What is the difference between bracketed and unbracketed citations in Princpia Mathematica

Question

I have been reading Whitehead and Russell's Principia Mathematica and I was wondering what the difference is between the justifications provided in brackets and the justifications outside of brackets. For example, in *4.4 Russell begins the proof with *3.2 but later he uses the notation [*3.48] to justify line 1. What is the difference between these 2 citations?

Answer 1

In general, between square brackets the authors write the "instruction" to produce the formula.

For a simple example, see the derivation of $\text {*2.01}$ (page 100): the instruction amounts to replacing $p$ with $\lnot p$ into $\text {Taut}$.

Thus, in modern terms, they are meta- expressions about the derivation (formal proof) and not part of it.

When they write e.g. "$\vdash . \text {*2.31}$" (see the derivation of $\text {*2.04}$, page 106), they are using a simple abbreviation that means:

$\vdash :p \lor (q \lor r ) . \supset .(p \lor q) \lor r$.

The derivation of $\text {*4.4}$ is a little bit tricky. We have to start from:

$\vdash [p \supset (q \supset (p \land q))] \supset ([p \supset (r \supset (p \land r))] \supset \{ [p \supset (q \supset (p \land q))] \land [p \supset (r \supset (p \land r))] \})$

that is very long instance of $\text {*3.2}$ (in schematic form: $\varphi \supset (\psi \supset (\varphi \land \psi))$).

Now, using the two instances: $\vdash p \supset (q \supset (p \land q))$ and $\vdash p \supset (r \supset (p \land r))$ of $\text {*3.2}$ in turn, by Modus Ponens twice we get:

$\vdash [p \supset (q \supset (p \land q))] \land [p \supset (r \supset (p \land r))])$.

This is the first part of the derivation, that W&R abbreviate in the single line:

$. \vdash \text {*3.2} . \ \ \ \supset \vdash [p \supset (q \supset (p \land q))] \land [p \supset (r \supset (p \land r))])$.