There is an obvious conceptual connection between the modern concept of paraconsistency and the traditional term coincidentia oppositorum (coincidence of opposites) as the corner stone in the philosophy of Nicholas of Cusa (or Cusanus, 1401-64). When I was considering this connection, my attention was attracted by a couple of passages concerning Cusanus from some seminal recent books on paraconsistency. I have in mind especially the following three works: 1. Graham Priest, In Contradiction (1987), 2. Paraconsistent Logic, eds. G. Priest, R. Routley and J. Norman (1989), 3. Graham Priest, Beyond the Limits of Thought (1995). In (1) Cusanus is only mentioned among "the number of philosophers who have consciously believed explicit contradictions"; in (2) he is included into the Christian tradition of Neo-Platonism, and his coincidentia oppositorum is represented with a famous passage from Cusanus’ major work De docta ignorantia ("Of Learned Ignorance", 1440) where the coincidence between maximum and minimum is stated:

In the third (3) of the mentioned books, a whole section of the first chapter (1.8) is devoted to Cusanus’ thought, considered from the point of limits of expression and (in)comprehensibility of God. Priest states that in Cusanus’ philosophy we have a paradoxical, and ─ as he argues ─ also a "dialetheic" situation (Priest defines "dialetheia" as a true contradiction), since Cusanus "accepts this contradiction about God [i.e. incomprehensibility vs. comprehensibility] as true"; Priest points out that also in this case, as in many other philosophical cases, both contradictory claims, named by him "Transcendence" and "Closure", are true:

In general, I agree with Priest’s conclusions ─ however, I think something more (or, maybe better to say, less) should be said concerning the "dialetheism" of Cusanus, so the main object of this paper is to put forward this distinction. In the following discussion I prefer to use the traditional term dialectic(s), adv. dialectical, because I think that Priest’s term "dialetheism" has, at least from the epistemological point of view which I am concentrated on, almost the same or very close meaning as the historical concept of (Hegelian) dialectic: the contemporary "dialetheism" is supposed to be a logical reconstruction of classical philosophical dialectics, revival of dialectical methods of thinking and formalization of them by means of modern nonclassical logics.

One more introductory remark has to be put here: in recent literature of paraconsistency there is no quite unanimous, among paraconsistent logicians generally accepted distinction between paraconsistent and dialectical logical systems. Following Priest, we will say that a logical system is paraconsistent, if and only if its relation of logical consequence is not "explosive", i.e., iff it is not the case that for every formula P and Q, P and not-P entails Q; and we will say a system is dialectical, iff it is paraconsistent and yields (or "endorses") true contradictions, named "dialetheias" (I take over this term from Priest, because it has no adequate classical equivalent). A paraconsistent system enables to model theories which in spite of being (classically) inconsistent are not trivial, while a dialectical system goes further, since it permits dialetheias, namely contradictions as true propositions. Still following Priest, semantics of dialectical systems provide truth-value gluts (its worlds or set-ups are overdetermined); however, truth-value gaps (opened by worlds or set-ups which are underdetermined) are considered by Priest to be irrelevant or even improper for dialectical systems. Beside that, sometimes the distinction is drawn between weak and strong paraconsistency, the latter considered as equivalent with dialectics. A reader of recent literature in this field may have an impression that dialectics as strong paraconsistency is more a question of ontology than of logic itself, namely that it states the existence of "inconsistent facts" (in our actual world) which should verify dialetheias. But it remains an open question whether, for example, semantical paradoxes express any "inconsistent facts".

Now let us go to Nicholas of Cusa. The question is: can we claim that Cusanus is a dialectical philosopher, can we say that his coincidentia oppositorum is a precursor of Hegelian dialectic and eo ipso of contemporary dialectical logic, formally (re)constructed by Priest and other paraconsistent and/or dialectical logicians? In the following discussion I am arguing that the epistemological attitude of Cusanus, expressed by himself as docta ignorantia, precludes any simple (or "categorical") affirmation of contradictions, as well as, of course, Cusanus does not accept the simple negation of them in the manner of the classical (Aristotelian) logic. This point can be expressed also in this way: docta ignorantia does not affirm contradictions just simpliciter, but ambigue – namely, Cusanus’ opposita, forming an "endorsed" contradiction, are both true or both false, depending on how we understand them. The term "dialetheia", when applied to Cusanus, should be taken - differently from Priest - in a double sense, applied not only to truth-value gluts, but also to truth-value gaps: a contradiction as coniunctio oppositorum is true not only if its opposites are both true, but also if they are both false. (Formally, this revised concept of dialetheia means that the rejection of the Law of Non-Contradiction entails the rejection of the Law of Excluded Middle.) Indeed, a typical Cusanus’ dialetheia, for example the conjunction of "Transcendence" (P) and "Closure" (not-P), mentioned above, has always two sides, like Janus’ head: from its "positive side" (leading to the "positive way", traditionally called via positiva), its opposites are both true (i.e., the propositional conjunction 'P and not-P' is true); but if we consider dialetheia from its "negative side" (leading to the "negative way", traditionally called via negativa), its opposites are both false (i.e., the propositional binegation 'neither P nor not-P' is true). My point here is that just this ambiguity of dialetheias is essential for understanding the "middle way" of Cusanus – the way directed by his basic epistemological insight and maxim: docta ignorantia. We will return to this point later.

We always meet difficulties when we try to interpret an ancient informal wisdom with our modern formal means. Cusanus’ coincidentia oppositorum has not been written in the formal language, even less it presented a well-defined logical system. So it is certainly difficult to determine its "underlying" logic, since "it is only in contemporary times that a clear conception of a formal or semantical system has developed." Nevertheless, we can surely claim that the underlying logic of Cusanus’ philosophy is not Aristotelian, but (at least) paraconsistent ─ in the sense, outlined above, namely that the relation of logical consequence (albeit informal one) in Cusanus’ philosophical thought is not "explosive": docta ignorantia surely admits a philosophical theory which is inconsistent and non-trivial - such a theory is Cusanus’ philosophical "system" itself. Let us call it (the system of) Docta Ignorantia (DI) and ask: is (DI), being paraconsistent, also dialectical? The answer is not so obvious as it seems from Priest’s passages concerning Cusanus. In order to see the problem more clearly, we have to examine some relevant passages from Cusanus’ great work De docta ignorantia.

When we try to understand Cusanus’ philosophy from the point of view of modern logic(s), we must not forget the following: God, named as maximum, is, by coincidentia oppositorum, also minimum, however, this concidentia is incomprehensible for human reason (ratio), for our discursive, logical thinking ─ yet it is in an unthinkable transcendent way present to our mind (mens, intellectus), namely by an intellectual intuition, philosophical contemplation. The incomprehensibility of coincidentia oppositorum for human reason (for our logical, even dialectical thinking) is considered by Cusanus to be essential for his philosophy. Here are two relevant passages:

From the point of view of Cusanus it would be a mistake to think "positively" (or simpliciter) the coincidence of opposites ─ since reason, using the principle of non-contradiction, actually cannot think coincidentia oppositorum which is supra omnem rationis discursum (i.e., "beyond the limits of thought"); and that is why it cannot be rationally decided whether opposites are both true or both false. This point is very important for understanding Cusanus’ docta ignorantia.

However, on the other hand, Cusanus is not a mystic, he is a great philosophical thinker who ─ like his brothers in spirit: Plotin, Eriugena, Kant, Wittgenstein, Nagarjuna and others ─ "manages to say a good deal about what cannot be said". How does Cusanus manage to do it?

In his last work De apice theoriae ("Of the Summit of Contemplation", 1464), as well as many times before, Cusanus wrote:

However, what does it mean ─ videre mentis? It is easier to say what it does not mean as what it actually means. (Needless to remark, this is one of the most difficult classical philosophical questions.) For Cusanus, "to see by mind" means neither a rational cognitive act nor just sitting and contemplating in silence. Mens (and/or intellectus, the distinction between them is not sharply outlined in Cusanus’ works) by contemplating "sees" symbols which "transfer" mind from their positive, finite meaning (being immanent in the world, articulated in language) to infinite transcendence, beyond any positive meaning and distinction.

And here Cusanus is especially interesting: for him, the most important philosophical "symbols" are provided by mathematics (mostly by geometry as the dominant mathematical discipline in those times). Cusanus based his metaphysical "intuitions" on geometrical symbolic models. Of course, he considered mathematics in its ancient (Platonic and Pythagorean) sense, namely as the clearest reflection of the universal order, of the World of Forms, ─ nevertheless, his idea that in the mirror of mathematics as "symbolic thinking" metaphysical and/or theological truths can be "seen" by the intellectual intuition, is new in the pre-Renaissance philosophy, and it is inspiring nowadays as well. Cusanus wrote:

And in this symbolic way of contemplating God’s incomprehensible and infinite being mathematics play a very important role:

One of the most famous mathematical "figures" of Cusanus which he used for symbolic representation of coincidentia oppositorum is the coincidence of ("the maximal") circle and a straight line (tangent); this coincidence is the "incomprehensible" limit of the sequence of larger and larger circles. Let’s quote Cusanus’ comment to this "figure":

This model ("symbol") of coincidentia oppositorum can be advanced by including triangles: the Triangle with "the maximal angle" coincides with the straight line and with the Circle; this is supposed to be a reductio ad perfectionem of geometrical objects, since: Circulus est figura perfecta unitatis et simplicitatis. (Doc. ign., ch. 21; "The circle is a perfect figure of unity and simplicity.", op. cit., p. 46), and just in the "infinite circle" the coincidence of opposites reveals itself in the most manifest, although still "symbolic" way:

We could go on with Cusanus in his geometrical symbolism by introducing the infinite Sphere instead of the Circle: "...centrum maximae sphaerae aequatur diametro et circumferentia..." (Doc. ign., ch. 23), but for our purpose the Circle will do. Let us denote this "maximal" Circle whose centrum est in ipso circumferentia with Greek capital letter Ω, and - making a sort of thought experiment ─ suppose that Ω can be an object of thought (an idea in the Lockean sense, without any heavy ontological commitment); then we put a pair of Kantian questions which lead to an antinomy, similar to Kant’s first antinomy:

(Q) Is Ω finite?

(Q’) Is Ω infinite?

Of course we might object that reductio ad absurdum is not applicable in such limit cases, but here I have in mind another point: in case we argued otherwise (quite symmetrically, following via negativa), we could answer NO to both questions (Q) and (Q’), asserting negatively that Ω is neither finite nor infinite. The point of docta ignorantia, relevant for our context, is that in such "limit questions" (which contain the ideas of the Ω-type) no human reason (not even dialectical mind in Hegel’s sense) can decide whether both opposite answers are true or both false. This point sounds very like Kant’s "solution" of antinomies, but there is an important difference: Kant "solved" his antinomies so that, to say shortly, he negated the legitimacy of both opposites on the theoretical level (by another reductio, since in classical logic both opposites cannot be demonstrated as true), and by rejecting both opposites Kant consequently rejected also the concept of actual infinity on the theoretical level of "pure reason", but he accepted infinity in another sense, namely on the level of "practical reason", as a "dialectical idea". Figuratively said, Kant solved the Gordian knot by cutting it into two separate ends, and so he managed to preserve classical logic. On the contrary, Cusanus in his "system" of Docta Ignorantia has not tried to preserve classical (Aristotelian) logic – at least on the level of maximum – since he did not accept the "simple" (and in his time already traditional) abolition of paradox by cutting its knot into two separate ends. Moreover, Cusanus endeavored to sustain an apparent absurdity: coincidence of opposites. And I think that just this preserving of opposita ut opposita is the most precious stone of his wisdom.

So, if we return to the main question of this paper: can we consider Cusanus’ Docta Ignorantia (DI) a dialectical theory in the sense of admitting true contradictions (dialetheias)? Yes, provided dialetheias are not limited to conjunctions of opposita which are both true, but include also opposita which are both false. The main argument for this claim is docta ignorantia itself as the principal Cusanus’ epistemological and methodological maxim: the essence of the learned ignorance is that in principle it can not (and will not) decide which way to the highest knowledge ("knowledge beyond knowledge") is the right way: via positiva or via negativa. They both are right, but none of them separately – and they both, taken together, are also wrong, since there is a "middle way" which transcends them both: docta ignorantia.

If we want to express the "underlying logic" of Cusanus’ (DI) by means of modern (nonclassical) logic, the best approach is to take as the basic matrix four-valued semantics, known from Michael Dunn’s semantics for First Degree Entailment (Dunn 1976, see also: Entailment, Vol. II, 1992, § 50). Four values are: true (T), false (F), both true and false (B), neither true nor false (N); T and F are classical truth-values, B and N may be called dialectical (and eo ipso paraconsistent) truth-values. Of course there are intuitive problems with both dialectical values, however, problems concerning N are in no respect more difficult than problems concerning B. Otherwise said, truth-value gaps are from the intuitive point of view equally (un)problematic as truth-value gluts. Dunn discusses this intuitive symmetry of gaps and gluts in the following passage:

The important point in this passage is the emphasis that four-valued semantics which includes the value N, does not mean that there are in fact such sentences which are neither true nor false (any more than that are because of the value B in fact such sentences which are both true and false). I understand this point also as an admonition that "factual" dialetheias do not actually exist, neither in gluts nor in gaps, since real facts (on the ontological level) cannot contradict themselves, and consequently, true "factual" sentences which correspond to facts by adaequatio, cannot be in fact both true and false or neither. However, the issue of this discussion depends on how we understand facts, adaequatio, truth etc., and so it is far out of the scope of this paper. I can just agree with Dunn, saying: "By the way, we are painfully aware of the strangeness of some of our remarks motivating the [four-valued] semantics." (Ibid.)

But on the other hand we should not forget that by applying the four-valued semantics as the most appropriate "underlying logic" to Cusanus’ (DI), we do not apply it to "factual" sentences, but to sentences which are "beyond the limits of thought"(or at least very near these limits). In the quotation above, Dunn presumes that truth-value gaps can emerge due, for example, to failure of denotations of singular terms. We may ask: is Cusanus’ term maximus circulus, whose centrum est in ipso circumferentia, which we denoted with Ω – a token of such a failure? Does singular term Ω denote anything at all? From the "factual" point of view, this term seems to be empty. Cusanus, of course, knew it, but it was not his point, since it is perfectly clear that maximus circulus makes sense only in the limiting process of thought, in "transfer" from comprehensible finite figures to the incomprehensible maximum (cf. above: rationes correspondenter ad infinitas tales figuras fransferre…). So, here there is no "denotation failure" – on the contrary: maximum which (who) is to be "denoted", can only be denoted by negative way, i.e., by failure of positive denotation.

If four-valued semantics is the most adequate "underlying logic" of Cusanus’ (DI), we may still ask: which of the four values T, F, B, N is (are) designated? Provided we accept the classical designation for the first two values, namely T* (designated) and F (undesignated), we have to decide which of two dialectical values (B, N) is/are designated and which not. Here again we must take into account Cusanus’ principal stance: docta ignorantia. (DI) precludes the decision (better, abstains from decision) which of two dialectical values is/are designated – the learned ignorance can only say that both are designated (B* and N*) or none of them (B and N). Let me give an example, borrowed from Priest (see above): he argues that "Transcendence" and "Closure" are both true in Cusanus’ teaching, and this is supposed to be one of reasons why his philosophy is dialectical (Priest says "dialetheic"). Right, but from the point of (DI), "Transcendence" and "Closure" can be considered as both false as well. They only cannot be of different truth-values, i.e., "Transcendence" true and "Closure" false, or vice versa, since this would not make sense in dialectical thinking of Cusanus, for in this case coincidentia oppositorum would be resolved into its positive and negative counterparts – via positiva and via negativa – like the "Gordian knot", cut into two separate ends of rope, or "Janus’ head" into two flat faces.

Still another question may be put here, namely: does the proposed symmetrical designation of truth-values nevertheless lead to iteration of ever-higher values, that is, into infinite regress? I think this is not the case, since (DI) effectively stops the further choice (iterated "oscillation") between two dialectical values, as well as between (B* and N*) and (B and N), by excluding (B* and N) and (B and N*), and by "identifying" (B* and N*) with (B and N). We can take, metaphorically, docta ignorantia as a "fixed point" for coniunctio oppositorum. Of course, formally we could go on and evaluate (DI) itself with a "fifth truth-value", say DI, but this would mean just something like "The Rest" (beside truth, falsity, both and none) in the four-valued scheme of the Buddhist philosopher and dialectician Nagarjuna. The fifth value does not mean that another truth-value in the proper sense is added to the four, since "The Rest" is out of them all, the upper limit of iteration.

After having written the major part of this paper, I have discovered that my interpretation of Cusanus’ coincidentia oppositorum is close to some ideas of Lorenzo Peña concerning the same subject. Peña puts the question: "In which sense the contradictions in God do not contradict themselves?" And he answers that contradictions in God, according to Cusanus, remain contradictions, they do not disappear simpliciter, for "in God, opposition and non-opposition of contradictions coincide". Peña’s main reference here is Cusanus’ statement from De visione Dei (III, 150): "In infinitate est oppositio oppositorum sine oppositione." We may ask, how can the "opposition without opposition" be rationally conceived at all? In a sense, it can. Peña introduces next to contradiction its opposite (better, symmetrical) concept, "neutrodiction":

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