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An index combines a possible world with other factors relevant to the truth value of a sentence.
More specifically, what is considered is the truth value of a sentence (or proposition, or the like).
This makes them poor candidates for accounting for the cognitive value of a sentence, as they lead to violations of Accept.
Modal logic, therefore, is intensional: in general, the truth value of a sentence is determined by something over and above its form and the extensions of its components.
In discussing Tarski's theory of truth in section 2, we focused on how these determine the truth value of a sentence.
According to Frege, the semantic value of a sentence is just its truth-value, and the semantic value of a proper name is its referent (that is, the object to which it refers).
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Although the truth-value of a sentence depends on the referents of terms, the cognitive value depends on the senses attached to the terms.
Bolzano's most important innovation was the method of variation, which involves considering what happens to the truth-value of a sentence when a constituent term is substituted by another.
Characteristically, his theory of supposition provides an account not only of the truth-values of a sentence, but also of its meaning; it is not therefore simply a theory of reference, but a sort of complex analysis of language viewed as a semiotic system whose unique interpretative model was the reality itself.
Treating the language this way, Kripke shows up to build up a very plausible extension and anti-extension for Tr, typically written E and A. The important property of the new extended model ⟨M0,⟨E,A⟩⟩ is that the truth value of any sentence A and Tr ⌜A⌝) are exactly the same.
A semantics for a language provides, in particular, an account of how the truth value of a given sentence of the language is determined in a model by the meanings of its semantically significant component parts, notably, the meanings of its names, predicates, and quantifiers.
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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