Patrick Grim

Formal systems are standardly envisaged in terms of a grammar specifying well-formed formulae together with a set of axioms and rules. Derivations are ordered lists of formulae each of which is either an axiom or is generated from earlier items on the list by means of the rules of the system; the theorems of a formal system are simply those formulae for which there are derivations. Here we outline a set of alternative and explicitly visual ways of envisaging and analyzing at least simple formal systems using fractal patterns of infinite depth. Progressively deeper dimensions of such a fractal can be used to map increasingly complex wffs or increasingly complex 'value spaces', with tautologies, contradictions, and various forms of contingency coded in terms of color. This and related approaches, it turns out, offer not only visually immediate and geometrically intriguing representations of formal systems as a whole but also promising formal links (1) between standard systems and classical patterns in fractal geometry, (2) between quite different kinds of value spaces in classical and infinite-valued logics, and (3) between cellular automata and logic. It is hoped that pattern analysis of this kind may open possibilities for a geometrical approach to further questions within logic and metalogic.

This is an exploration of a cluster of related logical results. Taken together these seem to have something philosophically important to teach us: something about knowledge and truth and something about the logical impossibility of totalities of knowledge and truth. The book includes explorations of new forms of the ancient and venerable paradox of the :Liar, applications and extensions of Kaplan and Montague's paradox of the Knower, generalizations of Godel's work on incompleteness, and new uses of Cantorian diagonalization. Throughout, the concern is with metaphysical and epistemological implications of these results: what such results may have to teach us about knowledge, truth, and possibility.

Philosophical modeling has a long and distinguished history, but the computer offers new and powerful prospects for the creation and manipulation of models. It seems inevitable that the computer will become a major tool in future philosophical research. Here I offer an overview of explorations in philosophical computer modeling that we in the Group for Logic and Formal Semantics at SUNY Stony Brook have undertaken: explorations regarding (1) the potential emergence of cooperation in a society of egoists, (2) self‐reference and paradox in fuzzy logic, (3) a fractal approach to formal systems, and (4) on‐going explorations with models for the emergence of communication. The attempt is not to give a detailed report on any of these but to give a bird s‐eye view of them all, emphasizing the general form that the models take, the philosophical motivations behind them, and the kinds of philosophical results that they may have to offer.

‘The problem with simulations is that they are doomed to succeed.’ So runs a common criticism of simulations—that they can be used to ‘prove’ anything and are thus of little or no scientific value. While this particular objection represents a minority view, especially among those who work with simulations in a scientific context, it raises a difficult question: what standards should we use to differentiate a simulation that fails from one that succeeds? In this paper we build on a structural analysis of simulation developed in previous work to provide an evaluative account of the variety of ways in which simulations do fail. We expand the structural analysis in terms of the relationship between a simulation and its real-world target emphasizing the important role of aspects intended to correspond and also those specifically intended not to correspond to reality. The result is an outline both of the ways in which simulations can fail and the scientific importance of those various forms of failure.

'Post-critical,' as Echeverria rightly notes, "is one of those catchy, yet slippery phrases which seem to crop up in so many places that they take on a power of their own." But for the purposes of this study, at least, he manages to circumscribe the topic with remarkable clarity: "... the traditional meaning of a critique of knowledge excluded reflection on one's own historical context as an essential trait of philosophical theories, and relegated that context to psychology. 'Post-critical' philosophy wants to reinstate that historical context as one that must itself necessarily be included in the very idea of theory or, as the case may be, of philosophical criticism."

Let us sum up. The paradox of the Knower poses a direct and formal challenge to the coherence of common notions of knowledge and truth. We've considered a number of ways one might try to meet that challenge: propositional views of truth and knowledge, redundancy or operator views, and appeal to hierarchy of various sorts. Mere appeal to propositions or operators, however, seems to be inadequate to the task of the Knower, at least if unsupplemented by an auxiliary recourse to hierarchy. But the cost of hierarchy appears to be an abandonment of any notion of all truth or of omniscience. What the contradictions of the Knower seem to demand, then, is at least an abandonment of these. As noted in the introduction, the argument is complicated enough that one must be wary of dogmatic and precipitate conclusions. One may legitimately wonder whether some new response, or some variation on an old one, will yet offer a way out.Far too often, however, it is asked what has gone wrong with paradox rather than what paradox may have to teach us. What the Knower may have to teach us, I think, is that there really can be no coherent notion of all truth and really can be no coherent notion of omniscience. In its own way that conclusion is perhaps as humbling as is any traditional notion of God.

Debates concerning the nature of mind and consciousness are active and ongoing, with implications for philosophy, psychology, artificial intelligence and the neurosciences. This book collects interviews with some of the foremost philosophers of mind, focusing on open questions, promising projects, and their own intellectual histories. The result is a rich glimpse of the contemporary debate through some of the people who make it what it is. Interviews with Lynne Rudder Baker, David Chalmers, Daniel Dennett, Fred Dretske, Owen Flanagan, Samuel Guttenplan, Valerie Gray Hardcastle, John Heil, Terence Horgan, Douglas Hofstadter, Frank Jackson, Jaegwon Kim, William Lycan, Alva No , Hilary Putnam, David Rosenthal, John Searle, Steven Stich, Galen Strawson, Michael Tye.

Since the sixties, computational modeling has become increasingly important in both the physical and the social sciences, particularly in physics, theoretical biology, sociology, and economics. Sine the eighties, philosophers too have begun to apply computational modeling to questions in logic, epistemology, philosophy of science, philosophy of mind, philosophy of language, philosophy of biology, ethics, and social and political philosophy. This chapter analyzes a selection of interesting examples in some of those areas.

A version of this paper was presented at the IEEE International Conference on Computational Intelligence, combined meeting of ICNN, FUZZ-IEEE, and ICEC, Orlando, June-July, 1994, and an earlier form of the result is to appear as "The Undecidability of the Spatialized Prisoner's Dilemma" in Theory and Decision . An interactive form of the paper, in which figures are called up as evolving arrays of cellular automata, is available on DOS disk as Research Report #94-04i . An expanded version appears as chapter 6 of The Philosophical Computer.