# Deontic Logics

## V. DEONTIC LOGICS

**G. H. von WRIGHT**

THE existing systems of (prepositional) deontic logic can conveniently be divided into two groups. I shall call them the *monadic *systems and the *dyadic *systems.

*vocabulary:*

* a.** *Variables *p, q, r, *. . .an unlimited supply, representing some "proposition-like" entities. * b.** *Truth-connectives ~, &, v, →, and ↔ for negation, conjunction, disjunction, material implication, and material equivalence respectively.

*c.*Two deontic operators. In the monadic logics

*these are P and*

*O.*In the dyadic logics they are

*P*(-/-) and

*O*(-/-).

*d.*Brackets.

* e. *An auxiliary constant *t *which stands short for an arbitrary tautology of prepositional logic (PL), e.g., for *pv/*~*p.* To save brackets we adopt the following convention concerning the “binding force” of the connectives: ~ is the weakest and then follow in order of increasing strength &, v, →, and ↔. By virtue of this convention we can, e.g., for ( ( (~(*p*)&*q*)v*r*→*s*) ↔ *u *write simply *~p&q*v*r*→*s* ↔ *u*. We also have the convention that when *O* and *P *are followed by a variable, the variable need not be in brackets. Thus, e.g., for *P(p) *we write simply* Pp.* The *expressions *(well-formed formulae) of a deontic logic we (here) define as follows:* a*. An expression of the form “*P(*-*)*”or “*O*(-)” or “*P*(-/-)” or *O*(-/-)" is well-formed when the place of “-“ is taken by an expression of PL. * b*. Truth-functional compounds of well-formed expressions are well-formed.

When the expressions of a deontic calculus are defined in this way, “mixed” expressions, i.e., truth-functional compounds of expressions of PL and of expressions of the deontic calculus, do not count as well-formed. Nor do expressions count as well-formed which *iterate *a deontic operator. These restrictions on the well-formed formulae can be removed. We shall not, however, in this paper consider the more general systems of deontic logic which admit "mixed" expressions and iteration of deontic operators. '*P*' may be read as “it is permitted that” and *'O' *may be read as “it is obligatory that´” or, alternatively, “it is forbidden that not.” Another way of reading the operators would be “it is permitted to see to it that” and “it is obligatory to see to it that” or, alternatively, “it is forbidden to see to it that not.” Under the second reading, the permitted and obligatory things will be those *actions *which result in the *states of affairs *described by the expressions of PL within the scope of the deontic operator.¨ There is a monadic deontic logic with the following axioms:^{1} A0M. A set of axioms of PL (with well-formed expressions substituted for the variables).

*Pp*↔

*~O~p.*

A2M. *P(p*v*q)* ↔*PP *v* Pq.*

*Pp*v

*P~p.*

R2. Detachment *(modus ponens).*

R3. A Rule of Extensionality to the effect that provably equivalent formulae of PL are intersubstitutable in the wffs of the deontic calculus. A dyadic deontic logic which, for reasons to be given later, may be called the "counterpart" of

^{1}This is, from the formal point of view, substantially the same as the system of my paper “Deontic Logic,” *Mind, *vol. 60 (1951), pp. 1-15. What is here A1M was there regarded as a *definition *of the one deontic operator in the terms of the other (and negation).

^{2}

AoD. A set of axioms of PL (with well-formed expressions substituted for

the variables).

AlD.

*P*(

*p*/

*q*)

*→*~

*O*(~

*p*/

*q*)

*.*

A2D.

*P*(

*p*v

*q*/

*r*v

*s*)

*↔*

*P*(

*p*/

*r*)v

*P*(

*p*/

*s*)v

*P*(

*q*/

*r*)v

*P*(

*q*/

*s*)

*.*

A

_{3}D.

*P*(

*p*/

*q*)v

*P*(

*p*/~

*q*)v

*P(*~

*p*/

*q*) v

*P*(

*~p*/~

*q*)

*.*

By virtue of the distribution-axioms A2, we could also write A3Min the form

*Pt*and A3D in the form

*P*(

*t*/

*t*).

It seems that in all known systems of deontic logic on can prove certain formulae which will strike one as counterintuitive or somehow ¨paradoxical." I shall here mention a few such paradoxes. They all arise in the monadic system which was outlined in the preceding section—and reappear, *mutatis mutandis, *in the dyadic system.

We easily prove that *O**p**→O*(*p*v*q*)*. *This says that an obligation to see to it that (it is the case that) *p *entails an obligation to see to it that (it is the case that) *p or q. *So, for example, if I ought to mail a letter, I also ought either to mail *or *to burn it. This seems odd. The oddity was first pointed out by the Danish jurist-philosopher Alf Ross in a paper which may be called a trenchant criticism of the very idea of a "deontic logic."^{3} I shall refer to this oddity under the name of Ross's Paradox.

This paradox has often been met by the following argument (or by some argument similar to it): If I see to it that *p, *and thereby fulfill the first of the two obligations above, then I also, by the laws of ordinary logic, see to it that it is the case that *p or q. *But from this it cannot, by the laws of any logic ("ordinary" or "deontic") be concluded that it were obligatory or even permitted for me to see to it that *q. *To say that *Op *entails *O(p*v*q) *is really no more paradoxical than to say that *p *entails *p*v*q. *Beginners in logic may find the last a little bit hard to swallow too, but they will soon get over the difficulty.

This argument has seemed satisfactory to many people. I have tried myself

to be pleased with it, but never quite successfully. I always felt that there was more to Ross's Paradox than can be met by the above argument. I hope to be able to show that this feeling is justified. (But I shall not follow Ross in concluding that deontic logic is therefore impossible.)

We also prove that ~*Pp**→*~*P*(*p&q*)*. *This says that if it is forbidden (not permitted) that it be the case that *p*, then it is also forbidden (not permitted) that it be the case that *p and *something else. This seems odd, particularly if *q *is something which we ought to see to that is the case, when *p* is the case. This last oddity is sometimes called the Paradox of the Good Samaritan.

A further thing which we can prove is that ~*Pp**→*~*O*(*p**→**q*)*. *This says that, if it is forbidden that it be the case that *p, *then it is our duty to see to it that it is the case that *q, if *it is the case that *p*—and this no matter what state of affairs *q* is. This oddity is known as a Paradox of Derived Obligation or of Commitment.^{4}

It is of some interest to note that, if we accept the "interdefinability" of the deontic operators as laid down in A1M, then the three formulae *"Op**→ **O*(*p*v*q*)*" *and *"*~*Pp**→*~*P*(*p*&*q*)*" *and *"*~*Pp**→O*(*p**→**q*)*" *are but variations of one and the same logical structure.

Finally, I mention the following perplexity. On the ordinary understanding of "it is permitted that," *P*(*p*v*q*)seems to entail *Pp*&*Pq. *If I say to somebody "you may work or relax" I normally mean that the person addressed has my permission to work and also my permission to relax. But, in what may be called the "standard" systems of deontic logic, *P(p*v*q) *does *not *entail *Pp*&*Pq. *If it did, we would moreover get strange consequences. We could then, in the system presented in Section I, easily derive as a theorem that *Op**→**Pq, *which says

^{2} The variety of dyadic systems seems to be greater than that of monadic systems. The, as far as I know, historically first dyadic system was proposed in my "A Note on Deontic Logic and Derived Obligation," *Mind, *vol. 65 (1956), pp. 507-509. A different system is due to N. Rescher, "An Axiom System for Deontic Logic." *Philosophical Studies, *vol. 9 (1958), pp. 24-30. The system A0D-A3D which is given here was first presented by me in "A New System of Deontic Logic," *The Danish Yearbook of Philosophy, *vol. I (1964) and "A Correction to a New System of Deontic Logic," *ibid. *vol. 2 (1965). In these papers, the axioms were stated in terms of the obligation operator "O(-/-)" and A2D was split up in two principles, *viz., O*(*p & q*/*r*)*↔ **O*(*p*/*r*)& *O*(*g*/*r*)and *O*(*p*/*q *v* r*)*↔ **O*(*p/q*)& *O*(*p*/*r*)*. *The truth of A1D was regarded as definitional.

^{ 3 }Alf Ross, "Imperatives and Logic," *Theoria, *vol. 7 (1941), pp. 53-71.

^{ 4} See A. N. Prior, "The Paradoxes of Derived Obligation." *Mind, *vol. 63 (1954.), pp. 64-65, and my reply "A Note on Deontic Logic and Derived Obligation," *Mind, *vol. 65 (1956), pp. 507-509. A challenging expression of dissatisfaction with the situation in deontic logic with regard to this paradox is R. M. Chisholm, "Contrary-to-Duty Imperatives and Deontic Logic," *Analysis, *vol. 24 (1963), pp. 33-36.

137

that if we ought to see to it that *p*, then we are permitted to see to it that* q*, no matter what this second state of affairs is. And from this result we could derive a contradiction in the system.

One could meet this paradox by saying that a disjunctive permission is *compatible *with the permissibility of both the disjuncts, but that it does not *entail *their permissibility. One could then admit that *normally P*(*p*v*q*)goes together with *P**p&Pq *but that this is no (logical) *necessity. *I have tried to meet the paradox in this way in past writings of mine.^{5} But I no longer think the defense entirely successful.

III

The occurrence of "paradoxes" in intuitively seemingly sound systems of monadic deontic logic, and the vagueness and conflicting nature of the intuitions upon which the dyadic systems build^{6 }make urgent the need of giving to all those systems a sound semantic foundation. In a recent paper,^{7 }Nicholas Rescher has made an important contribution toward satisfying this need.

Rescher's contribution immediately concerns the dyadic system. He proposes four different interpretations of the notion of a conditional permission, *P*(*p*/*q*), in the terms of relations between sets of *state-descriptions *or "possible worlds." I shall here exploit Rescher's idea for showing how a variety of (monadic and dyadic) systems of deontic logic may become generated from a common pattern. The systems, thus generated, are decidable and semantically complete with regard to a specified criterion of logical truth.

I introduce the following piece of new terminology: An expression of the form "P(*s*/*s'*)," where '*s*' and '*s**'*'are state-descriptions, may be read "in the possible world *s' *the possible world *s *is permitted (as an alternative world to *s'*)."Similarly, an expression of the form "*O*(*s*/*s'*)" may be read "in the possible world *s' *the possible world *s *isobligatory (as an alternative world to *s')" *or also as "in the possible world *s' *any possible world other than *s *is forbidden (as an alternative world to *s')." *The expression *"P*(*s*/*s'*)*" *thus says that a world of the description '*s'*'* may *become changed to a world of the description '*s*'. The expression "*O*(*s*/*s'*)" says that a world of the description '*s'*' *ought to *be thus changed, i.e., *must not *be changed to anything but this (nor left unchanged).

IV

I define four concepts of conditional permission as follows:^{8}

"*P*_{1}(*p*/*q*)"will mean that in *some *possible world in which it is true that *q some *possible world is permitted in which it is true that* p.*

"*P*_{2}(*p*/*q*)"will mean that in *all *possible worlds in which it is true that *q some *possible world is permitted in which it is true that *p*.

"*P _{3}*(

*p*/

*q*)"will mean that in

*some*possible world in which it is true that

*q all*possible worlds are permitted in which it is true hat

*p.*

“

*P*(

_{4}*p*/

*q*)”will mean that in

*all*possible worlds in which it is true that

*q all*possible worlds are permitted in which it is true that

*p.*To the four concepts of conditional permission there answer four concepts of conditional obligation, "

*O*

_{1}(

*-*/-)", "

*O*

_{2}(-/-)", "

*O*

_{3}(-/-)", and "

*O*

_{4}(-/-)":

"O

"O

_{1}(

*p*/

*q*)" which will mean that in

*all*possible worlds in which it is true that

*q all*possible worlds are forbidden in which it is

*not*true that

*p*;

"O

"O

_{2}

_{(}

*p*/

*q*)" which will mean that in

*some*possible world in which it is true that

*q all*possible worlds are forbidden in which it is

*not*true that

*p*;

"O

"O

_{3}(

*p*/

*q*)"which will mean that in

*all*possible worlds in which it is true that

*q some*possible world is forbidden in which it is

*not*true that

*p*; and

"O

"O

_{4}(

*p*/

*q*)" which will mean that in

*some*possible world in which it is true that

*q some*possible world is forbidden in which it is

*not*true that

*p.*

We set up four dyadic calculi which we shall refer to as the

*weak P*

_{1}

*-,*

*P*

_{2}

*-,*

_{ }

*P*

_{3}

*-,*and P

_{4}-calculi.

^{5 }"Deontic Logic," *Mind, *vol. 60 (1951), p. 7 n.

^{ 6} See the exchange of thoughts between Resdher and Anderson in the papers: N. Rescher, "An Axiom System for Deontic Logic," *Philosophical Studies, *vol. 9 (1958), pp. 21-30; A. R. Anderson, "On the Logic of 'Commitment'," *Philosophical Studies, *vol. 10 (1959), pp. 23-29; N. Rescher, "Conditional Permission in Deontic Logic," *Philosophical Studies, *vol. 13 (1962), pp. 1-6; and A. R. Anderson, "Reply to Mr. Rescher," *Philosophical Studies, *vol. 14 (1963), pp. 6-8. See also J. Robison, "Further Difficulties for Conditional Permission in Deontic Logic" *Philosophical Studies*, vol. 18 (1967).

^{ 7 }N. Rescher, "Semantic Foundations for Conditional Permission", *Philosophical Studies, *vol. 18 (1967).Philosophical Studies, vol 18 (1967). For a semantic approach to monadic deontic logic see W. H. Hanson, "Semantics for Deontic Logic" *Logique et Analyse*, vol. 8 (1965) pp. 177-I90.

^{ 8}** **These four concepts of conditional permission are not the same as those of Rescher.

138

The vocabulary and expressions are defined as in Section I.

All four calculi have this axiom in common:

Ao DP_{1}. A set of axioms of PL (with well-formed expressions substituted for the variables). All four calculi also have an axiom of the form:

AI DP_{1}. *P _{i}*(

*p*/

*q*)

*↔*

*~O*(~

_{i}*p*/

*q*)

*,*where for '

*i*' is substituted 'I' in the

*P*

_{1}-calculus, and . . . and '4' in the

*P*

_{4}-calculus.

Each calculus has its own characteristic

*distribution axiom*:

A2 DP

_{1}.

*P*

_{1}(

*p*v

*q*/

*r*v

*s*)

*↔*

*P*

_{1}(

*p*/

*r*) v

*P*

_{1}(

*p*/

*s*) v

*P*

_{1}(

*q*/

*r*) v

*P*

_{1}(

*q*/

*s*).

A2 DP

_{2}.

*P*

_{2}(

*p*v

*q*/

*r*v

*s*)

*↔*(

*P*

_{2}(

*p*/

*r*) v

*P*

_{2}(

*q*/

*r*)) & (

*P*

_{2}(

*p*/

*s*) v

*P*

_{2}(

*q*/

*s*) ).

A2 DP

_{3}.

*P*3(

*p*v

*q*/

*r*v

*s*)

*↔*(

*P*

_{3}(

*p*/

*r*) &

*P*

_{3}(

*q*/

*r*)) v (

*P*

_{3}(

*p*/

*s*) &

*P*

_{3}(

*q*/

*s*) ).

A2 DP

_{4}.

*P*

_{4}(

*p*v

*q*/

*r*v

*s*)

*↔*

*P*

_{4}(

*p*/

*r*) &

*P*

_{4}(

*p*/

*s*) &

*P*

_{4}(

*q*/

*r*) &

*P*

_{4}(

*q*/

*s*).

The rules of inference of all lour calculi are the rules RI-R3 (Section I).

Comment on the truth of the axioms: AI may be regarded as following from a proposed definition of the one deontic operator in the terms of the other (and negation). The distribution axioms A2 again are the syntactical equivalents of truths which in the semantics of the calculi follow directly from the definitions of the various permission-concepts.

The expressions of each calculus have what I propose to call a

*P*-normal form. This we obtain as follows: If the expression is a truth-functional compound which contains constituents of the form "

_{i}*O*(-/-)" we replace these constituents by constituents of the form "

_{i}*P*(-/-)" by virtue of AI. Thereupon we replace, by virtue of R3, the expressions to the left and to the right of '/' in the constituents by their disjunctive normal forms in terms of

_{i}*all*the variables which occur in the whole expression.

^{9}Finally, we distribute the constituents by virtue of the appropriate A2 as far as possible. The result of the distribution is the

*P*-normal form of the expression.

_{i}We adopt the following

*criterion of logical truth*in a weak dyadic deontic logic:

*An expression of a*

*P*(

_{i}-calculus is logically true, if and only if, its P_{i}-normal form is a tautology of prepositional logic*PL*).

By transforming a given expression into its normal form we can immediately see whether it is a logical truth or not. It is obvious that all formulae which are, on the set criterion, logically true in the

*P*-calculus arc also provable in the

_{i}*P*calculus (by virtue of A0) and that all formulae which are provable in the

_{i}-*P*-calculus are, on the set criterion, logically true. Thus, relative to the truth-criterion, the calculi are semantically complete.

_{i} To the axioms of each one of the weak calculi which we described in the preceding section we add a fourth axiom:

A_{3} DP_{1}. *P _{i}*(

*t*/

*t*). (I<

*i*<4.)

The four calculi which we then get I shall call the

*strong*(dyadic)

*P*

_{1}-,

*P*

_{2}-,

*P*

_{3}-, and

*P*

_{4}-caIculus. A

_{3}I shall call the Principle of Permission.

^{10}

"

*P*

_{1}(

*t*/

*t*)" says that in

*some*possible world

*some*possible world is permitted.

*"*

*P*

_{2}

*(*

*t*/

*t*)"says that in

*all*possible worlds

*some*possible world is permitted.

"

*P*

_{3}(

*t*/

*t*)¨ says that in

*some*possible world

*all*possible worlds are permitted.

"

*P*

_{4}(

*t*/

*t*)" says that in

*all*possible worlds

*all*possible worlds are permitted.

The expressions of each calculus have a

*P*-normal form. It is identical with the

_{i}*P*

_{i}_{-}normal form of the corresponding weak calculus.

The

*criterion of logical truth*in a strong dyadic deontic logic is the following one:

*An expression*

*'E'*

*of a strong P*

_{i}-calculus is logically true, if and only if, the P_{i}-normal form of the expression “P_{i}(t/t)*→E” is a*

*tautology of propositional logic (PL).*

The strong dyadic deontic calculi too are decidable and complete relative to their truth-criteria.

In the strong

*P*

_{2}-calclulus we prove as a theorem that

*P*

_{2}

*(t/p)*or that "under all circumstances something is permitted.''

In the strong

*P*

_{3}-calculus we prove that

*P*

_{3}(

*p*/

*t*) or that "under some circumstances everything is permitted."

In the strong

*P*

_{4}-calculus we prove that P

_{4}(

*p*/

*q*) or that "under all circumstances everything is permitted."

It may be thought that, because of these theorems, the

*strong*

*P*

_{3}- and

*P*

_{4}-calculi are absurd. But they are not "illogical." The strong dyadic

*P*

_{1}-calculus is the same as the dyadic system presented in Section I.

^{ 9 }If the expression of PL to the left or to the right of '/' happens to be a contradiction, it has no disjunctive normal form. In that case, replace the expression by '~*t*' and add '~*t*' as a disjunct to all the other disjunctive normal forms.

^{10 }See "Deontic Logic", *Mind*, vol. 60 (1951), p. 9.

The dyadic calculi which we have so far been considering accept the "interdennabilily"-principle that P_{i}(*p*/*q*) ↔~*O _{i}*(~

*p*/

*q*). This principle, however, may become challenged.

We split up the principle in two implications, viz., the principle that P

_{i}(

*p*/

*q*)→~

*O*(~

_{i}*p*/

*q*) and the principle that ~

*O*(~

_{i}*p*/

*q*)→

*P*(

_{i}*p*/

*q*). The first says that if, given that

*q*, it is permitted that

*p*then it is not the case that, in the same circumstances, it is obligatory that ~

*p*. This seems extremely plausible. For, is not to say that it is obligatory that ~

*p*the same as saying that it is forbidden that ~

*p*? If this is so, then to deny the first implication would be tantamount to admitting that, under given circumstances, one and the same thing could be both permitted and forbidden. But this seems contrary to the very

*meaning*of "permitted" and "forbidden."

The second implication says that if, given that

*q*, it is not obligatory that ~

*p*, then it

*is*permitted that

*p*. This is much less obvious than the first principle. For, it seems to say that what is not forbidden is

*ipso facto*permitted. But can this be regarded as true by virtue of the

*meaning*of "permitted" and "forbidden" alone?

The idea that what is not forbidden is

*ipso facto*permitted is related to the principle of legal philosophy "

*nullum crimen sine lege*." I shall, for convenience, refer to the principle that ~

*O*( ~

_{i}*p*/

*q*)→

*P*(~

_{i}*p*/

*q*) as the

*nullum crimen*idea. If we talte the view that the

*nullum crimen*principle does not embody a logical truth concerning the notions of permission, prohibition, and obligation, we must also refuse to regard the "interdefinability"-principle as logically true.

Deontic logics which reject AI wholesale would hardly be of interest. But systems which accept the principle that

*P*(

_{i}*p*/

*q*)→~

*O*(~

_{i}*p*/

*q*) but reject the

*nullum crimen*principle are certain to be of interest.

^{11}

If the

*nullum crimen*principle is rejected, we can no longer in the calculi derive distribution laws for the notion of obligation from those for the notion of permission. The distribution laws for obligation have to be derived from the definitions of the various obligation-concepts and stated as axioms in the calculi.

Deontic logics which do not assume that the deontic operators are interdefinable, I shall call

*P*-calculi. The

_{i}O_{i}*P*-calculi which correspond to the

_{i}O_{i}*weak*

*P*-calculi have the following axioms:

_{i}A0 DP

_{1}O

_{1}. As before.

AI DP_{1}O_{1}. *P _{i}*(

*p*/

*q*)→~

*O*( ~

_{i}*p*/

*q*). (I≤i≤4)

A2a DP

_{1}O

_{1}. Same as A2 DP

_{1}.

A2b DP

_{1}O

_{1}.

*O*

_{1}(

*p*&

*q*/

*r*v

*s*) ↔

*O*1(

*p*/

*r*) &

*O*

_{1}(p/

*s*) &

*O*

_{1}(

*q*/

*r*) &

*O*

_{1}(q/s)

A2b DP

_{2}O

_{2}.

*O*2(

*p*&

*q*/

*r*v

*s*) ↔

*O*2(

*p*/

*r*) &

*O*

_{2}(

*q*/

*r*) ) v

*O*

_{2}(

*p*/

*s*) &

*O*2(

*q*/

*s*) ).

A2b DP

_{3}O

_{3}.

*O*

_{3}(

*p*&

*q*/

*r*v

*s*) ↔ (

*O*

_{3}(

*p*/

*r*) v

*O*(

_{3}*q*/

*r*) ) & (

*O*

_{3}(

*p*/

*s*) v

*O*

_{3}(

*q*/

*s*) ).

A2b DP

_{4}O

_{4}.

*O*

_{4}(

*p*&

*q/r*v

*s*) ↔

*O*

_{4}(

*p*/

*r*) v

*O*

_{4}(

*p*/

*s*) v

*O*

_{4}(

*q*/

*r*) v

*O*

_{4}(

*q*/

*s*).

The principles of inference are RI-R3. The expressions of each calculus have what I propose to call a

*P*-normaI form. It is obtained in the following way: In the constituent P

_{i}O_{i}*-expressions we replace the expressions to the left and to the right of '/' by their disjunctive normal forms in terms of*

_{i}*all*the variables of the whole expression. In the constituent

*O*-expressions we replace the expressions to the left of '/' by their conjunctive and the expressions to the right of '/' by their disjunctive normal forms in terms of

_{i}*all*the variables of the whole expression.

^{12}Thereupon we distribute the constituents according to the (appropriate) axioms A2a and A2b. The result is the

*P*-normal form.

_{i}O_{i}The criterion of logical truth in the weak

*P*-calculi is: An expression '

_{i}O_{i}*E*' is logically true, if and only if, the

*P*-normal form of the expression

_{i}O_{i} is a tautology of PL. *c _{μ}* is a state-description in terms of the variables of

*E*.

*d*is the negation of

_{μ}*c*,

_{μ}in front of an expression of a certain form means the conjunction of all expressions of this form which we obtain by successively replacing an index-numeral by numerals I, . . .,

*k*.

The

*strong*

*P*-calculi have an additional axiom: A3 DP

_{i}O_{i}_{1}O

_{1}. Same as A3 DP

_{1}. The criteria of logical truth for the strong calculi are obtained from those of the weak calculi by adding as a conjunct "

*P*(

_{i}*t*/

*t*) to the antecedent of the implication.

^{11} One such system is explored in some detail in my book *Norm and Action *(London, Routledge & Kegan Paul 1963).^{12 }If the expression in question has no conjunctive normal form, write in its place '*t*' as and add '*t*' as a conjunct to all the other conjunctive normal forms. If again the expression has no disjunctive normal form, write in its place '~*t*' and add '~*t*' as a disjunct to all the other disjunctive normal forms.

*monadic counterparts*for the dyadic calculi.

The monadic expression *"P*_{1}*p" *will mean the same as the dyadic expression "*P*_{1}(*p*/*t*)" and hence, by the definition of the latter, it will mean that in *some *possible world *some *possible world is permitted in which it is true that *p*.

"*P*_{2}*p*" will mean the same as "*P*_{2}(*p*/*t*)" and hence it will mean that in *all* possible worlds *some* possible world is permitted in which it is true that *p*.

"*P*_{3}*p*" will mean the same as "*P*_{3}(*p*/*t*)" and hence it will mean that in* some* possible world *all* possible worlds arc permitted in which it is true that *p*.

"*P*_{4}*p*" will mean the same as "*P*_{4}(*p*/*t*)" and hence it will mean that in *all* possible worlds *all* possible worlds are permitted in which it is true that *p*.

The axioms of the *weak *monadic *P _{i}*-calculi are:

A0 MP_{1}. A set of axioms of PL (with well-formed formulae substituted for the variables).

A1 MP_{0}. *P _{i}p* ↔ ~

*O*. (I ≤

_{i}~p*i*≤4.)

A2 MP_{1}. *P*_{1}(*p*v*q*) ↔ *P*_{1}*p* v *P*_{1}*q*.

A2 MP_{2}. *P*_{2}(*p*v*q*) ↔ *P*_{2}*p* v *P*_{2}*q*.

A2 MP_{3}. *P*_{3}(*p*v*q*) ↔ *P*_{3}*p* v *P*_{3}*q*.

A2 MP_{4}. *P*_{4}(*p*v*q*) ↔ *P*_{4}*p* v *P*_{4}*q*.

_{1}and A2 MP

_{2}have the same structure. But they have different meanings. Their difference in meaning, however, reveals itself only in their dyadic translation. Hence we could say thai the P

_{1}and P

_{2}-calculi are "monadically identical." The same is true of the P

_{3}- and P

_{4}-calculi.

A3 MP_{1}. *P _{i}t*. (I≤

*i*≤4.)

The strong monadic P_{1}-calculus is identical with the monadic deontic logic which was described in Section I.

A monadic calculus too may reject the* nullum crimen* principle. We then get monadic *P _{i}O_{i}*-calculi.

The axioms of the *weak *monadic *P _{i}O_{i}*-calculi are:

A0 MP_{1}O_{1}. Same as Ao MP_{1}.

A1 MP_{1}O_{1}. * P*_{1}*p* →~*O _{i}~p*. (I ≤

*i*≤4.)

A2a MP_{1}O_{1}. Same as A2 MP_{1}.

A2bMP_{1}O_{1}. *O*_{1}(*p*&*q*) ↔ *O*_{1}*p* & *O*_{1}*q*.

A2b MP_{2}O_{2}. *O*_{2}(*p*&*q*) ↔ *O*_{2}*p* & *O*_{2}*q*.

A2b MP_{3}O_{3}. *O*_{3}(*p*&*q*) ↔ *O*_{3}*p* & *O*_{3}*q*.

A2b MP_{4}O_{4}. *O*_{4}(*p*&*q*) ↔ *O*_{4}*p* & *O*_{4}*q*.

The axioms of the strong monadic *P _{i}O_{i}*-calculi, finally, are those of the weak calculi with the addition:

A3 MP_{1}O_{1}. Same as A3 MP_{1}.

Again the monadic *P*_{1}*O*_{1}-calculus is "monadically identical" with the monadic *P*_{2}*O*_{2}-calculus, and the *P*_{3}*O*_{3}-calculus with the* P*_{4}*O*_{4}-calculus.

*P*-calculi has a

_{i}*P*-normal form. The normal forms are obtained by replacing expression of PL by their disjunctive normal forms in

_{i}O_{i}*P*-expressions and conjunctive normal forms in

_{i}*O*

_{i}-expressions and

*P*

_{i}-calculi we first replace

*O*

_{i}-expressions by the equivalent

*P*

_{i}-expressions.)

The criteria of logical truth of an expression '*E*' are:

In the weak monadic *P*_{i}-calculi: that the *P*_{i}-normal form of '*E*' be a tautology of PL.

In the strong monadic *P*_{i}-calculi: that the *P*_{i}-normal form of "*P*_{i}*t*→*E*' be a tautology of PL.

In the weak monadic *P _{i}O_{i}*-calculi: that the

*P*-normal form of

_{i}O_{i}

be a tautology of PL.

In the strong monadic *P _{i}O_{i}*-calculi: that the

*P*-normal form of

_{i}O_{i}

be a tautology of PL.

The four *P*-concepts which we distinguished in Section IV are not logically unrelated. As seen from their definitions, *P*_{4}(*p*/*q*) entails* P*_{3}(*p*/*q*) and *P*_{2}(*p*/*q*) and *P*_{1}(*p*/*q*); *P*_{3}(*p*/*q*) and *P*_{2}(*p*/*q*) both entail *P*_{1}(*p*/*q*); *P*_{3}(*p*/*q*) and *P*_{2}(*p*/*q*) are independent of one another. These relations carry over to the monadic concepts.

To the four *P*-concepts there correspond four *O*-concepts. *O*_{1}(*p*/*q*) entails *O*_{2}(*p*/*q*) and *O*_{3}(*p*/*q*) and *O*_{4}(*p*/*q*); *O*_{2}(*p*/*q*) and *O*_{3}(*p*/*q*) both entail *O*_{4}(*p*/*q*); *O*_{2}(*p*/*q*) and *O*_{3}(*p*/*q*) are independent.

It is of some interest to note that to the *weakest *concept of permission, there answers the *strongest* concept of obligation—and to the strongest obligation the weakest permission.

There is no objection to forming "mixed" truth-functional compounds of *P _{i}*-and/or

*O*

_{i}-expressions

with different indices *i*—nor to forming compounds of dyadic and monadic expressions. When contemplating the logical features of such "mixed" expressions we must be clear over which axioms we assume for the various *P*- and *O*-concepts, viz., whether we accept or reject the Principle of Permission and whether we accept or reject the *nullum crimem* principle for them. We may accept these principles for some of the *P*- and/or *O*-concepts involved and reject them for others.

The *P _{i}*- and/or

*P*-normal forms of the "mixed" expressions we get by transforming each one of the constituent expressions into

_{i}O_{i}*its*

*P*- or

_{i}*P*-normal form.

_{i}O_{i}In formulating criteria of truth for a calculus containing "mixed" expressions we have to observe the following equivalences which reflect the dependence between the various *P*- and *O*-concepts:

*P*

_{4}(

*p*/

*q*)→P

_{3}(

*p*/

*q*) (2)

*P*

_{4}(

*p*/

*q*)→P

_{2}(

*p*/

*q*)

(3)

*P*

_{4}(

*p*/

*q*)→P

_{1}(

*p*/

*q*) (4)

*P*

_{3}(

*p*/

*q*)→P

_{1}(

*p*/

*q*)

(5)

*P*

_{2}(

*p*/

*q*)→P

_{1}(

*p*/

*q*) (6)

*O*

_{1}(

*p*/

*q*)→

*O*

_{2}(

*p*/

*q*)

(7)

*O*

_{1}(

*p*/

*q*)→

*O*

_{3}(

*p*/

*q*) (8)

*O*

_{1}(

*p*/

*q*)→

*O*

_{4}(

*p*/

*q*)

(9)

*O*

_{2}(

*p*/

*q*)→

*O*

_{4}(

*p*/

*q*) (10)

*O*

_{3}(

*p*/

*q*)→

*O*

_{4}(

*p*/

*q*)

We can now deal afresh with the "paradoxes" of deontic logic which were mentioned in Section II. They all arise, we said, in the monadic system presented in Section I. This system is identical with the* strong* monadic *P*_{1}-calculus. It is easily shown, however, that the very same paradoxes arise already in the *weak* monadic *P*_{1}-calculus and that they occur in the weak and strong *P*_{1}*O*_{1}-calculi too. For their derivations depend only upon the distributive properties of the concepts *P*_{1} and *O*_{1} and are independent of acceptance of the Principle of Permission and the* nullum crimen* principle.

*P*

_{3}- and

*P*

_{3}

*O*

_{3}-calculi.

Ross's paradox does not arise, since *O*_{3}*p* does not entail *O*_{3}(*p*v*q*) in any of the calculi. Thus, in particular, if to mail a letter is *O*_{3}-obligatory, it does not follow that to mail this letter *or* to burn it is *O*_{3}-obligatory.

Since ~*P*_{3}*p* does not entail *O*_{3}(*p*→*q*) nor ~*P*_{3}(*p*&*q*), the paradoxes of derived obligation and of "the good Samaritan" do not arise in the *P*_{3}- or in the *P*_{3}*O*_{3}-calculi.

In the monadic *P*_{3}- and *P*_{3}*O*_{3} -calculi the equivalence *P*_{3}(*p*v*q*) ↔ *P*_{3}*p*&*P*_{3}*q* holds. This is not here connected with any paradox, not to speak of contradiction.

Shall we, on the basis of these findings, conclude that the *P*_{3}*- *and *P*_{3}*O*_{3}-calculi are those which best agree with our intuitive ideas of permission and obligation? The answer is that we must *not* conclude this. For, in these calculi too "paradoxes" arise. Thus we can in them prove, e.g., the formula *P*_{3}*p*→*P*_{3}(*p*&*q*)*.* It says that if it is permitted that a state of affairs obtains, then it is also permitted that any conjunction of this state with another obtains. Thus, in particular, a permission to enjoy life would entail a permission to enjoy life *and *commit murder (though not a* permission* to commit murder). This paradox may be said to correspond in the *P*_{3}-calculi to Ross's Paradox in the *P*_{1}-calculi. One can meet the paradox with similar arguments to those which have traditionally been used to meet Ross's Paradox in the standard systems of deontic logic. (See above Section II.) But one can also refuse to let those arguments be the end of the story and look for a calculus in which the paradoxical formula simply cannot be proved. Such a calculus is the (weak and strong) *P*_{1}-calculus.

Thus, in a way, the answers to the paradoxes which arise in the *P*_{1}-calculi are given in the *P*_{3}-calculi—and the answers to the paradoxes which arise in the *P*_{3}-calculi are given in the *P*_{1}-calculi.

The moral to be drawn from all this is that there are *several concepts* of permission and obligation and that the "paradoxes" of deontic logic arise from a confusion of them. When the concepts have become clearly distinguished and systematized, there are no "paradoxes."

To this insight should be added an observation concerning ordinary use of language. It seems to be the case that "or" in connection with the phrase "permitted that *p *or *q" *is normally so used chat it suggests a* **P*_{3}-permission, viz., a permission the sense of which is that anything which falls under (entails) the alternative *"p *or *q"* is permitted. And it also seems to be the case that "and" in connection with the phrase "obligatory that *p *and *q"* is normally so used that it suggests an *O*_{1}-obligation, viz., an obligation the meaning of which is that anything under which the compound *"p *and *q*" falls, i.e., anything entailed by it, is thereby also obligatory. Thus conjunctive obligations and disjunctive permissions are "naturally" understood

142

in the strongest sense of "obligation" and "permission." But since the correlative in a deontic logic to the strongest obligation-concept is the weakest permission-concept, it takes different "deontic logics" to do justice to all ou intuitions in the matter. Appeal to "intuition" in discussions of the acceptability or validity of various formulae of monadic or dyadic deontic logic is on the whole futile. When the question of validity is at stake, it should be framed as follows: In which deontic logic, if in any, is this formula valid? To each deontic logic answers different concepts of permission and obligation. The question can therefore also be formulated: For which concept of per- mission and/or obligation is this formula valid? We can trivially construct permission- and obligation-concepts, and therewith deontic logics, so that all formulae except those which can become refuted in some non-deontic branch of logic, become valid. But not all such constructions are interesting and worthwile.

*The Academy of Finland*

Received April 15, 1966

Received April 15, 1966

* *