Paradox of Self-Amendment by Peter Suber 7. The distinction between veridical and falsidical paradoxes is spurious under the law of excluded middle in a standard two-valued logic. This is seen by the by the equivalency in such a logic of describing Zeno's purpose to be the falsification of a common belief and the establishment of its uncommon negation. I will not use the distinction, but neither will I rely on the law of excluded middle. For a selection of modern essays on Zeno's paradoxes, and an excellent bibliography, see Wesley C. Salmon (ed.), Zeno's Paradoxes, Bobbs-Merrill, 1970. 8. Without the characteristic of leading to genuine paradox if affirmed, it must be admitted that the Barber-type paradox would be an ordinary contradiction distinguished from other contradictions only by the plausibility or apparent innocuousness of the "paradoxical" statement. 9. The same alternatives apply to Zeno's paradoxes. We are free to choose between affirming the logic which makes our experience of motion illusory, and affirming the veridicity of our experience, at least of motion, and upsetting the logic that would deny it. Proposed solutions to Zeno's paradoxes have traditionally been variations on the latter theme. The fact that we are apparently free to deny Zeno and revise our logic in the light of experience supports the theories of "mutable" logic cited below in note 17. 10. Hans Kelsen originally took this position in his General Theory of Law and the State, Harvard University Press, 1949, pp. 363, 375, and in his Pure Theory of Law, University of California Press, 1967, pp. 74, 328. Kelsen partially abandoned this position in his essay, "Derogation," in Ralph A. Newman (ed.), Essays in Jurisprudence in Honor of Roscoe Pound, Bobbs-Merrill, 1962, pp. 339-55, at pp. 351ff, holding that conflicting norms do not automatically imply repeal, although he also says (p. 351) that such conflict should not be compared to logical contradiction. See H.L.A. Hart, "Kelsen's Doctrine of the Unity of Law," in his Essays in Jurisprudence and Philosophy, Oxford University Press, 1983, pp. 309-42. See also Eduardo Garcia Maynez, "Some Considerations on the Problem of Antinomies in the Law," Archiv für Rechts- und Sozialphilosophie, 49, 1 (1963) 1-14. 11. David Daube, "Greek and Roman Reflections on Impossible Laws," Natural Law Forum, 12 (1967) 1-84 at p. 4. 12. Quine, op. cit. at p. 84 13. As an example one may cite the 1897 action of the Indiana House of Representatives in voting unanimously to legislate an incorrect value of pi, and to charge non-residents of Indiana a royalty for use of the value. The bill was approved on its first reading in the state Senate, but on the second reading indefinitely postponed. Petr Beckman, A History of Pi, St. Martin's Press, 1971, pp. 174-79. However, if lawyers often lack the logical scruples of logicians, then as Alf Ross the logician will demonstrate despite Alf Ross the jurist, logicians just as often lack the legal scruples of lawyers. 14. See the remainder of Part One, and Section 21.B. By "logical system" I will mean a formal, abstract system built on, or around, the principle of non-contradiction, as opposed to dialectical logics which embrace and employ contradiction. Paradigm examples of logical systems are those built by Aristotle, Frege, and by Whitehead and Russell. Philosophers in the dialectical tradition, preeminently Hegel, have developed logical systems that contain and depend on contradiction. None of the primary modern expositions of dialectic in Hegel, Marx, Engels, or Lenin addresses the logical paradoxes specifically. But as a logic that embraces contradiction dialectic should at least appropriate the malevolence of the paradoxes, if not "solve" them. Such an argument is made for modern dialectic by Henri Wald in his Introduction to Dialectical Logic, B.R. Grüner B.V., 1975, pp. 228-30. Ancient or Platonic dialectic differs in many ways from Hegelian and Marxist dialectic. But we know that Zeno of Citium (the founder of Stoicism, not the inventor of the paradoxes of motion) believed that the logical paradoxes could be solved by dialectic. See Plutarch, "On the Contradictions of the Stoics," 8:1034E, Moralia, vol. XIII of the Plutarch series of the Loeb Classical Library, Harvard University Press, 1976. 15. Recognizing the utility of dialectic is only one reason to resist the temptation. 16. The discussion of this paradox has been limited to virtually one man. See Alf Ross, "On Self-Reference and a Puzzle in Constitutional Law," Mind, 78 (1969) 1-24; earlier formulations appeared in his On Law and Justice, London, 1958, Section 16, pp. 80-84, his Dansk Statsforfatningsret [Danish Constitutional Law], Copenhagen: Nyt Nordisk Forlag, 2 vols., 1966, Sections 41 and 46, and in his Theorie der Rechtsquellen, F. Deuticke, 1929, Chapter XIV. Danish responses to his earlier formulations, which Ross took into account in his latest formulation, are cited in the Mind article, p. 7.n.1. See also Joseph Raz, "Professor Ross and Some Legal Puzzles," Mind, 81 (1972) 415-21; Norbert Hoerster, "On Alf Ross's Alleged Puzzle in Constitutional Law," Mind, 81 (1972) 422-26; J.M. Finnis, "Revolutions and Continuity of Law," in A.W.B. Simpson (ed.), Oxford Essays in Jurisprudence, Second Series, Oxford University Press, 1973, pp. 44-76 esp. pp. 53ff. Note that what is often called "Ross's Paradox" is not the paradox of self-amendment, but another paradox discovered by Alf Ross: if "'p' implies 'p or q'", then why do we hesitate to affirm that "'Smith is obligated to do p' implies that 'Smith is obligated to do p or q'"? See Jaakko Hintikka, "The Ross Paradox as Evidence for the Reality of Semantical Games," Monist, 60 (1977) 370-79; Azizah Al-Hibri, "Understanding Ross's Paradox," Southwestern Journal of Philosophy, 10 (1979) 163-70. Ross is as well-known for his contribution to deontic logic as for his jurisprudence. 17. Logical rules may be optional even though immutable or eternal. They are optional in the sense that the choice of initial axioms is either arbitrary or arbitrary within limits, and different initial axioms determine different subsequent rules. They are eternal in the sense that, once options are exercised and a system of valid rules is determined, the validity of the rules is not subject to change over time. Another way to make the same point is to say that experience, or events in time, cannot affect the validity of logical rules. A strong statement of this position is to be found in Wittgenstein: 8