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