Standard Mathematical Logic
Fatemeh Shirmohammadzadeh Maleki
Abstract
In this article, first we define a Kripke semantics for normal modal Logic with a binary operator and we introduce a system K^2 which is sound and complete for this semantics. Then, we will introduce two translations and show that binary normal modal logic K^2, and unary normal modal logic K, i.e. modal ...
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In this article, first we define a Kripke semantics for normal modal Logic with a binary operator and we introduce a system K^2 which is sound and complete for this semantics. Then, we will introduce two translations and show that binary normal modal logic K^2, and unary normal modal logic K, i.e. modal logic with one binary operator, are very closely related by these two translations. We call a translation a faithful interpretation if provability is preserved in both directions. So, with this terminology we will show that these two translations are faithful interpretation of K into K^2 and vice versa. A logic extending K will be a set of formulas containing K closed under its rules and uniform substitution. A logic extending K^2 is similarly defined. Finally, we will prove that the classes of logics extending K and K^2 are closely related as well and there is a 1-1-correspondence between the logics extending K and extending K^2.
Fatemeh Shirmohammadzadeh Maleki
Abstract
Our main goal in this paper is to find modal companions for some subintuitionistic logics introduced by de Yongh and Shirmohammadzadeh. They introduced two types of neighbourhood frames, N-neighbourhood frames and NB-neighbourhood frames, in order to prove the completeness of these subintuitionistic ...
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Our main goal in this paper is to find modal companions for some subintuitionistic logics introduced by de Yongh and Shirmohammadzadeh. They introduced two types of neighbourhood frames, N-neighbourhood frames and NB-neighbourhood frames, in order to prove the completeness of these subintuitionistic logics. The structure of N-neighbourhood frames are similar to the neighborhood frames for non-normal modal logics. But the structure of NB- neighbourhood frames was introduced with a somewhat more complex definition than the neighbourhood semantics for non-normal modal logics. So in order to find out the modal companions of these subintuitionistic logics, we consider two types of translation, one from the language of intuitionistic propositional logic to the language of modal propositional logic, and the other from the language of intuitionistic propositional logic to the language of binary modal propositional logic, and compare the provability of a formula and its translation. Finally, using these two types of translations, we obtained the modal companions of desired subintuitionistic logics.
Fatemeh Shirmohammadzadeh Maleki
Abstract
Intuitionistic logic is a non classical logic obtained by omitting the axiom of excluded middle from classical logic. This logic was created by philosophical motivation towards the foundation of mathematics. There are several semantics for intuitionistic logic (such as Kripke semantics, neighborhood ...
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Intuitionistic logic is a non classical logic obtained by omitting the axiom of excluded middle from classical logic. This logic was created by philosophical motivation towards the foundation of mathematics. There are several semantics for intuitionistic logic (such as Kripke semantics, neighborhood semantics and topological semantics) that are sound and complete. In this paper, we first present two new neighborhood semantics for propositional intuitionistic logic (IPC). Then we establish soundness and completeness of IPC with respect to these new neighborhood semantics. The relation between neighborhood and topological semantics are also investigated. One of these new neighborhood semantics is introduced with a somewhat more complex definition than the usual neighborhood semantics which was introduced before. This semantics is called NB-neighborhood semantics. In order to establish completeness with respect to NB-neighborhood semantics for IPC, first we need to introduce a system WF of subintuitionistic logic, weaker than Corsi's basic subintuitionistic system F.
