Tutorial on the Theory of Abstract Objects

This tutorial should help you to read the monograph Principia Metaphysica. It will give you some explanation of:

We cover these topics in order. In what follows, we refer to specific items in Principia Metaphysica using the following format: SectionTitle/ItemName. Every item in this monograph occurs in a unique section, and the items in a section are always uniquely named.*

The Language of the Theory

In order to understand the proper axioms of the theory of abstract objects, we must have an understanding of the language in which they are couched. These are defined precisely in the monograph in the Section called `The Language'. After reading through the definitions of the primitive terms, atomic formulas, complex formulas, and complex terms, return here to see some Examples .

The Logic of the Theory

In order to derive consequences of the proper axioms of the theory, we appeal to a precise logic. This logic consists of logical axioms and rules of inference, and from these axioms and rules alone, we can derive a wide range of logical theorems and other useful rules of inference. After examining the Section entitled "The Logic" in the monograph, return here for some examples (see below). In these examples we adopt the following convention:

The notion of a proof utilized in these conventions is defined in the item Logic/Derivability and Theoremhood as follows: In the examples which follow, this definition is illustrated and applied and so are the following definitions concerning theoremhood:

Examples: Logical Axioms, Theorems and (Derived) Rules of Inference

The Proper Axioms of the Theory

The way the theory is formulated in Principia Metaphysica, there are only 3 proper axioms. After examining these axioms in the monograph, return here for some Examples.

* In this tutorial, we can not refer to items in Principia Metaphysica by number because the monograph is frequently being modified and expanded and the numbering of items is constantly changing. In the LaTeX sourcefile of this monograph, these item numbers are not hard-wired to the each item, but are generated by a key system. LaTeX processes the document twice---on the first pass, it assigns numbers to the key words, on the second pass, it replaces the key words with the numbers assigned to them on the first pass. Since there is no simple way to similarly key the present HTML document to the items in Principia Metaphysica, we shall refer to items in the manner specified above.