I've come across something odd. If one wants to logically formulate trichotomy the following formulation is incorrect: $$ (\alpha \oplus \beta) \oplus \gamma $$
For all WFF's being $ T$ one gets that the entire formula is also $T$ (although we would like it be False).
The correct formulation for trichotomy is quite long - you have to have the 3 disjunctions and each one to affirm one case and deny the other two.
Yes, intuitively, since:
1) the two-place XOR can be interpreeted as 'exactly one'
and
2) the XOR is associative, and can thus be generalized to work with any number of argument
and
3) the AND and OR are associative as well, and can be generalized to 'all' and 'at least one' as applied to any number of arguments,
it would stand to reason that the generalzed XOR would be the 'exactly one' operator for any number of arguemnts.
But no!
OK, so what does the generalized XOR mean? Well, an XOR applied to any number of arguments would be true iff an odd number of them are true . Note that this works for two arguments as well. So, it does generalize. Just not to what you would initially think.