Modal logic is a type of formal logic that extends classical propositional and predicate logic to include modalities, which are expressions that convey modality. The most common modalities involve notions of necessity and possibility. In modal logic, statements are often expressed using modal operators, typically represented as: - **□ (box)**: This operator is used to indicate that a statement is necessarily true. For example, if \( P \) is a proposition, then \( □P \) means "it is necessary that P.
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A modal logic extends propositional connectives by operators such as and its dual . An interpretation determines their meaning: necessity, knowledge and formal provability lead to different axiom systems. In a relational semantics, holds at a world when holds at every accessible world. Provability logic interprets the box through arithmetically coded proofs.