Some preservation theorems in an intermediate logic. (English) Zbl 1100.03014

The author presents first steps in the model theory of a logic intermediate betweeen intuitionistic and classical logic, called semi-classical logic, in which \(x=y \vee x\neq y\) is an axiom. Models for semi-classical theories are special kinds of Kripke models whose frame is a linear order. Beside the usual notion of submodel, there are notions of left and right submodels, and thus of left and right model completeness. A theory turns out to be left (right) model-complete if and only if every formula is equivalent under \(T\) to an existential (universal) one. Similar preservation theorems are proved for left and right inductive theories.


03B55 Intermediate logics
03C40 Interpolation, preservation, definability
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[1] Categoricity and quantifier elimination for intuitionistic theories. To appear in Lecture Notes in Logic. · Zbl 1106.03034
[2] Bagheri, Math. Logic Quarterly 49 pp 479– (2003)
[3] Corsi, Studia Logica 51 pp 317– (1992)
[4] Logic and Structure (Springer, 1997).
[5] Görnemann, J. Symbolic Logic 36 pp 249– (1971) · Zbl 0276.02013
[6] A Shorter Model Theory (Cambridge University Press, 1997). · Zbl 0873.03036
[7] Marković, Publication de l’institut mathématique, Nouvelle série 30 pp 111– (1981)
[8] Cours de théories des modèles (Nour-al-Ma’rifaht Val-Mantigh, 1988).
[9] Very intuitionistic theories and quantifier elimination. Preprint. · Zbl 1303.03070
[10] Smorynski, J. Symbolic Logic 38 pp 102– (1973)
[11] and , Constructivism in Mathematics, vol. I (North-Holland, 1988). · Zbl 0653.03040
[12] Visser, Arch. Math. Logic 40 pp 277– (2001)
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