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By Edward N. Zalta
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Frames constitute the purely structural component of models. We say that the model M is based on the frame F iff both WM = WF and RM = RF . We may now define another sense of validity that is relative to a frame: ϕ is valid on the frame F (in symbols: F |= ϕ) iff for every model M based on F, |=M ϕ. We say that a schema is valid on frame F iff every instance of the schema is valid in every model based on F. , if |= ϕ, then F |= ϕ, for any F). But consider now the following claim, which looks similar to the ‘converse’ of the theorem in (23), and which does hold for the Si and Pi in (23): 25) Theorem: For any frame F, if F |=Si , then RF satisfies Pi .
As such, they contain all the tautologies, and contain the tautological consequences of any combination of formulas they contain. We turn next to a class of modal logics defined so as to capture the most basic, modally correct, forms of reasoning. These logics contain not only the tautologies and propositionally correct consequences of formulas they contain, but also both all the other valid formulas and all of the modally correct consequences of any combination of formulas they contain. These are the normal modal logics, the weakest of which is the logic K.
Prove that {ϕ | Γ Σ ϕ} is a modal logic, that {ϕ | Γ Σ ϕ} contains Σ ∪ Γ, and that if modal logic Σ contains Σ ∪ Γ, then {ϕ | Γ Σ ϕ} ⊆ Σ ). §4: Consistent and Maximal-Consistent Sets of Formulas Some readers may have already encountered the idea that a set of formulas Γ is consistent (relative to logic Σ) just in case there is no formula ϕ such that both ϕ and ¬ϕ are deducible from Γ (in Σ). But the following theorem shows this to be equivalent to saying that a set Γ is consistent (relative to Σ) just in case the falsum is not derivable from Γ (in Σ) 50) Theorem: Γ Σ⊥ iff there is a formula ϕ such that Γ Σϕ & ¬ϕ.