In his Philosophical Remarks, L. Wittgenstein wrote prophetically: “I predict a time when there will be mathematical investigations of calculi containing contradictions, and people will actually be proud of having emancipated themselves from contradictions.”
The import of this statement is deeper than you may think, because it puts into question not only what is the standard mode of thought of Western mathematics but also what is the rock bottom “foundation” of common-sense “philosophy.”
The most superficial review of the notion of contradiction in mathematics, logic and philosophy will show a very solid consensus which is deemed essential for the scientific method, out of which only nonsense can exist. Within this consensus, the mere idea that logical contradictions can be the object of study is considered as something outdated (for example “harking back” to Hegelian or Marxist dialectics) if not plainly wrong and laughable.
The view that there are true contradictions, or “dialetheia” as these are called in Philosophy, “flies in the face of what most philosophers take to be common sense” according to the Stanford Encyclopaedia of Philosophy (http://plato.stanford.edu/entries/dialetheism/ ).
The Encyclopaedia explains: “A dialetheia is a sentence, A, such that both it and its negation, ¬A, are true […] Assuming the fairly uncontroversial view that falsity just is the truth of negation, it can equally be claimed that a dialetheia is a sentence which is both true and false.”
And we also learn, that, despite of “the majority view, there are some dialetheists in the history of Western Philosophy. Moreover, since the development of paraconsistent logic in the second half of the twentieth century, dialetheism has now become a live issue once more.”
So it is true that there are people who believe that there are “true contradictions,” and, for the sake of full disclosure, I want to state here that I go further and say that there are complexes of contradictions, which are modalities of action and thought, ever present in every organisation and human institution.
Logicians tend to limit the concept of “contradiction” to “sentences” or “statements” (also called speech acts), but it is more useful to see contradictions as part of opposition structures –of discourses or modalities– which are correlated, simultaneously present but also clearly irreducible to each other.
This is none other than a theory of multipolarity similar to the one advanced by Professor Richard Jung in his essay “A Quaternion of Metaphors for the Hermeneutics of Life” and other texts. (See: http://www.richardjung.cz/0b.pdf ). By multipolarity I mean a philosophical position where essential logical oppositions are seen and gathered as constitutive of reality. Similar approaches are those of Stephen C. Pepper (See: “World Hypotheses,” 1962) and Magoroh Maruyama (See: “Mindscapes and Science Theories,” Current Anthropology, Vol. 21, 1980).
(Other terms appropriate for this approach are “polycontexturality” –as in the studies of Gunther Gotthard and Rudolf Kaehr; and “polythetics” –as formulated by Donald Kunze.)
A great start to study the logical theories of dialetheism is also the work of Professor Manuel Bremer, of Düsseldorf University, as his work is particularly clear regarding the strong logical values of dialetheism and the fact that the theories associated with it are not trivial. (See: M. Bremer´s work here: http://www.mbph.de/ )
Bremer errs in discussing logic only in terms of assertions of beliefs, but he makes a very useful suggestion showing how degrees of assertion or belief can be set into a table listing values of truth, falsity, “only falsity,” “only truth”, inconsistency, consistency and indeterminacy. As a consequence of this clever approach, truth appears as an *operator*, at the same level as the operator for “inconsistency,” which in turn clarifies the discussion.
Because of his concern with the accusations of triviality against dialetheism, Bremer shows how these degrees of assertion or belief are actually very practical. It is only a small step from there, to understand that these arrangements of oppositions are in fact part of the worldliness of every discourse and of all “structures” of discourse.
For in human life, contradiction is not only a matter of an individual “holding” a contradiction in mind, but actually that contradictions and dualities are the structure of worldliness itself.
Bremer´s approach allows us to see that “T” and “F” (truth and falsity) are operators at the same as others, but his argumentation fails when he says that “no-one (including dialetheists) can have pragmatic contradictions: speech acts being bodily movements that either occur or do not, there is no pragmatic parallel to naming it both ways, i.e. not(Ap and notAp).”
This stems –with all due respect– from the logician´s symbolic world, where he is prohibited by his method of any insight that goes beyond the notion of “logical statement.” In reality, every “bodily” assertion is simultaneously a “contradiction,” or, better said, it arises from the opposite and contradictory (joint) movement of the body and the mind. Furthermore, every speech act is incomplete, due to the inability of articulated language to express the full state of the body. Every assertion is an implicit, silent negation. Every negation is an assertion. The stance of affirmation (the speech act) is articulated as a result of a neurological selection, where the negative accompanies or even is at the root of the affirmation.
In other words, bodily and mental “movements” are always contradictory. And this happens not only at the level of the individual, for in fact, there is no “social” statement (or better no statement articulated in a social context or through a social context) that is not embedded in a mesh of contradictions and oppositions: Every assertion has a “socially manifest” negation (i.e. an opposition).
Bremer makes a very good point when he says that “a main motivation of dialetheia is universality.” Indeed, only dialetheism and the theories that can be associated with it are able to grasp the “polyocular” complexity of organisational and social life (using Maruyama´s terminology). A serious limitation would be to see the universal aim as focused on “language” or “cognition”, because polycontextural thinking is not limited to language or mental conceptions and understands non-cognitive as well as non-linguistic aspects of the multipolar world.
Bremer seems to recognise that logical contradiction is a social affair when he mentions among the motivations of dialetheism the fact that “people contradict each other,” but should also have noted that people contradict each other even when they agree wholeheartedly. It is only the naturel of a polythetic method to understand not only that there are true contradictions, but that there no contradiction exhausts reality (i.e. there is no universal contradiction or, as Bremer says: “representation of reality is not guaranteed”).
In other works, Bremer associates dialetheism with “internal realism.” He describes how internal realism rests on “two dualities,” those of internal and external perspective, and those of coherence and correspondence. Interestingly he says that these aspects are “irreducible.”
If –with Bremer– one assumes that truth is epistemological or at least “non non-epistemological,” we have to arrive at the notion of “internal perspective.” By taking the position of the Agent, and limiting his elaboration to “speech acts” Bremer is unable to complete his scheme based on the combinations of perspectives and coherence/correspondence.
According this a possible interpretation of his scheme, there must be Internal / External perspectives and justifications based on Coherence or Correspondence. Therefore, there must be Internal approaches that are either focused on Coherence or Correspondence. I call the first Semantic, the second Pragmatic. In parallel with this, there are External approaches which are either focused on Coherence or Correspondence. The first I call Syntactic, and the second Factual or Statistical.
While Bremer aligns three terms within his schema: Semantic = Gültig, Syntactic = Beweisbar and Pragmatic = Behauptbar, he does not have a label for those External approaches that are focused on correspondence. I don´t think that this is a major error, but only reveals a lack of completeness of Bremer´s model and could easily be corrected.
A major achievement of his work though is to call attention to these possible combinations or modes of Truth, which probably is a good summary of what this theory entails.
Bremer discusses how the pragmatic refers to the factual (die Tatsachen) but due to his focus on speech acts does not consider that while the pragmatic can refer to the factual, it cannot be “supported” by the factual as well as the pragmatic can never “exhaust” any given fact. This is only equivalent to saying that the Pragmatic and the Factual are not reducible to each other.
As a summary of this we can say that this valuable theory of Dialetheism further nurtures the meta-linguistic (metalogical) theories of multipolarity, while these give the context for Dialetheism, and both disciplines are bound to enhance each other.
For these references to Bremer´s work, see in particular:
“The Logic of Truth in Paraconsistent Internal Realism,” Studia Philosophica Estonica, 2008.
“Believing and Asserting Contradictions,” Logique et Analyse, 2007.
“Why and how to be a Dialetheist,” Studia Philosophica Estonica, 2008.