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Abstract:
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The thesis consists of four chapters. Chapter 1 contains a general framework for deductive systems and contains a sequence-conclusion natural deduction system for classical first order logic. A sequence NLC_n of intermediate propositional logics is considered in Chapter 2. It is shown that the sequence NLC_n contains three different systems only. These are the classical calculus NLC_1, Dummett's system NLC_2 and the logic NLC_3, an extension of the Heyting propositional logic by the axiom (A⇒B)∨(B⇒C)∨(C⇒A) . It is also shown that the logic NLC_3 is separable. In the sequel, the completeness of NLC_3 with respect to the corresponding Kripke type models having the property that ∀x∀y∀z(xRy∨yRz∨zRx) is proved, as well as its decidability and the independence of logical connectives. It is shown that some subsystems of NLC_3 are separable and that the limits of the considered systems is the Heyting propositional calculus. The logic of the weak law of excluded middle, an extension of the Heyting logic by ¬A∨¬¬A, is considered in Chapter 3. An embedding of classical logic into this logic is described and it is proved that this logic is the minimal one having this property. A Hilbert-type formulation of implication fragment of the Heyting propositional logic formalizing the deducibility relation, is presented in Chapter 4, enabling to define a decision procedure based on a kind of cut-elimination theorem. |