[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
>
> Steve Bayne

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