Steve
At 12:16 PM 3/15/99 +1030, you wrote:
>Chris is right to say that "is a brother of" is not symmetric. It therefore
>cannot be a counterexample to the proof of reflexivity from transtivity and
>symmetry, that Chris spells out, answering Steve's point. Steve seems to
>assume that if we use different letters, we are referring to different
>things, but this is a use-mention confusion. "The evening star" refers to
>the same body as "The morning star" refers to. Take the relation "is at
>least as big as", which is symmetric and transitive. "aRB and bRc" can in
>this instance be ""The evening star is as least as big as the morning star,
>and the morning star is as least as big as the evening star, therefore the
>evening star is as least as big as the morning star". In saying that "is a
>brother of" is only semi-transitive, I was only making explicit the
>assumption of non-identity between a, b & c that Steve assumes. I can't
>understand Peano's objection that Chris refers to.
>
>So we have two reasons for taking "is a brother of" as not a very fortunate
>counter-example to the claim that symmetry and tyransitivity entail
>reflexivity. It is not symmetric, and it is only transitive between
>distinct individuals.
>
>>Four points:
>>1. Sexism is showing - the relation 'brother to' is transitive but not
>>symmetrical because I am brother to my sister but she is not brother to me.
>>2. it is often held that relexivity is deducible from transitivity and
>>symmetry as follows
>>by symmetry we have aRb and bRa
>>then by transitivity if aRb and bRa then aRa.
>>According to Russell's discussion (Principles of Mathematics pp218ff) Peano
>>objected to this that there may not be a 'b' so the proof falls.
>>3. In one of his examples Russell seems prepared to accept a man can be his
>>own brother but this seems to me to be mad so I think this is a good
>>counter-example to the generality of deducing reflexivity.
>>4. In relation to the more general discussion Russell argues as follows:
>>As an axiom I hold the principle of abstraction according to which "Every
>>transitive symmetrial relation, of which there is at least one instance ,
>>is analysable into joint possession of a new relation to a new term, the
>>new relation being such that no term can have this relation to more than
>>one term, but that its converse does not have this property."....in common
>>language that transitive symmetrial relations arise from a common
>>proerty....It gives precise statement to the principle that symmetrical
>>transitive relations always arise from identity of content.
>>However it is clear to me that in our case this common property is that of
>>value in the first instance, and additional argument is required to assert
>>the substance of value is labour. As it stands this value could be the
>>result of the intersection of preference schedules.
>>In my view the key move is when with the capital form we reach a relation
>>in whcih value is its own end.
>>
>>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
>
>
>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