Re: [lojban] A or B, depending on C, and related issues

["Bob LeChevalier (lojbab)" <lojbab@lojban.org>]

At 02:22 PM 8/9/01 -0400, pycyn@aol.com wrote:
> 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.

Actually it did.  lu'a for selecting individuals from a set came *directly* 
from your posing this problem to me back in 1989 or so.  The other members 
of lu'a were added later.  I believe that Athelstan then demonstrated that 
we could match all 3 and 4 place truth functional connective truth table 
with no obvious limit to what we could handle in larger sizes being 
found.  The form translated as "1 from the set {coffee, tea} AND 1 from the 
set {sugar, cream} is an example of this solution.  sumti sets can include 
sets of propositions by using du'u or la'elu/li'u, which I think solves the 
first problem.

>   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,"

Which was precisely how we implemented it.

> 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 sure it would be interesting to note which multiple connective 
combinations actually have a use, and giving the set selection equivalents 
for them.

lojbab
--
lojbab                                             lojbab@lojban.org
Bob LeChevalier, President, The Logical Language Group, Inc.
2904 Beau Lane, Fairfax VA 22031-1303 USA                    703-385-0273
Artificial language Loglan/Lojban:                 http://www.lojban.org