Is Propositional Tautologousness a Modal Notion? (Raisins at Dawn #3)

After writing what appears below, I found a treatment-in-passing of the very same issue by von Wright in a rich paper called 'Modal Logic and the Tractatus'.

Discuss: is PC-tautologousness a modal notion?

'Can't come out F'.

But, having a populated abstract space of valuations:

'Doesn't come out F in any row/valuation'.

But then what goes into the conception of 'all valuations'? Isn't this the same as, or tantamount to, 'all possible valuations'?

But what if one tries to gainsay that?

'No, they're all actual.' - One can give non-modal conditions for 'permissible valuation', to be sure. Picking them out of a larger space. But how do we get the notion of the larger space?

We can say: there are valuations. A valuation is a mapping from atoms to {0, 1}. Any mapping from atoms to {0, 1} is a valuation.

We get, as it were, modal effects from a population of abstracta. And it seems we can conceive these abstracta without modal notions. That is, 'mapping from "p" to {0, 1}' doesn't seem like a modal notion.

But what about 'all such mappings'? Is there even any basis for decision? Is there a determinate answer? (Open texture.) Or is there a need to further fix concepts here? More than one legitimate way of going?

If so, we could have something like 'weakly modal' vs. 'strongly modal' so that 'all such mappings' is weakly but not strongly modal. But how do we draw the line?

For can't we, at least superficially, pull the same trick out in all cases, or a great many more than I was dimly thinking of above? The trick, that is, of trading modality - or strong modality - in for abstracta? Isn't ersatzism just such a move? Can we find a principled way of refuting ersatzism but allowing the move in the PC case?

I'm dimly seeing two distinctions:

One:

The case where everything under discussion is abstract, part of the abstract population.

vs.

The case where, as it were 'concrete configurations' are under discussion. So that to be a possible concrete configuration is to correspond to one of the abstract population. E.g. a switchboard.

Another:

The case where the abstract population is well-defined vs. not.

Also:

Abstract population discrete vs. continuous.

Before examining the crossings of these distinctions and trying for examples of each combination: is (1) really about abstract vs. concrete? Or is it rather about whether the object under discussion is itself part of the abstract population? You might, e.g., think of 'actual songs' as non-spatial but temporal and created, and thus hold an actual song to be distinct from its "abstract nature" or whatever, the latter being platonic (uncreated, non-temporal) - a space of "song-patterns".