Why can't ✳1.1 be expressed symbollically in Whitehead and Russell's PM?

Question

✳1.1. Anything implied by a true elementary proposition is true. Pp.

In the follow passage, it says, "we cannot express the principle symbolically, partly because any symbolism in which p is variable only gives the hypothesis that p is true, not the fact that it is true."*

*The footnote refers to Russell's Principles of Mathematics, §38, but I don't understand how exactly Russell solved Lewis Carroll's puzzle, "What the Tortoise said to Achilles."

Answer 1

About :

✳1.1. Anything implied by a true elementary proposition is true. Pp.

this is (a not very clear) formulation of modus ponens.

Modus ponens is an inference rule, so it is not "at the same level" of the laws of the calculus; in modern term, formula like $p \supset q$ are expressions in the language, but modus ponens is a rule fomulated in the meta-language, usually with schemata.

If you try to "download" mp in the languge, you will have :

$\vdash (p \cdot (p \supset q) ) \supset q$

and this is valid also when $p$ is false and $q$ is true.

We want instead, that the rule license a sound argument. What we need is something like :

$\frac {\vdash p \quad \vdash p \supset q}{\vdash q}$

The reference to Lewis Carroll is about the issue, discussed for the first time in "What the Tortoise said to Achilles", exactly about the role of "rules"; if I remeber well, was Carroll that argumented about the necessity of "postulating" rules that are not themselves "inferred", because in order to make a sound inference ... you need to apply a rule; this is the core of his paradox (a case of regressus). See the passage :

"Readers of Euclid will grant, I suppose, that Z follows logically from A and B, so that any one who accepts A and B as true, must accept Z as true?"

"Undoubtedly! The youngest child in a High School -- as soon as High Schools are invented, which will not be till some two thousand years later -- will grant that."

"And if some reader had not yet accepted A and B as true, he might still accept the sequence as a valid one, I suppose?"

"No doubt such a reader might exist. He might say 'I accept as true the Hypothetical Proposition that, if A and B be true, Z must be true; but, I don't accept A and B as true.' Such a reader would do wisely in abandoning Euclid, and taking to football."

"And might there not also he some reader who would say 'I accept A and B as true, but I don't accept the Hypothetical '?"