Re: Conjunctions, conjunctive adverbs, subordinators

On 3/18/06, Eldin Raigmore <eldin_raigmore@...> wrote:
> On Thu, 16 Mar 2006 19:30:07 -0500, Patrick Littell <puchitao@...>
> wrote:
> >On 3/16/06, Patrick Littell <puchitao@...> wrote:
> >>You can make do with just IF and NOT;
> [snip]
> >It's amazing that we can get the entirety of sentential logic from
> >just these three.
> [snip]
>
> Actually we can "get them all" with just _one_; and there are two choices
> for the one; we can "get them all" with NAND, and we can "get them all"
> with NOR.
Actually, Eldin, we already covered NAND and NOR a few messages ago.

Incidentally, those "three" are axioms, not operators; when you say we
can "get them all" with one you're "getting" a totally different
thing.  (You may have been a bit too hasty with your snipping -- I
didn't say that "we can get all of SL with those three" meaning "IF
and NOT"...  That doesn't even make sense.)  You are correct in saying
NAND is sufficient for expressing a *truth-functionally* complete
system and so is NOR, but that's not what the above is talking about.

Instead, what we can do with the axioms I gave is derive every true
sentence of SL by universal substitution and modus ponens.  When you
claim that NAND and NOR can "get them all" in the same way as
Lukasiewicz's axioms... well, that's conflating two very different
sorts of completeness.

Now, we *can* axiomatize SL using only the Sheffer Stroke (NAND), and
there are several ways to do it with only one axiom.  It's one long
axiom, however; no matter how you do it I don't believe it can be
expressed in less than 23 characters (excluding parentheses).  Here's
one of our options, from Mordchaj Wajsberg:

  ( p | ( q | r ) ) | ( ( ( s | r ) | ( ( p | s ) | ( p | s ) ) ) | (
p | ( p | q ) ) )

Our inference rule is as follows: From p | ( q | r ) and p, infer r.

-- Pat