Meditations on Dialectical Logic I: Should the Law of Non-Contradiction be a Theorem of Dialectical Logic

(This will be the first in a series of posts which will deal with various aspects of Dialectical Logic).

The question to consider is whether the Law of Non-Contradiction, hereinafter the (LNC), should be a theorem of a dialectical logic. Before we begin, let us have some preliminary understanding of what a dialectical logic is supposed to be. In what follows, we will understand a dialectical logic to be any logic that is paraconsistent, simply inconsistent, and contradictorial. Allow me to explain what these terms mean:

1. A paraconsistent logic is any logic which does not contain the Spread Rule, viz. A & ~A / B. I prefer to use the term "Spread Rule" here because, as we will see, there are some dialectical logics which include EFQ as a theorem, viz. (A & ~A) -> B. It is quite reasonable to expect a dialectical logic to be paraconsistent, since if it weren't, we would be lead at once to Trivialism.

2. An simply inconsistent logic is one which includes theorems of the form A & ~A. Thus, we might also say that a simply inconsistent logic is one wherein that are theorems which are both true and false at the same time and in the same respect. 

3. A contradictorial logic is any inconsistent logic which has the Adjunction rule, viz. A, ~A / A & ~A. This precludes a number of paraconsistent logics, such as non-adjunctive systems and preservationist logic, from being dialectical logics;  since these systems only allow for distributive contradictory statements, while simple contradictions on these systems immediately explode.

So, with that being said, which paraconsistent logics can count as dialectical logics? Well, that would be the Logics of Formal Inconsistency (LFIs), the many-valued paraconsistent systems, and the deep relevant logics. Clearly, all of these logics are both paraconsistent and contradictorial, and they all can perfectly well be simply inconsistent. The many-valued paraconsistent systems are simply inconsistent by design (due to the inclusion of a paradoxical truth-value), but we can ensure the simple inconsistency of the LFIs & the deep relevant systems by the inclusion of a determinate contradictory thesis within the axioms, such as p & ~p. 

So we have our categorization of dialectical logics, now we need to get clear at what our question exactly is. What do we mean by the LNC? For the purposes of this essay, we will be considering the LNC in its syntactical formulation, i.e. we will be asking the question whether ~(A & ~A) should be a theorem of dialectical logic.

So to begin, let us consider the reasons why someone might think the LNC should not be a theorem. Newton Da Costa, one of the pioneers of paraconsistent logics, included this as one of the adequacy criteria for a dialectical logic. If we're ready to countenance some sentences of the form A & ~A, then it does at first glance seem reasonable to conclude that ~(A & ~A) should therefore not be a part of our dialectical logic. When we begin to dig into the motivation behind this worry, it seems that the operating assumption here is that negation must function radically differently under dialectical logic.

What is more, if we are particularly interested in providing formal analyses of for example Hegelian dialectics, meaning we want to adhere as closely as we can to what the man himself thought, then it might only seem natural that we should reject the LNC as a theorem. For Hegel himself explicitly rejects this principle in the Science of Logic, so shouldn’t a formalized Hegelian dialectical logic also reject the LNC? It is a similar story for trying to formalize Buddhist logic. For, as we have discussed in previous posts, the Catuskoti explicitly rejects both the LNC and the Law of Excluded Middle. So it seems that a dialectical logic without LNC would also be the right tool to use in this scenario as well.

There is also a third argument we can give. Namely, as dialectitians we might be concerned with limiting the amount of contradictions in our theory. For if we do have the LNC as a theorem, then for any contradictory thesis of the form A & ~A, we will have another contradictory thesis of the form (A & ~A)& ~(A & ~A). But since this is a new contradictory thesis, we will have yet another thesis of the form ((A & ~A) & ~(A & ~A)) & ~((A & ~A) & ~(A & ~A)), and so on, ad infinitum. One might find this result objectionable, and thus rejecting the LNC as a theorem would be a natural way to contain it.

Now let us consider the reasons why a dialectician might want to include the LNC in his logic. The first and most obvious reason is that we want to ensure that the contradictions we are making true are actual contradictions; and the best way to do this is to ensure that the negation in our logic is a contradictory forming operator. To make this more concrete, let us consider the familiar example of the square of opposition. Recall that in the traditional square, the diagonal corners form a contradictory relationship, typically explained as the impossibility of the opposite corners having the same truth values in the same way at the same time; or, in symbols, ~(A & ~A). This seems to be as solid an understanding of contradiction as one is going to find. So if we want to include contradictory theses in our system, we had better ensure that such theses really are contradictory.

But there is also a second reason why we should include the LNC as a theorem. Namely, the thought that the inclusion of theses of the form A & ~A in our logic necessitates a rejection of ~(A & ~A) is nothing more than the Consistency Assumption. More precisely, it embodies the belief that if we have accepted a certain thesis A, then we are thereby obliged to reject ~A. But this sort of reasoning is just what we have rejected in formulating a dialectical logic, so it would seem that we have a nice reductio argument on our hands.

Similarly, if we have fully rejected the Consistency Assumption, then we should have no qualms with the infinite number of contradictory theorems that a ground-level contradiction will produce. What we as dialectitians should really be focused on is limiting the spread of explosion, which is already taken care of by the paraconsistent nature of the logic.

So with all that being said, which logics among the LFIs, deep relevant systems, and the many-valued paraconsistent systems would fit our criteria for an adequate dialectical logic? Well, the LFIs are can be immediately ruled out, since one of the central features of these logics is that the LNC is not a theorem. On the other hand, the many-valued paraconsistent systems, (LP, RM3, A3) all count as adequate dialectical logics under out criteria. What about the deep relevant systems? There are many such logics, not all of which meet our standards. The crucial feature of the ones that do is the inclusion of LEM (A v ~A) as an axiom, from which the LNC can then be derived.

This is a complex issue, and I don’t pretend to have resolved it here. But, it is my considered view that an adequate dialectical logic will include the LNC as a theorem. Make no mistake about it, those dialectical logics which do not include the LNC are most interesting indeed and certainly much more adequate than Classical Logic, but in my mind they don’t go far enough.