cbo@utai.UUCP (Calvin Bruce Ostrum) (10/21/85)
Richard Carnes writes: | Arrow's Theorem is one of the most striking results ever obtained by | an economist. The theorem originally appeared in Kenneth Arrow's | book *Social Choice and Individual Values*. | ...The proof is not very | long or difficult, but I will refrain from typing it in. Arrow's Theorem is certainly striking, but the proofs aren't *that* easy. Actually, most all "proofs" of Arrow's Theorem, including Arrow's original proof and his subsequent fixes, and the textbook versions (e.g. Sen's "Collective Choice and Individual Welfare" version) are incorrect. The errors in the proofs involve faulty attempts to use the axiom regarding independence of irrelevant alternatives. In fact, Arrow doesn't appear to use this axiom at all in his earliest proofs. The appeal to it is often clouded in so much informality that the error is not detected. The place to look for a correct proof (unfortunately, the theorem is true!) is in Richard Routley's article in Notre Dame Journal of Formal Logic, Oct79, "Repairing Proofs of Arrow's General Impossibility Theorem and Enlarging the Scope of the Theorem". Routley also has what appears to be some very interesting applications of this theorem in a paper in Logique et Analyse [sic!] sometime in 1980. The paper is titled "On the Impossibility of an Orthodox Social Theory and of an Orthodox Solution to Environmental Problems". I haven't read this last paper (yet); I would warn potential readers that Routley tends to be a little unusual. He is a staunch advocate of relevance logic (as an inordinate number of logicians from Australia seem to be!), and has claimed in a book that classical logic derives its major problems from being based upon capitalist ideology (I can dig out that reference for anyone whoso desires). -- Calvin Bruce Ostrum, University of Toronto Department of Computer Science uucp: { utzoo linus ihnp4 floyd allegra uw-beaver ubc-vision cornell watmath hcr decwrl }!utcsri!utai!cbo