Rudy's Diamond Strategies

This complementary Blog to the Chinese Challenge Blog is presenting studies to a mathematical theory of Diamonds. Diamond theory is studying for the first time, tabular categories as an interaction of categories and saltatories.

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Tuesday, July 10, 2012

Notes on the Tabularity of Polycontextural Logics

Bifunctoriality and Tabular Notation for Polycontextural Logics

Some new developments in the formalization of tabular logics as attempts to a non-hierarchical and not-tree-based paradigm of formal thinking.

Western Logic and Trees
Logic is easily connected with trees. Raymond Smullyan started the movement of “Logic with Trees” (Colin Howson), Melvin Fitting, the master of all trees dedicates his book “First-order Logic and Automated Theorem Proving" "To Raymond Smullyan who brought me into the trees".

The tree or tableaux method is highly elaborated by Melvin Fitting as the ultimate tableaux method, used today as a proof method for nearly all kinds of logical systems. There had been predecessors, as usual, like Evert Beth and Jaako Hintikka, or the Dialog Logic approaches of Paul Lorenzen and E. M. Barth.

Tree-thinking goes back to the Porphyry of Tyre with his Porphyrian tree. Tree-thinking is fundamental for Western thinking. Chinese thinking in contrast is based not on trees but on grids (Yang Hui (楊輝, c. 1238 - c. 1298)).

The tableaux approach to logic seems to be very natural. Its emphasis is focussed on a structure with a singular beginning (root) and (mostly) binary decision procedures for the prolongations of the tree build on the base of such a root and its branching. The established hierarchy between the root and its nodes is perfectly stabilized by the success of its applications and its lucid rationality rooted in classical Western thinking of Porphyrian tree-ontology and its re-invention in the Semantic Web, too.

It is believed, historically and actually, that non-rooted and non-hierarchical systems of thought and action are leading for short or long into chaos.

Postmodernist thinking believes that such arguments of and against hierarchical organizations are obsolete. Even the smallest kid experiences and knows how much we all are connected and taking part in massive networks where there is no beginning and no end and everything is nevertheless working fine. What’s a correct impression for kids is not necessary the truth of the adult game.

With or without clouds, the internet connections are strictly hierarchically mathematized, programmed, organized, regulated, governed and policed.

The mass of data and “contents” are blinding the fact of the covered simple hierarchical form of organization of the deep-structure of the Web. Not just ICANN and the reduction to uni-directional communication but also the reduction of any sign system to techniques and ideologies of digitalism is determining the structural poverty of the overwhelming possibilities on an informational data-level.

Towards Matrix-based Logics

For whatever reasons I never could find any enthusiasm for such an ultimate tree.

To stay in the context of the established form of rationality I prefer to live with/in forests instead of singular trees. I don’t see any reason why a node might not change into a root and a root not becoming a node of a different, equally fundamental tree.

Traditional trees are not just defined by their uniqueness and hierarchy but by the their definitive lack of interchangeability, chiasm or proemiality of the ‘fundamental’ terms, like nodes and root.

In fact, trees don’t come in plural. All the singular and factual trees, say of logic, are dominated by the concept and methods of a single, unique and ultimate idea of a tree.

A first, and simple approach to surpass such limitations is proposed with the idea and some elaborations of forest-based polycontextural logics.

Hence, nothing is wrong with “Logic with Trees”. I opt to just disseminate such ultimate trees. This, as such and alone, wouldn’t be specially interesting. What makes the forest approach interesting is the possibility of interactions between the plurality of such simultaneously existing ultimate trees. A forest is not the sum of singular trees but the interactivity between trees.

Forests of Tableaux-Trees

For the case of just one singular but ultimate tree we don’t have to know much about the structure of the place it is planted. Because of its uniqueness the knowledge of its ground(ing) can freely be omitted. For a forest, the loci of the trees becomes crucial. Disseminated trees are indexed to localize them in the grid of the ground. A ground and locus of a tree is not itself a tree. Hence, any logical characterization of the loci of the trees, that is building of a matrix and a grid, is obsolete. The matrix of the dissemination of logic-trees is defined by a a-logical or pre-logical structure. This pre-logical and pre-semiotic structure is covered by the methods of kenogrammatics. Thus, the grid of the forest is the kenomic matrix.

Again, the game starts again. There is no necessity to suppose a static hierarchy between the grid and the forests.

Trees in formal languages are reduced to the simple structure of “append”  and “remove” of “items”. Hence, disseminated trees are indexed, in this case, double-indexed to define a matrix of trees, and are defined by the similarly simple operations of “leave” a tree, ‘horizontally’ for replications (reflection) and “leave” a tree vertically for transpositions (transjunctions).

Other operations between trees, like permutation, reduction and iteration of trees of a forest, are easily introduced and implemented into the formal game of forest-logics. Forests are not static. They might grow or shrink and change their patterns.

From a more mathematical point of view, forests and their interplays are well ruled by the polycontextural concept of interchangeability, i.e. a generalization and subversion of the category-theoretic concept of bifunctoriality.

Without any big deviations and dangerous revolutions a move from the tree-culture to a forest-world of thinking and acting seems to be a fairly save and sane step of evolution even for the timid Western searcher of truth and computational efficiency.

In earlier papers about tree-farming I proposed contextural forests as forests of colored trees. This time, coloring has to wait for the paint.