Notes

*

   Doctoral Dissertation of J. Jay Zeman, The University of Chicago, 1964.
Preface:    

ii.1.

   C. S. Peirce, The Collected Papers of C. S. Peirce, ed. Charles Hartshorne and Paul Weiss, Cambridge: Harvard, Vols. I-VI.
     

Introduction:

    
001.1.   Murray G. Murphey, The Development of Peirce's Philosophy, (Cambridge: Harvard, 1961), p. 357.
001.2.   Peirce thus subtitles what is Book 2 of Volume IV of The Collected Papers, At this point we shall note again what we mentioned in the Preface, that citations from this collection shall generally be listed in the text of the paper without footnote, employing the decimal notation normally employed in Peirce scholarship; thus "4.327 " for example, will mean paragraph 327 of Vol. IV of the Collected Papers.
013.1.   At this point we will mention a "categorical rule of transformation" which is listed by Peirce, but which we will find no occasion to use in our work with the graphs. This is the rule which reads "Any graph well-understood to be true may be scribed unenclosed" (4.507). This rule may be considered a rule for the introduction of "extralogical" premises or hypotheses onto the sheet of assertion in order to reason about special problems.
015.1.   The rules of beta iteration and deiteration are somewhat difficult to state in a compact and perspicuous manner, as Peirce's own statements of them testify. Let us give this intuitive picture of how to do it. Given a graph Y containing a subgraph X which we wish to iterate, first copy the whole of Y on a separate sheet of paper. Now take a pair of scissors and cut the subgraph X out of the new copy of Y. In cutting X out of Y, we are not allowed to cut across any cuts, but we may cut through lines of identity which bind X to other parts of the whole graph Y. Also, the portion of Y thus cut out must be all one piece of paper. Take the piece of paper thus cut out and glue it onto the original graph Y, either in the same area within which the original subgraph X occurs, or in an area enclosed by  at least all the cuts enclosing that original occurrence of X. Now if there are any points [belonging to X] of lines of identity standing outside all cuts in X, any such point in the original occurrence of X may be connected to the corresponding point in the new occurrence of X by "geodesic line of identity" (this term will be defined in chapter II). Or, if a "geodesic line of identity" connected to a point belonging to the original X but outside all cuts in the original X already goes from that point to within the area in which the new X has been "pasted," the the corresponding point in the new X may be connected directly to the already existing line of identity. But the line of identity connections are optional.
024.1.   Synechism is a metaphysical doctrine developed by Peirce in his later years and in which he set great store. An "instant characterization" of synechism might be that it is a doctrine asserting "the reality of continua and the continuity of reality." But this is, of course, rather oversimplified. For further reference on this topic, see Murphey, pp.379 ff.
025.1.   Ibid., p. 406; Murphey here quotes from an unpublished paper of Peirce.
025.2.   Ibid., p. 405.
026.1.   Ibid., p. 387.
041.1.   Quasi-quotes: and are used as in Quine 1958.
045.1.   Martin Davis, Computability and Unsolvability (New York: McGraw Hill, 1958), p. 117 ff.
053.1.   A. N. Prior, Formal Logic (2nd ed; Oxford, Oxford, 1962), p. 64 ff.
065.1.   Ibid., p. 306.
066.1.   Davis, p. 119.
083.1.   Annotation, June 2002: Note that color available for the current publication of this work makes the lines of identity much clearer and easier to read (and their use is consistent with Peirce's own practice, which can be seen in color reproductions of the Mss). Color and other devices available in our technology also might be profitably incorporated into the formal structure of this system of diagrams (also a very Peircean move) but that is part of a story yet to be told; the current use of color will just help to illuminate the formal material set down by me in 1964. But much profitable extension is possible!
085.1.   Cf. Willard Van Orman Quine, Mathematical Logic (rev. ed:; Cambridge: Harvard, 1958), p. 70. Note the similarity between Quine's, "curved line formulas" and the beta graphs which will constitute the range of g¢.
093.1.   Alonzo Church, Introduction to Mathematical Logic (Princeton: Princeton, 1956, I, pp, 180-81.
104.1.   The perspective of decades reveals this speculation to be untrue: Peirce indeed developed a complete algebraic quantificational logic, as early as 1885. See, for example, my paper "Peirce's Philosophy of Logic," originally presented at a conference on "The Birth of Mathematical Logic" at Fredonia College, SUNY in March 1983. This paper appeared in the Transactions of the Charles S. Peirce Society 22 (1986), 1-22.
105.1.   Cf. . Prior., p . 366.
107.1.   Quine, p. 79.
107.2.   Note that since vacuous quantifiers are among those allowed to be introduced in the schema (2), Quine's *102 would also be easily provable in this system. We shall not, however, make use of this in the present development.
122.1.   Church, p. 55.
138.1.   Davis , p. 119.
141.1.   John M. Anderson and Henry. W. Johnstone, Jr., Natural Deduction (Belmont, California: Wadsworth, 1962), p. 130.
144.1.   An excellent quick reference to these well-known formulations may be found in Prior, pp. 312 ff.
147.1.   For details on this system, see Prior, pp. 208-209, and his Time and Modality, pp. 1 ff.
159.1.   This theorem-schema holds as stated in the systems S4, S4.2, and S5; at this point we will make no claims one way or the other for the other Lewis-modal systems. I am not aware of a general proof. for this thesis in the literature, so I have added to this paper, a short appendix which discusses the question of the deduction theorem in S4, S4.2, and S5, and shows that this schema holds in these systems.
161.1.   E. J. Lemmon, "New Foundations for the Lewis Modal Systems," Journal of Symbolic Logic 22 (1956)., 176-86.
164.1.   As we noted earlier, see the Appendix for the proof of this.
171.1.   But cf.., for :example, C, I, Lewis and C. H. Langford, Symbolic Logic (New York: Dover, 1959) , p. 501.
173.1.   Prior, Formal Logic, p . 312.
176.1.   J. Jay Zeman, "Bases for S4 and S4.2 without Added Axioms," Notre Dame Journal of Formal Logic 4 (1963), 227-30.
180.1.   Church.. p..87.
181.1.   Cf. ibid., p. 196.
181.2.   Prior, Formal Logic, p . 312.
181.3.   Zeman, NDJFL.
183.1.   Church, pp . 88-89