Explosion and Cut Required
Joseph Vidal-Rosset
joseph.vidal.rosset at gmail.com
Mon Jun 6 04:40:57 EDT 2022
Le dim. 06/05/22 juin 2022 à 06:52:15 , "Tennant, Neil"
<tennant.9 at osu.edu> a envoyé ce message:
> Jo,
> The relevant notion of validity is relevant validity.
> A,~A:# is relevantly valid. A,~A:B is not.
> Neil
Hello Neil, hello everyone,
This is an irrelevant answer, because it is not logical relevance that
is at stake here. Nobody contests that A,~A:# is relevantly valid and
that A,~A:B is not. My point about these sequent is that, with
Heyting's logic, both are claimed, while with Johansson's logic, only
the former is valid, which means also A,~A:# cannot be said a
"subsequent" of A,~A:B, because the assertion of such a relation is
finally based on the use of the intuitionistic rule of absurdity.
Now, reading "Core logic" p. 316, I see that you will reply that ~A,A:#
is *perfectly* valid and that is this "more exigent notion of sequent
validity" that is Core logician's. "Perfectly" for you means
"relevantly". The trouble is it is indisputable that we find again,
even with this more "exigent notion of sequent validity", a difference
in treatment between what concerns sequents with an inconsistent set
of premises, and others. I quote the end of page 316:
#+begin_quote
3. If \(\Delta\) is unsatisfiable but every proper subset of
\(\Delta\) is satisfiable, then the sequent \(\Delta : \bot\) is
perfectly valid.
4. If \(\Delta : \psi\) is perfectly valid, then some atomic sentence
occurs within some member of \(\Delta\) and within \(\psi\).
#+end_quote
If I understand correctly your English, it seems to me that with
claim 4, \(\Delta : \psi\) is said "perfectly valid", because some
atomic sentence occurs within some member of \(\Delta\) and within
\(\psi\), by contrast with \(\Delta : \bot\) that is is perfectly
valid, "tout court". And I guess that you would reject the validity
(perfect or not) of \(\bot : B\) or that of \(\lnot A, A : B\). But
now, I wonder how the Core logician can justify the theorem of set
theory according to which \(\{\} \subset C\) for every set /C/?
Last, I repeat that it seems to me that such presupposed distinction
between consistent set of premises and inconsistent set of premises do
not allow to the Core logician to state consistency theorems
independently of Gentzen's, for example, and that such a distinction
is not in agreement with the usual rule of sequents systems, where
nothing is said about the consistency or inconsistency of set of
premises.
All the best,
Jo.
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