## An Introduction to Substructural LogicsThis book introduces an important group of logics that have come to be known under the umbrella term 'susbstructural'. Substructural logics have independently led to significant developments in philosophy, computing and linguistics. An Introduction to Substrucural Logicsis the first book to systematically survey the new results and the significant impact that this class of logics has had on a wide range of fields.The following topics are covered: * Proof Theory * Propositional Structures * Frames * Decidability * Coda Both students and professors of philosophy, computing, linguistics, and mathematics will find this to be an important addition to their reading. |

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### Contents

Irs Ands and Ors | 9 |

Modalities | 47 |

Hilbert Systems | 73 |

Theories | 89 |

Gentzen Systems | 105 |

Formulae as Types Proofs as Terms | 127 |

Defining Propositional Structures | 155 |

Manipulating Propositional Structures | 189 |

Logics with Distribution | 235 |

Logics Rejecting Distribution | 275 |

Frame Constructions | 295 |

Decision Procedures | 313 |

Undecidability | 323 |

Using Substructural Logic | 339 |

349 | |

365 | |

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### Common terms and phrases

A A B A V B accessibility relation antecedent arrow atomic axiom behaviour bisimulation Boolean negation Chapter closure frame closure operator coherence space commutative complete conjunction connectives consecution consequent conservative extension consider construct corresponding defined DEFINITION desired disjunctive syllogism element elimination entails equivalent example extended finite follows formulae function functor fusion Gentzen system given gives h-pair hence Hilbert system holds homomorphism identity implication induction inference intensional interpretation intuitionistic logic Kleene star Lambek calculus language LEMMA Lindenbaum algebra linear logic metavaluation modal logic Morgan negation natural deduction natural deduction system partial order point set positive modality premises prime theory proof theory propositional structure provable prove punctuation mark relevant logics result Rxyz satisfies semantics Similarly string structural rules subformulae subset substructural logics Suppose term of type THEOREM true undecidable valid variables