Re: what most every language is missing :-)

[Guy Steele - Sun Microsystems Labs <address@hidden>]

   Date: Fri, 28 Mar 2003 10:36:56 -0600
   From: Trevis Rothwell <tjr@acm.org>
   To: Noel Welsh <noelwelsh@yahoo.com>, ll1-discuss@ai.mit.edu
   Subject: Re: what most every language is missing  :-)
   
   
   Maybe if instead of giving a definite "true", it would work like
   this:
   
   	schrodinger's-cat =? dead -> "hmmm...  could be..."
   
   and
   
   	schrodinger's-cat =? alive -> "hmmm...  could be..."
   
   Either way, is there a semantic difference between saying "the cat
   might be alive" and "the cat might not be alive"?  Certainly the
   first one sounds more positive and the second one more negative,
   but is there an actual difference?
   
There can be.  Suppose you are doing evaluation with
a possible-world semantics, so that the value of
an expression is actually the *set* of possible
values that the expression might take on under a
"normal" semantics.  (This is a standard strategy;
think, for example, of the standard technique for
converting a nondeterministic finite-state automaton
into a deterministic one: the states of the DFA
correspond to *sets* of states of the NDFA, and the
rest is fairly straightforward.)

So, under this possible-world evaluation, the result of a
boolean expression could be {true}, {false}, {true,false}, or {}.

Then "the cat might be alive" is true if the possible-worlds
result of the expression "the cat is alive" is either {true}
or {true,false}, whereas "the cat might not be alive" is true
if the possible-worlds result of the expression "the cat is alive"
is either {false} or {true,false}.  So in this context the two
phrases do indeed have distinct meanings.  They describe two
different mappings from the possible-worlds domain

	{{}, {true}, {false}, {true,false}}

to the ordinary boolean domain

	{true, false}

All this is in turn related to the variety of modal logic
that introduces the "possibly" and "necessarily" operators
(which are usually symbolized by a square diamond and a square,
respectively).

--Guy