Skip to main content
SpringerLink
Log in
Book cover

International Conference on Theory and Application of Diagrams

Diagrams 2018: Diagrammatic Representation and Inference pp 549–564Cite as

A Weakening of Alpha Graphs: Quasi-Boolean Algebras

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10871))

Abstract

Peirce introduced the Alpha part of the logic of Existential Graphs (egs) as a diagrammatic syntax and graphical system corresponding to classical propositional logic. The logic of quasi-Boolean algebras (De Morgan algebras) is a weakening of classical propositional logic. We develop a graphical system of weak Alpha graphs for quasi-Boolean algebras, and show its soundness and completeness with respect to this algebra. Weak logical graphs arise with only minor modifications to the transformation rules of the original theory of egs. Implications of these modifications to the meaning of the sheet of assertion are then also examined.

M. Ma—The work supported by the Project Supported by Guangdong Province Higher Vocational Colleges & Schools Pearl River Scholar Funded Scheme (2017).

A.-V. Pietarinen—The work supported by the Estonian Research Council Personal Research Grant PUT 1305 (Abduction in the Age of Fundamental Uncertainty) and Nazarbayev University Social Policy Grant 2018–2019.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    The reference R is to Peirce’s manuscripts by the Robin number [19].

  2. 2.

    He presented the 1880 system as an instrument for the analysis of the logical consequence relation, and thus came close to a sequent-style calculus in that presentation.

  3. 3.

    The De Morgan law \(\lnot (a\wedge b)\le \lnot a\vee \lnot b\) with intuitionistic negation \(\lnot \) does not hold in Heyting algebras while its converse \(\lnot a\vee \lnot b\le \lnot (a\wedge b)\) does. However, both De Morgan laws hold in quasi-Boolean algebras.

  4. 4.

    Peirce knew well this point: “[I]f an operation is thoroughly commutative ..., it is necessarily associative; and associative property is a mere corollary from its commutative property” (R 482).

  5. 5.

    To the opposite direction, namely making it explicit that the sheet is precisely that of the assertions, see [3, 4].

  6. 6.

    Instead of a weakening of the classical Alpha to get the proposed quasi-Boolean graphs, we could strengthen the implication-free fragment of the intuitionistic system of Alpha [13] for the same effect. There the scroll would only represent involution and there is no implication. A new rule is

    figure d

References

  1. Bellucci, F., Pietarinen, A.-V.: Existential graphs as an instrument of logical analysis: part I. Alpha. Rev. Symb. Logic 9, 209–237 (2016)

    Article  MathSciNet  Google Scholar 

  2. Bellucci, F., Pietarinen, A.-V.: From Mitchell to Carus: 14 years of logical graphs in the making. Trans. Charles S. Peirce Soc. 52(4), 539–575 (2017a)

    Article  Google Scholar 

  3. Bellucci, F., Pietarinen, A.-V.: Assertion and denial: a contribution from logical notations. J. Appl. Logics 25, 1–22 (2017b)

    Article  MathSciNet  Google Scholar 

  4. Bellucci, F., Chiffi, D., Pietarinen, A.-V.: Assertive graphs. J. Appl. Non-Class. Logics (2018). https://doi.org/10.1080/11663081.2017.1418101

  5. Belnap Jr., N.D.: A useful four-valued logic. In: Dunn, J.M., Epstein, G. (eds.) Modern Uses of Multiple-Valued Logic, pp. 5–37. Springer, Dordrecht (1977). https://doi.org/10.1007/978-94-010-1161-7_2

    Chapter  Google Scholar 

  6. Brünnler, K.: Deep Inference and Symmetry in Classical Proofs. Logos, Berlin (2003)

    MATH  Google Scholar 

  7. Dunn, J.M.: A relational representation of quasi-Boolean algebras. Notre Dame J. Formal Logic 23(4), 353–357 (1982)

    Article  MathSciNet  Google Scholar 

  8. Guglielmi, A.: Deep Inference. http://alessio.guglielmi.name/res/cos/

  9. Ma, M., Fei, L.: Sequent calculi for semi-De Morgan and De Morgan algebras. Stud. Logica (2017). https://doi.org/10.1007/s11225-017-9755-7

  10. Ma, M., Pietarinen, A.-V.: Peirce’s sequent proofs of distributivity. In: Ghosh, S., Prasad, S. (eds.) ICLA 2017. LNCS, vol. 10119, pp. 168–182. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-54069-5_13

    Chapter  Google Scholar 

  11. Ma, M., Pietarinen, A.-V.: Proof analysis of Peirce’s alpha system of graphs. Stud. Logica 105(3), 625–647 (2017)

    Article  MathSciNet  Google Scholar 

  12. Ma, M., Pietarinen, A.-V.: Gamma graph calculi for modal logics. Synthese (2017). https://doi.org/10.1007/s11229-017-1390-3

  13. Ma, M., Pietarinen, A.-V.: A graphical deep inference system for intuitionistic logic. Log. Anal. (in press)

    Google Scholar 

  14. Oostra, A.: Los gráficos Alfa de Peirce aplicados a la lógica intuicionista. Cuad. Sist. Peirceana 2, 25–60 (2010)

    Google Scholar 

  15. Peirce, C.S.: On an improvement in Boole’s calculus of logic. Proc. Am. Acad. Arts Sci. 7, 250–261 (1867)

    Google Scholar 

  16. Peirce, C.S.: On the algebra of logic. Am. J. Math. 3(1), 15–57 (1880)

    Article  Google Scholar 

  17. Peirce, C.S.: On the algebra of logic: a contribution to the philosophy of notation. Am. J. Math. 7(2), 180–196 (1885)

    Article  MathSciNet  Google Scholar 

  18. Peirce, C.S.: Letter to Christine Ladd-Franklin, 9 November 1900. (R L 237)

    Google Scholar 

  19. Peirce, C.S.: Manuscripts in the Houghton Library of Harvard University, as identified by Richard Robin (1967). Cited as R followed by manuscript number

    Google Scholar 

  20. Pietarinen, A.-V.: Signs of Logic: Peircean Themes on the Philosophy of Language, Games, and Communication. Springer, Dordrecht (2006). https://doi.org/10.1007/1-4020-3729-5

    Book  MATH  Google Scholar 

  21. Pietarinen, A.-V.: Is non-visual diagrammatic logic possible? In: Gerner, A. (ed.) Diagrammatology and Diagram Praxis. College Publications, London (2010)

    Google Scholar 

  22. Pietarinen, A.-V.: Exploring the Beta quadrant. Synthese 192, 941–970 (2015)

    Article  MathSciNet  Google Scholar 

  23. Roberts, D.D.: The existential graphs and natural deduction. In: Studies in the Philosophy of C. S. Peirce, pp. 109–121. University of Massachusetts Press (1964)

    Google Scholar 

  24. Roberts, D.D.: The Existential Graphs of Charles S. Peirce. Mouton, The Hague (1973)

    Google Scholar 

  25. Shin, S.-J.: The Iconic Logic of Peirce’s Graphs. MIT Press, Cambridge (2002)

    Book  Google Scholar 

  26. Sowa, J.: Knowledge Representation. Brooks, Pacific Grove (2000)

    Google Scholar 

  27. Zeman, J.: The Graphical Logic of Charles S. Peirce. Ph.D. dissertation, University of Chicago (1964)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Logic and Cognition, Sun Yat-sen University, Guangzhou, China

    Minghui Ma

  2. Tallinn University of Technology, Tallinn, Estonia

    Ahti-Veikko Pietarinen

  3. Nazarbayev University, Astana, Kazakhstan

    Ahti-Veikko Pietarinen

Authors
  1. Minghui Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ahti-Veikko Pietarinen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ahti-Veikko Pietarinen .

Editor information

Editors and Affiliations

  1. Edinburgh Napier University, Edinburgh, UK

    Peter Chapman

  2. University of Brighton, Brighton, UK

    Gem Stapleton

  3. Tallinn University of Technology, Tallinn, Estonia

    Amirouche Moktefi

  4. Mitre Corporation, McLean, VA, USA

    Sarah Perez-Kriz

  5. University of Bologna, Bologna, Italy

    Francesco Bellucci

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Ma, M., Pietarinen, AV. (2018). A Weakening of Alpha Graphs: Quasi-Boolean Algebras. In: Chapman, P., Stapleton, G., Moktefi, A., Perez-Kriz, S., Bellucci, F. (eds) Diagrammatic Representation and Inference. Diagrams 2018. Lecture Notes in Computer Science(), vol 10871. Springer, Cham. https://doi.org/10.1007/978-3-319-91376-6_50

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-91376-6_50

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-91375-9

  • Online ISBN: 978-3-319-91376-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation