## Partiality, Truth, and PersistenceIn recent years, 'semantical partiality' has emerged as an important concept in philosophical logic as well as in the study of natural language semantics. Despite the many applications, however, a number of mathematically intriguing questions associated with this concept have received only very limited attention. Partiality, Truth, and Persistence is a study in spatial model theory, the theory of partially defined models. First, with the introduction of truth value gaps in semantics, there are many ways to generalize the classical truth definition for the sentences of a first order predicate language. We know what it means for a sentence to be true or false in a classical, complete model, but how do we extend this relation when partial models are introduced? Various alternatives exist, and a detailed comparison is carried out between them. Since these studies concern a full first order predicate language, many distinctions appear that do not arise in the case of pure propositional logic. A condition of monotonicity or 'persistence' of truth relative to partial models has a prominent position among conditions that are not expressible in the framework of standard, complete model theory. The final chapter investigates the relation between such conditions and expressibility properties in general. These discussions culminate with a combined Lindstrom and persistence characterization theorem. Tore Langholm is a research fellow in mathematics at the University of Oslo. He is a co-author of Situations, Language and Logic. |

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### Contents

Propositional Logic | 13 |

The Strong Kleene Truth Definition | 43 |

Alternative Truth Definitions | 69 |

Extending the Language | 109 |

Further Directions | 133 |

139 | |

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alternative truth definitions assume atomic formulas Barwise chapter classical logic classical models classically equivalent clauses coherent compl complete models compositional connectives consequence relation contr corresponding countably compact cut-and-glue defined determinable and persistent existential quantification existential sentence expressive power extension of domain falsity-persistent finite sets finite subsets follows formula ip framework function Hence implies induction hypothesis ip[A isomorphism Jon Barwise Kleene truth definition language last step Lecture Notes Lemma Lindstrom theorem Lowenheim property maximal consistent set model theory monomorphism negation negatively equivalent non-empty notion occur free order logic partial models partial structures persistent iff persistent sentence positively equivalent predicate logic preserved under extension Proof propositional logic prove relation symbols relative saturation theorem satisfies the Lowenheim sentence ip signed sentences similarity type Skolem property sscl strong Kleene truth strongly equivalent substitution successor ordinal supervaluation trivial truth functional truth table variable assignment verification tree w-partial wscl