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A theory is
trivial if one can deduce from it any formula of its underlying
language, is inconsistent if from it one can deduce a contradiction,
and is paraconsistent if it is inconsistent yet non-trivial. It is
obvious that every trivial theory is inconsistent; we also know that for
a classical theory the converse holds equally true. Therefore our surprise
to find Wittgenstein, already at the 1930s, in comments and lectures given
on the foundations of mathematics, as well as in other writings, counseling
a certain tolerance on what concerns the presence of contradictions in a
mathematical system. "Contradiction. Why just this spectre? This is
really very suspicious." (Philosophical Remarks III-56)
It is utile thus to investigate the hypotheses that Wittgenstein would have
formulated the idea of a paraconsistent logic, or that one could reasonably
read this idea from Wittgenstein's works. It is also worth checking whether
the paraconsistent logic could be held as a justification of certain Wittgensteinian
ideas about the relatively inoffensive character of contradictions, and which
points are common between these ideas and those expressed on the project
of paraconsistency. Furthermore, we will understand Wittgenstein's position
regarding contradiction and inconsistency even better if we investigate its
origins, as well as its consequences for the mathematical praxis.
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