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 of the Theory: the language in which the theory is couched, and some of the notions using in that language,
• The Axioms of the Theory: the axioms of the theory of abstract objects
• The Deductive System: the deductive apparatus by which theorems (consequences of the axioms) are derived, and
• Metarules GEN and RN:the theorems derivable from the axioms using the deductive apparatus.

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. Consequently, we shall not 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.

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 a brief summary and some examples:

Metadefinition: The Language of the Theory.

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 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. (You can always turn proper axioms into logical axioms, i.e., axioms true under every interpretation of the language, by defining interpretations of the language so that only those structures that make all the proper axioms true count as interpretations; that would turn all the proper axioms into logical axioms. So any distinction is mostly arbitrary.) Moreover, the foundational notions of logic, namely those concerning predication (such as exemplification and encoding) are already entangled in metaphysical domains (such as the domain of relations and the domain of individuals). So we present all of the axioms together as a group and, for simplicity, we'll divide them into groups that characterize the ‘logic’ of the notion in question.

Axioms

• Propositional Logic of Negations and Conditionals (Classical)
• Quantificational Logic (Summary: Classical predicate logic for constants, variables, and λ-expressions in which the variables bound by the λ don't appear in encoding position in the matrix of the λ-expression; however, there is a free logic for definite descriptions and all other λ-expressions)
• Logic of Identity (= Unrestricted Substitution of Identicals): where α and β are any distinct variables of the same type, β is substitutable for α in φ, and φ′ is the result of replacing 0 or more free occurrences of α by β in φ, then it is axiomatic that α=β → (φ → φ′)
• Modal Logic (S5 modal logic + an axiom asserting that there might have been a concrete object that isn't actually concrete)
• Logic of Actuality (Classical: including a modally-fragile (★) axiom, i.e., one that can't be necessitated)
• Logic of Rigid Definite Descriptions (Classical): y=ιxφ → ∀x(𝒜φ ≡ x=y). In other words, y is identical to the individual that is (in fact) such that φ just in case all and only those individuals that are actually such that φ are identical to y.
• Logic of Complex Predicates (Relational λ-Calculus)
• Logic of Encoding

The Deductive System

Rule of Inference and (Modally Strict) Derivations and Proofs

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.

Modus Ponens (MP): This is the sole rule of inference:

• From φ and φ → ψ, infer ψ.

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, a member of Γ, or follows from two previous members of the sequence by Modus Ponens (or a derived rule of inference). When such a sequence exists, we write Γ ⊢ φ. However, a proof of φ is any derivation of φ from Γ when Γ is empty.

It is important to distinguish those derivations and proofs that depend on (i.e., are derived, somewhere in the complete chain of reasoning, from) a modally fragile axiom, i.e., an axiom whose necessitation is not assertible. (For example, one of the axioms governing the actuality operator, namely 𝒜φ → φ, is such a modally fragile axiom.) Consequently, we say that a derivation or proof is modally strict when it doesn't depend on a modally fragile axiom anywhere in the chain of reasoning. We write Γ ⊢_□ φ to indicate that a derivation of φ from Γ is modally strict, and sometimes write and ⊢_□ to indicate that a proof of φ is modally strict.

However, instead of marking all of the derivations proofs that are modally strict, it is useful to instead use a star (★) to mark those theorems whose proof depends on a modally fragile axiom; this will indicate that such formulas are not subject to the Rule of Necessitation (see below). However, modally strict theorem, which are derived without any dependence on a modally fragile axiom and which are subject to the Rule of Necessitation, will not be marked in any way.

Metarules: GEN and RN

The Rule of Generalization (GEN) and the Rule of Necessitation (RN) are derived as metarules; these tell us that if we have a derivation meeting certain conditions, then there exists a derivation of with a certain, desirable conclusion.

The Rule of Generalization (GEN)

This metarule states that if there is a derivation of φ from the premise set Γ and the variable α doesn't occur free in any formula in Γ, then there is a derivation of ∀αφ from Γ:

• If Γ ⊢ φ and α doesn’t occur free in any formula in Γ, then Γ ⊢ ∀αφ.

When Γ is the empty set ∅, then GEN asserts that if there is a proof of φ, then there is a proof of ∀αφ, i.e.,

• If ⊢ φ, then ⊢ ∀αφ

Note that there are variants of Rule GEN in which the antecedent and the consequent of the metarule both use ⊢_□ instead of ⊢.

Rule of Necessitation (RN)

The Rule of Necessitation asserts that if there is a modally strict derivation of φ from zero or more premises, then there is a (modally strict) derivation of □φ from the necessitations of the premises.

To formulate this statement precisely, we first introduce a metadefinition:

• For any set of formulas Γ, we use the notation □Γ to refer to { □ψ | ψ ∈ Γ}.

So □Γ is the set of formulas that results when a □ is prefixed to every formula in Γ.

Then RN may be stated as follows:

• Rule of Necessitation (RN): If Γ ⊢_□ φ, then □Γ ⊢_□ □φ.
• Rule of Necessitation (Weaker Form): If Γ ⊢_□ φ, then □Γ ⊢ □φ.

When Γ = ∅, RN reduces to:

• If ⊢_□ φ, then ⊢_□ □φ
• If ⊢_□ φ, then ⊢ □φ (Weaker Form)