Paraconsistency
Justification logic offers a well-principled approach to paraconsistency, which looks for noncollapsing logical ways of dealing with contradictory sets of assumptions, e.g.,
The following obvious observation shows how to convert any set of assumptions
into a logically consistent set of sentences while maintaining all the intrinsic structure of Γ.
Informally, instead of (perhaps inconsistently) assuming that Γ holds, we assume only that each sentence A from Γ has a justification, i.e.,
It is easy to see that for each Γ, the set
is consistent in what will be our basic justification logic J.
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