## Abstract Objects: An Introduction to Axiomatic MetaphysicsIn this book, I attempt to lay the axiomatic foundations of metaphysics by developing and applying a (formal) theory of abstract objects. The cornerstones include a principle which presents precise conditions under which there are abstract objects and a principle which says when apparently distinct such objects are in fact identical. The principles are constructed out of a basic set of primitive notions, which are identified at the end of the Introduction, just before the theorizing begins. The main reason for producing a theory which defines a logical space of abstract objects is that it may have a great deal of explanatory power. It is hoped that the data explained by means of the theory will be of interest to pure and applied metaphysicians, logicians and linguists, and pure and applied epistemologists. The ideas upon which the theory is based are not essentially new. They can be traced back to Alexius Meinong and his student, Ernst Mally, the two most influential members of a school of philosophers and psychologists working in Graz in the early part of the twentieth century. They investigated psychological, abstract and non-existent objects - a realm of objects which weren't being taken seriously by Anglo-American philoso phers in the Russell tradition. I first took the views of Meinong and Mally seriously in a course on metaphysics taught by Terence Parsons at the University of Massachusetts/Amherst in the Fall of 1978. Parsons had developed an axiomatic version of Meinong's naive theory of objects. |

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Meta-magick gets founded here, as applied by the NuChwezi school.

### Contents

1 THEORY DATA AND EXPLANATION | 1 |

2 THE ORIGINS OF THE THEORY | 6 |

ELEMENTARY OBJECT THEORY | 15 |

1 THE LANGUAGE | 16 |

2 THE SEMANTICS | 19 |

3 THE LOGIC | 28 |

4 THE PROPER AXIOMS | 32 |

5 AN AUXILIARY HYPOTHESIS | 37 |

3 MODELLING LEIBNIZS MONADS | 84 |

4 MODELLING STORIES AND NATIVE CHARACTERS | 91 |

5 MODALITY AND DESCRIPTIONS | 99 |

THE TYPED THEORY OF ABSTRACT OBJECTS | 107 |

1 THE LANGUAGE | 109 |

2 THE SEMANTICS | 113 |

3 THE LOGIC | 121 |

4 THE PROPER AXIOMS | 124 |

APPLICATIONS OF THE ELEMENTARY THEORY | 40 |

1 MODELLING PLATOS FORMS | 41 |

2 MODELLING THE ROUND SQUARE ETC | 47 |

3 THE PROBLEM OF EXISTENCE | 50 |

APPENDIX TO CHAPTER II | 52 |

THE MODAL THEORY OF ABSTRACT OBJECTS WITH PROPOSITIONS | 59 |

2 THE SEMANTICS | 61 |

3 THE LOGIC | 68 |

4 THE PROPER AXIOMS | 73 |

THE APPLICATIONS OF THE MODAL THEORY | 77 |

2 MODELLING POSSIBLE WORLDS | 78 |

APPLICATIONS OF THE TYPED THEORY 1 MODELLING FREGES SENSESI | 126 |

2 MODELLING FREGES SENSES II | 140 |

3 MODELLING IMPOSSIBLE AND FICTIONAL RELATIONS | 145 |

4 MODELLING MATHEMATICAL MYTHS AND ENTITIES | 147 |

CONCLUSION | 154 |

MODELLING THE THEORY ITSELF | 158 |

MODELLING NOTIONS | 167 |

NOTES | 172 |

187 | |

190 | |

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### Common terms and phrases

4-objects A-expressions abbreviate abstract objects abstract properties actual world asserts atomic formula believes Chapter clause Consequently context Crime and Punishment defined definite description DICTO elementary theory encodes a property encoding formulas English descriptions example existing objects expressions extranuclear fact fails to exemplify free object free object variable Frege given interpretation intuitions Lauben Leibniz logical axioms logical functions logically true Meinong metaphysical modal monads n-place relation term n-tuple names necessarily notion nuclear object encodes object exemplifies object term object which encodes object which exemplifies objects of type Parsons philosophers Plato possible world primitive principle proof proper axioms propositional attitude propositional formula quantifiers Raskolnikov reading respect restricted rules of inference Russellian satisfies schema schemata Section semantics sentences Socrates story suppose term of type theorem theory of abstract translate truth type theory unique vacuous properties variable ranging weak correlate wife of Tully