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In not positing facts, it does not posit any single object to which a true proposition or sentence might correspond.
That a proposition (or sentence) is actually accepted, i.e., that a judgement is made, must therefore be indicated by an additional sign like Frege's judgement-stroke or it remains implicit in the assertive use of a declarative sentence.
If a proposition (or sentence, statement) for which we claim truth is indeed true, it is so because it accurately refers to existing objects, or accurately represents actual states of affairs albeit objects and states of affairs about which we can state facts only under descriptions that depend on our linguistic resources.
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When John expects the stock market to fall, he stands in a certain relation to the proposition or sentence-meaning "the stock market will fall"; and when Mary wants to be a doctor, she stands in a different relation to the proposition or sentence-meaning "Mary will be a doctor".
The formal semantics officially consists of constructing models using sets of various kinds and functions from those sets to propositions or sentences (or other objects representing propositions or sentences: other sets do nicely).
The now dominant view is that propositions or sentences are the objects of belief, and that judgements occur when beliefs are acquired, manifested, or changed.
As a matter of logic, one will need some bridge principles to connect propositions or sentences about consciousness with those that do not mention it.
A fifth strategy is to reject the gap entirely, and to simply agree that there is no gap that divides either propositions or sentences.
Logical inferences are then defined as relations between propositions or sentences, abstracting from the mental attitudes that go along with them.
A Boolean algebra is just a set of e.g., propositions or sentences that is closed under the classical logical operators and negation.
In keeping with current orthodoxy, we assume that being metaphysically necessary (a property of propositions or sentences) is not to be identified with being a logical truth, being an analytic truth, being a conceptual truth, or being an a priori truth.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com