# Calc. Smith set

Bruce Anderson landerso at ida.org
Fri Apr 19 21:47:08 PDT 1996

```On Apr 19,  1:07am, Mike Ossipoff wrote:
> If "Any alternative is a member of the Smith set if it beats or
> ties a member of the Smith set" is vague, then could you tell us
> more than 1 meaning that it could have. If not, then it's you who
> are being vague.
>-- End of excerpt from Mike Ossipoff

As a declarative sentence, the quote above is not vague.  As a declarative
sentence, it's accurate, precise, and concise -- I couldn't have said it better
myself.  However, as an instruction in a procedure, it's vague -- it doesn't
clearly indicate what action the procedure should take at this point.  To
repeat, the context is:

On Apr 16,  4:55am, Mike Ossipoff wrote:
> Subject: Smith procedure briefly stated
> I'd like to state, more briefly & clearly, the procedure that I suggested
> for determining the Smith set:
>
> 1. Order the alternatives according to how many alternatives they're
> beaten by.
>
> 2. The alternative beaten by fewest alternatives is a member of the Smith set
> (as are any several that tie for that distinction).
>
> 3. Any alternative is a member of the Smith set if it beats or ties
> a member of the Smith set, or if it's beaten by no more alternatives
> than is a member of the Smith set.
>
> So admit alternatives to the Smith set according to these rules till
> there are no more that qualify.
>-- End of excerpt from Mike Ossipoff

The potential problem here is as follows.  Candidate i might not beat or tie any
member of the current Smith set, so i would not be admitted.  But then the next
candidate in the order, say j, might be admitted, and i might beat j.  So a
proper algorithm might have to go back and reconsider all previously rejected
candidates every time a new candidate was admitted.  I think that this is quite
a lot to have to read into the "state[ed] more briefly & clearly" procedure
above, which is why I called it vague.

On Apr 19,  1:07am, Mike Ossipoff wrote:
> You said that "Any alternative is a member of the Smith set if it's
> beaten by no more alternatives than is some member of the Smith set"
> is "...a tautology at best". What does that mean?
>-- End of excerpt from Mike Ossipoff

Since any alternative is beaten by no more alternatives than it (itself) is
beaten by, this statement seems to me to be logically equivalent to the
statement that "any alternative is a member of the Smith set if it's
a member of the Smith set".  That's a pretty good example of a tautology.

Bruce

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