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).
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 (closure under modus ponens);
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 = IPC + ¬¬p → p = IPC + ((p → q) → p) → p
the logic of the weak excluded middle (KC, Jankov's logic, De Morgan logic): IPC + ¬¬p ∨ ¬p
Gödel–Dummett logic (LC, G): IPC + (p → q) ∨ (q → p) = IPC + (p → (q ∨ r)) → ((p → q) ∨ (p → r))
Kreisel–Putnam logic (KP): IPC + (¬p → (q ∨ r)) → ((¬p → q) ∨ (¬p → r))
Medvedev's logic of finite problems (LM, ML): defined semantically as the logic of all frames of the form for finite sets X ("Boolean hypercubes without top"), not known to be recursively axiomatizable
realizability logics
Scott's logic (SL): IPC + ((¬¬p → p) → (p ∨ ¬p)) → (¬¬p ∨ ¬p)
Smetanich's logic (SmL): IPC + (¬q → p) → (((p → q) → p) → p)
logics of bounded cardinality (BCn):
logics of bounded width, also known as the logic of bounded anti-chains (BWn, BAn):
logics of bounded depth (BDn): IPC + pn ∨ (pn → (pn−1 ∨ (pn−1 → ... → (p2 ∨ (p2 → (p1 ∨ ¬p1))).
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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). A superintuitionistic logic is a set L of propositional formulas in a countable set of variables pi satisfying the following properties: 1. all axioms of intuitionistic logic belong to L; 2.
Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Joyal. It was first conceived for modal logics, and later adapted to intuitionistic logic and other non-classical systems. The development of Kripke semantics was a breakthrough in the theory of non-classical logics, because the model theory of such logics was almost non-existent before Kripke (algebraic semantics existed, but were considered 'syntax in disguise').
Non-classical logics (and sometimes alternative logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models of logical consequence and logical truth. Philosophical logic is understood to encompass and focus on non-classical logics, although the term has other meanings as well.
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