There is no inconsistency either in or about Core Logic.
joseph.vidal.rosset at gmail.com
joseph.vidal.rosset at gmail.com
Sat Jan 16 11:30:28 EST 2021
Le sam. 01/16/21 janv. 2021 à 02:54:27 , "Tennant, Neil"
<tennant.9 at osu.edu> a envoyé ce message:
> Jo,
>
> You have claimed now that
>
> "in your system [of Core Logic] you cannot show in one proof that
> (3) ~A -> (A -> B), ~ A, A |- B
> is provable and that
> (2) ~A -> (A -> B)
> is a theorem."
>
> My response is: So what?!
Dear Neil, fair enough, but what I claim is always the same point: the
*occurrence* of Core theorem (2) i.e. (~A -> (A -> B)) in the proof of
sequent (3) is unprovable in Core logic.
So what?
First, I believe that this idiosyncrasy of Core logic is in
contradiction with the usual definition of what is a theorem in natural
deduction which is not based, as far as I know, on the definition of
normal proof. But of course, "In logic, there are no morals", to quote
the famous Carnap's word, and you are free to define your rules and to
give your own definition of what is a Core theorem.
Nevertheless, and it is my second and last point, it seems to me that the
composition of proofs and their normalization when these proofs are not
normal is a basic property of natural deduction and of mathematical
logic in general. The Core logician can prefer breaks this composition
property, but I am quite convinced that many other logicians, like
Bartleby the Scrivener, would prefer not to.
All the best,
Jo.
PS: Because, I admit after discussion that it can be considered as a
fallacy to call about an "inconsistency" about Core logic, I am going to
delete this note on my blog, as I promised, and I will try maybe to
write another text that even you, the Pope of Core logic, will not
contest. It was yesterday the World Logic Day and I believed that it was
finally also the Core Logic Death... I was wrong. (Sorry, I was too
proud to make this bad pun in a language that is not my native
language! I was only joking.)
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