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«1. Peirce’s Method for Teaching Logic Peirce’s existential graphs (EGs) are the simplest, most elegant, and easiest-to-learn system of logic ever ...»

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A.4 Extensions for Gamma Graphs Throughout his long career, Peirce experimented with a variety of notations for logic and a wide range of semantic extensions that went far beyond ordinary first-order logic. In his article of 1885, in which he presented his most complete version of the algebraic notation, he used the terms first-intentional logic for quantifiers that range over simple individuals and second-intentional logic for quantifiers that range over relations. In that article, he used second intentional logic to define equality x=y by a statement that for every relation R, R(x) if and only if R(y). Ernst Schröder translated Peirce’s terms to erste Ordnung and zweite Ordnung, which Bertrand Russell translated back to English as first order and second order. Peirce also introduced notations for three-valued logic, modal logic, and metalanguage about logic. Roberts (1973) summarizes the various graphical and algebraic notations and cites the publications and manuscripts in which Peirce discussed them.

Peirce used the term Gamma graphs for the many variations of EGs that went beyond first-order (or firstintentional) logic. As early as 1898, he used the following example of a metalevel statement in EGs:

The sentence inside the oval could be expressed in EGIF as a proposition or medad with a name enclosed in double quotes. The line of identity and phrase outside the oval could be expressed by a defining node and a monadic relation with an enclosed name. But neither EGIF nor CGIF as defined by ISO/IEC 24707 can represent a line of identity linked to an oval. A proposed extension to Common Logic called IKL (Hayes and Menzel 2006) can support such constructs. An extension to the CLIF dialect with the IKL semantics uses an

operator that followed by a CLIF sentence to denote the proposition stated by the sentence:

("is much to be wished" (that ("You are a good girl")))

An equivalent extension to CGIF or EGIF would use the following notation:

[*x ("You are a good girl")] ("is much to be wished" ?x) Either the EG or its translation to CLIF, CGIF, or EGIF could be read That you are a good girl is much to be wished. A syntactic extension to EGIF for such expressions could be represented with an optional EG in the

rule for DefiningNode:

DefiningNode = '[', DefiningLabel, [EG] ']';

When a line of identity represents a proposition, the bound label prefixed with # could be used in the type position of a medad to assert the proposition: (#?x). It could then be negated in the usual way, ~[(#?x)], to say that it is false that you are a good girl. In some writings, Peirce used ovals with colors or dotted boundaries to represent modality. EGIF can use identifiers such as Possible or Necessary for relations applied to propositions. The first line of the following EGIF defines x as the proposition that you are a good girl and y as its negation. The second line says that if x is necessary, then y is not possible. With the double negation erased by rule 3e, that line would say it’s false that x is necessary and y is possible.

[*x ("You are a good girl")] [*y ~[(#?x)]] ~[(Necessary ?x) ~[ ~[(Possible ?y)] ]] The semantics of EGIF is formally defined by the model theory of Common Logic or the IKL extensions. For Alpha and Beta graphs, the EGIF semantics seems to be consistent with what Peirce had intended.

Determining exactly what he had intended for his many variations of Gamma graphs is still a research project, for which EGIF can be a useful tool.

References Barwise, Jon, and John Etchemendy (1993). Tarski’s World. Stanford, CA: CSLI Publications.

Dau, Frithjof (2006). Some notes on proofs with Alpha graphs. In Conceptual Structures: Inspiration and Application, (LNAI 4068), H. Schärfe, P. Hitzler, and P. Øhrstrom (eds.), 172-188. Berlin: Springer.

Dau, Frithjof (2010). Ligatures in Peirce’s Existential Graphs. Semiotica [This issue].

Fine, Kit (1985). Reasoning with Arbitrary Objects. Oxford: Basil Blackwood.

Frege, Gottlob (1879). Begriffsschrift. In From Frege to Gödel, J. van Heijenoort (ed.) (1967), 1-82. Cambridge, MA:

Harvard University Press.

Gentzen, Gerhard (1934). Untersuchungen über das logische Schließen [Investigations into logical deduction]. In The Collected Papers of Gerhard Gentzen, M. E. Szabo (ed. and translator), (1969), 68-131. Amsterdam: North-Holland Publishing Co.

Hayes, Patrick, and Chris Menzel (2006) IKL Specification Document, http://www.ihmc.us/users/phayes/IKL/SPEC/SPEC.html (accessed 15 November 2009).

Henkin, Leon (1961). Some remarks on infinitely long formulas. In Infinitistic Methods, (Proceedings of symposium on foundations of mathematics), 176-183. London: Pergamon Press.

Hilpinen, Risto (1982). On C. S. Peirce’s theory of the proposition: Peirce as a precursor of game-theoretical semantics.

The Monist 65: 182-188.

Hintikka, Jaakko (1973). Logic, Language Games, and Information. Oxford: Clarendon Press.

ISO/IEC (1996). Extended BNF, (IS 14977). Geneva: International Organisation for Standardisation.

ISO/IEC (2007). Common Logic (CL) — A Framework for a family of Logic-Based Languages, (IS 24707). Geneva:

International Organisation for Standardisation.

Majumdar, Arun K., and John F. Sowa (2009). Two paradigms are better than one, and multiple paradigms are even

better. In Proceedings of ICCS 2009, (LNAI 5662), S. Rudolph, F. Dau, and S.O. Kuznetsov (eds.), 32-47. Berlin:


Ockham, William of (1488 [1323]). Summa Logicae. Paris: Johannes Higman. (The edition owned by C. S. Peirce.) Peano, Giuseppe (1889). Aritmetices principia nova methoda exposita [Principles of arithmetic presented by a new method]. Torino: Bocca.

Peirce, Charles Sanders (1869). Grounds of validity of the laws of logic. Journal of Speculative Philosophy 2: 193-208.

http://www.peirce.org/writings/p41.html (accessed 15 November 2009).

Peirce, Charles Sanders (1878). How to Make Our Ideas Clear. Popular Science Monthly 12: 286-302.

http://www.peirce.org/writings/p119.html (accessed 15 November 2009).

Peirce, Charles Sanders (1880). On the algebra of logic. American Journal of Mathematics 3: 15-57.

Peirce, Charles Sanders (1882). Letter to O. H. Mitchell. In Writings of Charles S. Peirce, (1982-1993), 4: 394-399.

Bloomington: Indiana University Press.

Peirce, Charles Sanders (1885). On the algebra of logic. American Journal of Mathematics 7:180-202.

Peirce, Charles Sanders (1898). Reasoning and the Logic of Things, (The Cambridge Conferences Lectures of 1898), K. L. Ketner (ed) (1992). Cambridge, MA: Harvard University Press.

Peirce, Charles Sanders (1909). Manuscript 514. Transcribed by Michel Balat with commentary by J. F. Sowa.

http://www.jfsowa.com/peirce/ms514.htm (accessed 15 November 2009).

Peirce, Charles Sanders (1931-1958 CP). Collected Papers of C. S. Peirce, C. Hartshorne, P. Weiss, and A. Burks (eds.), 8 vols., 1931-1958. Cambridge, MA: Harvard University Press.

Pietarinen, Ahti-Veikko (2006). Signs of Logic: Peircean Themes on the Philosophy of Language, Games, and Communication. Dordrecht: Springer. Semiotica Roberts, Don D. (1973). The Existential Graphs of Charles S. Peirce. The Hague: Mouton.

Robinson, J. Alan (1965). A machine oriented logic based on the resolution principle. Journal of the ACM 12: 23-41.

Simons, Peter (2004). Judging correctly: Brentano and the reform of elementary logic. In The Cambridge Companion to Brentano, D. Jacquette (ed.), 45-65. Cambridge: Cambridge University Press, Cambridge.

Sowa, John F. (1992 [1987]). Semantic networks. In Encyclopedia of Artificial Intelligence,, S. C. Shapiro (ed.). New York: Wiley.

Sowa, John F. (2000). Knowledge Representation: Logical, Philosophical, and Computational Foundations. Pacific Grove, CA: Brooks/Cole Publishing Co.

Stewart, John (1996). Theorem Proving Using Existential Graphs, MS Thesis, Computer and Information Science.

Santa Cruz: University of California.

Tarski, Alfred (1933). Pojęcie prawdy w językach nauk dedukcynych [The concept of truth in formalized languages].

In Logic, Semantics, Metamathematics, A. Tarski (1982), 2nd ed, 152-278. Indianapolis: Hackett.

Tarski, Alfred (1936). Über den Begriff der logischen Folgerung [On the concept of logical consequence]. In Logic, Semantics, Metamathematics, A. Tarski (1982), 2nd ed, 409-420. Indianapolis: Hackett.

Whitehead, Alfred North, and Bertrand Russell (1925 [1910]) Principia Mathematica, 2nd ed. Cambridge: Cambridge University Press.

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