Logica Universalis

, Volume 5, Issue 2, pp 177–203

# On Pairs of Dual Consequence Operations

Open Access
Article

## Abstract

In the paper, the authors discuss two kinds of consequence operations characterized axiomatically. The first one are consequence operations of the type Cn + that, in the intuitive sense, are infallible operations, always leading from accepted (true) sentences of a deductive system to accepted (true) sentences of the deductive system (see Tarski in Monatshefte für Mathematik und Physik 37:361–404, 1930, Comptes Rendus des Séances De la Société des Sciences et des Lettres de Varsovie 23:22–29, 1930; Pogorzelski and Słupecki in Stud Logic 9:163–176, 1960, Stud Logic 10:77–95, 1960). The second kind are dual consequence operations of the type Cn that can be regarded as anti-infallible operations leading from non-accepted (rejected, false) sentences of a deductive system to non-accepted (rejected, false) sentences of the system (see Słupecki in Funkcja Łukasiewicza, 33–40, 1959; Wybraniec-Skardowska in Teoria zdań odrzuconych, 5–131, Zeszyty Naukowe Wyższej Szkoły Inżynierskiej w Opolu, Seria Matematyka 4(81):35–61, 1983, Ann Pure Appl Logic 127:243–266, 2004, in On the notion and function of rejected propositions, 179–202, 2005). The operations of the types Cn + and Cn can be ordinary finitistic consequence operations or unit consequence operations. A deductive system can be characterized in two ways by the following triple:
$$\begin{array}{ll}{\rm by\,the\,triple}:\hspace{1.4cm} (+ , -)\hspace{0,6cm}<S, Cn^{+},Cn^{-}> \\ {\rm or\,by\,the\,triple}:\hspace{1.0cm} (-, +)\hspace{0,6cm} <S, Cn^{-}, Cn^{+}>.\end{array}$$
We compare axiom systems for operations of the types Cn + and Cn , give some methodological properties of deductive systems defined by means of these operations (e.g. consistency, completeness, decidability in Łukasiewicz’s sense), as well as formulate different metatheorems concerning them.

## Mathematics Subject Classification (2010)

Primary 03B22 Secondary 01A60 03B99

## Keywords

Axiom systems of theories of deductive systems consequence operations unit consequence operations a rejection consequence operation a dual consequence operation asserted system refutation system

## Notes

### Acknowledgments

We would like to thank the Referees for their useful remarks and comments which have contributed to improvement of our paper.

## References

1. 1.
Bonikowski Z. (2010) Unit operations in approximation spaces. In: Szczuka M., Kryszkiewicz M., Ramanna S., Jensen R. (eds). Rough Sets and Current Trends in Computing LNAI, vol 6086. Springer, Berlin, pp 337–346Google Scholar
2. 2.
Bonikowski, Z., Wybraniec-Skardowska, U.: A generalization of certain set-theoretical operations. In: Beziau, J.-Y., Caleiro, C., Costa-Leite, A., Ramos, J. (eds.) UniLog 2010, Book of Abstracts, World Congress and School on Universal Logic (3rd edition), April 18–25, 2010, Monte Estoril, Portugal. Instituto Superior Tecnico, Departamento de Matematica, p. 59 (2010)Google Scholar
3. 3.
Borkowski, L.: Logika Formalna. PWN, Warszawa (1970) (English translation: Formal Logic, Akademie-Verlag, Berlin (1977))Google Scholar
4. 4.
Bryll, G.: Kilka uzupełnień teorii zdań odrzuconych (Some supplements of the theory of rejected propositions). In: [45], 133–154 (1969)Google Scholar
5. 5.
Bryll, G.: Zwia̧zki logiczne pomiȩdzy zdaniami nauk empirycznych (Logical relations between sentences of empirical sciences). In: [45], 155–216 (1969)Google Scholar
6. 6.
Bryll, G.: Metody odrzucania wyrażeń (Methods of Rejection of Expressions). Problemy Współczesnej Nauki. Teoria i Zastosowania. Informatyka. Akademicka Oficyna Wydawnicza PLJ, Warszawa (1996)Google Scholar
7. 7.
Kuratowski K.: Surl’operation Ādel’ analysys situs. Fundam. Math. 3, 182–199 (1922)
8. 8.
Łukasiewicz, J.: Logika dwuwartościowa (Two-valued logic), Przegla̧d Filozoficzny 23, 189–205 (1921) (English translation: Two-valued logic. In: [13], 89–109 (1970))Google Scholar
9. 9.
Łukasiewicz, J.: O sylogistyce Arystotelesa (On Aristotle’s syllogistic). Sprawozdania z Czynności i Posiedzeń Polskiej Akademii Umiejȩtności 44 (1939). (Reprinted in: Z Zagadnień Logiki i Filozofii. Pisma Wybrane (Problems in Logic and Philosophy. Selected Papers (J. Słupecki (Ed.)), PWN, Warszawa, 220–227 (1961)Google Scholar
10. 10.
Łukasiewicz, J.: Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic. Oxford University Press, Oxford (1951) (Second edition in (1955))Google Scholar
11. 11.
Łukasiewicz J.: On the intuitionistic theory of deduction. Indagationes Math. Ser. A 3, 202–212 (1952)Google Scholar
12. 12.
Łukasiewicz, J.: A system of modal logic. J. Comput. Syst. 1(3), 111–149 (1953) (reprinted in [13], 352–360 (1970))Google Scholar
13. 13.
Łukasiewicz, J.: Selected Works (L. Borkowski, Ed.). North-Holland Publication, Amsterdam (1970)Google Scholar
14. 14.
Pogorzelski W.A., Słupecki J.: Podstawowe własności systemów dedukcyjnych opartych na nieklasycznych logikach (Basic properties of deductive systems based on nonclassical logics), part I. Stud. Logic. 9, 163–176 (1960)
15. 15.
Pogorzelski W.A., Słupecki J.: Podstawowe własności systemów dedukcyjnych opartych na nieklasycznych logikach (Basic properties of deductive systems based on nonclassical logics), part II. Stud. Logic. 10, 77–95 (1960)
16. 16.
Pogorzelski, W.A., Wojtylak, P.: Completeness theory for propositional logics. Series: Studies in Universal Logic. Birkhäuser Verlag AG, Basel (2008)Google Scholar
17. 17.
Skura T.: A complete syntactical characterization of the intuitionistic logic. Rep. Math. Logic 23, 75–80 (1989)
18. 18.
Skura T.: Refutation calculi for certain intermediate propositional logics. Notre Dame J. Formal Logic 33, 552–560 (1992)
19. 19.
Skura T.: Some results concerning refutation procedures. Acta Univ. Wratislaviensis Logik. 15, 83–95 (1993)Google Scholar
20. 20.
Skura T.: Syntactic refutation against finite models in modal logic. Notre Dame J. Formal Logic 35, 573–582 (1994)
21. 21.
Skura T.: A Łukasiewicz-style refutation system for the modal logic S5. J. Philos. Logic 24, 573–582 (1995)
22. 22.
Skura, T.: Refutation and proofs in S4. In: Wansing, H. (ed.) Proof Theory of Modal Logic. Kluwer, Dordrecht (1996)Google Scholar
23. 23.
Skura, T.: Aspects of refutation procedures in the intuitionistic logic and related modal systems. Logika 20(2190); Acta Universitatis Wratislaviensis, Wrocław (1999)Google Scholar
24. 24.
Skura T.: A refutation theory. Log. Univers. 3(2), 293–302 (2009)
25. 25.
Supecki, J.: Z badań nad sylogistyka̧ Arystotelesa (Invistigations on Aristotle’s Syllogistic). Prace Wrocławskiego Towarzystwa Naukowego (B) 6, (1948)Google Scholar
26. 26.
Słupecki, J.: Funkcja Łukasiewicza (The Łukasiewicz function). Seria B. No. 3, Zeszyty Naukowe Uniwersytetu Wrocławskiego (Publications of the Wrocław University), Matematyka-Fizyka-Astronomia II, 33–40 (1959)Google Scholar
27. 27.
Słupecki J.: Ł-decidability and decidability. Bull. Sect. Logic. Polish Acad. Sci. 1(3), 38–43 (1972)Google Scholar
28. 28.
Słupecki J., Bryll G., Wybraniec-Skardowska U.: Pewna teoria równoważna teorii systemów dedukcyjnych Tarskiego (A certain theory equivalent to Tarski’s theory of deductive systems), Zeszyty Naukowe Wyższej Szkoly Pedagogicznej w Opolu, Seria A, Matematyka. Logika Matematyczna 10, 61–67 (1970)Google Scholar
29. 29.
Słupecki J., Bryll G., Wybraniec-Skardowska U.: Theory of rejected propositions. I. Stud. Logic. 29, 76–123 (1971)Google Scholar
30. 30.
Słupecki J., Bryll G., Wybraniec-Skardowska U.: Theory of rejected propositions, II. Stud. Logic. 30, 97–107 (1972)
31. 31.
Sochacki, R.: Metody refutacyjne w badaniach nad systemami logicznymi (Refutation Methods in Studies on Logical Systems), Wydawnictwo Naukowe Uniwerstetu Opolskiego. Studia i Monografie, Opole (2010)Google Scholar
32. 32.
Spasowski M.: Some connections between Cn, Cn −1, dCn. Bull. Sect. Logic. Polish Acad. Sci. 2(1), 53–56 (1973)
33. 33.
Tarski, A.: Funamentale Begriffe der Methodoligie der deductiven Wissenschaften. Monatshefte für Mathematik und Physik, vol. 37, 361–404 (1930) (English translation: Fundamental concepts of the methodology of the deductive sciences. In: [35], 60–109 (Second edition edited and introduced by John Corcoran)Google Scholar
34. 34.
Tarski, A.: Über einige fundamentale Begriffe der Metamathematik. Comptes Rendus des Séances De la Société des Sciences et des Lettres de Varsovie 23, 22–29 (1930) (English translation: On some fundamental concepts of metamathematics. In: [35], 30–37)Google Scholar
35. 35.
Tarski, A.: Logic, Semantics, Metamatematics. Papers from 1923 to 1938. First edition published by Oxford University Press, Oxford (1956) (translated by J. H. Woodger). Second edition edited and introduced by John Corcoran, Hackett Publishing Company, Indianapolis, Indiana (1983)Google Scholar
36. 36.
Woleński J.: On comparison of theories by their contents. Stud. Logic. 48(4), 617–622 (1989)
37. 37.
Wójcicki R.: Dual counterparts of consequence operation. Bull. Sect. Logic. Polish Acad. Sci. Wrocław 2(1), 54–57 (1973)Google Scholar
38. 38.
Wybraniec-Skardowska, U.: O roli dedukcji w naukach empirycznych (On role of deduction in empirical sciences). Sprawozdania Wrocławskiego Towarzystwa Naukowego, Wrocław, 19–20 (1965)Google Scholar
39. 39.
Wybraniec-Skardowska, U.: Teoria zdań odrzuconych (The theory of rejected propositions). In: [45], 5–131Google Scholar
40. 40.
Wybraniec-Skardowska, U.: Badania Jerzego Słupeckiego nad sylogistyka̧ Arystotelesa i ich rezonans we współczesnej logice (Jerzy Słupecki’s investigations on Aristotle’s syllogistic and their response in the contemporary logic), Zeszyty Naukowe Wyższej Szkoły Inżynierskiej w Opolu, Seria: Matematyka 4(81), 35–61 (1983)Google Scholar
41. 41.
Wybraniec-Skardowska U.: Foundations for the formalization of metamathematics and axiomatizations of consequence theories. Ann. Pure Appl. Logic 127, 243–266 (2004)
42. 42.
Wybraniec-Skardowska U. (1989) On generalization of approximation spaces. Bull. Polish Acad. Sci. XXVII, 33–48Google Scholar
43. 43.
Wybraniec-Skardowska U.: Unit operation. Zeszyty Wyższej Szkoy Pedagogicznej w Opolu. Ser. Matematyka 27, 113–129 (1992)
44. 44.
Wybraniec-Skardowska, U.: On the notion and function of rejected propositions. Logika 23 (M. Magdziak, J. Zygmunt; Eds.) Acta Universitatis Wratislaviensis, No. 2754, 179–202 (2005)Google Scholar
45. 45.
Wybraniec-Skardowska, U., Bryll, G.: Z badań nad teoria̧ zdań odrzuconych (Investigations on the Theory of Rejected Propositions). Zeszyty Naukowe Wyższej Szkoły Pedagogicznej w Opolu, Seria B, Studia i Monografie, vol. 22, Opole (1969)Google Scholar