In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent superintuitionistic logic. Thus consistent superintuitionistic logics are called intermediate logics (the logics are intermediate between intuitionistic logic and classical logic).

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  • In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent superintuitionistic logic. Thus consistent superintuitionistic logics are called intermediate logics (the logics are intermediate between intuitionistic logic and classical logic). Specifically, a superintuitionistic logic is a set L of propositional formulas in a countable set of variables pi satisfying the following properties: all axioms of intuitionistic logic belong to L; if F and G are formulas such that F and F → G both belong to L, then G also belongs to L; if F(p1, p2, ... , pn) is a formula of L, and G1, G2, ... , Gn are any formulas, then F(G1, G2, ... , Gn) belongs to L (closure under substitution). Such a logic is intermediate if furthermore L is not the set of all formulas. There exists a continuum of different intermediate logics. Specific intermediate logics are often constructed by adding one or more axioms to intuitionistic logic, or by a semantical description. Examples of intermediate logics include: intuitionistic logic (IPC, Int, IL, H) classical logic (CPC, Cl, CL): IPC + p ∨ ¬p the logic of the weak excluded middle: IPC + ¬¬p ∨ ¬p Gödel–Dummett logic (LC, G): IPC + (p → q) ∨ (q → p) Kreisel–Putnam logic (KP): IPC + (¬p →) → (∨) Medvedev's logic of finite problems (LM or ML): defined semantically as the logic of all frames of the form <math>\langle\mathcal P(X)\setminus\{X\},\subseteq\rangle</math> for finite sets X ("Boolean hypercubes without top"), as of 2008 not known to be recursively axiomatizable realizability logics Scott's logic: IPC + (→) → (¬¬p ∨ ¬p) Smetanich's logic: IPC + (¬q → p) → (→ p) The tools for studying intermediate logics are similar to those used for intuitionistic logic, such as Kripke semantics. For example, Gödel–Dummett logic has a simple semantic characterization in terms of total orders.
  • 中间逻辑是在直觉逻辑和经典逻辑之间的中介，这是在它们包含在直觉逻辑中不可证明的定理，而不导致完整的经典逻辑的意义上的。这种逻辑也叫做超直觉或次经典逻辑。 有连续统个不同的中间逻辑，通常是向直觉逻辑增加一个或多个公理而获得的。 这种逻辑的例子有： 直觉逻辑 (IPC, Int, IL, H) 经典逻辑 (CPC, Cl, CL): IPC + P ∨ ¬P 弱排中律逻辑（KC, Jankov 逻辑，德·摩根定律逻辑）: IPC + ¬¬P ∨ ¬P 哥德尔-Dummett 逻辑 (LC): IPC + (P → Q) ∨ (Q → P) Kreisel-Putnam 逻辑：IPC +（¬P → (Q ∨ R)）→（(¬P → Q) ∨ (¬P → R)) Medvedev 有限问题的逻辑 可实现性逻辑 Scott 逻辑：IPC + (→) → (¬¬P ∨ ¬P) Smetanich 逻辑：IPC + (¬Q → P) →（（(P → Q) → P）→ P）研究中间逻辑的工具类似于直觉逻辑所使用的，比如Kripke语义。例如，Gödel-Dummett 逻辑有依据全序特征化的一个简单语义。
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  • In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent superintuitionistic logic. Thus consistent superintuitionistic logics are called intermediate logics (the logics are intermediate between intuitionistic logic and classical logic).
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  • Intermediate logic
  • 中间逻辑
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