The Projective Beth Property and Interpolation in Positive and Related Logics

L. L. Maksimova

Algebra i logika, Tome 45 (2006) no. 1, pp. 85-113.

Résumé

We look at the interplay between the projective Beth property in non-classical logics and interpolation. Previously, we proved that in positive logics as well as in superintuitionistic and modal ones, the projective Beth property PB2 follows from Craig's interpolation property and implies the restricted interpolation property IPR. Here, we show that IPR and PB2 are equivalent in positive logics, and also in extensions of the superintuitionistic logic KC and of the modal logic Grz.2.

Détail

Keywords: projective Beth property, restricted interpolation property, positive logic, superintuitionistic logic, modal logic.

Bibliographie

[1] L. L. Maksimova, “Proektivnoe svoistvo Beta v modalnykh i superintuitsionistskikh logikakh”, Algebra i logika, 38:3 (1999), 316–333 | MR | Zbl

[2] L. L. Maksimova, “Superintuitsionistskie logiki i proektivnoe svoistvo Beta”, Algebra i logika, 38:6 (1999), 680–696 | MR | Zbl

[3] L. L. Maksimova, “Intuitionistic logic and implicit definability”, Ann. Pure Appl. Log., 105:1–3 (2000), 83–102 | DOI | MR | Zbl

[4] L. L. Maksimova, “Neyavnaya opredelimost v pozitivnykh logikakh”, Algebra i logika, 42:1 (2003), 65–93 | MR | Zbl

[5] L. L. Maksimova, “Opredelimost v normalnykh rasshireniyakh logiki $S4$”, Algebra i logika, 43:4 (2004), 387–410 | MR | Zbl

[6] L. L. Maksimova, “Teorema Kreiga v superintuitsionistskikh logikakh i amalgamiruemye mnogoobraziya psevdobulevykh algebr”, Algebra i logika, 16:6 (1977), 643–681 | MR | Zbl

[7] L. Maksimov, “Restricted interpolation in modal logics”, Advances in modal logic, 4, King's College Publications, London, 2003, 297–311 | MR | Zbl

[8] G. Kreisel, “Explicit definability in intuitionistic logic”, J. Symb. Log., 25 (1960), 389–390 | DOI

[9] L. L. Maksimova, “Analog teoremy Beta v normalnykh rasshireniyakh modalnoi logiki $K4$”, Sib. matem. zh., 33:6 (1992), 118–130 | MR

[10] W. Craig, “Three uses of Herbrand-Gentzen theorem in relating model theory”, J. Symb. Log., 22:3 (1957), 269–285 | DOI | MR | Zbl

[11] H. Rasiowa, R. Sikorski, The mathematics of metamathematics, PWN, Warszawa, 1963 | MR

[12] E. Hoogland, “Algebraic characterisations of various Beth definability properties”, Stud. Log., 65:1 (2000), 91–112 | DOI | MR | Zbl

[13] L. L. Maksimova, “Modalnye logiki i mnogoobraziya modalnykh algebr: svoistvo Beta, interpolyatsiya i amalgamiruemost”, Algebra i logika, 31:2 (1992), 145–166 | MR | Zbl

[14] J. Czelakowski, D. Pigozzi, “Amalgamation and interpolation in abstract algebraic logic”, Models, algebras and proofs, Sel. papers X Latin Am. Symp. math. logic held Bogota, Lect. Notes Pure Appl. Math., 203, eds. X. Caicedo, C. H. Montenegro, Marcel Dekker, New York, 1999, 187–265 | MR | Zbl

[15] I. Sain, “Beth's and Craig's properties via epimorphisms and amalgamation in algebraic logic”, Algebraic logic and universal algebra in computer science, Lect. Notes Comput. Sci., 425, eds. C. H. Bergman, R. D. Maddux, D. I. Pigozzi, Springer-Verlag, Berlin, 1990, 209–226 | MR

[16] L. L. Maksimova, “Interpolyatsiya i opredelimost v rasshireniyakh minimalnoi logiki”, Algebra i logika, 44:6 (2005), 726–750 | MR | Zbl

[17] L. L. Maksimova, “Razreshimost proektivnogo svoistva Beta v mnogoobraziyakh geitingovykh algebr”, Algebra i logika, 40:3 (2001), 290–301 | MR | Zbl

[18] L. L. Maksimova, “Ogranichennaya interpolyatsiya i proektivnoe svoistvo Beta v ekvatsionalnoi logike”, Algebra i logika, 42:6 (2003), 712–726 | MR

[19] D. M. Gabbay, L. Maksimova, Interpolation and De.nability: Modal and Intuitionistic Logics, Oxford Univ. Press, Oxford, 2005 | MR | Zbl

[20] W. Blok, Varieties of interior algebras, PhD Thesis, Univ. Amsterdam, 1976