[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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