Logic

Basic Concepts In Modal Logic by Edward N. Zalta

By Edward N. Zalta

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9) The schema D is not valid with respect to the class of symmetric transitive models (frames). Remark 1 : There is a another important reason for proving these facts besides that of developing our intuitions. , in establishing that there are theorems of logic Σ that are not theorems of logic Σ . We shall not spend time in the present work investigating such questions about the independence of logics, but simply prepare the reader for such a study, indicating in general how these facts play a role.

N . , is |= (ϕ1 & . . & ϕn ) → ψ, whenever ψ is a tautological consequence of ϕ1 , . . , ϕn ? 33) Consider the following relationship between ϕ and Theorem: If |= ϕ, then |= £ϕ £ϕ: Remark : This relationship grounds the Rule of Necessitation (RN). It proves to be important to the definition of ‘normal’ modal logics. RN allows us to suppose that £ϕ is a theorem of a normal modal logic whenever ϕ is a theorem of the logic. The present (meta-)theorem tells us that RN preserves validity. Exercise 1 : Show that this rule preserves truth in a model but not truth at a world in a model.

Remember here that the formula ϕ1 → . . → ϕn → ϕ is shorthand for the formula ϕ1 → (. . (ϕn → ϕ) . ). Note also that this latter formula is tautologically equivalent to the defined notation: (ϕ1 & . . & ϕn ) → ϕ. The biconditional having these two formulas as the conditions is therefore a tautology, and so an element of every logic Σ. Thus MP guarantees that the latter is in Σ iff the former is. So it is an immediate consequence of our definitions that Γ Σ ϕ iff ∃ϕ1 , . . , ϕn ∈ Γ (n ≥ 0) such that Σ (ϕ1 & .

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