Deontic Logic Arquivo 61

7/28/2019 Deontic Logic Arquivo 61

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Deontic Logic

G. H. von Wright

Mind, New Series, Vol. 60, No. 237. (Jan., 1951), pp. 1-15.

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[January, 1951

M I N D A Q U A R T E R L Y R E V I E W

PSYCHOLOGY A N D PHILOSOPHYI. DEONTIC LOGIC

1. So called modal concepts might conveniently be dividedinto three or four main groups. There are the alethic modes ormodes of truth. These are concepts such as the necessary (thenecessarily true), the possible (the possibly true), and thecontingent (the contingently true). There are the epistemicmodes or modes of knowing. These are concepts such as theverified (that which is known to be true), the undecided, and thefalsified (that which is known to be false). There are the deonticmodes or modes of obligation. These are concepts such as theobligatory (that which we ought to do), the permitted (that whichwe are allowed to do), and the forbidden (that which we must notdo). As a fourth main group of modal categories one mightadd the existential modes or modes of existence. These areconcepts such as universality, existence, and emptiness (ofproperties or classes).There are essential similarities but also characteristic differencesbetween the various groups of modalities. They all deserve,therefore, a special treatment. The treatment of the existentialAmodes is usually known as quantification theory. The treatmentof the alethic modes covers most of what is traditionallyknown as modal logic. The epistemic modes have not to anygreat extent and the deontic modes hardlv at all been treateduby logicians.In the present paper an elementary formal logic of the deonticmodalities will be outlined.

For the term " deontic " I am indebted to Professor C. D. Broad.1


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2 G. H. VON WRIGHT :2. First a preliminary question must be settled. What are the

" things "which are pronounced obligatory, permitted, forbidden,etc. ?We shall call these " things " acts.The word " act ", however, is used ambiguously in ordinarylanguage. It is sometimes used for what might be called act-qualifying properties, e.g. theft. But it is also used for theindividual cases which fall under these properties, e.g. the in-dividual thefts.The use of the word for individual cases is perhaps moreappropriate than its use for properties. For the sake of verbalconvenience, however, we shall in this paper use "act " forproperties and not for individuals. We shall say that theft,murder, smoking, etc. are acts. The individual cases that fallunder theft, murder, smoking, etc. we shall call act-individuals.I t is of acts and not of act-individuals that deontic words arepredicated.The performance or non-performance of a certain act (by anagent) we shall call performance-values (for that agent). Anact will be called a performance-function of certain other acts. ifits performance-value for any given agent uniquely depends uponthe performance-values of those other acts for the same agent.The concept of a performance-function is strictly analogousto the concept of a truth-function in propositional logic.Particular performance-functions can be defined in strictcorrespondence to the particular truth-functions.Thus by the negation (-act) of a given act we understandthat act which is performed by an agent, if and only if he doesnot perform the given act. Fcr example : the negation of theact of repaying a loan is the act of not repaying it. If A denotes(is the name of) an act, -A will be used as a name of its negat'ion(-act).Similarly, m7e can define the conjunction-, disjunction-,implication-, and equivalence-act of two given acts. (Theimplication-act, e.g., of two given acts is the act which is per-formed by an agent, if and only if it is not the case that thefirst act is performed and the second act is not performed bythe agent in question.) If A and B denote acts, A & B will beused as a name of their conjunction, A v B as a name of theirdisjunction, A +B as a name of their implication, and A++B asa name of their equivalence.Finally, we can defing the tautology- and contradiction (-act)of n given acts. The first is the act which is ~erformed nd thesecond the act which is not performed by an agent, whatever be


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3EONTIC LOGICthe performance-values of the n given acts for the agent inquestion.We shall call -A the negation-name of A, and A & B theconjunction-, A v B the disjunction-, A +B the implication-, andA t+ B the equivalence-name of A and B.A name of an act which is neither the negation-name of anothername of an act, nor the conjunction-, disjunction-, implication-, orequivalence-name of two other names of acts we shall call anatomic name.By a molecular complex of n names of acts m7e understand :(i) Any one of the n names themselves and any one of theirnegation-names.(ii) The conjunction-, disjunction-, implication-, and equiv-alence-name of any two of the n names.(iii) The negation-name of any molecular complex of the nnames, and the conjunction-, disjunction-, implication-, andequivalence-name of any tm7o molecular complexes of the nnames.The n names are called constituents of their molecular com-plexes. If they are atomic names, they are called atomicconstituents.As to the use of brackets we adopt the conventioi that thesymbol & has a stronger combining force than v, +, and++ ;the symbol v than + and t - t ; and the symbol + than ++.Thus, e.g., we write for (((A& B) vC) -t D)t+ E simply

A & R V C + D t + E.The symbols -, &, V, ->, and t-+will be used for truth-functions as well as for performance-functions. This ambiguitydoes not easily lead to confusion and is, therefore, to be preferredto the introduction of two special sets of symbols.3. As an undefined deontic category we introduce the conceptof permission. It is the only undefined deontic category whichm7e need.If an act is not permitted, it is called forbidden. For instance :Theft is not permitted, hence it is forbidden. We are not allowed

t o steal, hence m7emust no t stea1.lIf the negation of an act is forbidden, the act itself is calledobligatory. For instance : it is forbidden to disobey the law,hence it is obligatory to obey the law. We ozqht to do that whichwe are not allowed not to do.If an act and its negation are both permitted, the act is called

I t need hardly be stressed that the question of validity of variousdeontic propositions (other than those which are true on formal grounds)does not concern us in this paper.


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4 G . H. VON WRIGHT :(morally) indifferent. For ihstance : in a smoking compartmentwe may smoke, but we may also not smoke. Hence smoking ishere a morally indifferent form of behaviour.It should be observed that indifference is thus a narrowercategory than permission. Everything indifferent is permitted,but everything permitted is not indifferent. For, what isobligatory is also permitted, but not indifferent.(The difference between the permitted and the indifferentamong the deontic modes is analogous to the difference betweenthe possible and the contingent among the alethic modes.)The above deontic concepts apply to a single act (or perfor-mance-function of acts). There are also deontic concepts which

apply to pairs of acts.Two acts are morally incompatible, if their conjunction isforbidden' (and compatible if it is permitted). For instance :giving a promise and not keeping it are (morally) incompatibleacts.Doing one act commits us to do another act, if the implicationof the two acts is obligatory. For instance : giving a promisecommits us to keep it.The proposition that the act named by A is permitted willbe expressed in symbols by PA.The proposition that the act named by A is forbidden, is thenegation of the propdsition that it is permitted. I t can thus besymbolized by -(P A).The proposition that the act named by A is obligatory, is thenegation of the proposition that the negation of the act ispermitted. It can thus be symbolized by - (P-A). We shallalso use the shorter expression 0 A.The proposition that the act named by A is (morally) indifferentcan be symbolized by (PA) & (P-A).The proposition that the acts named by A and by B are(morally) incompatible can be symbolized by -(P A & B).The proposition that the performance of the act named by Acommits us to perform the act named by B can be symbolizedby 0 A -t B. But 0 A -t B means the same as - ( P- (A +-B)),and this means the same as - (PA &-B). Commitmentcan thus be explained in terms of compatibility.P and 0 are called the'deontic operators. Sentences of thetype " P ", where a name of an act (or a molecular complexof names of acts) has to be inserted in the blank, we shall callP-sentences. Similarly, we shall call sentences of the type"0 " 0-sentences.As to the use of brackets it should be remarked that P- and


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5EOXTIC LOGIC0-s en ten ces as co nsti tue nts of m olecular com plexes of se ntencesshould be enclosed within brackets in order to avoid confusion.I t should further be observed that a deontic operator before amolecular complex of na m es of ac ts refers to t h e whole com plexa n d not to i ts f irst consti tuenb only. Thu s, e .g., P A v B meanstha t the act named b y A v B is permitted.Th e system of Deon tic Logic, wl-hich we a re o utlinin g i n th ispaper, studies propositions (an d truth-fu nction s of propositions)about the obligatory, permitted, forbidden, and other (deri-vative) deontic chara cters of ac ts (and performance-functions ofacts) .w e shall cal l th e proposit ions which a re th e object of s tu dydeontic propositions. Th e sentences, in which th ey ar e expressedin our sy stem, are P - a nd 0-sentences or m olecular complexes ofsuch sentences.4. A tas k of particu lar impo rtance which Deon tic Logic set sitself is to develop a technique for deciding, whether the pro-positions it studies ar e logically tr u e or no t. (T he decisioilr~rob lem.Som etimes molecular complexes of P - an d 0-sentences ex presstr u th s of logic for reasons which hav e nothing to do with t h especific ch ara cter of deontic conc epts. F o r ins tan ce : If A i sperm itted, if B is perm itted, t he n B is forbidden, if A is forbidden.I n s y m b o l s : ((PB ) + (PA ) ) -t ( N (PA ) +N (PB)). Thisis a t r u t h of logic. I t is a n application of a va rian t of t h e socalled modus tollens which is valid for anv sentences. whetherdeontic or not . I t is, therefore , a t r iv ia l t r u th f rom th e pointof view of our Deontic Logic.Sometimes, hox~~ever,molecular complexes of P - an d 0-sentences express tr u th s of logic for reasons which depen d u po nt h e specific (logical) ch ara cte r of deontic conc epts. F o r inst an ce :If A is obligatory a n d if doing A commits us to do B, the n B isobligatory too. In symbo l s : (0A ) & (0A -t B) -t (0B ) . I tis intuit ivelv obvious th a t th is is a tr u th of logic. i .e. som eth im" , "which is valid on purely formal grounds. It i s , hox~~ever ,o ta n application of a n y scheme which is valid for any sentences,whether deon tic or not. Th e existence of logical tr u th s whichare peculiar to deontic concepts is what makes the study ofDeontic Logic interesting.If a mo lecular com plex of P - an d 0-senten ces expresses logicaIt r u t h for reasons which ar e inde pen den t of t h e specific na tur e ofdeontic concepts, the n i ts t ru th can be established or proved in atruth-table of propositional logic.I f , however, a molecular complex of P - an d 0-sentences


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6 G. H. VON WRIGHTexpresses logical truth for reasons which depend on the specificnature of deontic concepts, then its truth cannot be establishedby the means of propositional logic alone. The question there-fore arises : What is the necessary and sufficient criterionwhich a molecular complex of P- and/or O-sentences must satisfyin order to express a logically true proposition ?5. Let us call " permitted " and " forbidden " the two deonticvalues.An act will be called a deontic function of certain other acts,if the deontic value of the former uniquely depends upon thedeontic values of the latter.I t is easy to see that not any act which is a performance-function of certain other acts is also a deontic function of them.(Otherwise the logic of deontic concepts would be trivial.)Consider first the negation of a given act. From the fact that

A is performed, we can conclude to the fact that -A is not per-formed. But from the fact that A is permitted, we can concludenothing as to the permitted or forbidden character of - A .Sometimes - A is permitted. sometimes not. If A is what wehave called indifferknt, then' - A is also permitted, but if Ahappened to be obligatory as well as permitted, then - A wouldbe forbidden. In the smoking compartment, e.g., not-smokingis permitted and also smoking. But in the non-smoking com-partment, not-smoking is permitted and smoking forbidden.Consider next the conjunction of two acts. From the factthat A and B are both performed, it follows that A & B is per-' formed. But from the fact that A and B are both permitted,it does not follow that A & B is permitted. Sometimes A &Bis permitted, sometimes not. For, A and B may both bepermitted, but doing either of them may commit us not to do theother. I may be free to promise and also not to promise to givea certain thing to a person, and free to give and also not to givethis thing to him, but forbidden to promise to give and yet notgive it.Consider, finallv, the disiunction of two acts. From the factthat a t least one if the twi act s A and B is performed, it followsthat A v B is performed, and from the fact that none of the twoacts A and B is performed, it follows that A v B is not performed.Similarlv. from the fact that at least one of the acts is ~ermitted.i t follo;~' that their disjunction is permitted, and fro& the factthat both acts are forbidden, it follows that their disjunction isforbidden. In other words: the disjunction of two acts ispermitted, if and only if at least one of the acts is permitted.Speaking loud or smoking is permitted in the reading-room, if


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DEONTIC LOGIC 7a nd only if speaking loud is perm itted or smoking is perm itted.lThus deontic functions are similar to performance-functions(and truth-functions) in regard to disjunction, but not similarin regard to negation and conjunction. The similarity can belaid down as a Principle of Deontic Distribution :

If an act is the diyhnction of two other acts, then the propositionthat the disjunction is permitted i s the disjunct ion of the propositionthat the$rst act is permitted and the proposition that the second acti s permitted.(This principle can, naturally, be extended to disjunctionswith an y number n of members.)I n v irtu e of familiar principles of formal logic, any molecularcom plex of n nam es of ac ts has what we propose to call a perfectdisjunc tive normal form. This is a 0-, I-, or more-than-l-term ed disjunction-name of n-termed conjunction-nam es. Ea chof the n original name s or it s negation-name occurs in every oneof the conjunction-names.I n virtue of the above Principle of Deontic Distribution, anymolecular complex of n nam es of ac ts denotes a deontic function ofthe acts denoted by the conjunction-names in its perfect dis-junctive normal form.Consider now a P-sentence P c, where c stand s for (an atomicna m e of a n ac t or) a molecular com plex of names of ac ts. Let c,,. . . c , stand for the conjunction-names in the perfect dis-junctive normal form of c. The sentences P c,, . . . , P c,we shall call th e P -constitue nts of P c.Since, in virtue of t he Principle of D eontic Distribution, cdenotes a deontic function of t h e acts named b y c,, . . . , C k ,it follows th at P c expresses a truth -function of th e propositionsexpressed by P c,, . . . , P c,. Generally speaking : a P-sen-tence expresses a tru th-fu nc tion of th e propositions expressedby its P-constituents.

Consider n nam es of ac ts A,, . . . , A,. There are in all 2"conjunction-names which can be formed by selecting rn (05 m5 n ) out of the n names an d takin g th e negation-names of th e

The meaning of" or " in ordinary language is not quite settled. When 'we say that we are permitted to do A or B, we sometimes mean, by im-plication, that we are allowed to do both. Sometimes, however, we meanthat we are allowed t o do one and one only of the two acts. Whichmeaning the " or " conveys by implication depends upon the materialnature of th e individual case, in which i t is used. It ought to be stressedthat our use of " or " in thi s paper is neutral with regard to such material,differences in the individual situations. That we are permitted t o doA or B means here tha t we are permitted to do a t least one of the two acts,an d neither excludes nor includes, by implication, the permission to do both.


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8 G. H. VON WRIGHT :remaining n-rnnames. (The order of the namcs in a conjunction-name is irrelevant.) By the deontic units in the deontic realmof the acts named by A,, . . . ,A, we shall understand thepropositions that the respective acts named by those 2n con-junction-names are permitted. By the deontic realm itselfwe shall understand the disjunction of all the deontic units.Thus, e.g., the deontic units of the deontic realm of the soleact named by A are the propositions expressed by P A andP- A.The deontic realm itself is the proposition expressed by( P A ) v (P -A). The deontic units of the deontic realm of theacts named by A and B are the propositions expressed byPA&BandPA&-BandP-A&BandP-A&-B . Etc.

The deontic units of the deontic realm of given acts arelogically independent of one another, meaning that they can betrue or false in anv combination of truth-values. There is.however, one point a t which this independence might be ques-tioned. Could all the deontic units be false ?Let A be the name of an act. That all (both) the deonticunits in the deontic realm of this act are false means that theact itself and its negation are both forbidden. In symbols :- (PA) &- (P-A). Since the act or its negation is per-formed by any agent whenever he acts, the falsehood of all thedeontic units means that we are forbidden to act in any waywhatsoever.Is such a prohibition illogical ? Its counterpart in the logicof the alethic modalities would be the case, when a propositionand its negation are both impossible, and its counterpart in thelogic of the epistemic modalities would be the case, when aproposition and its negation are both known to be false. Thesecases are obviously logical impossibilities. On the other hand,in the logic of the existential modalities the corresponding caseis, when a property and its negation are both empty. This isnot an impossibility, since the Universe of Discourse may haveno members. The question, therefore is, whether the deonticmodes at this point resemble the alethic and the epistemicmodes or whether they resemble the existential modes.Ordinary language and our common sense logical intuitionsseem at first not to provide us with a clear answer. A simplelogical transformation, however, will help us to make up ourminds.That the negation of an act is forbidden means tha t the actitself is obligatory. Thus we can for - (P - A ) write 0 A.That an act and its negation are both forbidden means the sameas that the act itself is both obligatory and forbidden.


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DEONTIC LOGIC 9At th is point a n appeal to o rdinary language will, I think, bedecisive. We seem prepared t o reject a use of the words,according to which one and th e sam e act could be tru ly called b oth

obliga tory and f0rbidden.l If , however, we reject this use, wemust also reject the idea that all the units in a deontic realmcould be false.Thbs, on the point at issue, the deontic modalities appear toresemble the alethic and th e epistemic modalities rather th an th eexistential ones.The restriction on th e logical independence of th e deo ntic units,which we are forced to accept, can be laid down as a Principleof Permission :Any given act i s either itself permitted or its negation i s permitted.There are alte rnativ e formulations of th e principle. We mightalso have said : If th e negation of an act is forbidden, then t h eact itself is permitted. And this again is equivalent to saying :If an act is obligatory, the n i t is permitted.6. Which truth-function of its P-cons tituen ts a P-sentenceexpresses can be investigated and decided in truth- tab les .We shall here construct a truth-table for the following P-sentences : P A and P -A and P A & B and P A v B and

P A - t B and P A c + B and P A v - A . The perfect dis-junctive norm al form of A (in term s of A an d B ) is A & B vA & - B . T heno rm alfo rm of - A i s - A & B v - A . & - B .The tab le Iooks as follows :

P A & B PA&-B P-A&B P-A&-BI P A P-A PA-BA v BA & B PAt+B PA"-A--- -------T T T T T T T T T T TT T T F T T T T T T TT T F T T T T T T T TT T F B' T P T T T T TT F T T T T T T T T TT F T F T T T T T T TT F . F T T T T T T T TT F B' F T F T T T T TF T T T T T F T T T TF T T F T T F T T F TF T F T T T F T T T TF T F F T F F T F F TF F T T B ' T B ' T T T TF F T F F T F T T F TB' F F T F . T F F T T T-For the " relativity " of deontic propositions of. below, p. 15.


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10 G. H. V O N WRIGHT:The norm al form of A & B is A & B . The normal form of A v Bis A & B v A & - B v - A & B . The normal form of A - t Bi s A & B v A & B v - A & - B . T henorm alfo rm of A t - t BA & B v - A &- B . The norm al form of A v - A is A & Bv A & - B v - A & B v - A & - B . Thus the seven P-sentences have in all four P-constituents, viz. P A & B andP A & - B a n d P - A & B a n d P - A & - B . They expressth e deontic units of t he deontic realm of th e two a cts named byA and by B.In distributing truth-values over the deontic units (or theP-co nstituen ts) we have to observe the restriction imposed by t hePrinciple of Perm ission. The subsequent calculation of tr u th -values for the seven deontic propositions (or the seven P-sentences) depends only on the Principle of Deontic Distribution.What is the truth-table for P A & -A ? The perfect dis-junctive normal form of A &-A is " empty ", i .e . a 0-termeddisjunction. Thus P A & - A too is a 0-term ed disjunction ofP-constituents. It might be argued that a disjunction is true,if an d only if a t least one of it s members is tru e, and th a t a 0 -termed disjunction, since it has no members, is never true(always false). If, however, P A & - A is always false, it snegation - ( P A & - A) is always true. Bu t - (PA & - A )means the same as 0 A v - A. Thus, on th e above criterion forth e tr u th of a 0-term ed disjunction, it follows th a t 0 A v -Ais a deontic tautology.It might, however, be questioned whether i t can be regarded asa tr u th of logic th a t a tautologous ac t is obligatory (and acontradictory act forbidden). The corresponding proposition inth e logic of t he alethic modalities is t h a t a tautologous pro-position is necessary (and a contrad ictory proposition impossible),a n d t h e corresponding proposition in t h e logic of th e existentialmodalities is that a tautologous property is universal (anda contradictory property empty). These corresponding casesar e obviously logical tru th s. On th e other hand, the corres-ponding propositions in t h e logic of th e epistemic modalitiesis th a t a tautologous proposition is verified (and a contradictoryproposition falsified). Th i s is not a logical tr u th . For, aproposition may be tautologous (contradictory) without ourknowing it . The question therefore is, whether the deonticmodes at this point resemble the alethic and the existentialmodes, or w hether the y resemble th e epistemic modes.Ordinary language and our common sense logical intuitionsseem no t to provide us with any clear answer. I t appears,moreover, that no further logical considerations can help us to


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D E O N T I C L O G I Cdecide on th e issue. I t may be thought " awkward " to permitcontra dic tory actions bu t it is difficult to conceive of a ny logicalarg um en t against th is permission. Fro m th e point of view oflogic, therefore, th e m ost plausible course seems t o be t o regardP A & N A and 0 A v N A as expressing contingen t propositionswhich can be either tru e or false.Thus, on the point at issue, the deontic modalities appear toresemble the epistemic rather tha n the alethic and th e existentialmodalities.We suggest th e following Principle of De ont ic Contingency :A tautologous act i s not necessarily obligatory, and a contradictorya t s not necessarily forbidden.7. Le t us consider a m olecular complex of P - and/o r 0-sentences. 0-sentences can be regarded as abbreviations fornegation-sentences of cer tain P-se ntences. (Cf. above, p. 8.)If th e m olecular complex happened t o contain 0-sentences, wereplace them by negation-sentences of P-sentences. Thu s wege t a new m olecular complex, al l th e constituents of which ar eP-sentences.We now turn our attention to the (molecular complexes of)nam es of ac ts which follow afte r th e modal operato rs in th is new

molecular complex of P -sentences. We make a n inclusive listof a ll atomic names which are constituen ts of a t least one of t h e(molecular complexes of) nam es of act s in question. Thereupo nwe transform these (molecular complexes of) names of ac ts in toth ei r perfect disjunctive normal forms in terms of a ll atom icna m es which occur in our list. The respective conjunction-nam es in these normal forms preceded by th e deontic operator Pwe shall call th e P-co nsti tue nts of t h e initially given molecularcom plex of P-and/o r 0-sentences. (Cf. th e example given below).We know already that any P-sentence expresses a truth-func tion of th e propositions expressed by it s P-con stituen ts.Since any molecular complex of P- and/or 0-sentences expressesa truth-fu nctio n of th e propositions expressed by t h e P- and/or0-sentences themselves, it follows that any molecular complexof P- and/o r 0-sentences expresses a truth -func tion of th epropositions expressed by i ts P-co nstitue nts.W hich truth-fu nc tion of t h e propositions expressed by its P-constituents a molecular complex of P - and/or O-sentencesexpresses can be investigated and decided in a truth-table.

Contradictory acts should n ot b e confused with (morally) indifferent.acts. The former are acts which, by definition, are never performed by a n,agent. The latter are act s which we are permitted t o perform, but alson o t t o perform.


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12 G. H. VO N WRIGHTThis fact constitutes a solution of the decision problem for thesystem of Deontic Logic which we are outlining in this paper.The technique of constructing truth-tables in Deontic Logicwill be illustrated by an example.Let the molecular complex be (0A) & (0A -tB ) -t (0B).(Cf. above, p. 5.)0 A is an abbreviation for - (P -A) , 0 A -tB for - (PA &-B), and 0 B for - (P- B ). By replacing 0-sentencesby P-sentences in our initial complex, we get the new complex- ( P - A ) & - ( P A & - B ) + - ( P - B ) .The atomic names of acts w h ch a re constituents of (a t leastone of) th e molecular complexes " nside " the operator P are A

and B . The perfect disjunctive normal form of -A in term s ofA a n d B i s - A & B v - A & - B . T h e n o rm al f o rm of A &- B i s A & - B . Thenormal fo rmof -BisA&-Bv-A& -B . The P-constituents of th e initially given molecularcomplex, therefore,are P-A & B and P A &-B and P -A&-B .Since the P-constituents do not represent all the deonticunits of the deontic realm of th e acts named b y A a nd by B(cf. above p. 10) the Principle of Permission does not hereimpose any restrictions upon the combinations of truth-values.The calculation of tru th-va lue s depends only upon th e Principleof Deontic Dist ribu tion (and principles of propositional logic).The. tab le looks as follows :

It is seen th a t the m olecular complex which we are investigating(indicated by " -t " n th e column t o th e extreme right) expressesth e tautology of th e propositions expressed by i ts P-constituents.8. A molecular complex of P- and/or 0-sentences whichexpresses th e tautology of th e propositions expressed by it sP-constituents, is said t o express a tr u th of Deontic Logic or adeontic tautology.


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DEONTIC LOGIC 13A (true) proposition to the effect that a certain molecularcomplex of P- and/or 0-sentences expresses a deontic tautologywill be called a law of Deontic Logic.We mention below some examples of such laws. When wecall two molecular complexes of P- and/or 0-sentences identical,we mean that their equivalence-sentence expresses a deontictautology. When we say that (the proposition expressed by)one molecular complex of P- and/or 0-sentences entails (theproposition expressed by) another, we mean that their implication-sentence expresses a deontic tautology. The - propositionsexpressed by the molecular complexes of sentences given below(or by the equivalence- or implication-sentences in question)

are easily shown by truth-tables to be tautologies.' (i) Two laws on the relation of permission to obligation, andvice versa :

a P A s identical with - (0-A), i.e. (PA) t - t- (0 -A)expresses a deontic tautology.b 0 A entails P A, i.e. (0A) -t ( P A ) expresses a deontictautology.The second of these laws should not be confused with theabove (alternative formulation of the) Principle of Permission

(p. 9). In the proof of (i)b this principle is already assumed.(ii) Four laws for the " dissolution " of deontic operators : a 0 A & B is identical with (0A) & (0B). b P A v B is identical with ( P A) v (PB). c (0A) v (0B) entails 0 A v B. d P A & B entails ( P A) & (PB). The second of these laws should not be confused with the Principle of Deontic Distribution (p. 7). In the proof of (ii)bthis principle is already assumed.(iii) Six laws on " commitment " :a (0A) & (0A -tB) entails 0 B. If doing what we oughtto do commits us to do something else, then this new act is alsosomething which we ought to do. (This was the exampleof a deontic tautology which we discussed above.)b (P A) & (0A -tB) entails P B. If doing what we are freeto do commits us to do something else, then this new act isalso something which we are free to do. In other words : doingthe permitted can never commit us to do the forbidden.c N (PB) & (0A -tB) entails - (PA). This is but a newversion of the previous law. If doing something commits usto do the forbidden, then we are forbidden to do the first thing.For instance : if i t is obligatory to keep one's promises and if we


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14 G. H. VON WRIGHT :promise to do something which is forbidden, then the act ofpromising this thing is itself forbidden.

d (0A -+ B v C) & -- (PB) & -- (PC) entails -- (PA). Thisis a further version of the two previous laws. An act whichcommits us to a choice between forbidden alternatives is for-bidden.

e - ((0A v B) & -- (PA) & -- (PB)). It is logically im-possible to be obliged to choose between forbidden alternatives.1f (0A) & (0A & B I- C) entails 0 B -t C. If doing twothings, the first of which we ought to do, commits us to do athird thing, then doing the second thing alone commits us to dothe third thing. " Our commitments are not affected by our(other) obligations.". g 0 --A -+A entails 0 A. If failure to perform an actcommits us to perform it, then this act is obligatory.The t,ruth of all these laws follows from our intuitive notions ofobligation and permission. Not all of the laws themselves,E.owever, are intuitively obvious. In the case of some of thelaws, moreover, i t is not intuitively clear whether their t ruth is amatter of logic or a matter of moral code. This proves that thedecision procedure of Deontic Logic which we have outlinedis not void of philosophical interest.Any molecular complex of P- a n d / ~ r -sentences has whatwe propose to call an absolutely perfect disjunctive normal form.This we get by replacing every one of the P- and/or 0-sentencesby a disjunction-sentence of P-constituents of the complex andtransforming the molecular complex of P-sentences thus obtainedinto its perfect disjunctive normal form. If the normal formcontains the conjunction of the negation of all P-constituents,we omit i t from the normal form.

The absolutely perfect disjunctive normal form shows withwhich ones of the possible combinations of truth-values in itsP-constituents the molecular complex in question expressesagreement and with which ones it expresses disagreement. Ifi t agrees with all possibilities, it expresses a deontic tautology, i .e .is a truth of Deontic Logic.Aquinas several times refers to the laws d and e. He distinguishesbetween a man's being perplexus simpliciter and his being perplexussecundum quid. The former is the case, if he is, as such, obliged to choosebetween forbidden alternatives. The latter is the case, if he by a previous

wrong act commits himself to a choice between forbidden alternatives.Aquinas rightly denies that a man could be perplexus simpliciter (e) andaffirms th at a man might be perplexus secundum quid (d). Cf. De Veritate,Q. 17, art. 4 ; Su m m a Theologica, ajjar, Q . 1 9 , art. 6 ; Su m m a Theologica,jiia, Q . 64 , art. 6. For these observations I am indebted to'Mr. P. Geach.


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15EONTIC LOGIC9. There is one relevant respect, in which the deontic modalitiesdiffer from the alethic, epistemic, and existential modalities.

It can be illustrated as follows : If a proposition is true, then itis possible, and if a proposition is true, then i t is not falsified,and if a property is true of a thing, then the property exists.But if an act is performed (or not performed), then nothingfollows as regards its obligatory, permitted or forbidden character.There is thus a n important sense in which the deontic modalitiesunlike the alethic, epistemic, and existential ones have no logicalconnexions with matters of fact (truth and falsehood). This isa point about deontic categories which has often been stressed bymoral philosophers.

10. In this paper deontic propositions have been treated as" absolute ". They can, however, be made " relative " inseveral ways.First of all, it might be argued that deontic propositions aresometimes, or perhaps always, relative to some so-called moralcode. What is obligatory within one moral code, may beforbidden within another.Secondly, instead of simply considering whether an act isobligatory, permitted or forbidden, we may consider propositionsof the following type : x is permitted to do A, or x permits yto do A. Introducing quantifiers we then get propositions of thetype : somebody is permitted to do A, or somebody permitseverybody to do A, etc. The logical systems which we getby such extensions are of considerable complexity. Theirdecision-problem can be solved for many interesting cases, butnot for all cases.CAMBRIDGEUNIVERSITY.

Deontic Logic Arquivo 61

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