Download Logic, deontic. The study of principles of reasoning pertaining to

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) 6 ~A & ~B) and reflexivity (~A 6 A). Of these, reflexivity is obviously false under the
deontic interpretation. Replacing it by the weaker formula ~A 6 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 6 B implies the provability of
~A 6 ~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 6 ~~A,
though it also risks eliminating formulae like ~(~A 6 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 ¬(~Av~¬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).
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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 ºukasiewicz 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.
ºukasiewicz'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 ºukasiewicz 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 (6) 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
))3))
1 * 0
½ * ½
0 * 1
A * B * A6 B
))3)))3))))
1 * 1 * 1
1 * ½ * ½
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 6 ¬P) 6 ¬P is a logical truth
by these tables; if (as ºukasiewicz intended) only 1 is designated then it is not. With ºukasiewicz's
understanding that P w Q abbreviates (P 6 Q) 6 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 ºukasiewicz system has been generalized, axiomatized, reinterpreted, modified and
otherwise studied.
ºukasiewicz himself considered generalizations permitting more than one intermediate truth
value. ¬A gets truth value 1 minus the truth value of A; A w 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 ºukasiewicz'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 ºukasiewicz'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).
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