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There are two notions of logically possible. I will label them as the positive sense and the negative sense. The positive sense is that a set of sentences S is possible if and only if there exists a possible world where all sentences in S are true. The negative sense is that no logical contradiction can be deduced from S. Certainly, the positive sense implies the negative sense. But, I wonder, is there a distinction between them? And also, what have philosophers said about this topic?

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  • Logically possible worlds are exactly those realizing logically consistent sets of sentences by definition. You can define possible worlds in some other way, but that will produce a kind of possibility other than logical, see SEP, Varieties of Modality.
    – Conifold
    Commented Apr 11 at 22:04

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But, I wonder, is there a distinction between them?

one is a syntactic notion, and the other is a semantic one, besides one of them belonging/being applicable to a more restricted class of logics

There are two notions of logically possible. [...] The negative sense is that no logical contradiction can be deduced from S.

in case S ⊬ ⊥, people generally just call such an S "consistent" or "a consistent set of sentences"; note that this makes sense for logics lacking modal operators in their languages

The positive sense is that a set of sentences S is possible if and only if there exists a possible world where all sentences in S are true.

this makes sense in the more specific context of kripke semantics for alethic modal logics, but not much for, say, usual semantics for classical logic

it may be that some people colloquially/informally speak of "logical possibility" in the first case by thinking that S ⊢ ⊥ somehow 'means' that "S is logically impossible", but it's just that, a non-technical use of the terms

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