Correction to a paper on modal logic

Extract

In a recent paper (here referred to as “IDP”) the writer outlined a decision procedure for Lewis's system S4 of modal logic. One of the clauses in definition 3.1 of IDP requires correction. Clause II of 3.1 (2) should read as follows.

II. Some constituent of the form ◊β, of degree n₁ ≤ n, has the value T in Row (i), and some constituents of the forms ◊δ₁, … ◊δ_h, and ◊η₁, …, ◊η_m, all have the value F in Row (i) (h ≥ 0, m ≥ 0, h+m ≥ 1), where β → (δ₁ ∨ … ∨ δ_h ∨ ◊η₁ ∨ … ∨ ◊η_m) is an (n₁ − 1)-tautology of S4.

This change is required in order to carry out the proof of Metathcorcm 3.19. In particular, the change guarantees the following. If the expression η of the second paragraph of 3.20 is of degree n − 1, then the antecedent λ of formula ζ on page 210 of IDP is also of degree n − 1; and consequently the formula ζ of 3.19 is of degree n (since ◊λ is a constituent of ζ). (If we fail to make the correction, then it might be the case that both ζ and (3) are of degree n − 1, in which case Row (i) would not satisfy clause II as originally stated, contrary to the claim at the end of 3.20.) The proofs for the remaining cases of 3.20 can then be carried out, using the revised clause II, in the way originally indicated.

The proof of Metatheorem 3.2 requires only trivial corrections for case 2.