[BLML] Cheshire cat [SEC=UNOFFICIAL]

Alain Gottcheiner agot at ulb.ac.be
Fri Sep 10 10:22:28 CEST 2010


  Le 9/09/2010 19:42, Thomas Dehn a écrit :
> Alain Gottcheiner<agot at ulb.ac.be>  wrote:
>>    Le 9/09/2010 17:35, Jean-Pierre Rocafort a écrit :
>>> Alain Gottcheiner a écrit :
>>>>     Le 9/09/2010 15:54, Robert Park a écrit :
>>>>>      On 9/9/10 9:27 AM, Alain Gottcheiner wrote:
>>>>>>       Le 9/09/2010 15:09, Thomas Dehn a écrit :
>>>>>>> Alain Gottcheiner<agot at ulb.ac.be>      wrote:
>>>>>>>>        Le 9/09/2010 7:43, richard.hills at immi.gov.au a écrit :
>>>>>>>>> Richard Hills, 11th April 2005:
>>>>>>>>>
>>>>>>>>>> It is easy for apparently "simple logic" to conceal invalid
>>>>>>>>>> logical operations.
>>>>>>>>>>
>>>>>>>>>> For example:
>>>>>>>>>>
>>>>>>>>>> (Axiom 1) Nothing is better than eternal happiness.
>>>>>>>>>> (Axiom 2) A ham sandwich is better than nothing.
>>>>>>>>>> (Conclusion) Therefore, a ham sandwich is better than
>>>>>>>>>> eternal happiness.
>>>>>>>> AG : it's not logic that we should consider the culprit here ; it's
>> the
>>>>>>>> extreme ambiguity of words, especially English words.
>>>>>>> One would also have to prove mathematically that "is better than"
>>>>>>> is a transitive operation.
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>> It is usually considered that preferences are transitive.
>>>>>> For example, all of votation theory and 90% of economics and marketing
>>>>>> fall down if one uses a non-transitive preference relation (NB :
>>>>>> relation, not operation)
>>>>>>
>>>>>> It is an axiom, rather than a theorem, so no proof, sorry.
>>>>> Wrong. See, for example, "Mathematical Games," by Martin Garner,
>>>>> Scientific American, Oct 1974.
>>>>>
>>>>> As a simple example, he demonstrates an asymetrical dice game with 4
>>>>> dice (either there of in a related article) where we play for a dollar
>> a
>>>>> point and I let you choose your die first. Whichever die you choose, I
>>>>> can choose a better one.
>>>>>
>>>>>
>>>> AG : BTA this isn't a preference relation. A preference relates to what
>>>> a human being likes (as in the above problem), not to winning games.
>>>>
>>>> Seems to me that both attempted counterexamples just show that some
>>>> people don't understand that, when an object doesn't meet a definition,
>>>> it is normal that it doesn't share the attributes of those which do.
>>>> Would you consider butter as a counterexample of "stone is hard" because
>>>> it isn't hard ?
>>> agreed but, in the first place you were the one to restrict the question
>>> to stones, i mean to preferences. all relations of the form "is better
>>> than" are not connected to somebody's feelings.
>>>
>> AG : indeed. Let's put it another way.
>> All relations of the type "better than" are of the transitive type.
> Consider objects that have three different scores assigned
> to them.
>
> Object A has scores:  green 10, blue 20, red 30
> Object B has scores:  green 20, blue 30, red 10
> Object C has scores:  green 30, blue 10, red 20.
>
> An object is "better" than another object if it beats
> that object in two out of three categories. Doesn't that look
> like a reasonable definition of "better"?
>
>
AG : you could define it as such. Nobody can compel you.  But it won't 
satisfy the definition of such a relation, in classical mathematics, 
i.e. a (possibly partial) ordering relation.

I know the problem very well ; it is intended as an illustration of the 
so-called "Condorcet's paradox", which in fine tells us that we should 
be wary about what "better" in fact means.


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