Abstract
We give completeness results — with respect to Kripke's semantic — for the negation-free intermediate predicate calculi:
and the superintuitionistic predicate calculus:
The central point is the completeness proof for (1), which is obtained modifying Klemke's construction [3].
For a general account on negation-free intermediate predicate calculi — see Casari-Minari [1]; for an algebraic treatment of some superintuitionistic predicate calculi involving schemasB andD — see Horn [4] and Görnemann [2].
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References
E. Casari andP. Minari,Negation-free intermediate predicate calculi Bollettino della Unione Matematica Italiana (6) 2-B (1983), pp. 499–536.
S. Görnemann,A logic stronger than intuitionism Journal of Symbolic Logic 36 (1971), pp. 249–261.
D. Klemke, Ein Henkin-Beweis für die Vollständigkeit eines Kalküls relativ zur Grzegorczyk-SemantikArchiv für matematische Logik 14 (1971), pp. 148–161.
A. Horn,Logic with truth values in a linearly ordered Heyting algebra Journal of Symbolic Logic 34 (1969), pp. 395–408.
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Minari, P. Completeness theorems for some intermediate predicate calculi. Stud Logica 42, 431–441 (1983). https://doi.org/10.1007/BF01371631
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DOI: https://doi.org/10.1007/BF01371631