Skip to main content
SpringerLink
Log in

An Application of Peircean Triadic Logic: Modelling Vagueness

  • Published:
Journal of Logic, Language and Information Aims and scope Submit manuscript

Abstract

Development of decision-support and intelligent agent systems necessitates mathematical descriptions of uncertainty and fuzziness in order to model vagueness. This paper seeks to present an outline of Peirce’s triadic logic as a practical new way to model vagueness in the context of artificial intelligence (AI). Charles Sanders Peirce (1839–1914) was an American scientist–philosopher and a great logician whose triadic logic is a culmination of the study of semiotics and the mathematical study of anti-Cantorean model of continuity and infinitesimals. After presenting Peircean semiotics within AI perspective, a mathematical formulation of a Peircean triadic set is given in relationship with classical and fuzzy sets. Using basic logical operators, all possible respective implication operators, bi-equivalence operators, valid rules of inference, and associative, distributive and commutative logical properties are derived and verified through the truth function approach. In order to suggest practical directions, aggregation operators for Peirce’s triadic logic have been formulated. A mathematical formulation of a medical diagnostic problem and ER diagram of a library management system using Peirce’s triadic relation show potential for further applications of the proposed triadic set and triadic logic. Alongside, a classical AI game—The Wumpus World—is implemented to show practical efficacy in comparison with binary implementation. Besides giving some preliminary formulations for trichotomous set theory and definition of finite automaton, development of hybrid architectures for intelligent agents and evolutionary computations are discussed as potential practical avenues for Peirce’s triadic logic.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Notes

  1. Though Peirce connects this definition with philosophical logic, it is applicable to Triadic logic as well.

  2. For more details about axiomatic systems, continuity, infinitesimals, and topology see Shield (1981), Bell (2005), Lane (1999) and Eisele (1976).

References

  • Agler, D. W. (2010). Vagueness and its boundaries: A Peircean theory of vagueness. Ph.D. thesis, Faculty of the University Graduate School in partial fulfillment of the requirements for the degree Master of Arts in the Department of Philosophy, Indiana University.

  • Agler, D. W. (2013). Peirce and the specification of borderline vagueness. Semiotica, 2013(193), 195–215.

    Article  Google Scholar 

  • Akhtar, J., Koshul, B. B., & Awais, M. M. (2013). A framework for evolutionary algorithms based on Charles Sanders Peirce’s evolutionary semiotics. Information Sciences, 236, 93–108.

    Article  Google Scholar 

  • Annoni, M. (2006). Implications of synechism: Continuity and second-order vagueness. Cognitio-Estudos: Revista Electronica de Filosofia, 3(2), 96–108.

    Google Scholar 

  • Augusto, L. M. (2011). Putting the horse before the cart: A pragmatist analysis of knowledge. Trans/Form/Ação, 34(2), 135–152.

    Article  Google Scholar 

  • Beg, I., & Khalid, A. (2012). Belief aggregation in fuzzy framework. Journal of Fuzzy Mathematics, 20(4), 911–924.

    Google Scholar 

  • Bell, J. L. (2005). The continuous and the infinitesimal in mathematics and philosophy. Milan: Polimetrica S.A.S.

    Google Scholar 

  • Belnap, N. D. (1970). Conditional assertion and restricted quantification. Noûs, 4, 1–12.

    Article  Google Scholar 

  • Black, M. (1937). Vagueness. An exercise in logical analysis. Philosophy of Science, 4(4), 427–455.

    Article  Google Scholar 

  • Branting, L. K., & Aha, D. W. (1995). Stratified case-based reasoning: Reusing hierarchical problem solving episodes. In Proceedings of the 14th international joint conference on Artificial intelligence (Vol. 1, pp. 384–390). Los Altos: Morgan Kaufmann.

  • Buchanan, B. G., Shortliffe, E. H., et al. (1984). Rule-based expert systems (Vol. 3). Reading, MA: Addison-Wesley.

    Google Scholar 

  • Buckley, J. J. (2006). Fuzzy probability and statistics. Heidelberg: Springer.

    Google Scholar 

  • Ciucci, D., & Dubois, D. (2013). A map of dependencies among three-valued logics. Information Sciences, 250, 162–177.

    Article  Google Scholar 

  • Coniglio, M., Esteva, F., & Godo, L. (2013). Logics of formal inconsistency arising from systems of fuzzy logic. Preprint. arXiv:1307.3667.

  • Cooper, W. S. (1968). The propositional logic of ordinary discourse 1. Inquiry, 11(1–4), 295–320.

    Article  Google Scholar 

  • Detyniecki, M. (2001). Fundamentals on aggregation operators. This manuscript is based on Detyniecki’s doctoral thesis.

  • Dietrich, F., & List, C. (2010). The aggregation of propositional attitudes: Towards a general theory. Oxford Studies in Epistemology, 3(215), 34.

    Google Scholar 

  • Domshlak, C., Hüllermeier, E., Kaci, S., & Prade, H. (2011). Preferences in AI: An overview. Artificial Intelligence, 175(7), 1037–1052.

    Article  Google Scholar 

  • Dubois, D., Ostasiewicz, W., & Prade, H. (2000). Fuzzy sets: History and basic notions. In D. Dubois & H. Prade (Eds.), Fundamentals of fuzzy sets (pp. 21–124). Berlin: Springer.

    Chapter  Google Scholar 

  • Dubois, D., & Prade, H. (2001). Possibility theory, probability theory and multiple-valued logics: A clarification. Annals of Mathematics and Artificial Intelligence, 32(1), 35–66.

    Article  Google Scholar 

  • Duddy, C., & Piggins, A. (2013). Many-valued judgment aggregation: Characterizing the possibility/impossibility boundary. Journal of Economic Theory, 148(2), 793–805.

    Article  Google Scholar 

  • Eisele, C. (1976). The new elements of mathematics (Vol. 4). Berlin: Mouton.

    Google Scholar 

  • Eze, U. F., Etus, C., & Uzukwu, J. E. (2014). Database system concepts, implementations and organizations–A detailed survey. International Journal of Scientific Engineering and Research (IJSER), 2(2), 22–34.

    Google Scholar 

  • Fisch, M., & Turquette, A. (1966). Peirce’s triadic logic. Transactions of the Charles S. Peirce Society, 2(2), 71–85.

    Google Scholar 

  • Friedman, A., & Feichtinger, E. (2017). Peirce’s sign theory as an open-source R package. Signs-International Journal of Semiotics, 8, 1.

    Google Scholar 

  • Goldsmith, J., & Junker, U. (2009). Preference handling for artificial intelligence. AI Magazine, 29(4), 9.

    Article  Google Scholar 

  • Grosan, C., & Abraham, A. (2007). Hybrid evolutionary algorithms: Methodologies, architectures, and reviews. In A. Abraham, C. Grosan, & H. Ishibuchi (Eds.), Hybrid evolutionary algorithms (pp. 1–17). Berlin: Springer.

    Google Scholar 

  • Haack, S. (1978). Philosophy of logics. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Havenel, J. (2008). Peirce’s clarifications of continuity. Transactions of the Charles S. Peirce Society: A Quarterly Journal in American Philosophy, 44(1), 86–133.

    Google Scholar 

  • Hussain, M. (2010). Fuzzy relations. Master thesis, Department of Mathematics and Science, School of Engineering, Blekinge Institute of Technology, Sweden.

  • Hyde, D. (2008). Vagueness, logic and ontology. Farnham: Ashgate Publishing, Ltd.

    Google Scholar 

  • Hyde, D. (2011). The Sorites paradox. In G. Ronzitti (Ed.), Vagueness: A guide (pp. 1–17). Berlin: Springer.

    Google Scholar 

  • Khatchadourian, H. (1962). Vagueness. The Philosophical Quarterly, 12(47), 138–152.

    Article  Google Scholar 

  • Kockelman, P. (2007). Agency. Current Anthropology, 48(3), 375–401.

    Article  Google Scholar 

  • Kruse, R., Schwecke, E., & Heinsohn, J. (1991). Uncertainty and vagueness in knowledge based systems. New York: Springer.

    Book  Google Scholar 

  • Lalka, N., & Jain, S. G. (2015). Fuzzy logic for medical diagnosis. Ph.D. thesis.

  • Lane, R. (1999). Peirce’s triadic logic revisited. Transactions of the Charles S. Peirce Society, 35(2), 284–311.

    Google Scholar 

  • Lane, R. (2001). Triadic logic. In J. Queiroz & R. Gudwin (Eds.), Digital encyclopedia of Charles S. Peirce. http://www.digitalpeirce.fee.unicamp.br/lane/trilan.htm.

  • Lane, R. (2007). Peirce’s modal shift: From set theory to pragmaticism. Journal of the History of Philosophy, 45(4), 551–576.

    Article  Google Scholar 

  • List, C. (2012). The theory of judgment aggregation: An introductory review. Synthese, 187(1), 179–207.

    Article  Google Scholar 

  • Ma, Z., & Yan, L. (2010). A literature overview of fuzzy conceptual data modeling. Journal of Information Science and Engineering, 26(2), 427–441.

    Google Scholar 

  • McLaughlin, A. L. (2009). Peircean polymorphism: Between realism and anti-realism, Transactions of the Charles S. Peirce Society: A Quarterly Journal in American Philosophy, 45(3), 402–421.

    Article  Google Scholar 

  • Moktefi, A., & Shin, S.-J. (2012). A history of logic diagrams. In D. M. Gabbay, F. J. Pelletier, & J. Woods (Eds.), Handbook of the history of logic (Vol. 11, pp. 611–682). Amsterdam: North-Holland.

    Google Scholar 

  • Nikravesh, M. (2007). Evolution of fuzzy logic: From intelligent systems and computation to human mind. In M. Nikravesh, J. Kacprzyk, & L. A. Zadeh (Eds.), Forging new frontiers: Fuzzy pioneers I (pp. 37–53). Berlin: Springer.

    Chapter  Google Scholar 

  • Ochs, P. (1993). Continuity as vagueness: The mathematical antecedents of Peirce’s semiotics. Semiotica, 96(3–4), 231–256.

    Google Scholar 

  • Pawlak, Z. (1995). Vagueness and uncertainty: A rough set perspective. Computational Intelligence, 11(2), 227–232.

    Article  Google Scholar 

  • Pawlak, Z. (1997). Rough set approach to knowledge-based decision support. European Journal of Operational Research, 99(1), 48–57.

    Article  Google Scholar 

  • Pigozzi, G., Slavkovik, M., & van der Torre, L. (2009). A complete conclusion-based procedure for judgment aggregation. In F. Rossi & A. Tsoukis (Eds.), Algorithmic decision theory (pp. 1–13). Berlin: Springer.

    Google Scholar 

  • Rosser, J. B., & Turquette, A. R. (1954). Many-valued logics. British Journal for the Philosophy of Science, 5(17), 80–83.

    Google Scholar 

  • Royce, J. (1892). The spirit of modern philosophy: An essay in the form of lectures. Boston: Houghton Mifflin Company.

    Book  Google Scholar 

  • Russell, B. (1923). Vagueness. The Australasian Journal of Psychology and Philosophy, 1(2), 84–92.

    Article  Google Scholar 

  • Russell, S., & Norvig, P. (2003). Artificial intelligence: A modern approach. New Delhi: 2/E, Pearson Education India.

    Google Scholar 

  • Shield, P. (1981). On the logic of numbers. Ph.D. thesis, in partial fulfillment of the requirements for the degree Doctor of Philosophy in the Department of Philosophy, Fordham University.

  • Shin, S.-J. (2002). The iconic logic of Peirce’s graphs. Cambridge: MIT Press.

    Book  Google Scholar 

  • Shin, S.-J. (2011). Peirce’s alpha graphs and propositional languages. Semiotica, 2011(186), 333–346.

    Article  Google Scholar 

  • Shin, S.-J., & Hammer, E. (2016). Peirce’s deductive logic. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy, winter 2016 Edition. Metaphysics Research Lab, Stanford University.

  • Sobociński, B. (1952). Axiomatization of a partial system of three-value calculus of propositions. New York: Institute of Applied Logic.

    Google Scholar 

  • Sorensen, R. (2001). Vagueness and contradiction. Oxford: Oxford University Press.

    Google Scholar 

  • Turquette, A. R. (1967). Peirce’s Phi and Psi operators for triadic logic. Transactions of the Charles S. Peirce Society, 3(2), 66–73.

    Google Scholar 

  • Turquette, A. R. (1976). Minimal axioms for Peirce’s triadic logic. Mathematical Logic Quarterly, 22(1), 169–176.

    Article  Google Scholar 

  • Turquette, A. R. (1978). Alternative axioms for Peirce’s triadic logic. Mathematical Logic Quarterly, 24(25–30), 443–444.

    Article  Google Scholar 

  • Van Deemter, K. (2010). Not exactly: In praise of vagueness. Oxford: Oxford University Press.

    Google Scholar 

  • van der Lubbe, J., & Nauta, D. (1993). Semiotics, pragmatism and expert systems. Pragmatics in Language Technology, 1993, 6.

    Google Scholar 

  • Van Hees, M. (2007). The limits of epistemic democracy. Social Choice and Welfare, 28(4), 649–666.

    Article  Google Scholar 

  • Vernon, D. (2014). Artificial cognitive systems: A primer. Cambridge: MIT Press.

    Google Scholar 

  • Vernon, D., Metta, G., & Sandini, G. (2007). A survey of artificial cognitive systems: Implications for the autonomous development of mental capabilities in computational agents. IEEE Transactions on Evolutionary Computation, 11(2), 151–180.

    Article  Google Scholar 

  • Wooldridge, M., & Jennings, N. R. (1995). Agent theories, architectures, and languages: A survey. In M. Wooldridge & N. R. Jennings (Eds.), Intelligent agents (pp. 1–39). Berlin: Springer.

    Chapter  Google Scholar 

  • Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353.

    Article  Google Scholar 

  • Zadeh, L. A. (1989). Knowledge representation in fuzzy logic. IEEE Transactions on Knowledge and Data Engineering, 1(1), 89–100.

    Article  Google Scholar 

  • Zadeh, L. A. (1994). Fuzzy logic: Issues, contentions and perspectives. In 1994 IEEE international conference on acoustics, speech, and signal processing, 1994. ICASSP-94 (Vol. 6, pp. VI–183). New York: IEEE.

  • Zadeh, L. A. (2001). A new direction in AI: Toward a computational theory of perceptions. AI Magazine, 22(1), 73.

    Google Scholar 

  • Zadeh, L. A. (2002). A new direction in AI toward a computational theory of perceptions. In B. Bouchon-Meunier, J. Gutierrez-Rios, L. Magdalena, & R. R. Yager (Eds.), Technologies for constructing intelligent systems (Vol. 1, pp. 3–20). Heidelberg: Springer.

    Chapter  Google Scholar 

  • Zadeh, L. A. (2008). Is there a need for fuzzy logic? Information Sciences, 178(13), 2751–2779.

    Article  Google Scholar 

  • Zadeh, L. A. (2012). Outline of a restriction-centered theory of reasoning and computation in an environment of uncertainty and imprecision. In IRI.

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science and Software Engineering, International Islamic University, Islamabad, Pakistan

    Asim Raza

  2. Department of Computing and Technology, Abasyn University, Islamabad, Pakistan

    Asim D. Bakhshi

  3. Department of Humanities and Social Sciences, Lahore University of Management Sciences, Lahore, Pakistan

    Basit Koshul

Authors
  1. Asim Raza
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Asim D. Bakhshi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Basit Koshul
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Asim D. Bakhshi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Raza, A., Bakhshi, A.D. & Koshul, B. An Application of Peircean Triadic Logic: Modelling Vagueness. J of Log Lang and Inf 28, 389–426 (2019). https://doi.org/10.1007/s10849-019-09287-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10849-019-09287-2

Keywords

Search

Navigation