| 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. |