Re: [lojban] Sets etc.

[Peter Moulder <reiter@netspace.net.au>]

From: Peter Moulder <reiter@netspace.net.au>

John Cowan <jcowan@reutershealth.com> writes:

> Which leads me to the following challenge, due to Quine.  A student at Yale
> may belong to zero or more clubs.  Some clubs are final clubs.  A final
> club is defined as "a club such that membership in it precludes membership in
> any other final club".
  ^^^
Does `any' above mean `all' or `some'?  I've taken it to mean `all'.

Consider the case where there are two or more clubs, and no club
precludes membership in any other club.  What answer would you want
the "definition" to give in this case?

Let's split the problem in two: decide on an answer in predicate logic
notation first, then translate to Lojban once we're satisfied with a
predicate logic answer.

Define a predicate final(C, E, F) such that if C is the set of clubs
at Yale, and E is the set of pairs of clubs (C1, C2) such that
membership of C1 precludes membership of C2, then final(C, E, F) is
true iff F is a valid set of final clubs for C, E.

Notation of the below is a mixture of LaTeX and Mercury/Prolog;
more detail is at end of message.

Domain of `final' is the set of (C, E, F) such that:
 C is a set of clubs,
 F \subseteq C,
 E is a set of pairs of clubs such that
  all[C1, C2]
   ((pair(C1, C2) \in E)
    => ((pair(C2, C1) \in E),
        (C1 \in C),
        (C2 \in C))).

For all values of (C, E, F) within the above domain:

 final(C, E, F) <=>
   (all[C1]
     ((C1 \in C)
      => ((C1 \in F)
          <=> (all[F1]
                ((F1 \in F)
                 => ((C1 = F1)
                     ; (pair(C1, F1) \in E)))))).


For any (C, E, F) within the domain, the above definition allows
deciding whether or not F is a valid set of final clubs for C, E.

pjm.


Notation (mixture of LaTeX and Mercury/Prolog):
  P1 => P2              ganai P1 gi P2
  P1 <=> P2             iff: go P1 gi P2
  P1, P2                and: ge P1 gi P2
  P1 ; P2               or:  ga P1 gi P2
  E1 \in S1             E1 is an element of S1: E1 cmima S1
  S1 \subseteq S2       S1 is a subset of (or equal to) S2:
                         ga S1 gripau S2 gi S1 du S2
     [I'm not sure whether {gripau} means a proper/strict subset or
      whether it means the same as \subseteq.]
  some[V1] P1           existential quantification: su'o V1 zo'u P1
  all[V1] P1            universal quantification: ro V1 zo'u P1
  all[V1, V2] P1        abbreviation of `all[V1] all[V2] P1'
  pair(E1, E2)          E1 ce'o E2

Notation used in the above notation explanations:
 V1, V2: any variable names
 P1, P2: any predicate expressions; can be any item of the Notation
          list other than the last (viz. pair(E1, E2)).
 S1, S2: any sets
 E1, E2: elements: pretty much any expressions.


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