# 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:

- the language in which the theory is couched,
- the logic (i.e., the deductive apparatus) by which theorems
(consequences of the axioms) are derived, and
- the proper axioms of the theory of abstract objects and their
consequences.

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.