The Formal Analysis of Normative Concepts by Alan Ross Anderson; Omar Khayyam Moore;A Reduction of Deontic Logic to Alethic Modal Logic by Alan Ross Anderson; The Logic ofNorms by Alan Ross AndersonReview by: A. N. PriorThe Journal of Symbolic Logic, Vol. 24, No. 2 (Jun., 1959), pp. 177-178Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2964772 .

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REVIEWS 177

ALAN Ross ANDERSON and OMAR KHAYYAM MOORE. The formal analysis of normative concepts. American sociological review, vol. 22 (1957), pp. 9-17.

ALAN Ross ANDERSON. A reduction of deontic logic to alethic modal logic. Mind, n.s. vol. 67 (1958), pp. 100-103.

ALAN Ross ANDERSON. The logic of norms. Logique et analyse (Louvain), n.s. vol. 1 (1958), pp. 84-91.

I shall refer to these papers as FANC, RDL, and LN respectively, and shall use throughout the notation of RDL.

All three concern a version of deontic logic in which the distinctive deontic operators O (It is obligatory that), P (It is permissible that), etc. are applied to propositional variables (rather than to "act-variables" as in von Wright's XVII 140 and XVIII 174) and defined in terms of ordinary modal operators supplemented by a propositional constant S representing a "sanction"; e.g. in LN, Op is defined as LCNpS ("the omission of p necessitates the sanction").

FANC commends the study of deontic logic to sociologists, particularly as making more precise and more readily applicable to complex cases the notion of the "con- sistency" of the set of norms adopted in a given society. The system of von Wright's XVII 140 is sketched; then one of the above type with S (presented as developing the central idea of Bohnert's XI 98), but using not the simple Op, etc., but forms O'p etc., which incorporate the contingency of the argument p as a conjunctive part of their meaning.

Special postulates for S are not given in FANC, but in LN it is pointed out that all commonly accepted deontic laws are deducible, given the definitions, from the one axiom MNS ("the sanction is avoidable"), superimposed on any ordinary system of modal logic. In RDL it is shown that even in quite a weak modal system (material detachment, intersubstitutability of material equivalents, p.c., and modal axioms CpMp, EMApqAMpMq, NMKPNP) this special axiom is dispensable if we replace the unanalysed S by KMNeA9, where E6 is a propositional constant with no associated axioms whatever. For even in so weak a modal system as this, MNKMNpp is a theo- rem. It is shown that this system meets some quite precise requirements for a "normal deontic logic," namely provability of APpPNP and EPApqAPpPq and non-prov- ability of CpPp, CPpP, and CMpPp.

But what can Ed be here? Anderson suggests that it too can be a "sanction," i.e., some "bad" state-of-affairs which a forbidden act or omission necessarily implies. Thus he reads the equivalence of Op with LCNpKMN9AP as an equating of "p is obligatory" with "the failure of p leads to a state-of-affairs SD which is 'bad' but avoidable." But then the avoidability of the "bad" state-of-affairs is no longer asserted in the system; for MN9A is not asserted (and Anderson's translation, just given, of LCNpKMNjAY is rather misleading - it could be, for all that the formula says, that MN9/ is not the case but is merely something, like be itself, that Np if true would "lead to"). Note also that if we wished to assert not only the avoidability but the possibility of the sanction S we could not replace the fuller axiom KMSMNS by a definition, for there is no formula a of any ordinary modal system such that KMoxMNox is a theorem (for all such systems remain consistent if Mp is replaced throughout asserted formulas by the plain p). The procedure of RDL also precludes (because it uses N to define S) the other economy, suggested in correspondence by A. Bausch, of starting from a modalised implicational calculus without negation, introducing S with the axiom CLSp, and defining Np as CpLS.

LN is primarily a philosophic defence of deontic systems based on S, it being insisted that the definition of Op as LCNpS does not commit one to, say, "teleological" as opposed to "deontological" views of ethics. Apparently counter-intuitive details are discussed, e.g. the assertion of CPpPApq although we would not ordinarily infer

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178 REVIEWS

"It is permitted either to smoke or to embezzle" from "It is permitted to smoke." Anderson says that in common speech "or" in such contexts means "and," so that the above true principle is confused with the false CPpPKpq. This seems over-simpli- fied; what PApq is commonly confused with is not PKpq but KPPPq, and it remains puzzling why it should be. Nor does the introduction of quantifiers, which. Anderson sees will remove some difficulties, seem to help here. "All ways of doing a or fi" is indeed equivalent to "All ways of doing a and all of doing fi"; but a (say my feeding this man) is usually considered permissible even if some ways of doing a (e.g. feeding him with poison) are not, so long as some are; and smoking is one way of smoking-or- embezzling. A. N. PRIOR

K. JAAKKO J. HINTIKKA. Quantifiers in deontic logic. Societas Scientiarum Fennica, Commentationes humanarum'litterarum, vol. 23 no. 4, Helsingfors 1957, 23 pp.

The author's aim is to show that standard formulations of deontic logic (e.g. in von Wright XVIII 174 and Prior's Formal logic) are badly deficient and that some remedy is to be found by introducing quantifiers over individual acts in addition to deontic pred- icates. Let a, b, c, . .. 'be free variables ranging over acts, x, y, z, . . . corresponding bound variables, A, B, C, ... free variables ranging over properties of acts (e.g. being an act of promising), 0, F, P the standard deontic proposition-forming operators on propositions, and /, g, . . . arbitrary formulas; formation-rules may be given as usual. That acts of type A are forbidden may now be expressed by (x) 0 -A (x). It is important to distinguish two types of obligation: (i) the obligation always to do A, expressed by (x) 0 A (x), for example the obligation to abstain from stealing; (ii) the obligation to do a certain act of type A, expressed by 0 (3x) A (x), for example the obligation to pay one's tax this year. That acts of type A are permitted is best expressed by (x) P A (x), rather than by (3x) P A (x) (corresponding to type (i) obli- gation) or by P (x) A (x) (corresponding to type (ii) obligation). Thus Hintikka's quantified symbolism allows us to record something of the complexity of deontic notions which is hidden in the propositional analyses. In addition, certain grave difficulties in the interpretation of Prior's symbolism are overcome by Hintikka's approach, as he shows. The author goes on to set up a deontic system with quantifiers, not axiomatically, but by giving rules for testing the satisfiability and validity of a set of deontic formulas (for the approach, cf. his XXII 361(2)). Roughly, we construct imagined states of affairs, in which what is permitted is actually done and where all our obligations still hold and are thought of as fulfilled, and test the description of these states for consistency; the idea here is that what is permitted is that which "can take place without violating any existing obligations or any obligations which arise through its taking place" (p. 12). There is a difficulty, however, about the 'can': for Hintikka it seems to be the 'can' of logical possibility; but surely this is too weak - murder can logically be performed without violating obligations (if our moral code is different) but is not in fact permitted. Nevertheless, Hintikka goes on in the last section to apply his technique revealingly to difficult deontic formulas. (a) He considers (1) (O-A (a) & (A (a) D 0 B(a))) D 0 B(a), accepted as valid by Prior, page 225, and shows its invalidity: we escape the obligation to do B simply by not fulfilling our obligation to do A; this is immoral, but not inconsistent, on our part. If 0 is prefixed to (1), however, a valid thesis results. (Hintikka also shows that (1) leads to the undesirable 0 A (a) D A (a).) (b) The logical form of commitment, according to Hintikka, is (x)(f D 0 (3y)g): e.g. all acts of promising imply an obligation to perform an act in fulfilment. This form must carefully be distinguished from 0 (x) (f D (3y)g). (c) The important formulas (2) P(3x) A (x) D (3x) P A (x) and (3) (3x) 0 A (x) D

0 (3x) A (x) are rejected by Hintikka on intuitive grounds, and the rejection of the latter is seen to involve a special restriction on the formal rules.

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