[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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