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Bucephalus.org is the title and address of a research project in logic.

All material (text and software) is presented at

www.bucephalus.org

For all interaction (blog, comments, forum) see

www-bucephalus-org.blogspot.com

news
September 5, 2013
The Canvas Handbook is published, along with a forum page for comments and remarks.
March 30, 2012
A new page for the Change Logic project is created and soon to be populated with new material.
March 25, 2010
A 5-pages paper called Change Logic and the Change in Logic has been added.
February 2, 2010
The Haskell PropLogic package is published along with related papers and other updates.
November 4, 2009
Descriptions of some major and minor projects have been added or improved (see list of projects).
October 5, 2009
This homepage has undergone a complete makeover. A couple of new small texts have been added already, some bigger and big items are awaiting a final touch and should be added soon as well.
September 28, 2009
To increase the interactivity of the whole project, a blog was created, using the easy way provided by blogger.com.
[project list]

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

  1. The Propositional Logic project

  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.
  4. The Change Logic project

  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.

Links and footnotes

[text list]

List of papers and books

October 2009

Meaning concept

IntroMeaning
Introduction to a logical concept of meaning, 20 pages
EinfBedeutung
Einführung eines logischen Bedeutungsbegriffs, 22 Seiten
Meaning
A logical concept of meaning, 59 pages

Theory algebras

TheoryAlgebras
introduction and overview
TheoAlgRelations
Theory algebra of relations, 178 pages
TheoAlgAxioms
Axioms of theory algebras, 10 pages
WorldAlg
World algebras, 10 pages

Algorithms

PNFCanon
Theory and implementation of efficient canonical systems for sentential calculus, based on Prime Normal Forms, 36 pages

The Haskell PropLogic package

PropLogic
overview
InstallPropLogic.html
Installing the PropLogic package
ExecPropLogic.html
A little program for propositional logic
IntroToPropLogic.html
Brief introduction to PropLogic

Bucanon Java applet

BucanonGuide
Guide to the bucanon software and documentation, HTML page
BucanonIntro
Bucanon introduction: How to do boolean algebra with the bucanon program, 4 pages
BucanonIntroContinued
Bucanon introduction, continued: Motivating theory algebras, 3 pages
BucanonManual
Bucanon Manual, 14 pages
BucanonSyntax
Bucanon syntax, 1 page
BucanonScreenshots
Screenshots and samples of the bucanon applet in action, HTML page

Hyper-digital logic

HyperDigitProject
Hyper-digtal logic, project overview, HTML page
HyperDigitSum
Hyper-digital logic: Summarized overview of its syntax and semantics, 3 pages
HyperDigitIntro
Introduction to hyper-digital logic, 24 pages
SetFieldLog
Set field logic and its embedding into hyper-propositional logic, 6 pages
HyperDigitAlg
Algebraic properties of hyper-digital logic, text under construction
MixedHyperDigit
Mixed hyper-digital logic, text under construction
HyperAndModal
Hyper-digital and modal logic, text under construction
HyperPropNorm
Hyper-propositional normalizations and canonizations, text under construction

Little Helper

Haskell98numbers
A picture on the numbers systems in the Haskell programming language, 1 page
[code list]

Available software

February 2010

Currently, there are two systems for propositional logic:

Bucanon
an online Java applet
PropLogic
a Haskell package for propositional logic

The HTML5 Canvas Handbook
CanvasHandbook.html is an introduction to the <canvas> tag and comprehensive reference to the according JavaScript objects.
This one file contains the whole document and printed, this is a book of about 80-110 pages.
There is also a forum page for all further remarks and user comments on this document and subject.
PropLogic
For a new perspective on the old subject of propositional logic, see here. For the practical translation into a versatile Haskell package, see here. For any comment, please check out the according blog entry.
bucanon applet
bucanon applet There is an online Java applet, a kind of pocket calculator for propositional logic. It provides common features like the evaluation of a formula or the construction of its truth table. But its real power is the generation of prime canonizations. Next to the applet itself, there is also a separate guide.
a book
A theory algebra is a structure that is not just a boolean algebra, but a typical interconnection of two (almost) boolean algebras: a syntactic and a semantic (quasi-)order structure. Theory algebras of relations presents a full exploration of these structures. Another text is on its way that shows how this provides an algebraization of predicate logic.
CodeDown
CodeDown is a simple and powerful approach to the auto-documentation of arbitrary programming languages.