$ (thus $φ \circ f$ from $A$ to $ℝ$ ): $(R, {⊙_{i}})$ corresponds to $(T, {α_{i}})$ through $φ \circ f$ .

$R$ has a regular homomorphism to $T$ : it has a homomorphism to $T$ and for every scales $(f, (T, {α_{i}}))$ , $(g, (T, {α_{i}}))$ , $f (a) = f (b) \Leftrightarrow g (a) = g (b)$ .

Scale type depends on the class of admissible transformations φ.

Absolute: φ(x) = x; Counting
Ratio: φ(x) = rx with r > 0; Mass
Interval: φ(x) = rx + s with r > 0; T° without absolute zero
Ordinal: strictly monotone increasing transformation; Ordinal preference, Mohs scale of hardness
Nominal: any bijection; Labels

4. Representation of complete preorders

4.1. Arithmetic and infinite sets

Successor of $s$ : $s ⋃ {s}$
Axiom of infinity: ∃ successor set A
ℕ: intersection of all successor sets in A
Permits induction, which we use to define addition of zero, then addition of one, …
Finite set: bijection with an element of ℕ

4.2. The ≥ relation

If R is a complete preorder and ℱ® is finite, it is homomorphic to ≥.
Necessary and sufficient conditions.
Can we relax finiteness?

Scale type: ordinal

5. When indifference is not transitive

5.1. Requirement of transitivity

R homomorphic to ≥ requires transitivity.
Define I as the symmetric part of R.
R homomorphic to ≥ requires I to be transitive.

5.2. Examples

Detection threshold

Coffee with sugar
Noise level

Incomparability

Pony VS bicycle (Armstrong, 1939)
Good job VS apartment
Reform social security: better for end-of-life VS better life expectancy

5.3. Representation

x ≥δ y: x ≥ y - δ
Constant threshold δ

Won’t do for incomparability: Pony* ≻ Pony, Bicycle ≽ Pony*, Bicycle* ≻ Bicycle but Pony ≽ Bicycle*.

5.4. Completed semiorder

≽ is a completed semiorder: weakly complete and reflexive and satisfies

S2: a ≽ b ∧ c ≽ d ⇒ c ≽ b ∨ a ≽ d
S3: a ≽ b ≽ c ⇒ a ≽ d ∨ d ≽ c

Covers: a W c iff (d ≽ a ⇒ d ≽ c) ∧ (c ≽ d ⇒ a ≽ d).

S2 and S3 are together equivalent to ∀a ≻ b ≽ c: a W c.
S2 and S3 are together equivalent to ∀a ≽ b ≻ c: a W c.

Equivalently:

S2 can be written ¬(a ≽ b ≻ c ≽ d ≻ a) and S3 can be written ¬(c ≻ d ≻ a ≽ b ≽ c), writing x ≻ y iff ¬(y ≽ x).
S2 can be written ¬(d ≽ a ≻ b ≽ c ≻ d) and S3 can be written ¬(d ≻ a ≻ b ≽ c ≽ d).
S2 can be written ∀a ≻ b ≽ c: (d ≽ a ⇒ d ≽ c) and S3 can be written ∀a ≻ b ≽ c: (c ≽ d ⇒ a ≽ d).
S2 can be written ¬(c ≻ d ≽ a ≻ b ≽ c) and S3 can be written ¬(b ≽ c ≽ d ≻ a ≻ b).
Renaming, S2 can be written ¬(d ≻ a ≽ b ≻ c ≽ d) and S3 can be written ¬(d ≽ a ≽ b ≻ c ≻ d).
Thus, S2 can be written ∀a ≽ b ≻ c: (c ≽ d ⇒ a ≽ d) and S3 can be written ∀a ≽ b ≻ c: (d ≽ a ⇒ d ≽ c).
By contrapositive, using these two equivalences, ¬(a W c) ⇒ ∀b: (b ≽ c ⇒ b ≽ a) ∧ (a ≽ b ⇒ c ≽ b), in other words, ¬(a W c) ⇒ c W a.

If ≽ is a completed semiorder, then W is a complete preorder. Proof. Transitive: If c ≽ d then by b W c ≽ d we get b ≽ d and by a W b ≽ d we get a ≽ d. It is strongly complete, as seen just above.

If ≽ is a completed semiorder, then c W b ⇒ c ≽ b (and b ≻ c ⇒ b W c). Proof. c W b ⇒ (c ≽ c ⇒ c ≽ b) ⇒ c ≽ b (and use contrapositive and completeness of W).

Thm. If ≽ is a completed semiorder, then a W b W c ~ a ⇒ a ~ b ~ c, writing x ~ y iff x ≽ y ≽ x. Pr. Rewrite the hypothesis as c ≽ a W b W c ≽ a and obtain c ≽ b ≽ a by definition of W and a ≽ b ≽ c using W ⊆ ≽.

5.5. Representation theorem

If ≽ is a completed semiorder, then there exists a function f from ℱ(≽) to ℝ such that $f (a) \geq_{1} f (b) \Leftrightarrow a ≽ b$ .

5.6. Related applications

Liberal to conservative politics
Psychological stages of development
Chronology of archeological artifacts