This paper was suggested by some work of Alan Ross Anderson on deontic logic. Anderson proposed in [1] that the various deontic modalities (‘it is obligatory that’, ‘it is forbidden that’, ‘it is permitted that’, etc.) should be defined in terms of the ordinary alethic modalities (‘it is necessary that’, ‘it is possible that’, etc.). The definition I shall consider is not in fact Anderson's own, though it is closely related to it. His definition, as set out in [1], involves the idea of an (unspecified) penalty or sanction. A forbidden action is defined as one whose commission entails the application of the sanction; an obligatory action is one whose omission entails the application of the sanction.

The purpose of this paper is to provide a partial logical analysis of a few concepts that may be classified as value concepts or as concepts that are closely related to value concepts. Among the concepts that will be considered are striving for, doing, believing, knowing, desiring, ability to do, obligation to do, and value for. Familiarity will be assumed with the concepts of logical necessity, logical possibility, and strict implication as formalized in standard systems of modal logic (such as S4), and with the concepts of obligation and permission as formalized in systems of deontic logic. It will also be assumed that quantifiers over propositions have been included in extensions of these systems.

Two questions were left open in [1]. 1) Do W(σ) and W(r) have the same many-one degree for any σ, τ such that ωη+1.2 ≤ σ < τ < ωη+2? 2) Is W(σ) many-one reducible to W(r) for any σ, τ such that ωη+1.2 ≤ σ < τ < ωη+2? We answered the second question affirmatively in [2]. In this paper we answer the first question in the negative. From the results of [1] [2] and this paper we have a complete picture of the many-one degrees of the predicates e ∈ W(σ)

This addendum supplies details of the proof in 3.6 of our paper Disjunction and existence under implication in elementary intuition-istic formalisms in this Journal, vol. 27 (1962), pp. 11–18, as promised in Footnote 8 of that paper (p. 16 lines 11 and 16, for “├” read “|”).

A congruence-substitution (or free substitution with change of bound variables) on a formula E with result F consists in, simultaneously, substituting for each of the free variables of E in all its free occurrences a respective term and replacing each bound occurrence of a variable in E by a respective variable, so that (a) each image in F of a free occurrence of a variable in E is free (IM p. 410) and (b) each image in F of a bound occurrence of a variable in E is bound by the corresponding quantifier (i.e. if the j-th quantifier in E binds the original occurrence, then the j-th quantifier in F binds the image).

In a recent paper Professor Wesley Salmon1 has shown that a certain class of c-functions, including the c-function c*. described by Rudolph Carnap2 yield incompatible results when applied to the same sentences in two languages systems which, though they have the same individual constants, do not have the same predicates. Each c-function of the class in question is characterized by a parameter λ which is a function of the number of Q-predicates in the language system in which the c-function is used.3 Taking c* as representative of this class of c-functions, I shall argue that Professor Salmon's results do not provide a reasonable basis for rejecting such c-functions in favor of others. More specifically, I shall argue (i) that such c-functions yield incompatible results in two languages because not both of the languages are sufficiently complete, (ii) that for any two languages in which such c-functions yield incompatible results there is a rule that will select either the more complete of the two languages or a language that is more complete than either of the two languages, and (iii) that it is impossible for such c-functions to yield incompatible results in two languages that are equally complete.

In Smullyan [1], where recursive enumerability is defined in terms of elementary dyadic arithmetics (dyadic EA's), it is shown that for any number n, the parallel definition in terms of n-adic EA′s is equivalent. This proof at one stage uses the deep result of Godei that plus and times form a sub-basis for the recursively enumerable attributes (sets and relations)2. The aim of this note is to prove this equivalence in more pedestrian fashion.