1. The game fractal outlined here can be thought of as a fractally embedded form of the familiar game tree. See for example A. K. Dewdney, The New Turing Omnibus, New York: Computer Science Press, 1993, esp. chapter 6, and A. L. Samuel, "Some studies in machine learning using the game of checkers," in Computers and Thought, ed. E. A. Feigenbaum and J. Feldman, New York: McGraw-Hill, 1968, pp. 71-108.
2. The following section offers a graphical representation of an approach for enumerating logical formulae mentioned in "Logic for Computer Science" by Steve Reeves and Michael Clark Addison- Wesley, 1990;
3. See for example Robert L. Devaney, Chaos, Fractals, and Dynamics, Menlo Park: Addison- Wesley, 1990; Heinz-0tto Peitgen, Hartman J?rgens, and Dietmar Saupe, Fractals for the Classroom, New York: Springer-Verlag, 1992; and A. J. Crilly, R. A. Earnshaw, and H. Jones, eds., Fractals and Chaos, New York: Springer-Verlag, 1991.
4. It is tempting--but would be mistaken--to try to use this schema as a representation not only for full propositional calculus, but for a full infinitary propositional calculus, allowing for infinite formulae involving infinite connectives by way of conjunction, disjunction, or Sheffer strokes. (Infinitary systems of this type appear in Leon Henkin, "Some Remarks on Infinitely Long Formulas," in International Mathematical Union and Mathematical Institute of the Polish Academy of Sciences, eds., Infinitistic Methods, New York: Pergamon Press, 167-183 and in Carol Karp, Languages with Expressions of Infinite Length, Amsterdam: North-Holland, 1964.) This is tempting for one reason because infinite disjunctions of sentence letters represented in this way might seem to offer nonperiodic binary decimals. A simple example would consist of the disjunction of all our atomic sentence letters, giving us the truth table 01111... , with no repetition of its initial zero. For a more interesting example, consider an infinite disjunction which leaves out some of the set of sentence letters. Leave out only the second sentence letter, as outlined above, and you would appear to get the disjunctive value 01011111... Leave out only the third and you would appear to get the pattern 01110111.... In general, leaving out the nth sentence letter from an infinite disjunction of all sentence letters would appear to introduce a zero in the (2n-1 + 1)th place. If every even sentence letter of the set were left out, so the reasoning goes, the result would be a classic non-periodic decimal in which 0's are separated by ever- increasing expanses of 1's. An interpretation of infinitely-extended truth-tables is also tempting because universal quantification can be thought of as an infinite conjunction, existential quantification as an infinite disjunction. Were this scheme interpretable in such a way, then, it would offer a model not only for propositional but predicate calculus. Restricted to finite connectives it can at best correspond only to arbitrarily large finite models for propositional calculus. The difficulty which blocks both of these tempting moves, however, is that the infinite extension of truth-tables outlined, although adequate for arbitrarily large finite complexes, cannot be thought of as adequate for genuinely infinite complexes. This becomes evident if one asks at what point in the table we will find a row which represents a '1' value for all of our sentence letters; it is clear that such a row can have no (finite) place in the scheme. A standard diagonal argument gives the same result: that there will be an infinite complex of our sentence letters which has no corresponding row in the table, and thus that the table will not be adequate for representation of all values in a genuinely infinitary system. For that we would require truth- tables somehow not merely of countably infinite but of uncountable length. 5. See Manfred Schroeder, Fractals, Chaos, and Power Laws, New York: W. H. Freeman and Co., 1991, esp. pp. 20-25.
6. See Gerald A. Edgar, Measure, Topology, and Fractal Geometry, New York: Springer-Verlag, 1990.
7. This second type of value solid first appears in Gary Mar and Paul St. Denis, "Chaos in Cooperation: Continuous-Valued Prisoner's Dilemmas in Infinite-Valued Logic," International Journal of Bifurcation and Chaos, 4 (1994), 943-958.
8. ukasiewicz himself outlined his system in terms of implication and negation. Here we take as a ukasiewicz 'or' the classical transform from implication: /p v q/ = /~p and q/), with 'and' by a similar transformation. See Nicholas Rescher, Many-valued Logic, New York: McGraw-Hill, 1969.
9. Ivan Amato, "Speculating in Precious Computronium," Science 253, August 1991, 856-857.