Existential Graphs: What a Diagrammatic Logic of Cognition Might Look Like

Abstract

This paper examines the contemporary philosophical and cognitive relevance of Charles Peirce's diagrammatic logic of existential graphs (EGs), the ‘moving pictures of thought’. The first part brings to the fore some hitherto unknown details about the reception of EGs in the early 1900s that took place amidst the emergence of modern conceptions of symbolic logic. In the second part, philosophical aspects of EGs and their contributions to contemporary logical theory are pointed out, including the relationship between iconic logic and images, the problem of the meaning of logical constants, the cognitive economy of iconic logic, the failure of the Frege–Russell thesis, and the failure of the Language of Thought hypothesis.

Acknowledgements

Supported by the University of Helsinki ‘Excellence in Research’ Grant (2023031, ‘Peirce's Pragmatistic Philosophy and Its Applications’, 2006–2008, Principal Investigator A.-V. Pietarinen). Further support received from the Academy of Finland, Peking University, Sun Yat-sen University and Wuhan University. Earlier versions have been presented in Applying Peirce: International Conference on Peirce's Thought and Its Applications, Helsinki, June 2007; Center for Logic, Language and Cognition Seminar, Peking University, August 2007; the 10th International Conference on Logic and Cognition, Sun Yat-sen University, Guangzhou, November 2008; School of Philosophy, Wuhan University, Wuhan, March 2009; Model-based Reasoning in Science Conference, Symposium on Existential Graphs, Campinas, December 2009; and in the Annual Meeting of the Korean Association for Logic, Seoul, August 2010. My thanks to Johan van Benthem, Risto Hilpinen, Jaakko Hintikka, Woosuk Park, Ju Shier, John Sowa, Zhou Beihai and Zhu Zhifang, as well as to a number of other organizers and commentators of these occasions.

Notes

¹Peirce's allusion is to Peano's pasigraphy: ‘Peano's system is no calculus; it is nothing but a pasigraphy; and while it is undoubtedly useful, if the user of it exercises a discrete freedom in introducing additional signs, few systems of any kind have been so wildly overrated, as I intend to show when the second volume of Russell and Whitehead's Principles of Mathematics appears’ (MS 499). For Peirce logic is ‘not intended for a plaything’; it is neither a universal system of expression nor a calculus in its limited sense: ‘This system [of logical algebras and graphs] is not intended to serve as a universal language for mathematicians or other reasoners, like that of Peano. [And this] system is not intended as a calculus, or apparatus by which conclusions can be reached and problems solved with greater facility than by more familiar systems of expression’ (CP 4.424, c.1903). To these two requisites Peirce adds that he has endeavoured to exclude any consideration of human psyche that could have been involved in those traits of thinking that led to the inventions of the signs employed in his systems of diagram logics.

²Though apparently not in Cambridge (see below)!

³The proof has been reconstrued in Pietarinen & Snellman (2006) and Pietarinen (2010a).

⁴I have detailed in Pietarinen (2006a) how EGs dawned upon Peirce in view of his work on quantificational theory at Johns Hopkins University in collaboration with Oscar H. Mitchell and other students of his. It is noteworthy how in the 1897 Monist paper he comes to suggest ‘Skolem normal forms’ for first-order logic, which accentuates the importance of quantifies dependencies, that is, the functional dependence of Σ s on Π s. He accomplishes the same feat in the beta part of EGs in terms of the nesting of identity lines.

⁵Such as in accepting metalogical approach to ascertain pragmatic purposes of deductive systems, and in agreeing with the centrality of the ‘significance of concepts’ rather than the ‘non-absolute character of truth’ (Lewis 1970: 12).

⁶Moreover, Lewis 1923 studies systems and possible worlds defining them in terms of closely reminiscent of what was later to be known as maximally consistent or Hintikka sets. The semantic notion of possible worlds for modal logics germinated in Peirce's studies (Pietarinen 2006b).

⁷The Welby–Ogden Correspondence, The Welby Fonds, York University Archives, Toronto. All quotations from the correspondence refer to copies received from the York University Archives (YUA).

⁸The original handwritten letter has a continuation which is omitted from the original typescript in the YUA: ‘But I find I have made a mistake and that when he refers in the letter to “my Monist exposition of Existential Graphs” he means the October’1906 number [Prolegomena to an Apology for Pragmaticism] and not the other [Some Amazing Mazes]: which is only a continuation and had to be understood by itself. I hope you will allow me to borrow that number for a short time as well? I am quite ashamed to be so greedy—but I much want to master the difficulty once and for all, and the single number which I have is of little use!’ Ogden soon gets hold of the Prolegomena issue as well and becomes ‘very busy with the Existential Graphs’, being ‘just beginning to see their meaning’, which he admits to be ‘rather a formidable undertaking’ (Ogden to Welby, 12 April 1911).

⁹More precisely, take sheets of assertions as layers in a 3D space and predicates as cylinders connecting those layers. An equivalent method is to take the lines of identities to have direction: for any two non-connected lines, the flow of semantic information can be both the standard outside-in (Peirce's ‘endoporeutic’) as well as the converse, inside-out direction.

¹⁰Transcriptions of MS 669 and MS 670 are available at www.helsinki.fi/~pietarin/courses/.

¹¹In 17 September 1911 Ogden writes to Welby to have seen ‘no signs of Dr Peirce's Book or of the volume Dr Slaughter is editing: or even of Dr Schiller's Logic’. Welby, who died in March 1912, is silent on these matters in their subsequent exchange.

¹²Quoted in Ogden and Richards 1923, p. 282.

¹³The abundance of typographical errors in Ogden's quotations testifies this.

¹⁴Let us add here Wittgenstein's own dismissive assessment of the worthiness of the entire project undertaken in The Meaning of Meaning.

¹⁵‘Whoever wishes a convenient introduction to the remarkable researches into the logic of mathematics that have been made during the last sixty years, and that have thrown an entirely new light both upon mathematics and upon logic, will do well to take up this book. But he will not find it easy reading. Indeed, the matter of the second volume will probably consist, at least nine-tenths of it, of rows of symbols’ (Peirce 1903: p. 308, 15 October, Review of Russell's Principles of Mathematics and Welby's What is Meaning). See also my ‘Notable Women of Logic and Semiotics: Ladd-Franklin and Welby’, forthcoming in Semiotica.

¹⁶The system of archegetic transformation rules is sound, since ‘the rules are so constructed that the permissible transformations are all those, and all those only, by which it is logically impossible to pass from a true graph to a false one’. This metalogical explanation itself ‘is no part of the rules, which simply permit, but do not say why’ (MS 478: 150). The system of rules moreover is what we can express in terms of semantic completeness, since ‘none of its rules follows as a consequence from the rest, while all other permissibilities are consequences of its rules’ (MS 478: 151).

¹⁷The gamma part here concerns the ‘potentials’ and not the broken-cut modal logics. Potentials give rise to higher-order graphs in which quantificational lines refer not to individuals but to what in Peirce's terms are ‘strange kinds’ of ‘proper names’ that refer to ‘substantive possibilities’ devoid of individualities (MS 508, ‘Syllabus B.6’).

¹⁸Peirce makes in his writings on EGs a number of very modern observations, including those pertaining to generalised quantifiers. For example, the quantification in ‘there are at least as many women as men’ is not first-order expressible, as it is represented in a gamma graph by a special enthickened LI identifying the two three-place directed spots ρ 2 (the ‘potentials’), (CP 4.470; see Pietarinen 2007).

²⁰ Pietarinen 2008 suggests EGs for the representation of commands and imperatives.

¹⁹See e.g. Pietarinen 2006a, Roberts 1973, Shin 2002 and Zeman 1964, as well as http://www.helsinki.fi/~pietarin/courses/ for explanations of the basic concepts not covered here.

²¹We can add anaphora here, too: ‘A dean dances in the park. He sings’=‘A dean dances in the park and is singing’. Aside from predication, identity, and existence, the same notation also takes care of coreference.

²² Sánches 1991 studies natural logic, an attempt to take logic closer to the actual structures of natural language. In that regard the goal is similar to Peirce's graphs, to take a step closer to the iconic structures of thought. I am indebted to Johan van Benthem for pointing out the relevance of Sánches's studies.

²³At least not unless we scribe diagrams on higher than three-dimensional SAs.

²⁴ Greimas and Courtés 1982 should function as a warning sign. It is not an occupational hazard that their entry on ‘Semiotics’ has no reference to Peirce, for example.