Gary Mar

The philosophical computer: exploratory essays in philosophical computer modeling

Philosophical modeling is as old as philosophy itself; examples range from Platos Cave and the Divided Line to Rawlss original position. What is new are the astounding computational resources now available for philosophical modeling. Although the computer cannot offer a substitute for philosophical research, it can offer an important new environment for philosophical research. – The authors present a series of exploratory examples of computer modeling, using a range of computational techniques to illuminate a variety of questions in philosophy and philosophical logic. Topics include self-reference and paradox in fuzzy logics, varieties of epistemic chaos, fractal images of formal systems, and cellular automata models in game theory. Examples in the last category include models for the evolution of generosity, possible causes and cures for discrimination, and the formal undecidability of patterns of social and biological interaction. The cross-platform CD-ROM provided with the book contains a variety of working examples, in color and often operating dynamically, embedded in a text that parallels that of the book. Source code of all major programs is included to facilitate further research. – Contents : Preface. Introduction. – 1. Chaos, Fractals, and the Semantics of Paradox; – 2. Notes on Epistemic Dynamics; – 3. Fractal Images of Formal Systems; – 4. The This Evolution of Generosity n a Hobbesian Model; – 5. Real-Valued Game Theory: Real Life, Cooperative Chaos, and Discrimination; – 6. Computation and Undecidability in the Spatialized Prisoners Dilemma. – Appendix A: Competitive Strategies Adequate for a Minsky Register Machine; – Appendix B: An Algebraic Treatment for Competitive Strategies. Afterword. Notes; Index.

CHAOS IN COOPERATION: CONTINUOUS-VALUED PRISONER’S DILEMMAS IN INFINITE-VALUED LOGIC

The Prisoner’s Dilemma (PD) has become a paradigm of the evolution of cooperation. The PD can be generalized to the continuous-valued case in which players are allowed to choose intermediate levels of cooperation. When continuous-valued PDs are played in the spatial context of cellular automata, generous strategies are favored. Continuous-valued PDs are naturally represented in infinite-valued logic. The infinite-valued logic allows us to prove that cooperative interactions between continuous-valued strategies are paradigmatically chaotic. Escape-time diagrams using a given level of mutual cooperation as a threshold produce fractal images. The sensitive dependence of chaotic dynamical systems models a practical unpredictability within the PD (even though only deterministic nonstochastic strategies are involved) that is characteristic of many real life choice situations.

What the Liar Taught Achilles

Zenos paradoxes of motion and the semantic paradoxes of the Liar have long been thought to have metaphorical affinities. There are, in fact, isomorphisms between variations of Zenos paradoxes and variations of the Liar paradox in infinite-valued logic. Representing these paradoxes in dynamical systems theory reveals fractal images and provides other geometric ways of visualizing and conceptualizing the paradoxes.

Self-reference and paradox in two and three dimensions

Abstract The Liar paradox is a familiar sentence that asserts its own falsehood. Just slightly less familiar is the Dualist, involving a pair of mutually referential sentences. Here we consider variations on the Dualist and Triplist, modelling their dynamical semantics using iteration within an infinite-valued logic. Strange attractors and fractal escape-time diagrams appear in two dimensions for the Dualist and in three dimensions for the Triplist.

Gödel’s Ontological Dreams

Godel, arguably the greatest logician of the 20th century, dreamed of establishing philosophical theses and ontological results with the rigour and precision of mathematics. His dream was to a remarkable extent fulfilled. Two modal principles that characterize properties of proof, time, and God reveal some intriguing logical interconnections among Godel’s Second Incompleteness Theorem, his argument for the unreality of time in the General Theory of Relativity, and his Ontological argument.

Chapter 16 – Self-reference and paradox in two and three dimensions

Publisher Summary This chapter focuses on variations of the Dualist conceived in an infinite-valued logic, in which sentences are allowed to take on any real truth-value in the [0, 1] interval. The dynamical semantics of self-reference can be mapped using iterated algorithms. The Liar paradox is a familiar sentence that asserts its own falsehood. The Dualist (slightly less familiar), involves a pair of mutually referential sentences. This chapter considers variations on the Dualist and Triplist, modelling their dynamical semantics using iteration within an infinite-valued logic. Strange attractors and fractal escape-time diagrams appear in two dimensions for the Dualist and in three dimensions for the Triplist.