The ℒ n m -propositional calculus

Carlos Gallardo; Alicia Ziliani

Mathematica Bohemica (2015)

  • Volume: 140, Issue: 1, page 11-33
  • ISSN: 0862-7959

Abstract

top
T. Almada and J. Vaz de Carvalho (2001) stated the problem to investigate if these Łukasiewicz algebras are algebras of some logic system. In this article an affirmative answer is given and the ℒ n m -propositional calculus, denoted by ℓ n m , is introduced in terms of the binary connectives → (implication), ↠ (standard implication), ∧ (conjunction), ∨ (disjunction) and the unary ones f (negation) and D i , 1 ≤ i ≤ n - 1 (generalized Moisil operators). It is proved that ℓ n m belongs to the class of standard systems of implicative extensional propositional calculi. Besides, it is shown that the definitions of L n m -algebra and ℓ n m -algebra are equivalent. Finally, the completeness theorem for ℓ n m is obtained.

How to cite

top

Gallardo, Carlos, and Ziliani, Alicia. "The ${\mathcal {L}}^m_n$-propositional calculus." Mathematica Bohemica 140.1 (2015): 11-33. <http://eudml.org/doc/269870>.

@article{Gallardo2015,
abstract = {T. Almada and J. Vaz de Carvalho (2001) stated the problem to investigate if these Łukasiewicz algebras are algebras of some logic system. In this article an affirmative answer is given and the $\{\mathcal \{L\}\}^\{m\}_\{n\}$-propositional calculus, denoted by $\{\ell ^\{m\}_\{n\}\}$, is introduced in terms of the binary connectives $\rightarrow $ (implication), $\twoheadrightarrow $ (standard implication), $\wedge $ (conjunction), $\vee $ (disjunction) and the unary ones $f$ (negation) and $D_\{i\}$, $1\le i\le n-1$ (generalized Moisil operators). It is proved that $\{\ell ^\{m\}_\{n\}\}$ belongs to the class of standard systems of implicative extensional propositional calculi. Besides, it is shown that the definitions of $L^\{m\}_\{n\}$-algebra and $\{\ell ^\{m\}_\{n\}\}$-algebra are equivalent. Finally, the completeness theorem for $\{\ell ^\{m\}_\{n\}\}$ is obtained.},
author = {Gallardo, Carlos, Ziliani, Alicia},
journal = {Mathematica Bohemica},
keywords = {Łukasiewicz algebra of order $n$; $m$-generalized Łukasiewicz algebra of order $n$; equationally definable principal congruences; implicative extensional propositional calculus; completeness theorem},
language = {eng},
number = {1},
pages = {11-33},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The $\{\mathcal \{L\}\}^m_n$-propositional calculus},
url = {http://eudml.org/doc/269870},
volume = {140},
year = {2015},
}

TY - JOUR
AU - Gallardo, Carlos
AU - Ziliani, Alicia
TI - The ${\mathcal {L}}^m_n$-propositional calculus
JO - Mathematica Bohemica
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 140
IS - 1
SP - 11
EP - 33
AB - T. Almada and J. Vaz de Carvalho (2001) stated the problem to investigate if these Łukasiewicz algebras are algebras of some logic system. In this article an affirmative answer is given and the ${\mathcal {L}}^{m}_{n}$-propositional calculus, denoted by ${\ell ^{m}_{n}}$, is introduced in terms of the binary connectives $\rightarrow $ (implication), $\twoheadrightarrow $ (standard implication), $\wedge $ (conjunction), $\vee $ (disjunction) and the unary ones $f$ (negation) and $D_{i}$, $1\le i\le n-1$ (generalized Moisil operators). It is proved that ${\ell ^{m}_{n}}$ belongs to the class of standard systems of implicative extensional propositional calculi. Besides, it is shown that the definitions of $L^{m}_{n}$-algebra and ${\ell ^{m}_{n}}$-algebra are equivalent. Finally, the completeness theorem for ${\ell ^{m}_{n}}$ is obtained.
LA - eng
KW - Łukasiewicz algebra of order $n$; $m$-generalized Łukasiewicz algebra of order $n$; equationally definable principal congruences; implicative extensional propositional calculus; completeness theorem
UR - http://eudml.org/doc/269870
ER -

References

top
  1. Almada, T., Carvalho, J. Vaz de, 10.1023/A:1013846725213, Stud. Log. 69 329-338 (2001). (2001) MR1874418DOI10.1023/A:1013846725213
  2. Berman, J., 10.1007/BF01836429, Aequationes Math. 16 165-171 (1977). (1977) Zbl0395.06007MR0480238DOI10.1007/BF01836429
  3. Blyth, T. S., Varlet, J. C., Ockham Algebras, Oxford Science Publications Oxford University Press, Oxford (1994). (1994) Zbl0835.06011MR1315526
  4. Boicescu, V., Filipoiu, A., Georgescu, G., Rudeanu, S., Łukasiewicz-Moisil Algebras, Annals of Discrete Mathematics 49 North-Holland, Amsterdam (1991). (1991) Zbl0726.06007MR1112790
  5. Burris, S., Sankappanavar, H. P., 10.1007/978-1-4613-8130-3_3, Graduate Texts in Mathematics 78 Springer, New York (1981). (1981) Zbl0478.08001MR0648287DOI10.1007/978-1-4613-8130-3_3
  6. Cignoli, R., Moisil Algebras, Notas de Lógica Mathemática 27 Instituto de Matemática, Universidad Nacional del Sur, Bahía Blanca (1970). (1970) Zbl0212.31701MR0345884
  7. Figallo, A. V., Gallardo, C., Ziliani, A., Weak implication on generalized Łukasiewicz algebras of order n , Bull. Sect. Log., Univ. Łód'z, Dep. Log. 39 187-198 (2010). (2010) Zbl1286.03180MR2817230
  8. Maronna, R., A characterization of Morgan lattices, Port. Math. 23 169-171 (1964). (1964) Zbl0133.27102MR0188119
  9. Rasiowa, H., An Algebraic Approach to Non-Classical Logics, Studies in Logic and the Foundations of Mathematics 78 North-Holland Publishing, Amsterdam; Elsevier Publishing, New York (1974). (1974) Zbl0299.02069MR0446968
  10. Sholander, M., 10.4153/CJM-1951-003-5, Can. J. Math. 3 28-30 (1951). (1951) Zbl0042.02704MR0038942DOI10.4153/CJM-1951-003-5
  11. Urquhart, A., 10.1007/BF00370442, Stud. Log. 38 201-209 (1979). (1979) Zbl0425.06008MR0544616DOI10.1007/BF00370442
  12. Carvalho, J. Vaz de, 10.1007/s11225-010-9236-8, Stud. Log. 94 291-305 (2010). (2010) MR2602577DOI10.1007/s11225-010-9236-8

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.