Which one of the following logical propositions is to be preferred?

Question

I'm trying to update the symbolism of Giuseppe Peano's "Arithmetices Principia", to make the translation freely available. Might I ask you, which of the following might be a correct mathematical reasoning? Since the original symbolism has bi-fold meanings. I suppose the symbolism should be updated as either one of the following:

• $(\alpha \xrightarrow[\forall x]{} \beta) = [(\overline{x} \in \alpha) \rightarrow (\overline{x} \in \beta)]$

• $(\alpha \underset{\forall \text{x}}\subset \beta) = [(\overline{x} \in \alpha) \rightarrow (\overline{x} \in \beta)]$

Is any of them non-sense, or could any of them be preferred?

If you are interested, the line I am trying to translate can be found as the line numbered with "63.", in the original manuscript on page "XII".

P.S. | Notation:

• $\overline{x} \in a$ means ... those $x$ such that $a$, or solutions, or roots of the condition $a$, indicates the class consisting of individuals which satisfy the condition $a$.

• The sign $\subset$ means ... is contained. Thus $a \subset b$ means the class $a$ is contained in the class $b$.

If the propositions $a,b$ contain the indeterminate quantity $x$, that is, express conditions on these objects, then:

• $a \xrightarrow[\forall x]{} b$ means ... whatever the $x$ from propositions $a$ one deduces $b$.

Many thanks for any advice.

Answer 1

Long comment

You have to note that Peano's "inverted-C" symbols plays a double role.

Regarding propositions :

$a \supset b$, with $a,b \in \mathsf P$ [where $\mathsf P$ stands for propositio]

has to be read as : "$a$ deducitur $b$", i.e. (more or less) our : $a \to b$ [see : page VIII].

Regarding classes :

$a \supset b$, with $a,b \in \mathsf K$ [where $\mathsf K$ stands for classis]

has to be read as : "$a$ continetur in $b$", i.e. our : $a \subseteq b$ [see : page XI].

The two "roles" are linked into (slightly "modernized") :

50. $\ a,b \in \mathsf K \ [ \ a \supset b . \equiv . (x \in a \supset_x x \in b) \ ]$.

We have to note that Peano uses the same "style" of variables both for proposition and classes.

In [see : page XII] :

63. $\alpha \supset_x \beta . \equiv . [x \in \alpha] \supset [x \in \beta]$

we have to be careful to read the "complex" symbol $[x \in \alpha]$ as denoting :

"ea $x$ quibus $a$" (more or less : "those $x$ such that $a$") i.e. "classem significat individuis constitutam, quae conditioni $a$ satisfaciunt"

i.e. "the class of all those individuals that satisfy $a$".

Thus, we have not to use $a$ in the "propositional context" : $a \supset b$ and in the "class context" : $x \in a$, without some care.

W&R used $\psi x$ to express the condition "involving" $x$ and $\hat x \psi x$ for the class of all those $x$ such that $\psi x$ holds.

Thus, a "possible translation" of 63 above can be :

$\alpha \to_x \beta . \equiv . (x \in \hat \alpha \subset x \in \hat \beta)$,

where I've "reversed" the "inclusion" symbol (following modern practice) and I've "modernized" the conditional sign into : $\alpha \supset_x \beta$.

I hope it can help ...