Operators in the paradox of the knower

Predicates are term-to-sentence devices, and operators are sentence-to-sentence devices. What Kaplan and Montague's Paradox of the Knower demonstrates is that necessity and other modalities cannot be treated as predicates, consistent with arithmetic; they must be treated as operators instead. Such is the current wisdom.A number of previous pieces have challenged such a view by showing that a predicative treatment of modalities neednot raise the Paradox of the Knower. This paper attempts to challenge the current wisdom in another way as well: to show that mere appeal to modal operators in the sense of sentence-to-sentence devices is insufficient toescape the Paradox of the Knower. A family of systems is outlined in which closed formulae can encode other formulae and in which the diagonal lemma and Paradox of the Knower are thereby demonstrable for operators in this sense.I am deeply indebted to Robert F. Barnes and Evan W. Conyers, without whom these ideas might not have germinated and certainly would not have grown. Many of the results offered here evolved in the course of mutual discussion and correspondence. I am also grateful to an anonymous reviewer forSynthese for many very helpful suggestions.The current paper contains the technical results promised in Footnote 25 of Grim (1988) and Footnote 26, Chapter 3, of Grim (1991).     

Fractal Images of Formal Systems

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.\looseness=-1     

Evolution of communication in perfect and imperfect worlds

We extend previous work on cooperation to some related questions regarding the evolution of simple forms of communication. The evolution of cooperation within the iterated Prisoner's Dilemma has been shown to follow different patterns, with significantly different outcomes, depending on whether the features of the model are classically perfect or stochastically imperfect (Axelrod, 1980a,b, 1984, 1985; Axelrod and Hamilton, 1981; Nowak and Sigmund, 1990, 1992; Sigmund, 1993). Our results here show that the same holds for communication. Within a simple model, the evolution of communication seems to require a stochastically imperfect world.     

Correction to: Rational social and political polarization

Philosophical Studies - In the original publication of the article, the Acknowledgement section was inadvertently not included. The Acknowledgement is given in this Correction.