# [hist-analytic] Proof Sketches: Kenneth Arrow's Lemmas

Baynesr at comcast.net Baynesr at comcast.net
Sun Aug 22 11:35:01 EDT 2010

```Kenneth Arrow gives proof sketches of a number of lemmas in his groundbreaking work:

Social and Individual Values
by Kenneth Arrow

Cowles Foundation #12 [1951] 1963 pp. 14-15.

Below, I fill out some of the details of a number of the proofs.

Axiom 1: (x)(y)(xRy v yRx) ('R' connected - Tarski)
Axiom 2: (x)(y)(z)(xRy & yRz -> xRz

Def. 1: xPy = df. ~yRx
Def. 2: xIy = df. xRy & yRx

Lemma b: xPy -> xRy

Proof of Lemma C: xPy & yPz -> xPz

1. xPy & yPz
(Assumption)

2. xPz
(1, Axiom II)

3. zRx
(Assumption)

4. xPy
(1, Simplification)

5. xRy
(4, Lemma b)

6. zRy
(3,5, Axiom II)

7. yPz
(1, Simplification)

8. ~zRy
(7, Df1: xPy =df ~yRx)

9. zRy & ~ zRy
(6,8, Conjunction)

10. zRx -> zRy & ~zRy
(Conditional Proof: 3-10)

11. zRx & ~zRy
(3,10 Modus Ponens)

12. ~zRx
(3-11, Reductio)

13. xPz
(12, Def. 1)

14. xPy & yPz -> xPz
(1-13 Conditional Proof)

Proof of Lemma d: xIy & yIz .-> xIz

1. xIy & yIz
(Assumption)

2. xIy
(1, Simplification)

3. xRy & yRz
(2, Def. 2)

4. xRz
(3, Axiom II)

5. yIz
(1, Simplification)

6. yRx
(2, Def. 2, Simplification)

7. zRy
(5, Def. 2, Simplification)

8. zRx
(6, 7, Axiom II)

9. xRz & zRx
(4, 8, Conjunction)

10. xIz
(9, Def. 2)

11. xIy & yIz .-> xIz
(1-10, Conditional Proof)

Proof of Lemma f: xPy & yRz .-> xPz

1. xPy & yRz
(Assumption)

2. zRx
(Assumption)

3. yRz
(1, Simplification)

4. yRx
(2,3, Axiom 2)

5. xPy
(1, Simplification)

6. ~yRx
(5, Df. 1)

7. yRx & ~yRx
(4,6 Conjunction)

8. ~zRx
(2-7, Reductio)

9. xPz
(8, Def. 1)

10. xPy & yRz .-> xPz
(1-9 Conditional Proof)

This is not standard format with bells and whistles, but it may be of some use to people
wading through Arrow's micro-explanations!

Steve Bayne
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