Aristotle’s Syllogistic as a Deductive System

Öz

Aristotle’s syllogistic is the first ever deductive system. After centuries, Aristotle’s ideas are still interesting for logicians who develop Aristotle’s work and draw inspiration from his results and even more from his methods. In the paper we discuss the essential elements of the Aristotelian system of syllogistic and Łukasiewicz’s reconstruction of it based on the tools of modern formal logic. We pay special attention to the notion of completeness of a deductive system as discussed by both authors. We describe in detail how completeness can be defined and proved with the use of an axiomatic refutation system. Finally, we apply this methodology to different axiomatizations of syllogistic presented by Łukasiewicz, Lemmon and Shepherdson.

Anahtar Kelimeler

• Aristotle’s logic
• syllogistic
• Jan Łukasiewicz
• axiomatic system
• axiomatic refutation
• completeness

Proje Numarası

SOA-2022-1120

Dedüktif Bir Sistem Olarak Aristoteles’in Kıyası

Öz

Aristoteles'in kıyası tarihteki ilk dedüktif sistemdir. Yüzyıllar sonra, Aristoteles'in fikirleri, çalışmalarını geliştirmeye devam eden, onun ulaştığı sonuçlardan ve hatta yöntemlerinden ilham alan mantıkçıların ilgisini hala çekmeye devam etmektedir. Makalede, Aristotelesçi kıyas sisteminin temel unsurlarını ve Łukasiewicz'in bunu modern formel mantığın araçlarına dayalı olarak yeniden inşasını tartışıyoruz. Her iki yazar tarafından da tartışıldığı gibi, dedüktif sisteminin eksiksizliği (completeness) kavramına özel önem atfediyoruz. Bir aksiyomatik çürütme sistemi kullanarak, eksiksizliğin nasıl tanımlanabileceğini ve ispatlanabileceğini ayrıntılı olarak tanımlıyoruz. Son olarak, bu metodolojiyi Łukasiewicz, Lemmon ve Shepherdson tarafından sunulan kıyasın farklı aksiyomlaştırmalarına uyguluyoruz.

Anahtar Kelimeler

• Aristoteles mantığı
• kıyas
• Jan Łukasiewicz
• aksiyomatik sistem
• aksiyomatik çürütme
• eksiksizlik.

Destekleyen Kurum

Bursa Uludağ Üniversitesi Bilimsel Araştırma Projeleri Birimi (BAP)

Proje Numarası

SOA-2022-1120

Teşekkür

Bu çalışma, SOA-2022-1120 kodlu, “Modern Formel Mantık Bakımından Aristoteles'in Kıyas Teorisi ve Din Bilimlerinde Kullanılmasının İmkanı” adlı Öncelikli Alan Araştırma Projesi (ÖNAP) kapsamında hazırlanmıştır ve Bursa Uludağ Üniversitesi Bilimsel Araştırma Projeleri Birimi tarafından desteklenmiştir.

Kaynakça

1. Smith, R. Aristotle’s Logic. In The Stanford Encyclopedia of Philosophy; Zalta, E.N., Ed.; Metaphysics Research Lab, Stanford University: Stanford, CA, USA, 2019.
2. Glashoff, K. Aristotelian Syntax from a Computational-Combinatorial Point of View. J. Log. Comput. 2005, 15, 949–973.
3. Steinkrüger, P. Aristotle’s assertoric syllogistic and modern relevance logic. Synthese 2015, 192, 1413–1444.
4. Crager, A. Meta-Logic in Aristotle’s Epistemology. Ph.D. Thesis, Princeton University, Princeton, NJ, USA, 2015.
5. Read, S. Aristotle’s Theory of the Assertoric Syllogism. Available online: https://philarchive.org/archive/REAATO-5 (accessed on 1 April 2020).
6. Moss, L. Completeness theorems for syllogistic fragments. In Logics for Linguistic Structures; Hamm, S.K., Ed.; Mouton de Gruyter: Berlin, Germany; New York, NY, USA, 2008; pp. 143–174.
7. Pratt-Hartmann, I.; Moss, L. Logics for the relational syllogistic. Rev. Symb. Log. 2009, 2, 1–37.
8. Glashoff, K. An Intensional Leibniz Semantics for Aristotelian Logic. Rev. Symb. Log. 2010, 3, 262–272.

9. Moss, L.S. Syllogistic Logics with Verbs. J. Log. Comput. 2010, 20, 947–967.
10. Kulicki, P. On a Minimal System of Aristotle’s Syllogistic. Bull. Sect. Log. 2011, 40, 129–145.
11. Moss, L.S. Syllogistic Logic with Comparative Adjectives. J. Log. Lang. Inf. 2011, 20, 397–417.
12. Rini, A. Aristotle’s Modal Proofs. Prior Analytics A8-22 in Predicate Logic; Springer: Berlin, Germany, 2011.
13. Kulicki, P. On minimal models for pure calculi of names. Log. Log. Philos. 2012, 1, 1–16.
14. Bellucci, F.; Moktefi, A.; Pietarinen, A. Diagrammatic Autarchy: Linear diagrams in the 17th and 18th centuries. In Proceedings of the First InternationalWorkshop on Diagrams, Logic and Cognition, Kolkata,India, 28–29 October 2012; Burton, L.C., Ed.; CEUR Workshop Proceedings: Kolkat a, India, 2013.
15. Pratt-Hartmann, I. The Syllogistic with Unity. J. Philos. Log. 2013, 42, 391–407.
16. Castro-Manzano, J. Re(dis)covering Leibniz’s Diagrammatic Logic. Tópicos Revista de Filosofía 2017, 52, 89–116.
17. Pietruszczak, A.; Jarmu˙ zek, T. Pure Modal Logic of Names and Tableau Systems. Stud. Log. 2018, 106, 1261– 1289.
18. Sautter, F.T.; Secco, G.D. A Simple Decision Method for Syllogistic. In Proceedings of the Diagrammatic Representation and Inference—10th International Conference, Diagrams 2018, Edinburgh, UK, 18–22 June 2018. Lecture Notes in Computer Science; Chapman, P., Stapleton, G., Moktefi, A., Pérez-Kriz, S., Bellucci, F., Eds.; Springer: Berlin, Germany, 2018; Volume 10871, pp. 708–711.
19. Endrullis, J.; Moss, L.S. Syllogistic logic with “Most”. Math. Struct. Comput. Sci. 2019, 29, 763–782.
20. Boche´ nski, I.M. Ancient Formal Logic; North-Holland: Oxford, UK, 1951.
21. Aristotle. Prior Analytics. Book I; Translated with an Introduction and Commentary by Gisela Striker; Clarendon Press: Oxford, UK, 2014.
22. Boche´ nski, I.M. The Methods of Contemporary Thought; D. Reidel: Dordrecht, The Netherlands, 1965. Lear, J. Aristotle’s compactness proof. J. Philos. 1979, 76, 198–215.
23. Scanlan, M. On finding compactness in aristotle. Hist. Philos. Log. 1983, 4, 1–8.
24. Smiley, T. What is a syllogism? J. Philos. Log. 1973, 2, 136–154.
25. Boger, G. Completion, reduction and analysis: Three proof-theoretic processes in Aristotle’s Prior Analytics. Hist. Philos. Log. 1998, 19, 187–226.
26. Adžic, M.; Došen, K. Gödel’s Notre Dame Course. Bull. Symb. Log. 2016, 22, 469–481.
27. Corcoran, J. Completeness of an ancient logic. J. Symb. Log. 1972, 37, 696–702.
28. Corcoran, J. Aristotle’s Natural Deduction System. In Ancient Logic and Its Modern Interpretations; Reidel Publishing Co.: Dordrecht, The Netherlands, 1974; pp. 1–100.
29. Frege, G. Sense and Reference. Philos. Rev. 1948, 57, 209–230.
30. Słupecki, J. Uwagi o sylogistyce Arystotelesa. Ann. UMCS 1946, I, 187–191.
31. Wedberg, A. The Aristotelian theory of classes. Ajutas 1948, 15, 299–314.
32. Menne, A. Logik und Existenz; Westkulturverlag Anton Hain: Berlin, Germany, 1954.
33. Shepherdson, J. On the Interpretation of Aristotelian Syllogistic. J. Symb. Log. 1956, 21, 137–147.
34. Geach, P.T. History of the Corruption of Logic (1968). In Logic Matters; University of California Press: Berkeley, CA, USA, 1980; pp. 44–61.
35. Ajdukiewicz, K. Logika Pragmatyczna; PWN: Warszawa, Poland, 1965.
36. Borkowski, L. Logika Formalna; PWN: Warszawa, Poland, 1970.
37. Grzegorczyk, A. Zarys Logiki Matematycznej; PWN: Warszawa, Poland, 1969.
38. Pogorzelski,W.A. Elementarny Słownik Logiki Formalnej; DziałWydawnictw Filii UniwersytetuWarszawskiego: Białystok, Poland, 1992.
39. Marciszewski, W. (Ed.) Mała Encyklopedia Logiki; PWN: Warszawa, Poland, 1988.
40. Waragai, T.; Oyamada, K. A System of Ontology Based on Identity and Partial Ordering as an Adequate Logical Apparatus for Describing Taxonomical Structures of Concepts. Ann. Jpn. Assoc. Philos. Sci. 2007, 15, 123– 149.
41. Słupecki, J.; Bryll, G.; Wybraniec-Skardowska, U. Theory of rejected propositions. I. Stud. Log. 1971, 29, 75–115.
42. Goranko, V.; Pulcini, G.; Skura, T. Refutation Systems: An Overview and Some Applications to Philosophical Logics. In Knowledge, Proof and Dynamics; Liu, F., Ono, H., Yu, J., Eds.; Springer: Singapore, 2020; pp. 173–197.
43. Wybraniec-Skardowska, U. Rejection in Łukasiewicz’s and Słupecki’s Sense. In The Lvov-Warsaw School. Past and Present; Garrido, Á., Wybraniec-Skardowska, U., Eds.; Springer International Publishing: Cham, Switzerland, 2018; pp. 575–597.
44. Kulicki, P. The Use of Axiomatic Rejection. In The Logica Yearbook 1999; Childers, T., Ed.; Filosofia: Prague, Czech Republic, 2000; pp. 109–117.
45. McKinsey, J. The Decision Problem for some Classes of Sentences without Quantifiers. J. Symb. Log. 1943, 8, 61– 76.
46. Słupecki, J. Z badan´ nad sylogistyka˛Arystotelesa; Wrocławskie Towarzystwo Naukowe: Wrocław, Poland, 1948. Lukasiewicz, J. Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic; Clarendon Press: Oxford, UK, 1952.
47. Lemmon, E.J. Quantifiers and Modal Operators. Proc. Aristot. Soc. 1958, 58, 245–268.
48. Kulicki, P. Aksjomatyczne Systemy Rachunku Nazw; Wydawnictwo KUL:Warszawa, Poland, 2011.
49. Prior, A.N. Formal Logic; Clarendon Press: Oxford, UK, 1962.