Once we realize that there is both dialectical and trialectical thinking then we can think about extending the series as I have done in my dissertation (emergentdesign.net). But it also draws attention that dialectical thinking takes two things and produces a third which is a synthesis of the two, and then leaps to a forth, which is neither of the two (See Hegel’s Logic). On the other hand trialectics is the mediation of three things by each other, so that each is an thesis, anti-thesis and non-thesis to the others and the non-thesis mediates between the other two. This can be seen worked out in Peirce’s semiotics which has the sign, the object of the sign, and the subjective interpretation of the sign as the three theses. It extends the duality of subject and object by creating a mediating realm of signs between them. Peirce defines the different kinds of sign based on his philosophical principles being applied to each other.
So we can only define dialectical thinking when we understand the series monolectical, dialectical, trialectical . . . and how each level allows us to understand things in a different way with our reason. Hegel formulates trialectics just before going into the nature of Spirit, and so we can assume that it is the doorway that opens up the realm of Spirit to us which is based on his understanding of the trinity.
Because dialectics became an ideological issue in the Cold War it was not developed as it should have been. Marxists clung to a very mechanical idea of the dialectic given to them by Marx, and those in non-communist countries avoided the topic because it was a loaded topic with political repercussions. However, the French intellegensia who were mostly marxist intellectuals kept trying to refine the idea. The best example of this is Sartre’s Critique of Dialectical Reason. Which for the first time to my knowledge tries to think dialectics dialectically, i.e. make dialectics something dynamic and process oriented rather than seeing it as a something mechanical. However, this does not result in the rediscovery of Hegel’s trialectics. As far as I know this was only developed by C.S. Peirce.