By Edward N. Zalta
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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. Proof : By (45), given that ⊥ and ϕ & ¬ϕ are tautological consequences of each other.
This may come as a surprise. Moreover, it is a consequence that if ¦ϕ is taken as a primitive formula of the language rather than deﬁned as ¬£¬ϕ, then in the deﬁnition of a normal logic we have to stipulate that normal logics, in addition to containing K and being closed under RN, contain all the instances of the valid schema ¦ϕ ↔ ¬£¬ϕ, for otherwise we could not preserve the interdeﬁnability of the £ and ¦ in normal systems. Exercise 1 : Prove that if Σ is a normal logic, then Σ contains the following theorems and is closed under the following rules:17 ¦¬ϕ ↔ ¬£ϕ ϕ → ψ/£ϕ → £ψ (‘RM’) ϕ ↔ ψ/£ϕ ↔ £ψ (‘RE’) ϕ → (ψ → χ)/£ϕ → (£ψ → £χ) (‘RR’) £(ϕ & ψ) → (£ϕ & £ψ) (‘M’) (£ϕ & £ψ) → £(ϕ & ψ) (‘C’) £(ϕ & ψ) ↔ (£ϕ & £ψ) (‘R’) (ϕ1 & .
So by conditional proof, if M1 M1 |=w1 £ϕ, then |=w1 ϕ, which means, by the Remark in (7), that 1 |=M w1 £ϕ → ϕ. Now assume (ii). Then, by antecedent failure, M1 1 |=w1 £ϕ → ϕ. So, in either case, we have |=M w1 £ϕ → ϕ. 1 Proof of (b): By (a) and the Fact , we know |=M w2 £ϕ → ϕ. So since M1 £ϕ → ϕ is true in both w1 and w2 , we have |= £ϕ → ϕ. Since it is clear that RM1 is not reﬂexive, we have established that every instance of the T schema is true in M1 , but RM1 is not reﬂexive. This counterexample shows that the ‘converse’ of the present theorem is false.