bucephalus.org

# List of projects

November 2009

Originally, Bucephalus [1] was the name of Alexander the Great's horse. In the centuries afterwards, this frequently served as a proper name example in many classic text books of philosophy and logic. Here and now, bucephalus is the title of an ongoing research project in logic, which went online as bucephalus.org in 1999.

## Major and minor projects

Up to present, the papers and software available were very bottom-up and mathematical in character, stand-alone pieces that offered proper formal solutions in themselves. But in fact, the whole endeavour is inspired by a much broader perspective and motivation, a rather philosophical change of view on the whole subject of logic itself. The intention is to work out an overview of this whole endeavour rather soon. But to give at least some idea of the whole plan, the ongoing work is dynamically structured as interconnected "major" and "minor" projects. Some of these projects are finished and available online, others are in progress or planned for some some future time.

## Current division into major projects

2. ### The Hyper-digital Logic project

This title denotes a recursive extension of propositional logic by building an infinite hierachy. The whole is effectively computable with an approach similar to the fast prime normal form construction we used before on propositional logic. This allows to implement very complex systems, e.g. a computable subsystem of general set theory.
3. ### The Foundation project

By now we have an effectively computable propositional logic system. This comes with an algebraic abstraction of propositional logic as a theory algebra, based on the semantic and syntactic order, rather than the usual and purely semantical reconstruction as a boolean algebra. In a next step, we generalize this bit-valued or propositional theory algebra to a general theory algebra of relations. This Foundation project is a foundation in two directions: first, we use the propositional logic system as the sole tool for a reconstruction of the computability concept. In other words, we develop a programming language, which is based on propositional algebra, similar to functional languages, which are based on the lambda calculus. This dynamic exploitation of propositional logic starts off with a theoretificationism, which shows how theories, theory functions and theory relations can be seen to be identical entities. In the other direction, we show how the more general theory algebras of relations provide a very appropriate and natural semantics for predicate logic. And there is a hierachy of computable subsystems of these theory algebras, which enable the direct implementation of more or less general model constructors for first-order theories. As a consequence of this approach, the theorem of Church and Turing for the undecidability of first-order predicate logic could appear in a new light.

5. ### The Formal Intelligence project

With the tools provided by the previous projects, we introduce a theory machine, i.e. a kind of finite automaton on propositional algebras. With the earlier introduced theoretificationisms, this is already a monotone learning system. It is also straight forward to consider these as neural networks. But only when we combine the monotone learning with some binary distinction mechanism, this learning becomes somehow creative, and able to guide its own choices without any additional preference mechanisms. We obtain a dualism which is similar to Piagets assimilation-accomodation dichotomy. By means of this Piaget learning and with our non-representational semantics, we may be able to describe, how a kind of language emerges from a principle of causality, mechanized by these theory machines. This could provide us with a formal and computable system which might behave in a way that shows many similarities to what is commonly called intelligent. In the end, we aim at a formal concept of intelligence.