Blogicisim #0 Principia *1

27 Nov 2024 -

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Last updated 1/28/25

Intro

My main goal in writing this series of posts is to leave myself an organized record of my thoughts to refer to later. I am posting them online since what I write may be useful to others. Principia has a reputation for being somewhat impenetrable, so I hope to de-mystify it somewhat with these notes.

My aim at first will be to show the most important results in principia’s notation, what I understand to be modern notation, and in english. Maybe I’ll discuss some proofs but probably not…

These notes will not be based soley on the Principia itself. According to my teacher Gregory Landini at the University of Iowa, the Principia cannot be understood in isolation. Russell was his own worst enemy when writing the text; some of his efforts have made the work more obscure than it needed to be. For example, the Principia suppresses notation that would indicate the type of propositions. To paraphrase Landini, the entire introduction of the Principia should be skipped as it is misleading. One should skip directly to the first chapter, $\pmast 1$ . In order to understand Principia, we must go through the proofs and consult secondary literature written by authors who have done the same and understood Russell’s intentions correctly. When preparing these notes, my references have included the following.

The Basic Laws of Arithmetic by Gottlob Frege
Principles of Mathematics by Bertrand Russell
Introduction to Mathematical Philosophy by Bertrand Russell
Logic and Knowledge by Bertrand Russell
The Foundations of Mathematics by F. P. Ramsey
Russell’s Metaphysical Logic by Bernard Linsky
Russell by Gregory Landini
Nice discussions of Principia, chapter by chapter, on YouTube by The Second Number-Class

Some Core Ideas and Motivations

On Frege

One of Russell’s motivations was to follow up on Gottlob Frege’s failed system, which he felled by showing that Basic Law Five causes a paradox.

On Math

When pure mathematicians are “doing math” they are studying relational structures
Math is the study of relations

On Numbers

We should avoid making metaphysical commitments to the existence of abstract objects like numbers
Numbers cannot be defined using counting because counting uses numbers
A number is not identical to a collection of things with that number of things in it
Peano’s axiomatization of the natural numbers can be simplified

On Classes (and Sets)

We should avoid making metaphysical commitments to the existence of abstract objects like classes
Math can make use of the idea of classes without assuming that such things as classes actually exist

Things You May Find Weird

A contributing factor to the impenetrability of Principia by modern readers is that Whitehead and Russell’s notation differs significantly from what is used in contemporary presentations of logic. Speaking for myself, I will say there is not much overlap between the notation Whitehead and Russell use for their propositional logic and what I learned in my introductory logic coursework in engineering and computer science. Even the book itself has a non-standard layout.

Principia is Unfinished - There are three published volumes of Principia and two editions. There was supposed to be a fourth volume (it is even referenced in volume III).
Chapter Structure - Instead of Chapter 1, 2, 3,…, Principia has $\pmast 1, \pmast 2, \pmast 3,\ldots$ . (Sub)Sections of the chapters are ordered by decimal places.
Function Notation - Usually we would read $\phi(x)$ as a function. In Principia, we would write $\phi \hat{x}$ to designate the function itself and write $\phi x$ to designate an ambiguous value of the function. These notational conventions are not used consistently in the text. Sometimes, you will see function expressions such as $f\{\pmdsc{x} (\phi x)\}$ where the “argument” is not actually an argument because it doesn’t actually exist.
Delimiters - Instead of parentheses and brackets, you fill find dots. Numbers of dots correspond to levels of scope. Among $\pmdottt, \pmdott, \pmdot$ , $\pmdottt$ is the broader scope and $\pmdot$ is the narrowest scope.
Definitions - Rather than use $:=$ or $=_{df}$ , a definition will be followed by $\pmdf$ . Similarly, a primitive will be followed by $\pmpp$
Proofs - Rather than begin a proof with Proof., Principia begins them with Demonstrations signified by $\pmdem$
Law of the Excluded Middle - Principia accepts the law of excluded middle, which states that a proposition is either true or false. You might say this is unreasonable because they hold that future events which are uncertain such as “In 50 years I will have a beard.”
Truth Values - True and False are not objects in the Principia. Truth values are used informally.

$\pmast 1$ - Primitive Ideas and Propositions

Summary and Highlights

Principia begins by characterizing a basic propositional logic which, other than the symbols used, is similar to what one would encounter in a basic logic course. This logic is supported by primitive ideas and propositions that are not (fully) defined. The reason for this is that a system employing definitions must either be circular or be founded upon undefined terms. Circularity trivializes a system, so the latter is preferable.

Later, in $\pmast 9$ , the propositions from here up to $\pmast 5$ will be generalized such that any proposition can be replaced by one with any number of quantifiers. In this way, quantification logic will be built.

Primitives Introduced

Elementary Proposition - Proposition with no variables or quantifiers
Elementary Propositional Function - Unsaturated statement such that, when saturated, yields an elementary proposition
Assertion of a Proposition - Saying that the proposition is true, rather than simply stating it. This is expressed as $\pmthm \pmdot p$ .
Assertion of a propositional function - Saying that the propositional function is true for general ambiguous argument.
Negation - $\pmnot p$ means $p$ is false.
Disjunction - $p \pmor q$ means “either $p$ is true or $q$ is true.”

Implication as Material Implication

Implication is a relation between propositions.

One proposition implying another is symbolized as $p \pmimp q \pmdot = \pmdot \pmnot p \pmor q \pmdf$ This definition of implication is called material implication and any feelings that this definition is odd are rightly justified. Since I am writing this post in Iowa, the material implication “If I am in London, then I am in France” is a true statement. There are other conventions for characterizing implication: enthymematic, deductive, and strict.

Primitive Propositions

$\pmast 1 \pmcdot 1 \text{ Anything implied by a true proposition is true} \pmpp$

$\pmast 1 \pmcdot 11$ Where $\phi x$ can be asserted, $\phi x \pmimp \psi x$ can be asserted, then $\psi x$ can be asserted, where $x$ is a real variable in each case. $\pmpp$

We get inference rules from 1.1 and 1.11. The statement of 1.11, although it does not include notation for type, gives some type safety.

$\pmast 1 \pmcdot 2$ (Principle of Tautology)

$\begin{aligned} & \pmthm \pmdott p \pmor p \pmdot \pmimp \pmdot p \pmpp \\ & \pmthm (p \pmor p) \implies p \end{aligned}$

Any proposition implies itself.

$\pmast 1 \pmcdot 3$ (Principle of Addition)

$\begin{aligned} & \pmthm \pmdott q \pmdot \pmimp \pmdot p \pmor q \pmpp \\ & \pmthm q \implies (p \pmor q) \end{aligned}$

A disjunction of a true proposition with another proposition is still true.

$\pmast 1 \pmcdot 4$ (Principle of Permutation)

$\begin{aligned} & \pmthm \pmdott p \pmor q \pmdot \pmimp \pmdot q \pmor p \pmpp \\ & \pmthm (p \pmor q) \implies (q \pmor p) \end{aligned}$

If a disjunction is true, the result after reversal of the terms is still true.

$\pmast 1 \pmcdot 5$ (Associative Principle)

$\begin{aligned} & \pmthm \pmdott p \pmor (q \pmor r) \pmdot \pmimp \pmdot q \pmor (p \pmor r) \pmpp \\ & \pmthm (p \pmor (q \pmor r)) \implies (q \pmor (p \pmor r)) \end{aligned}$

$\pmast 1 \pmcdot 6$ (Principle of Summation)

$\begin{aligned} & \pmthm \pmdott q \pmimp r \pmdot \pmimp \pmdott p \pmor q \pmdot \pmimp \pmdot p \pmor r \pmpp \\ & \pmthm (q \implies r) \implies ((p \pmor q) \implies (p \pmor r)) \end{aligned}$

In an implication, an alternative may be added to both premiss and conclusion without impairing the truth of the implication.

(Closure Properties)

$\pmast 1 \pmcdot 7\pmcdot 71$

If $p,q$ are elementary propositions then so are $\pmnot p$ and $p \pmor q \pmpp$

$\pmast 1 \pmcdot 72$ (Axiom of Identification of Real Variables)

If $\phi p,\psi p$ are elementary propositional functions which take elementary propositions as arguments, then $\phi p \pmor \psi p$ is an elementary propositional function $\pmpp$