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 (a) an understanding of the language in which they are couched, and (b) an understanding of the notions defined in terms of the primitive notions of the language. The axioms are stated in terms of the primitive and defined notions of the language. These are presented in the monograph in the Chapter called ‘The Language’. After reading through the definitions of the primitive terms, atomic formulas, complex formulas, and complex terms, and the defined notions defined in terms of these primitives, return here to see some Examples.

Note that existence and identity are defined in terms of the two basic kinds of predication and the distinguished predicate E!x, which allows us to define the distinction between ordinary and abstract objects.

The Proper Axioms of the Theory

The way the theory is formulated in Principia Metaphysica, there is no distinction between logical axioms and proper axioms. That's because the distinction between logic and metaphysics is not clear-cut. The foundational notions of logic, namely those concerning predication (such as exemplification and encoding) are already entangle in metaphysical domains (such as the domain of relations and the domain of individuals). So we present all of the axioms together.

Examples: Axioms

• Propositional Logic (Classical)
• Quantificational Logic (A Free Logic as Default, but Classical for Constants and Variables, and some λ-expressions)
• Logic of Identity = Unrestricted Substitution of Identicals
• Modal Logic (S5 modal logic)
• Logic of Actuality (Classical: including a ★-axiom, i.e., one that can't be necessitated)
• Logic of Definite Descriptions (Classical, for Rigidly Interpreted Descriptions)
• Logic of Complex Predicates (Relational λ-Calculus)
• Logic of Encoding

The Deductive System

In order to derive consequences of the axioms of the theory, we introduce a deductive system that is based on the single rule of inference Modus Ponens. The Rule of Generalization (GEN) and the Rule of Necessitation (RN) are derived as metarules, as are many other rules of inference.

In the usual manner, a derivation of a formula φ from a set of premises Γ is any sequence of formulas such that each member of the sequence is either an axiom or follows from two previous members of the sequence by a rule of inference. A proof of φ is any derivation of φ from no premises. However, we distinguish those derivations and proofs that depend on a ★-axiom from those that do not. The latter are modally strict and subject to the Rule of Necessitation. The former are neither.

* In this tutorial, we don't refer to the numbered items in Principia Metaphysica by number because the monograph is frequently being updated and expanded; 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.