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

David Fuenmayor davfuenmayor at gmail.com
Mon May 23 05:43:13 EDT 2022

```Regarding the recent discussion on the notions of EFQ and contradiction,
I'd like open a particular thread on their relation with paraconsistency

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
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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