A topic that has really been on my mind as of late is the question whether the Law of Noncontradiction, viz. ~(A & ~A) should be a theorem of a Dialectical Logic. For our purposes, let us define a Dialectical Logic as a logic which is Paraconsistent, contains Adjunction in rule form i.e. A, ~A / A & ~A, and which is simply inconsistent (i.e. contains at least one thesis of the form A & ~A.

Now clearly there are a range of logics which fit within this criteria, particularly Deep Relevant Logics, Paraconsistent Many-Valued Logics, and the Logics of Formal Inconsistency. The question under consideration is whether the LNC should be a theorem of a suitable Dialectical Logic? Note that the LNC is by default not a theorem of the Logics of Formal Inconsistency, while is it by default a theorem in the Paraconsistent Many Valued Logics. Deep Relevant Logics are a mixed bag; some of them contain the LNC as a theorem, while others do not.

To make things fully transparent, I should say that I am of the opinion that a decent Dialectical Logic will include the LNC as a theorem. But it is something I am still working through. But more importantly, why might someone think otherwise?

More precisely, why might someone think that the LNC should not be a theorem of Dialectical Logic? Well there is one obvious reason; namely, if we have already accepted theses of the form A & ~A, then it would seem to be the case that the LNC is already thereby rejected. Moreover, if we are interested in formalizing dialectics of the Hegelian or Meinongian varieties, then it would only seem natural to reject the LNC as a theorem, since that is what these men did.

But I have 2 overriding reasons why I think the LNC should be maintained as a theorem. Firstly, if the LNC is not a theorem, then this must mean that the negation operator under consideration is something radically different from what it is standardly taken to be; namely, a contradictory-forming operator. If the negation operator in our logic does not adhere to the contradictoriness relation familiar from the Traditional Square of Opposition, then just what exactly is it supposed to mean?

Secondly, the idea that if we have theses of the form (A & ~A) then we should reject ~(A & ~A) is based upon a Consistency Assumption. More precisely, the claim presumes that if a given proposition A is true, then it follows that ~A must be false. But this is exactly what we have rejected in formulating a Dialectical Logic. So it follows by Reductio Ad Absurdum that the original claim is false.

But I am aware that this is a complex issue and the Dialectical Logics which do not contain LNC as a theorem certainly have a lot going for them. What do you guys think?