Chris Arthur
>At 01:18 PM 3/12/99 +1030, you wrote:
>>Gil's example is a good example of a relation that is not fully transitive.
>>"is a brother of" is only semi-transtivie, ie, If A does not = B, and B
>>does not = C, then if A is brother of B and B is brother of C, then A is
>>brother of C. Logically, if a relsation is symmetric and transitive, then
>>it is also reflexive. It is a moot point whether "exchange' is only
>>partially transtyive in the way that "brother of" is. In any case, the
>>formal definition of (fully) transitive doies sustyain the inference from
>>symmetry and transitiveness to reflexivity.
>
>Could Ian please define more clearly what he means by semi-transitive. In
>the example above with the relation "is a brother of", it is clear that the
>relation is not reflexive, contrary to Ian's claim.
>
>Let a, b, c be three distinct individuals.
>Let R be the relation "is a brother of."
>
>Symmetry says: If aRb, then bRa, and in this case R satifies symmetry.
>
>Transitivity says: If aRb and bRc, then aRc, and again in this case R
>satifies transitivity.
>
>Reflexivity says: aRa, and we know directly that R does not satisfy
>reflexivity as a cannot be the brother of a.
>
>Thus, R is not an equivalence relation in this case (NB, this is beside the
>point that Gil and I have been arguing about exchange)
>
>Now, Ian implies that contrary to our direct understanding about R, that if
>symmetry and transitivity obtain, then reflexivity must. Something must
>give. Consider the following:
>
>By symmetry, if aRb, then bRa.
>By transitivity, if aRb and bRc, then aRc.
>
>And, by symmetry, if aRc, then cRa.
>
>There is no way to get to aRa, from symmetry and transitivity. Hence, the
>claim is wrong in general and in particular aRa fails in the case of R
>being "is a brother of".
>
>I don't think this should come as a surprise as these axioms have been used
>by hundreds of social choice theorists, among others, who are extremely
>careful in not wanting to use redundant axioms.
>
>Also, I don't know who, or even if someone did, first made the claim that
>reflexivity and symmetry imply transitivity but that is also immediately
>false. Reflexitivity and symmetry are binary relations between two
>distinct elements. Transitivity requires three distinct elements.
>
>Steve
>
>
>
>
>
>>While having some sympathy with Gil's pointsd about the insufficiency of
>>Marx's vol 1 arguument, I have no sympathy with his invocation of Birkoff
>>and McLean's definition of "identity" within the context of set theory (ie
>>identity of sets) to argue that Marx cannot have it right in saying that
>>commodities are 'identical" ( of course commodities are not normally
>>'identical' in the sense of the "same thing" since when X is exchanged for
>>Y, X and Y are not the same thing - but this is so obvious that Marx was
>>surely aware of it). Marx's point is that they have the 'same value' and
>>that there is a third thing -not identical to either commodity exchanged -
>>that is 'identical' in the exchange and explains why it occurs as it does.
>>I do not think that this follows immediately, as Marx seems to suggest,
>>from the fact that the exchangers equate the worth of the articles that
>>they exchange, but it is possible to ask what are the principal
>>determinants at a time or over time of the ifferent proportions in which
>>commodities exchange. It is not self-evident that there is 'third', as Marx
>>seems to claim, but it is not self-evident that there is not something that
>>explains the broad patterns of exchamnge at a time or changes in those
>>patterns over time.
>>
>>>In a recent reply to one of my posts, Alan reads me as saying that
>>>reflexivity can be inferred from symmetry and transitivity:
>>>
>>>>I begin with an apparently minor point: as Gil points out (4B) reflexivity
>>>can
>>>>be deduced from symmetry and transitivity. (proof: suppose aRb, then bRa by
>>>>symmetry, hence aRa by transitivity). Steve makes the same point.
>>>
>>>[As I mentioned earlier, I don't "point this out", I say to do so would be
>>>a confusion. I suggest why below.]
>>>
>>>>Only one conclusion follows from the above result, namely, we can reduce
>the
>>>>axiom set by one axiom.
>>>>
>>>>This is an excellent result. It shows we don't need to imagine things
>>>>exchanging with themselves, to reproduce Marx's argument. Consequently,
>this
>>>>argument doesn't depend logically on something that can't happen.
>Excellent.
>>>>Wish I could say the same for neoclassical general equilibrium.
>>>>
>>>>The question for me is: Why does Gil have a problem with that?
>>>
>>>I'll illustrate: let R be the relation "is a brother to", in the sense of
>>>blood relations.
>>>In this case R is symmetric (if A is a brother to B then B is a brother to
>>>A) and transitive (if A is B's brother and B is C's brother, then A is C's
>>>brother), but not reflexive (I'm not my own brother). But by Alan's
>>>reading, reflexivity follows from symmetry and transitivity, so if I have a
>>>brother, it follows that I am also my own brother. This example
>>>illustrates my problem with Alan's reading. Gil
>>
>>
>>Dr Ian Hunt,
>>Associate Professor in Philosophy,
>>Director, Centre for Applied Philosophy,
>>Philosophy Dept,
>>Flinders University of SA,
>>Humanities Building,
>>Bedford Park, SA, 5042,
>>Ph: (08) 8201 2054 Fax: (08) 8201 2556
>>
>>
>#############################################
>Stephen Cullenberg Office: 909-787-5037, ext. 1573
>Department of Economics Fax: 909-787-5685
>University of California Email: stephen.cullenberg@ucr.edu
>Riverside, CA 92521 www.ucr.edu/CHSS/depts/econ/sc.htm