Logic, deontic. The study of principles of reasoning pertaining to obligation, permission, prohibition, moral commitment and other normative matters. Although often described as a branch of logic, deontic logic lacks the "topic-neutrality" characteristic of logic proper. It is better viewed as an application of logic to ethical concepts, in much the same way as formal geometry is an application of logic to spatial concepts. Likewise, although hopes have been expressed that deontic logic might help to systematize the practical reasoning whereby we infer from general principles and observed facts what we ought to do, the most studied systems of deontic logic comprise mainly theoretical principles, expressing inferential relations among various ethical concepts. Several principles prominent in the current literature were noted by various medieval philosophers, and again by Leibniz and by Jeremy Bentham, but focused and sustained thought in the field is a twentieth century phenomenon, kindled largely by the writings of G. H. von Wright. Early work was motivated by analogies between the deontic concepts of obligation, permission and prohibition and the alethic concepts of necessity, possibility and impossibility. The first analogies to be noted concerned "interchange" principles. If and represent necessity and possibility, for example, then the formula A A says that to deny A is necessary is to assert not-a is possible. If they represent obligation and permission it says (equally plausibly) that to deny A is obligatory is to assert not-a is permitted. Similarly, A A and IA A (where I is either "impossible" or "forbidden") have equally plausible alethic and deontic readings. The development of complete formal systems of necessity led naturally to an effort to see how far the analogy can be extended. The weakest system in which can plausibly be regarded as a expressing some form of necessity is the system T, which contains, in addition to the interchange principles, principles of distribution ( (A & B) A & B) and reflexivity ( A A). Of these, reflexivity is obviously false under the deontic interpretation. Replacing it by the weaker formula A A (what is obligatory is permitted), yields what is sometimes called the standard system of deontic logic. The system T is known to be characterized by an interpretation according to which A is true at a world w exactly when A is true at all worlds that are possible relative to w, i.e., at all worlds at which all the necessary
truths of w are true. It follows that the standard system of deontic logic is characterized by an interpretation according to which A is true at w exactly when A is true in all worlds "deontically accessible" from w, i.e., all worlds in which the all obligations of w are fulfilled. Much of the contemporary work in deontic logic has been inspired by the deontic paradoxes, a collection of puzzle cases that have seemed to highlight deficiencies in the standard system. For example, according to a version of Chisholm's paradox, the following clauses should be mutually independent and jointly consistent: Dr. Jones ought to administer anaesthesia if she operates; she ought not to if she doesn't; she has an obligation to operate, which she fails to meet. But attempts to represent these sentences within the standard system yield inconsistencies or redundancies. According to a version of the good Samaritan paradox, Smith's repenting of a murder logically implies his committing the murder, but his obligation to repent does not imply his obligation to have committed it. Yet in the standard system, the provability of A B implies the provability of A B. One reaction to examples like these has been to take sentences like "Jones should administer anaesthesia if she operates" as exemplifying an irreducibly dyadic relation of conditional obligation. "A is obligatory given B" has been interpreted, for example, as saying that B is true in the "best" of the worlds in which A is. Another reaction has been to eschew the operator "It is obligatory that..." which attaches to sentences in favour of a predicate of obligation which attaches only to names of actions. This approach eliminates altogether awkward formulae like A A, though it also risks eliminating formulae like ( A A) which have been thought to express important truths. It raises interesting questions about the nature of combined actions like "a or b" and about the relations between general deontic statements ("smoking is prohibited") and their instances ("Smith's smoking here now is prohibited"). In recent years, there has been considerable discussion about the plausibility of the schema ( A A), which is provable in the standard system. The issue is whether there is a phenomenon of moral experience, ruled out by the schema, in which an agent is faced with irresolvable and tragic moral "dilemma" or "conflict." It has also been suggested that some of the shortcomings of the standard system can be remedied by a closer
attention to the ways in which obligation and permission depend on time, and that there might be fruitful connections among deontic logic, formal epistemology and logics for the verification of computer programs. Bibliography L. Åqvist, 'Deontic Logic', D. Gabbay and F. Guenthner (eds), Handbook of Philosophical Logic, Vol II (Dordrecht, 1984); C. Gowans (ed.) Moral Dilemmas, (Oxford, 1987); Hilpinen (ed.), Deontic Logic: Introductory and Systematic Readings (Dordrecht, 1971); Hilpinen (ed.), New Studies in Deontic Logic: Norms Actions and the Foundations of Ethics (Dordrecht, 1981). STK
Logic, many-valued. Logical systems in which formulas may be assigned truth values other than merely "true" and "false". The term is often used more narrowly to refer to many-valued tabular logics, in which the truth value of a formula is determined by the truth values of its subformulas. (This characteristic distinguishes many-valued logics from standard *modal logics.) The idea that logic ought to countenance more than two truth values arose naturally in ancient and medieval discussions of *determinism and was reexamined by C.S. *Peirce, Hugh MacColl, and Nikolai Vasiliev in the first decade of this century. Explicit formulation and systematic investigation of many-valued logics began with writings of Jan ukasiewicz and Emil Post in the nineteen twenties and D. Bochvar, Jerzy S upecki and Stephen Kleene in the late thirties. There has been some renewed interest in the subject recently, because of perceived connections with programming languages and artificial intelligence. ukasiewicz's work is inspired by a view of "future contingents" often attributed to Aristotle. There is a sense in which whatever happens in the present or past is now unalterable. This idea sometimes finds expression in the doctrine that sentences now true are unalterably true and those now false are unalterably false. But, although it seems that There will be a sea battle tomorrow is now either true or false, it does not seem unalterably true or unalterably false. Considerations like this led ukasiewicz to adopt the view that future contingent sentences are not either true or false, but have an intermediate truth value, "the possible". He constructed a formal language, taking the conditional ( ) and negation ( ) as primitive connectives and false (0), possible (½), and true (1) as truth values. Truth values of compound formulas are determined by the tables below. A * A A * B * A B ))3)) ))3)))3)))) 1 * 0 1 * 1 * 1 ½ * ½ 1 * ½ * ½ 0 * 1 1 * 0 * 0 ½ * 1 * 1 ½ * ½ * 1 ½ * 0 * ½ 0 * 1 * 1 0 * ½ * 1 0 * 0 * 1
To obtain a many-valued logic from a table like this, one specifies certain truth values as designated. The argument from set to formula A is logically valid if A gets a designated truth value under any assignment in which all the members of do. A is logically true if it gets a designated value under any assignment. For example, if 1 and ½ are both designated then (P P) P is a logical truth by these tables; if (as ukasiewicz intended) only 1 is designated then it is not. With ukasiewicz's understanding that P Q abbreviates (P Q) Q, the formula in question expresses the law of *excluded middle. It is doubtful that these (or any) truth tables capture precisely the kind of possibility exhibited by future contingents. Why, for example, should If there won't be a sea battle there will be one be considered true, while If 2+2=4 then there will be a sea battle is merely possible? Nevertheless, the original ukasiewicz system has been generalized, axiomatized, reinterpreted, modified and otherwise studied. ukasiewicz himself considered generalizations permitting more than one intermediate truth value. A gets truth value 1 minus the truth value of A; A B gets the greater of the truth values of A and B. Other many valued systems have been motivated by the idea that additional truth values might express the notion of a proposition's being paradoxical (its truth implying its falsity and its falsity implying its truth), of its having uncomputable truth value, of its being approximately true, and of its having failed presuppositions of various sorts. Nearly all the systems considered generalize classical logic in the sense that if truth values other than 0 and 1 are dropped, classical logic is obtained. Post formulated a technically advantageous system in which ukasiewicz's negation is replaced by a "cyclic" negation--the truth values are 0,1,..,m and the truth value of A is 0 if the truth value of A is m and it is 1+the truth value of A otherwise. Post's negation and disjunction are truth functionally complete: any connective in a finite-valued logic (including the conditional and negation of ukasiewicz's three valued logic discussed above) can be defined from them. This result has practical significance, for just as the formulas of classical propositional logic correspond to logic
circuits, the formulas of m-valued logics correspond to switching circuits in which inputs and outputs can assume m states. More recent investigations have examined the model theory and proof theory of general many-valued logic and of continuous logic, in which the truth values are assumed to have a topological structure. It is enlightening to see certain results of classical logic proved in a more general setting. The works cited below are a small sample of the large and varied literature on the subject, but they do contain references to much of the rest. Bibliography N. Rescher, Many-valued Logic, ( New York, 1969); A. Urquhart, 'Many-valued Logic', D. Gabbay and F. Guenthner (eds), Handbook of Philosophical Logic, Vol III, ( Dordrecht, 1986); R Wójcicki, Theory of Logical Calculi: Basic Theory of Consequence Operations, (Dordrecht 1988); R. Wolf, 'A Survey of Many-valued Logic (1966-1974)', in Dunn and Epstein (eds.), Modern Uses of multiple-valued logic' (Dordrecht, 1975). STK