Subject: A or B, depending on C, and related issues

From: [email-masked]
Date: Thu, 9 Aug 2001 14:22:54 EDT

There are 256 three-placed truth functions (8 lines each capable of bing 
filled in two ways, each line independently of the others). There are three 
times as many ways to join three sentences using two two-place truth 
functional connectives. So, it looks like there ought to be a way to express 
any three-place truth functional connective using only three sentences and 
two two-place connectives. But it doesn't work; too many of the reduced 
forms produce the same function. As a result, in Lojban, we have often to use 
three truth functions and four sentences (one repetition or denial) to 
represent some relations among three sentences. Indeed, we sometimes need 
even more complex forms.
>From time to time we have considered either devising three-place truth 
functional operators or working up non-truth-functional (officially) ways of 
dealing with larger cases (threes and on up). Neither of these projects has 
ever come to any official product that I can find. The first (three-place 
functions) runs into serious grammatical issues, not to mention logical ones 
of grouping and the like. The second typically involves a set of sentences 
and a selector of some sort: "exactly one of the following three," for 
example -- the generalization of exclusive "or" -- leaving the issue of what 
the members of the set are exactly and what the grammar is to be.
A third approach has been to look for the simplest versions of interesting 
cases might be. Surely among the first to be dealt with would be the two 
versions of "if P then Q, else R," which are also pleasantly simple:
(if P then Q) and (if not P then R)" and "(P iff Q) and (not P iff R)" While 
I am sure there are easier ways to show that these are the simplest forms for 
these functions, I confess to just having run all the possibilities from 
disjunctive normal forms on down.

I'm not quite sure what this has to do with anything, but it arose in some 
other discussion -- perhaps in a more general form -- and perhaps this 
answer, which is easily generalizable though with rapidly lengthening 
results, will help.