TeXipedia

lozenge

Renders a diamond-shaped symbol commonly used in modal logic and mathematical proofs.

Overview

Serves as a crucial operator in formal logic and mathematical reasoning, particularly in modal logic where it denotes possibility or necessity.

  • Frequently appears in formal proofs and logical arguments to indicate modal operators.
  • Used in theoretical computer science for expressing temporal logic properties.
  • Common in mathematical logic textbooks and academic papers discussing possibility theory.
  • Sometimes employed in discrete mathematics to denote special relations or operators.

Examples

Modal logic formula expressing possibility of a proposition P.

◊P→¬□¬P\lozenge P \rightarrow \neg\square\neg P 
\lozenge P \rightarrow \neg\square\neg P

Temporal logic formula showing eventual occurrence of event E.

◊E∧□(E→X)\lozenge E \land \square(E \rightarrow X) 
\lozenge E \land \square(E \rightarrow X)

Logical statement combining necessity and possibility operators.

□(P→Q)→(◊P→◊Q)\square(P \rightarrow Q) \rightarrow (\lozenge P \rightarrow \lozenge Q) 
\square(P \rightarrow Q) \rightarrow (\lozenge P \rightarrow \lozenge Q)