On EFQ, ECQ, and paraconsistency (with a FOM question)

[David Fuenmayor]

Regarding the recent discussion on the notions of EFQ and contradiction,
I'd like open a particular thread on their relation with paraconsistency
(with admin's permission ;)

A good thing about the literature on paraconsistent logics is that it has
served to clarify the distinction between EFQ (ex falso quodlibet) and ECQ
(ex contradictio quodlibet, aka. 'explosion').
To explain the difference I like to paraphrase EFQ as 'ex *falsum*
quodlibet', meaning that from a blatantly false/absurd statement (falsum)
everything follows.
Seen this way EFQ is always valid *by definition*, since this is how
'falsum' (which denotes emptyset/bottom in modal/algebraic logic
terminology) is defined.

Now, paraconsistent logics contest ECQ. They basically say that a
contradiction (when formalized using a paraconsistent negation '-') such as
A /\ -A does not necessarily entail falsum (or more algebraically: A /\ -A
noteq bottom).
More interesting is the case of the Logics of Formal Inconsistency (LFIs)
which in fact *conservatively* extend classical logic by a unary
'consistency' operator: 'o' (meaning that classical logic is a proper
fragment of any LFI: the classical negation '~' can in fact be defined in
terms of '-' and 'o', but we get no new theorems using the classical
signature).

Using 'o' you can selectively choose for which sentences A 'classicality'
is to be recovered (like when we can confidently assert that we have no
contradictory evidence for A in a database, see e.g.
https://philarchive.org/archive/CARTAP-8).
So in LFIs we can have, among others:

A /\ ~A = bottom (traditional ECQ using the classical negation '~')
A /\ -A noteq bottom
oA /\ A /\ -A = bottom (weaker varion of ECQ for paraconsistent negation
'-')
A noteq --A
oA --> (A = --A)

And now a FOM question:
Assume (some part of) mathematics formalized using this more expressive
framework. This can be done trivially by using the classical negation '~'
only, or, more interestingly, by employing the paraconsistent negation
'-' and e.g. assuming oA for all 'normal' or 'well-behaved' mathematical
sentences A.
Now, can you think of any sentence G in mathematics for which 'oG' may not
hold? What could be the implications?

Best
David
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