Published online by Cambridge University Press: 12 March 2014
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.
An earlier draft of this paper was presented as a retiring presidential address to the Association for Symbolic Logic; read before the Association at Atlantic City, New Jersey, December 27, 1961.
2 For example see A. R. Anderson, The formal analysis of normative systems. Technical Report No. 2, Contract No. SAR/Nonr-609(16), Office of Naval Research, Group Psychology Branch, 1956; also, by the same author, A reduction of deontic logic to alethic modal logic, Mind, n.s. vol. 67 (1958), pp. 100–103.
3 Such quantifiers can be introduced by methods analogous to those used in Barcan (Marcus), R. C., A functional calculus of first order based on strict implication, this Journal, vol. 11 (1946), pp. 1–16Google Scholar; The deduction theorem in a functional calculus of first order based on strict implication, ibid., pp. 115–118; and Fitch, F. B., Intuitionistic modal logic with quantifiers, Portugaliaq mathematica, vol. 7 (1948), pp. 113–118Google Scholar. See also, Carnap, R., Modalities and quantification, this Journal, vol. 11 (1946), pp. 33–64.Google Scholar
4 Burks, A. W., The logic of causal propositions, Mind, n.s. vol. 60 (1951), pp. 363–382.CrossRefGoogle Scholar
5 This theorem is essentially due to an anonymous referee of an earlier paper, in 1945, that I did not publish. This earlier paper contained some of the ideas of the present paper.
6 This result in slightly different form is to be found in the two papers by Anderson cited above. He uses it in constructing a model of deontic logic in alethic modal logic and attributes it to Parry, W. T., Modalities in the survey system of strict implication, this Journal, vol. 4 (1939), pp. 137–154Google Scholar, Theorem 22.8.
7 It is interesting to observe that 12–19 may be used to serve as postulates for an algebra like Boolean algebra but somewhat weaker, provided that the identity symbol is regarded as a symbol for equality in such an algebra and that (in place of II) there are added postulates to the effect that equality is symmetrical and transitive, and that the negates, conjuncts, and disjuncts of equal elements of the algebra are equal.
Also, there should be a postulate to the effect that there are at least two unequal elements of the algebra. Such an algebra provides an algebraic formulation for the Anderson-Belnap system of first degree entailments with quantifiers omitted (A. R. Anderson and N. D. Belnap, Jr., First degree entailments, Technical Report No. 10, ibid., 1961 , since the assertion that p entails q can be defined as the assertion that p equals the conjunction of p with q, or equivalently as the assertion that q equals the disjunction of q with p. This algebra was suggested to me by a list of theorems on page 21 of my paper, A system of combinatory logic, Technical Report No. 9, ibid., 1960, and in part also by some discussions with Anderson. It also bears a close relation to the system of my paper. The system CA of combinatory logic, Technical Report No. 13, ibid., 1962 (also forthcoming in this Journal). The system of first degree entailment including quantifiers was also arrived at independently by Miss Patricia A. James and myself as a modified form of the system of my book Symbolic logic (New York, 1952) prior to the Anderson-Belnap formulation of that system. This alternative approach to the system of first degree entailment is sketched on p. vii of Miss James's, doctoral dissertation. Decidability in the logic of subordinate proofs (Yale University, 1962)Google Scholar.
