An application of Peircean triadic logic: modelling vagueness. (English) Zbl 07099906

Summary: 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.


03-XX Mathematical logic and foundations
68-XX Computer science


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[1] 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.
[2] Agler, DW, Peirce and the specification of borderline vagueness, Semiotica, 2013, 195-215, (2013)
[3] Akhtar, J.; Koshul, BB; Awais, MM, A framework for evolutionary algorithms based on Charles Sanders Peirce’s evolutionary semiotics, Information Sciences, 236, 93-108, (2013) · Zbl 1284.68522
[4] Annoni, M., Implications of synechism: Continuity and second-order vagueness, Cognitio-Estudos: Revista Electronica de Filosofia, 3, 96-108, (2006)
[5] Augusto, LM, Putting the horse before the cart: A pragmatist analysis of knowledge, Trans/Form/Ação, 34, 135-152, (2011)
[6] Beg, I.; Khalid, A., Belief aggregation in fuzzy framework, Journal of Fuzzy Mathematics, 20, 911-924, (2012) · Zbl 1271.91036
[7] Bell, J. L. (2005). The continuous and the infinitesimal in mathematics and philosophy. Milan: Polimetrica S.A.S.
[8] Belnap, ND, Conditional assertion and restricted quantification, Noûs, 4, 1-12, (1970) · Zbl 1366.03023
[9] Black, M., Vagueness. An exercise in logical analysis, Philosophy of Science, 4, 427-455, (1937)
[10] 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.
[11] Buchanan, B. G., Shortliffe, E. H., et al. (1984). Rule-based expert systems (Vol. 3). Reading, MA: Addison-Wesley.
[12] Buckley, J. J. (2006). Fuzzy probability and statistics. Heidelberg: Springer.
[13] Ciucci, D.; Dubois, D., A map of dependencies among three-valued logics, Information Sciences, 250, 162-177, (2013) · Zbl 1321.03037
[14] Coniglio, M., Esteva, F., & Godo, L. (2013). Logics of formal inconsistency arising from systems of fuzzy logic. Preprint. arXiv:1307.3667. · Zbl 1405.03062
[15] Cooper, WS, The propositional logic of ordinary discourse 1, Inquiry, 11, 295-320, (1968)
[16] Detyniecki, M. (2001). Fundamentals on aggregation operators. This manuscript is based on Detyniecki’s doctoral thesis.
[17] Dietrich, F.; List, C., The aggregation of propositional attitudes: Towards a general theory, Oxford Studies in Epistemology, 3, 34, (2010)
[18] Domshlak, C.; Hüllermeier, E.; Kaci, S.; Prade, H., Preferences in AI: An overview, Artificial Intelligence, 175, 1037-1052, (2011)
[19] Dubois, D.; Ostasiewicz, W.; Prade, H.; Dubois, D. (ed.); Prade, H. (ed.), Fuzzy sets: History and basic notions, 21-124, (2000), Berlin · Zbl 0967.03047
[20] Dubois, D.; Prade, H., Possibility theory, probability theory and multiple-valued logics: A clarification, Annals of Mathematics and Artificial Intelligence, 32, 35-66, (2001) · Zbl 1314.68309
[21] Duddy, C.; Piggins, A., Many-valued judgment aggregation: Characterizing the possibility/impossibility boundary, Journal of Economic Theory, 148, 793-805, (2013) · Zbl 1275.91041
[22] Eisele, C. (1976). The new elements of mathematics (Vol. 4). Berlin: Mouton.
[23] Eze, UF; Etus, C.; Uzukwu, JE, Database system concepts, implementations and organizations-A detailed survey, International Journal of Scientific Engineering and Research (IJSER), 2, 22-34, (2014)
[24] Fisch, M.; Turquette, A., Peirce’s triadic logic, Transactions of the Charles S. Peirce Society, 2, 71-85, (1966)
[25] Friedman, A.; Feichtinger, E., Peirce’s sign theory as an open-source R package, Signs-International Journal of Semiotics, 8, 1, (2017)
[26] Goldsmith, J.; Junker, U., Preference handling for artificial intelligence, AI Magazine, 29, 9, (2009)
[27] Grosan, C.; Abraham, A.; Abraham, A. (ed.); Grosan, C. (ed.); Ishibuchi, H. (ed.), Hybrid evolutionary algorithms: Methodologies, architectures, and reviews, 1-17, (2007), Berlin
[28] Haack, S. (1978). Philosophy of logics. Cambridge: Cambridge University Press.
[29] Havenel, J., Peirce’s clarifications of continuity, Transactions of the Charles S. Peirce Society: A Quarterly Journal in American Philosophy, 44, 86-133, (2008)
[30] Hussain, M. (2010). Fuzzy relations. Master thesis, Department of Mathematics and Science, School of Engineering, Blekinge Institute of Technology, Sweden.
[31] Hyde, D. (2008). Vagueness, logic and ontology. Farnham: Ashgate Publishing, Ltd.
[32] Hyde, D.; Ronzitti, G. (ed.), The Sorites paradox, 1-17, (2011), Berlin
[33] Khatchadourian, H., Vagueness, The Philosophical Quarterly, 12, 138-152, (1962)
[34] Kockelman, P., Agency, Current Anthropology, 48, 375-401, (2007)
[35] Kruse, R., Schwecke, E., & Heinsohn, J. (1991). Uncertainty and vagueness in knowledge based systems. New York: Springer.
[36] Lalka, N., & Jain, S. G. (2015). Fuzzy logic for medical diagnosis. Ph.D. thesis.
[37] Lane, R., Peirce’s triadic logic revisited, Transactions of the Charles S. Peirce Society, 35, 284-311, (1999)
[38] 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.
[39] Lane, R., Peirce’s modal shift: From set theory to pragmaticism, Journal of the History of Philosophy, 45, 551-576, (2007)
[40] List, C., The theory of judgment aggregation: An introductory review, Synthese, 187, 179-207, (2012) · Zbl 1275.91049
[41] Ma, Z.; Yan, L., A literature overview of fuzzy conceptual data modeling, Journal of Information Science and Engineering, 26, 427-441, (2010)
[42] McLaughlin, AL, Peircean polymorphism: Between realism and anti-realism,, Transactions of the Charles S. Peirce Society: A Quarterly Journal in American Philosophy, 45, 402-421, (2009)
[43] Moktefi, A.; Shin, S-J; Gabbay, DM (ed.); Pelletier, FJ (ed.); Woods, J. (ed.), A history of logic diagrams, No. 11, 611-682, (2012), Amsterdam
[44] Nikravesh, M.; Nikravesh, M. (ed.); Kacprzyk, J. (ed.); Zadeh, LA (ed.), Evolution of fuzzy logic: From intelligent systems and computation to human mind, 37-53, (2007), Berlin
[45] Ochs, P., Continuity as vagueness: The mathematical antecedents of Peirce’s semiotics, Semiotica, 96, 231-256, (1993)
[46] Pawlak, Z., Vagueness and uncertainty: A rough set perspective, Computational Intelligence, 11, 227-232, (1995)
[47] Pawlak, Z., Rough set approach to knowledge-based decision support, European Journal of Operational Research, 99, 48-57, (1997) · Zbl 0923.90004
[48] Pigozzi, G.; Slavkovik, M.; Torre, L.; Rossi, F. (ed.); Tsoukis, A. (ed.), A complete conclusion-based procedure for judgment aggregation, 1-13, (2009), Berlin · Zbl 1260.68400
[49] Rosser, JB; Turquette, AR, Many-valued logics, British Journal for the Philosophy of Science, 5, 80-83, (1954)
[50] Royce, J. (1892). The spirit of modern philosophy: An essay in the form of lectures. Boston: Houghton Mifflin Company.
[51] Russell, B., Vagueness, The Australasian Journal of Psychology and Philosophy, 1, 84-92, (1923)
[52] Russell, S., & Norvig, P. (2003). Artificial intelligence: A modern approach. New Delhi: 2/E, Pearson Education India.
[53] 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.
[54] Shin, S.-J. (2002). The iconic logic of Peirce’s graphs. Cambridge: MIT Press.
[55] Shin, S-J, Peirce’s alpha graphs and propositional languages, Semiotica, 2011, 333-346, (2011)
[56] 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.
[57] Sobociński, B. (1952). Axiomatization of a partial system of three-value calculus of propositions. New York: Institute of Applied Logic.
[58] Sorensen, R. (2001). Vagueness and contradiction. Oxford: Oxford University Press.
[59] Turquette, AR, Peirce’s Phi and Psi operators for triadic logic, Transactions of the Charles S. Peirce Society, 3, 66-73, (1967)
[60] Turquette, AR, Minimal axioms for Peirce’s triadic logic, Mathematical Logic Quarterly, 22, 169-176, (1976) · Zbl 0331.02005
[61] Turquette, AR, Alternative axioms for Peirce’s triadic logic, Mathematical Logic Quarterly, 24, 443-444, (1978) · Zbl 0387.03008
[62] Van Deemter, K. (2010). Not exactly: In praise of vagueness. Oxford: Oxford University Press.
[63] Lubbe, J.; Nauta, D., Semiotics, pragmatism and expert systems, Pragmatics in Language Technology, 1993, 6, (1993)
[64] Hees, M., The limits of epistemic democracy, Social Choice and Welfare, 28, 649-666, (2007) · Zbl 1180.91251
[65] Vernon, D. (2014). Artificial cognitive systems: A primer. Cambridge: MIT Press.
[66] Vernon, D.; Metta, G.; Sandini, G., A survey of artificial cognitive systems: Implications for the autonomous development of mental capabilities in computational agents, IEEE Transactions on Evolutionary Computation, 11, 151-180, (2007)
[67] Wooldridge, M.; Jennings, NR; Wooldridge, M. (ed.); Jennings, NR (ed.), Agent theories, architectures, and languages: A survey, 1-39, (1995), Berlin
[68] Zadeh, LA, Fuzzy sets, Information and Control, 8, 338-353, (1965) · Zbl 0139.24606
[69] Zadeh, LA, Knowledge representation in fuzzy logic, IEEE Transactions on Knowledge and Data Engineering, 1, 89-100, (1989)
[70] 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.
[71] Zadeh, LA, A new direction in AI: Toward a computational theory of perceptions, AI Magazine, 22, 73, (2001)
[72] Zadeh, LA; Bouchon-Meunier, B. (ed.); Gutierrez-Rios, J. (ed.); Magdalena, L. (ed.); Yager, RR (ed.), A new direction in AI toward a computational theory of perceptions, No. 1, 3-20, (2002), Heidelberg
[73] Zadeh, LA, Is there a need for fuzzy logic?, Information Sciences, 178, 2751-2779, (2008) · Zbl 1148.68047
[74] Zadeh, L. A. (2012). Outline of a restriction-centered theory of reasoning and computation in an environment of uncertainty and imprecision. In IRI.
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