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

mdoctorow at ca.rr.com mdoctorow at ca.rr.com
Sun Aug 22 12:09:39 EDT 2010


Good to see this, Steve.  But as before, if I can't get a copy of my own posting to your group, I won't participate.  I never knew whether my posts were accepted or not.  Also, I couldn't get reader posts other than from you.  Hopefully, I'll get this one back (if I'm still registered with your group).

Osher Doctorow
mdoctorow at ca.rr.com

---- Baynesr at comcast.net wrote: 
> 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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