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  1.  11
    A Negationless Interpretation of Intuitionistic Theories. I.Victor N. Krivtsov - 2000 - Studia Logica 64 (3):323-344.
    The present work contains an axiomatic treatment of some parts of the restricted version of intuitionistic mathematics advocated by G. F. C. Griss, also known as negationless intuitionistic mathematics.Formal systems NPC, NA, and FIMN for negationless predicate logic, arithmetic, and analysis are proposed. Our Theorem 4 in Section 2 asserts the translatability of Heyting's arithmetic HAinto NA. The result can in fact be extended to a large class of intuitionistic theories based on HAand their negationless counterparts. For instance, in Section (...)
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  2.  37
    A negationless interpretation of intuitionistic theories.Victor N. Krivtsov - 2000 - Erkenntnis 53 (1-2):155-179.
    In a series of papers beginning in 1944, the Dutch mathematician and philosopher George Francois Cornelis Griss proposed that constructive mathematics should be developed without the use of the intuitionistic negation and, moreover, without any use of a null predicate. In the present work, we give formalized versions of intuitionistic arithmetic, analysis, and higher-order arithmetic in the spirit of Griss' "negationless intuitionistic mathematics'' and then consider their relation to the current formalizations of these theories.
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  3.  19
    An Intuitionistic Completeness Theorem for Classical Predicate Logic.Victor N. Krivtsov - 2010 - Studia Logica 96 (1):109-115.
    This paper presents an intuitionistic proof of a statement which under a classical reading is logically equivalent to Gödel's completeness theorem for classical predicate logic.
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  4.  17
    Semantical Completeness of First-Order Predicate Logic and the Weak Fan Theorem.Victor N. Krivtsov - 2015 - Studia Logica 103 (3):623-638.
    Within a weak system \ of intuitionistic analysis one may prove, using the Weak Fan Theorem as an additional axiom, a completeness theorem for intuitionistic first-order predicate logic relative to validity in generalized Beth models as well as a completeness theorem for classical first-order predicate logic relative to validity in intuitionistic structures. Conversely, each of these theorems implies over \ the Weak Fan Theorem.
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  5.  5
    A Negationless Interpretation of Intuitionistic Theories. II.Victor N. Krivtsov - 2000 - Studia Logica 65 (2):155-179.
    This work is a sequel to our [16]. It is shown how Theorem 4 of [16], dealing with the translatability of HA(Heyting's arithmetic) into negationless arithmetic NA, can be extended to the case of intuitionistic arithmetic in higher types.
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  6.  14
    A Negationless Interpretation Of Intuitionistic Theories.Victor N. Krivtsov - 2000 - Erkenntnis 53 (1-2):155-172.
    In a seriesof papers beginning in 1944, the Dutch mathematician and philosopherGeorge Francois Cornelis Griss proposed that constructivemathematics should be developedwithout the use of the intuitionistic negation1 and,moreover, without any use of a nullpredicate.In the present work, we give formalized versions of intuitionisticarithmetic, analysis,and higher-order arithmetic in the spirit ofGriss' ``negationless intuitionistic mathematics''and then consider their relation to thecurrent formalizations of thesetheories.
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  7.  45
    A negationless interpretation of intuitionistic theories. I.Victor N. Krivtsov - 2000 - Studia Logica 64 (1-2):323-344.
    The present work contains an axiomatic treatment of some parts of the restricted version of intuitionistic mathematics advocated by G. F. C. Griss, also known as negationless intuitionistic mathematics.Formal systems NPC, NA, and FIMN for negationless predicate logic, arithmetic, and analysis are proposed. Our Theorem 4 in Section 2 asserts the translatability of Heyting's arithmetic HAinto NA. The result can in fact be extended to a large class of intuitionistic theories based on HAand their negationless counterparts. For instance, in Section (...)
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  8.  38
    A negationless interpretation of intuitionistic theories. II.Victor N. Krivtsov - 2000 - Studia Logica 65 (1-2):155-179.
    This work is a sequel to our [16]. It is shown how Theorem 4 of [16], dealing with the translatability of HA(Heyting's arithmetic) into negationless arithmetic NA, can be extended to the case of intuitionistic arithmetic in higher types.
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  9.  17
    Creative subject, Beth models and neighbourhood functions.Victor N. Krivtsov - 1996 - Archive for Mathematical Logic 35 (2):89-102.
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  10.  13
    Note on extensions of Heyting's arithmetic by adding the “creative subject”.Victor N. Krivtsov - 1999 - Archive for Mathematical Logic 38 (3):145-152.
    Let HA be Heyting's arithmetic, and let CS denote the conjunction of Kreisel's axioms for the creative subject: \begin{eqnarray*} {\rm CS}_1.&&\quad \,\forall\, x (\qed_x A \vee \; \neg \qed_x A)\; ,\nn {\rm CS}_2. &&\quad \,\forall\, x (\qed_x A\to A)\; ,\nn {\rm CS}_3^{\rm S}. &&\quad A\to\,\exists\, x \qed_x A\; ,\nn {\rm CS}_4.&&\quad \,\forall\, x\,\forall\, y (\qed_x A & y \ge x\to\qed_y A)\; .\nn \end{eqnarray*} It is shown that the theory HA + CS with the induction schema restricted to arithmetical (i.e. not (...)
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