Explosion and Cut Required

[Joseph Vidal-Rosset]

Le 16/06/2022 à 16:42, Mario Carneiro a écrit :

> I'm quite sure that "identity of indiscernibles", applied the way you
> are trying to do, is false in Core logic. For example, it is possible to
> derive A & ~A |- A in core logic but not F |- A even though |- (A & ~A)
> <-> F is derivable.

Wait. I carefully wrote  "Indiscernibility of Identicals", that is an
indisputable principle, and not its converse, the "Identity of
Indiscernibles" that is far more controversial. The former says that if
x = y, then  for any property F, F(x) iff F(y) (See Papineau,
Philosophical Devices, p. 84).  I based my argument on this principle:
if it is provable that sequent A and sequent B are interderivable, i.e.
equivalent, then  A is invalid iff  B is invalid too. That seems quite
reasonable.

Anyway, I agree about your examples and I am not surprised, but it is a
very bad point for Core logic. It also means that, on the left of the
turnstile (F & A) i.e. the conjunction of the falsity constant and atom
A, cannot be reduced to F by the Core logician. Indeed,

~ A & A |- A

is also (philosophical reluctance apart) equivalent to

(A -> F) & A |- A

and is therefore also reducible to

(F & A) |- A

but not to

F |- A

even if, semantically (F & A) is  equivalent to F.

The conclusion is that if it is true that truth tables are respected by
Core logic ("Core logic", p. 185), it is maybe true on the right of the
turnstile, but certainly not on the left. That is another false claim
made by the Core logician.
Last, note that this point does not prevent the Core logician  to also
claim the completeness of his logical system. That is certainly true,
but you also need to change what means for a logical system to be
"complete".

What else?

All the best,

Jo.