Explosion and Cut Required - Small Touch-Up

[dennis.hamilton at acm.org]

Apparently UTF8 doesn't get through the list server unscathed.

-----Original Message-----
From: FOM <fom-bounces at cs.nyu.edu> On Behalf Of dennis.hamilton at acm.org
Sent: Thursday, June 16, 2022 09:56
To: 'Foundations of Mathematics' <fom at cs.nyu.edu>
Subject: RE: Explosion and Cut Required - Inference Systems

[orcmid] ...

[orcmid] > I am sorry to tell you that analogies, suggestions and feelings
cannot be considered as rebuttals.


In a logical system, given a claim,

      P1, p2, ..., pn |- s

If (p1) & (p2) & ... & (pn) is not satisfiable, there is *nothing*
**derivable**  from such premises.

It might as well be

                   bot |- s

since logical constants seem to be allowed and Tarski can be smiling down at
us.

And, of course, if the conjunction of premises is tautological, the claim
would be tantamount to

                     top  |- s

simply               |- s

I do not need to know the system of inference to know that, for a
recognizable logic that is sufficient for well-known propositional
situations, unsatisfiable conjunctions of premises do not justify deduction
of anything whatsoever.

Perhaps the system of inference (and associated logic?) is under-specified,
admitting such mishaps tantamount to bot |- s.  I do not know.  I am
reluctant to consider that Tennant is so careless.

I do know that there is a well-known label for computational procedures
(including inference rules taken as such) that do not reject such things.
It is called "garbage-in, garbage-out."

 I trust that I am being specific enough now.

Regards,

 - Dennis