Published online by Cambridge University Press: 12 March 2014
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, … ◊δh, and ◊η1, …, ◊ηm, all have the value F in Row (i) (h ≥ 0, m ≥ 0, h+m ≥ 1), where β → (δ1 ∨ … ∨ δh ∨ ◊η1 ∨ … ∨ ◊ηm) 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.
1 Improved decision procedures for Lewis's calculus S4 and von Wright's calculus M, this Journal, vol. 19 (1954), pp. 201–214Google Scholar.
2 I am indebted to Dr. Herman Rubin for pointing out tha t the proof as originally-given is incorrect. The statement of condition II given here is the same as tha t contained in the first draft of IDP, and in the abstract of IDP (this Journal, vol. 18 (1953), pp. 187–188). In the course of revising and condensing the first draft, the statement of condition II was inadvertently assimilated to the statement of condition II of 4.5 of IDP. The latter formulation, for von Wright's system M, still appears to be correct.
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