[OPE-L:7164] [OPE-L:671] Re: Re: I'm my own brother!

Ian Hunt (Ian.Hunt@flinders.edu.au)
Mon, 15 Mar 1999 12:16:02 +1030

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.
>>>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
>>>>>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
>>>>>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,
>>>>>argument doesn't depend logically on something that can't happen.
>>>>>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