Explosion and Cut Required

[Vaughan Pratt]

While I haven't engaged in earnest with the core of Core Logic, my previous
remarks taking Neil's side have been based on peripheral aspects of it.
(Full disclosure: while I have no other connection with Neil, he was kind
enough to fill in for me in June 1992 as an invited speaker at a conference
in the Netherlands taking place four weeks after my quintuple bypass
surgery.  The doctors were ok with my giving invited talks at LICS'92 in
Santa Cruz six weeks later, which Jurek Tiuryn kindly drove me to, and at
LFCS'92 in Tver, Russia eight weeks later where my whole family accompanied
me, by which time I felt fine.  Conference co-organizer Anil Nerode took
his family too and the eight of us shared a little cottage in a remote
resort for the Russian middle class hosting the conference on the Volga.)

Noticing Peter Smith's 2016 "Note on Core Logic" I asked Peter what he
thought of the present discussion and he said (in effect) that it sounded
as though things hadn't changed much in the intervening six years.

Based on page 3 of Peter's note (easily googled for), and writing CA for
A&~A, denoting a Contradiction involving A, it occurs to me that perhaps
there is a distinction between CA -->  and CA  |--   that could be drawn.
(I'm assuming A&~A |-- and A,~A |-- denote the same thing, and likewise
for  :  and |-- .)

CA is what Harvey calls an explosion, which I'll shorten to a bang.

Could it be that CA --> is a Little Bang while CA  |--  is a Big Bang?

And if so, how should we think about that distinction?

Little bangs can happen all over the place while other things are going on
independently and therefore don't notice them and aren't bothered by them.
Little bangs result from pursuing a local counterfactual that blows up and
can then be forgotten about.  They are the supernovas of the logical
universe if you will.

But the Big Bang (emphasis on "the") is applicable to the entire universe.
It can be viewed as the heat death of the whole universe which somehow is
the mirror image or dual of the beginning of another universe at an
insanely high temperature.

While Core Logic is fine with little bangs going off, it eschews the
possibility of the Big Bang ending everything.

If that's not how to think of this particular aspect of Core Logic I'll
keep trying to understand it better.

But if it is, then it would be nice to have a short proof showing that
while CA --> can arise in Core Logic, CA |-- cannot.  Or however Core Logic
distinguishes them.

Vaughan Pratt
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