1. In this and the next chapter we shall present the fundamentals of a formal Logic of Norms or Deontic Logic.

The ‘substructure’ of this logic has three layers viz. the (‘classical’) Logic of Propositions the Logic of Change which we sketched in Chapter II and the Logic of Action which we sketched in Chapter IV. The formal set-up and the principles of these three logics are incorporated and presupposed in our Logic of Norms.

The Logic of Propositions is a formal study of p-expressions our Logic of Change a formal study of T-expressions and our Logic of Action a formal study of df-expressions. The formalism of the Logic of Change employs in addition to the symbols of the Logic of Propositions one new symbol T. The formalism of the Logic of Action employs in addition to the symbols of the Logic of Propositions and the Logic of Change two new symbols d and f. An embellishment of the formalism of the Logic of Action with one further symbol will be made in Chapter IX.

In Chapter V we introduced the notion of the norm-kernel. The norm-kernel consists of the three components or parts of a norm which we call the character the content and the condition of application. As symbols for the two norm-characters we introduced the letters O and P. The symbols for norm-contents are df-expressions.

One of the several ways of dividing norms into classes which we mentioned in Chapter V is their division into categorical and hypothetical norms. The conditions of application of categorical norms we said (Ch. V Sect. 6) can be ‘read off’ from their contents. No new symbol is needed for stating the conditions of application of categorical norms. The conditions of application of hypothetical norms however cannot be ‘read off’ from their contents; a new symbol is needed for stating them. This new symbol is the embellishment of the formalism of the Logic of Action to which we referred above and which will be introduced in the next chapter.

The symbols of the norm-kernels of categorical norms are the atomic O- and P-expressions which we defined in Section 4 of Chapter V. A generalized notion of (atomic) O- and P-expressions will be defined in the next chapter in connexion with the introduction of a symbolism for the conditions of application of hypothetical norms.

The Logic of Norms which we are going to outline is a formal study of that ‘part’ of norms only which we call the norm-kernels (cf. Ch. V Sect. 1). This is a limitation of our Logic of Norms which future research into the subject ought to remove.

The norm-kernels we said in Section 1 of Chapter V may be regarded as the common parts of norms of all types. The Logic of Norms which we are here sketching is primarily conceived of as a logical theory of the norm-kernels of prescriptions. No explicit claim will be made on behalf of its validity for the kernels of other types of norm.

The Logic of Norms we also call Deontic Logic. The Greek verb δέομαι means in English to bind. Related to it is the impersonal verb δεῖν which may be translated by ought or to be necessary. A noun form of this impersonal verb is τό δέον which means that which ought to be or is duty or obligatory. The adverb δεόντως roughly means duly or as it should be.

2. The first problem confronting our attempt to build a logic of norms is whether the so-called truth-connectives or the symbols for negation conjunction disjunction etc. can be used for forming molecular complexes of (atomic) O- and P-expressions. It is important that we should see quite clearly the nature of the problem before us. For it is no doubt a somewhat confusing problem.

The ideas of negation conjunction etc. are primarily at home in descriptive discourse. In it sentences are used for making statements which express propositions. To say that the sentence ‘It is not raining’ is the negation (-sentence) of ‘It is raining’ is to say some such thing as this: The sentence ‘It is not raining’ expresses a proposition which is true if the proposition expressed by the sentence ‘It is raining’ is false and false if the proposition expressed by the sentence ‘It is raining’ is true.

That the truth-connectives can be used for forming molecular complexes of T-expressions and of d- and f-expressions is no more problematic than that they can be used for forming molecular complexes of p-expressions. For p- and T- and df-expressions all belong to (formalized) descriptive discourse. They are schematic forms of sentences which express propositions. As schematic forms of sentences which are used for giving prescriptions O- and P-expressions belong to prescriptive discourse. It is not clear that truth-connectives have a meaningful use in prescriptive discourse at all.

The words for truth-connectives in ordinary language are ‘not’ ‘and’ ‘or’ and a number of others. It is easy to note that these words have a use in prescriptive discourse too. ‘Shut the window and open the door’ ‘You may not park here’ ‘Stop smoking or leave the room’.

The mere fact however that the words ‘not’ ‘and’ etc. are used in prescriptive discourse does not settle the question whether truth-connectives can be used for forming molecular complexes of O- and P-expressions. Of course we can use the signs ~ & ∨ etc. for forming complexes of O- and P-expressions. But such use would challenge the question what the complexes thus formed mean and whether the meaning of ~ & etc. in prescriptive language is sufficiently like their meaning in descriptive language to warrant the use of the same symbols.

It is here relevant to point out that norms at least of the kind we call prescriptions are neither true nor false. If O- and P-expressions are schematic forms of sentences which are used for giving prescriptions then molecular complexes of such expressions would not express truth-functions of their constituent parts. This alone would mark them as logically different from molecular complexes of p- T- and df-expressions.

O- and P-expressions can be regarded as the ‘formalized’ equivalents of deontic sentences (O- expressions also as formalizations of imperative sentences). As we know (Ch. VI Sect. 9) deontic sentences in ordinary usage exhibit a characteristic ambiguity. Sometimes they are used as norm-formulations. We shall call this their prescriptive use. Sometimes they are used for making what we called normative statements. We call this their descriptive use. When used descriptively deontic sentences express what we called norm-propositions. If the norms are prescriptions norm-propositions are to the effect that such and such prescriptions ‘exist’ i.e. have been given and are in force (see Ch. VII Sect. 8).

In view of this ambiguity the question may be raised whether O- and P-expressions should be regarded as formalized norm-formulations or as formalized sentences expressing norm-propositions.

One way of answering the question would be to decide that O- and P-expressions shall be consistently understood prescriptively as norm-formulations. Then we should have to introduce if needed a special symbolism for sentences which express norm-propositions.

Another way of answering the question would be to let O- and P-expressions retain the same ambiguity as deontic sentences in ordinary language. Retaining the ambiguity needless to say must not lead to confusion. We should then have not two symbolisms but two interpretations of the same symbolism. I shall call them the prescriptive and the descriptive interpretation of O- and P-expressions. Prescriptively interpreted these expressions are (formalized) norm-formulations. Descriptively interpreted they are (formalized) sentences which express norm-propositions.

I shall here decide in favour of the second answer. It will save us the trouble of doubling our symbolism.

That truth-connectives can be used for forming molecular complexes of descriptively interpreted O- and P-expressions is clear and uncontroversial. The molecular complexes express truth-functions of the norm-propositions expressed by the atomic O- and P-expressions which occur in the complexes.

The question open to debate is whether truth-connectives can be used for forming molecular complexes of prescriptively interpreted O- and P-expressions.

We can settle this question in the affirmative only at the cost of introducing an ambiguity in the meanings of the truth-connectives. We should have to distinguish between a descriptive or truth-functional meaning of the signs ~ & etc. and a prescriptive or non-truth-functional meaning of them.

This distinction would be thoroughly sensible. The words of ordinary language ‘not’ ‘and’ etc. sometimes have a truth-functional meaning as e.g. in ‘The window is shut and the door is open’. Sometimes they have a non-truth-functional meaning as e.g. in ‘Shut the window and open the door’. If someone prefers to speak of ‘function’ or ‘use’ instead of ‘meaning’ I shall not object. One must not break one's head over the question whether ‘and’ means the same thing or not in the two sentences which we cited. But it is important to note that the first sentence constructed by means of the word ‘and’ from two other sentences expresses a truth-function of the propositions expressed by those other sentences whereas the second sentence constructed by means of ‘and’ does not do this.

We shall here decide to use ~ & etc. only in the truth-functional way. This means that we settle the above question in the negative. Truth-connectives cannot (will not) be used for forming molecular complexes of prescriptively interpreted O- and P-expressions. In other words: molecular complexes of O- and/or P-expressions will always be interpreted descriptively as schematic forms of sentences expressing norm-propositions.

The question may be raised whether this is a practical decision. Since we have decided to retain in the formalism the ambiguity of deontic sentences in ordinary usage why not retain in the formalism also the ambiguity of using the connectives sometimes truth-functionally sometimes non-truth-functionally? The practicality of the decision will have to show itself in the sequel. Be it only observed in this place that although we shall study also non-truth-functional uses of the connectives it will not be necessary for our purposes to duplicate the symbolism for the connectives.

The decision which we have taken answers (settles) the question which we raised at the beginning of the present section. But it also raises a number of new questions.

One such question is whether the Logic of Norms which we are building is a logical study and theory of descriptively or of prescriptively interpreted O- and P-expressions. I do not myself know what is the best answer to this question. The ‘fully developed’ system of Deontic Logic is a theory of descriptively interpreted expressions. But the laws (principles rules) which are peculiar to this logic concern logical properties of the norms themselves which are then reflected in logical properties of norm-propositions. Thus in a sense the ‘basis’ of Deontic Logic is a logical theory of prescriptively interpreted O- and P-expressions.

Another question is what relevance to the logic of norms the prescriptive use of the connectives ‘not’ ‘and’ etc. may possess. This too is a question which I do not know how to answer in straightforward terms. That the prescriptive use of the connectives is relevant will however be plain from the subsequent discussion.

3. We introduce the notion of a (self-) consistent norm. A norm will be called (self-) consistent if and only if the norm-content is consistent. Conversely a norm will be called inconsistent if and only if its content is inconsistent.

The conditions of consistency (and inconsistency) of df-expressions we have investigated earlier (Ch. IV Sects. 3 and 4). A handy way of laying down the conditions is to say that a df-expression is consistent if and only if it has a (not-vanishing) positive normal form. Atomic O- and P-expressions are thus consistent if and only if the df-expression which follows after the letter O or P is consistent.

The ontological significance of this notion of a consistent norm is not clear in itself. That a p-expression (formula of propositional logic) is consistent means (‘ontologically’) that the state of affairs which it describes can obtain. Or strictly speaking: it means that the state can obtain so far as the principles of the Logic of Propositions are concerned. There may however be other reasons of logic why the described state is impossible. Similarly that a T-expression is self-consistent means that the change which it describes can happen (take place)—as far as the principles of the Logic of Change are concerned. That a df-expression is self-consistent means that the action which it describes can be performed—as far as the principles of the Logic of Action are concerned.

Could the self-consistency of O- and P-expressions mean anything analogous to this? As norm-formulations (of prescriptions) such expressions do not describe anything. They prescribe i.e. order or permit certain actions. It is not clear by itself why a prescription should be called consistent if the prescribed action can be performed and inconsistent if it cannot be performed.

It is clear that it is logically impossible for one and the same agent to do and forbear the same thing on the same occasion. But is it logically impossible to command or permit an agent to do and forbear the same thing on the same occasion? If commanding and permitting consisted just in shouting out certain words to him then this would not be impossible. Surely I can address somebody with the words e.g. ‘Shut the window and leave it open’ and even threaten him with punishment if he does not obey. But does this mean that I have commanded him? The answer depends upon what we think that commanding is—wherein the giving of commands consists. The answer in other words depends upon the solution to what we called the ontological problem of norms (prescriptions).

We discussed this problem in the last chapter. We now begin to see the relevance of this discussion to the problems of formal logic with which we are dealing in this chapter.

We took the view that a prescription of O- character expresses or manifests a will to make agents do or forbear certain things and a prescription of P-character a will to let agents do or forbear certain things. We also took the view that the normative relationship in the existence of which the existence of the prescription consists cannot materialize unless the subject(s) of the prescription can do or forbear those things which the authority of the prescription wants to make or let him (them) do or forbear. If for reasons of logic these things cannot be done (and forborne) one cannot make or let agents do or forbear them. Therefore neither can one command or permit or prohibit them to agents. Such prescriptions cannot ‘exist’.

Our definitions of consistent and inconsistent prescriptions thus amount to saying that accepting a certain view of the ontology of norms consistent prescriptions are such as can exist and inconsistent prescriptions such as cannot exist—as far as logic is concerned.

4. It is a function of the word ‘not’ in descriptive language to negate i.e. to express propositions of the opposite truth-value to the propositions expressed by those sentences to which the word is being attached or added. This is not the only function of ‘not’ in descriptive language but it is perhaps its most important function.

In order to find an analogue to negation in prescriptive language we have to study how the word ‘not’ when attached to or inserted in sentences used for enunciating prescriptions affects or changes the meaning of the original sentence. In particular we have to consider whether the relationship between the meaning of a norm-formulation with and the meaning of a ‘corresponding’ norm-formulation without the word ‘not’ in it is sufficiently like the relation between a proposition and its negation to justify us in speaking about a prescription (norm) and its negation. That ‘not’ is used in prescriptive language as well as in descriptive language is easy to note. But from this alone it does not follow that the function of ‘not’ in prescriptive language is to negate nor is it at all clear what ‘negating’ means in prescriptive contexts.

Consider the atomic O-expression Od( ~pTp). We can think of it as enunciating a command to open a window. In ordinary language this command could be expressed in an imperative sentence ‘Open the window’ or in a deontic sentence ‘The window ought to be opened’ (‘You ought to open the window’).

What could be the meaning of the expression not-Od( ~pTp)? The only meaningful insertion of a negation into the imperative sentence ‘Open the window’ is to form of it the sentence ‘Don't open the window’. It expresses a prohibition to open the window. It thus answers to the symbolic form Of( ~pTp). Shall we say that the ‘negation’ of an order to do a certain thing is an order to forbear this same thing? It is soon seen on reflexion that even if Of( ~pTp) is a possible interpretation of not-Od( ~pTp) it is not the only possible interpretation of it and hardly the most plausible one.

The insertion of ‘not’ into the above deontic sentences yields ‘You ought not to open the window’ and ‘The window ought not to be opened’. They admit as norm-formulations of two interpretations. One is to understand them as expressing a prohibition to open the window. Then they answer to the symbolic form Of( ~pTp). The other is to understand them as expressing a permission to leave the window closed. Then they answer to the symbolic form Pf( ~pTp). Here we have a second candidate for the position of ‘negation’ of our original expression Od( ~pTp).

There is however still a third possibility to be considered. ‘You ought not to open the window’ could be understood to mean that you have not been ordered to open the window that no such command (prescription norm) has been given to you. When thus understood the deontic sentence with the ‘not’ in it is not a norm-formulation. It is a descriptive sentence which expresses a norm-proposition.

Consider next the atomic P-expression Pd( ~pTp). An instantiation of it could be a permission to open a window. In ordinary language the permission could be expressed in the words ‘You may open the window’ or ‘You are allowed to open the window’.

What could not-Pd( ~pTp) mean? In order to find an answer consider how we should understand the sentences ‘You may not open the window’ or ‘You are not allowed to open the window’.

It is obvious that there are two possible interpretations. The first is to regard the sentences with ‘not’ in them as enunciating a prohibition to open the window. The words ‘may not’ then mean the same as ‘must not’. The meaning of the negated formula could also be expressed by Of( ~pTp).

On the second interpretation the sentences with ‘not’ in them are not prescriptive but descriptive. They say that there is not a permission to open the window that no such permission has been given and is in force. Generally speaking not-Pd( ~pTp) then expresses a norm-proposition to the effect that there does not exist a permission to transform a ~p-world to a p-world.

By similar arguments we easily see that there are three candidates for the position of ‘negation’ of the atomic O-sentence Of( ~pTp) and two for the position of ‘negation’ of the atomic P-sentence Pf( ~pTp). The three possible meanings of not-Of( ~pTp) are given by the sentences ‘p ought to be done’ ‘p may be done’ and ‘There is no prohibition to the effect that p must not be done’. The two possible meanings of not-Pf( ~pTp) again are expressed by ‘p ought to be done’ and ‘There is no permission to the effect that p may be left undone’.

As we notice the role of negation in prescriptive language is bewildering. Sentences which originate from norm-formulations thanks to the insertion of the word ‘not’ in them are grammatically correct and well known from ordinary discourse. But their meaning is unclear. Or rather: the sentences exhibit characteristic ambiguities. Several ‘candidates’ as we have said for the position of the negation (of the meanings) of atomic O- and P-expressions emerge. We shall have to make a choice between them. Then the question what justifies the choice will arise. This question is but a special case of the general question of the criteria for calling one entity the ‘negation’ of another.

We cannot here discuss the problem of negation in general. As already observed in Section 2 the concept of negation is primarily at home in descriptive discourse and the realm of propositions. Even here it is a controversial notion. One way of defining it would be to lay down the following five requirements which the negation of a given proposition has to satisfy:

1. (i) The negation of a given proposition shall be a proposition.
2. (ii) Negation shall be unique i.e. there shall be one and only one negation of a given proposition.
3. (iii) Negation shall be reciprocal i.e. if a second proposition is the negation of a first proposition then the first is the negation of the second.
4. (iv) A given proposition and its negation shall be mutually exclusive i.e. it must not be the case that they are both true or both false.
5. (v) A given proposition and its negation shall be jointly exhaustive i.e. it must be the case that one or the other of the two is true.

(Logicians of the so-called intuitionist school would dispute that a proposition and its negation need be jointly exhaustive.)

If we apply mutatis mutandis these four requirements to the notion of the negation of a norm the first would say that the negation of a norm shall be a norm. And this would at once disqualify the interpretations of not-Od( ~pTp) etc. as expressing norm-propositions as possible candidates for the position of negations of the norms expressed by Od( ~pTp) etc.

The proposition expressed by the sentence ‘There is not an order to the effect that p ought to be done’ can correctly be said to be the negation of the proposition expressed by the sentence ‘There is an order to the effect that p ought to be done’. But we shall not call it the negation of the prescription (norm) expressed in the words ‘p ought to be done’. When not-Od( ~pTp) is interpreted descriptively as expressing a norm-proposition the part Od( ~pTp) in it must be interpreted descriptively too.

As indicated in Section 2 for the descriptive interpretation of not-Od( ~pTp) and for it only we shall use the symbol ~Od( ~pTp).

For the prescriptive interpretation of not-Od( ~pTp) we need no new symbols. The reason why we need no special symbol for ‘not’ in prescriptive language may be gathered from our discussion above of the possible prescriptive meanings of not-Od( ~pTp) etc. When not-Od( ~pTp) etc. were not interpreted as sentences expressing norm-propositions they were interpreted as identical in meaning with certain atomic O- and P-expressions.

We now return to the question of selecting ‘candidates’ for the position of a ‘negation’ of a norm (prescription).

Of the two candidates for the negation of the norm expressed by Od( ~pTp) we dismiss Of( ~pTp) and of the two candidates for the negation of Of( ~pTp) we dismiss Od( ~pTp). After these rejections the remaining candidate for the negation of the norm expressed by Od( ~pTp) is Pf( ~pTp) for the negation of Pd( ~pTp) it is Of( ~pTp) for the negation of the norm expressed by Of( ~pTp) it is Pd( ~pTp) and for the negation of the norm expressed by Pf( ~pTp) it is Od( ~pTp).

The reason for the rejections is that we want negation to satisfy the requirements of uniqueness and reciprocity ((ii) and (iii) above). The requirement that the negation of a norm shall be a norm we satisfy through a decision to stick to the prescriptive interpretation of the atomic O- and P-expressions throughout.

Now consider the two pairs:

Our suggestion is that the norms which the members of each pair of norm-formulations express are related to one another as a norm and its ‘negation’. On our suggestion the negation of a positive command is thus a negative permission and conversely and the negation of a negative command is a positive permission and conversely. In still other words: a command to do and a permission to forbear are related to one another as negations and so are a command to forbear and a permission to do.

This notion of a norm and its negation-norm can be generalized. We previously introduced the notions of external and internal negations of df-expressions i.e. of possible norm-contents (see Ch. IV Sect. 6). It is readily seen that the contents of the members of each pair of norms above are related to one another as internal negations. Their characters are ‘opposite’ i.e. one has the O-character and the other the P-character. Our generalized definition of the notion of a negation-norm now runs as follows:

A norm is the negation-norm of another norm if and only if the two norms have opposite character and their contents are the internal negations of each other.

Consider for example the norm expressed by O(d( ~pTp) ∨ f( ~pT ~p)). It says that one ought to produce the state of affairs described by p or let it happen depending upon the nature of the occasion. Its negation-norm is expressed by P(f( ~pTp) ∨ d( ~pT ~p). It says that one may leave the state of affairs described by p unproduced or suppress it.

Similarly the negation of the norm expressed by O(d( ~pTp) & d( ~qTq)) is the norm expressed by P(d( ~pTp) & f( ~qTq) ∨ f( ~pTp) & d( ~qTq) ∨ f( ~pTp) & f( ~qTq)). The first orders the production of two states of affairs. The second permits the leaving of at least one of the two states unproduced.

Does the concept of a negation-norm satisfy mutatis mutandis the requirements (iv) and (v) above? Are a given norm and its negation-norm mutually exclusive and jointly exhaustive?

Before we can answer these questions it ought to be made clear what mutatis mutandis should be understood by mutual exclusiveness and joint exhaustiveness in prescriptive discourse. It is near at hand to define the notions in a manner which is analogous to our definition in Section 3 of consistency. A proposition is consistent if it can be true a norm we said if it can exist. Similarly we could say that two norms are mutually exclusive if they cannot both exist i.e. co-exist and jointly exhaustive if at least one of the two must (will necessarily) exist.

In order to answer the question whether a norm and its negation-norm are in the sense defined mutually exclusive we ought to give criteria for the possible co-existence of norms. This we shall do in the next section. It will then be seen that the answer to our question is affirmative—though with an important qualification (cf. below Section 7).

The question whether a norm and its negation-norm are in the sense defined jointly exhaustive leads to the problem of necessary existence of norms. It too will be discussed later (Section 8). We shall find that the answer to the question concerning joint exhaustiveness is negative. By virtue of this the notion of negation in prescriptive discourse has a certain resemblance to the intuitionist notion of negation.1

5. Possibility of co-existence of norms we could say is the ontological aspect or significance of the formal notion of compatibility of norms. Before we turn to the ontological aspect we shall have to define and comment on the formal notion.

The compatibility of two or more norms we shall also call the mutual consistency of two or more norms. A set of compatible norms will be called a consistent set of norms.

It will be assumed throughout that the norms whose compatibility we are defining and discussing are (self-) consistent norms. How the problem of compatibility is to be treated for norms which do not satisfy the condition of (self-) consistency I shall not discuss in detail. The problem seems of minor importance. A set of norms at least one member of which is not (self-) consistent may on that account be called an inconsistent set.

The problem before us can now be put as follows: Which conditions should a set of (self-) consistent norms satisfy in order that the set be consistent the norms compatible?

We shall conduct the discussion in three steps. First we define consistency for a given set of norms all of which are norms of the O-character. Then we assume that a given set of norms contains only norms of the P-character. Finally we define consistency for a given set of norms some of which are of the O- and others of the P-character.

We can speak of the three kinds of sets of norms as an O-set a P-set and an O + P-set (or ‘mixed set’) respectively.

To the sets of norms there answer sets of norm-formulations i.e. (atomic) O- and/or P-expressions. It is convenient to conduct the investigation speaking in the first place of the expressions and their formal properties—not forgetting that the relevance of our talk is ultimately for the norms themselves. When talking of the O- and P-expressions it will throughout be assumed that the df-expressions in them are in the positive normal form. It will be assumed moreover that the normal forms are made uniform (Ch. IV Sect. 8) with regard to all atomic p-expressions (variables p q etc.) which occur in the entire set of O- and/or P-expressions.

(i) We consider a set of O-norms and a corresponding set of O-expressions.

We make a list of all the atomic p-expressions which occur in the O-expressions. Let the number of atomic p-expressions be n. Thereupon we list the 2ⁿ state-descriptions which answer to these n atomic p-expressions. Next we list the 2ⁿ×2ⁿ change-descriptions which answer to these 2ⁿ state-descriptions. These change-descriptions constitute a complete list of the conditions of application of the O-expressions i.e. norms in the set.

For each one of the conditions of application we make a list of those parts if any of the (uniformed) normal forms of the df-expressions in the respective O-expressions which answer to those conditions. These lists tell us what the individual norms require to be done under the respective conditions. Thereupon we form the conjunction of the members of each of these lists. These conjunctions tell us what the totality of norms requires to be done under the respective conditions. It is not certain that there are as many conjunctions as there are conditions of applications in the complete list. For it can happen that under some of the conditions none of the norms applies.

The conjunctions are df-expressions. We examine whether they are consistent. This can be done according to several methods. We can for example transform the conjunctions into their positive normal forms. If this is not-vanishing the conjunctions are consistent. The result of these transformations however can be immediately read off from the conjugated expressions themselves. These are parts of the uniformed positive normal forms of the df-expressions in the O-expressions of our set. They are thus disjunctions of conjunctions of elementary d- and f-expressions. The conjugated disjunctions are consistent if and only if they have at least one common disjunct i.e. conjunction of elementary d- and f-expressions. Otherwise they are inconsistent.

Assume that none of the conjunctions is inconsistent. Then and then only the O-set of norms is consistent its members compatible.

Assume that some (at least one) of the conjunctions are inconsistent. Then the O-set of norms is inconsistent its members incompatible.

Assume finally that all conjunctions are inconsistent. Then we have a special form of inconsistency and incompatibility which we shall call ‘absolute’.

When the conjunction which answers to a given condition of application of the norms is consistent we shall also say that the set of norms is consistent and its members compatible under those conditions. When a given conjunction is inconsistent we say that the set of norms is inconsistent and its members incompatible under those conditions.

Thus on our definitions a consistent set of O-norms is consistent under all conditions of application of the norms—but an inconsistent set may be consistent under some conditions of application.

Our definition of a consistent set of O-norms amounts to this: a set of commands is consistent (the commands compatible) if and only if it is logically possible under any given condition of application to obey all commands (collectively) which apply on that condition.

(ii) We next consider a set of self-consistent P-norms.

Such a set is ipso facto consistent. Permissions never contradict each other. This is one of the basic logical differences between commands and permissions. To take the simplest possible illustration: a command to do a certain thing is incompatible with a command to forbear this same thing on a given occasion (see Section 6). But a permission to do a certain thing is not incompatible with a permission to forbear this same thing on a given occasion. The ‘ontological’ significance of this difference between commands and permissions we shall discuss later.

(iii) We finally consider a mixed set of self-consistent O- and P-norms commands and permissions and the corresponding set of expressions.

To find and formulate the conditions of consistency of the set we first divide it into two parts or sub-sets. One consists of all the O-norms in the set the other of all the P-norms. We call the two sub-sets the O-part and the P-part of the mixed set. The P-part is ipso facto consistent. It is a condition of the consistency of the whole set that the O-part should be consistent. But this is not the sole condition of consistency.

We make a list of all the atomic p-expressions which occur in the O- and P-expressions of our mixed set. We then construct the corresponding lists of state-and change-descriptions. The list of change-descriptions includes all conditions of application of the norms in our mixed set.

Consider now the sub-set which consists of the O-part of the whole set and one of the members of the P-part.

For each of the conditions of application we list the parts if any of the normal forms of the df-expressions which occur in the O-expressions and the one P-expression of our sub-set (of expressions). We form the conjunctions of the members of each list and test the conjunctions for consistency.

If all conjunctions are consistent we say that the sub-set of norms is consistent and the members of the sub-set compatible. We then also say that the one P-norm is compatible with the (set of) O-norms.

If some (at least one) conjunction is inconsistent the sub-set of norms is inconsistent and in particular the one permissive norm incompatible with the (set of) commands.

If none of the conjunctions is consistent the sub-set is absolutely inconsistent. If none of the conjunctions which answer to the several conditions of application of the P-norm is consistent then the permissive norm is absolutely incompatible with the commands.

We repeat this procedure for all the members individually i.e. one by one of the P-part of our mixed set of norms. The definition of consistency of the mixed set is as follows:

A mixed set of norms is consistent its members compatible if and only if each one of the members of its P-part is individually compatible with its O-part.

If some member of the P-part of the set is incompatible with the O-part then the mixed set is inconsistent.

Our definition of consistency and compatibility also amounts to this: a set of commands and permissions is consistent (the norms compatible) if and only if it is logically possible under any given condition of application to obey all the commands collectively and avail oneself of each one of the permissions individually which apply on that condition.

6. We shall next mention and comment on some consequences of our definitions of compatibility and incompatibility of norms.

A first consequence is that a norm and its negation-norm (Section 4) are on our definition incompatible. This is seen as follows:

A norm and its negation have opposite characters. It can thus not happen that both are permissions. (Permissive norms never contradict one another.) Their contents are internal negations of one another. This entails that the two norms have the same conditions of application. The conjunction of those parts of the normal forms of the expressions for the contents of the two norms which answer to given conditions of application is inconsistent. (This follows from the definition of internal negation.) Hence the two norms are incompatible. Since moreover they are incompatible under all their conditions of application they are absolutely incompatible.

It follows at once from this that two norms of O-character whose contents are the internal negations of one another are (absolutely) incompatible. For example: the commands expressed by Od( ~pTp) and Of( ~pTp) are absolutely incompatible.

The above results concerning the incompatibility of norms hold also for the general case when the contents are internally incompatible and not only for the special case when the contents are the internal negations of one another. Thus two norms of opposite character whose contents are internally incompatible are (absolutely) incompatible. And two norms of O-character whose contents are internally incompatible are (absolutely) incompatible.

For example: The commands expressed by O(d(pTp) & d(qTq)) and O(d(pTp) & f(qTq)) are incompatible and so are the command expressed by O(d(pTp) & d(qTq)) and the permission expressed by P(d(pTp) & f(qTq)).

The results can easily be generalized to sets of norms. A set of commands is inconsistent if the contents of two of its members are internally incompatible. A permission is incompatible with a set of commands if the content of the permission is internally incompatible with the content of one of the commands. (These incompatibilities of norms are not necessarily absolute.)

It is important to observe that mere incompatibility of the contents of two commands or of a command and a permission does not on our definition entail an incompatibility of the norms. The incompatibility of the contents must be internal.

The case when there is external but not internal incompatibility between the norm-contents has sometimes interesting logical peculiarities. We shall here consider one such peculiarity. It will first be illustrated by means of an example.

Consider the two commands Od( ~pTp) and Od(pT ~p). We could think of the first as an order to open a window and of the second as an order to close this same window. Do the commands contradict each other? Are they incompatible? Perhaps in some special sense of ‘contradict’ and ‘incompatible’ but certainly not in the sense which we have here given to the terms. The reason why on our definition the norms are not incompatible although their contents contradict each other is that they have no common condition of application. The second command applies to a world in which the state of affairs described by p obtains and does not independently of action vanish; the first to a world in which this state does not obtain and does not independently of action come into existence.

Compare the above commands with Od( ~pTp) and Of( ~pTp) for example with an order to open a window and an order to leave this same window closed. They contradict each other on our definition because whatever an agent does on an occasion when both commands apply he will necessarily disobey one of them. On an occasion when a certain window is closed and does not open of itself an agent who masters the art of window-opening will necessarily either open this window or leave it closed. But he will not necessarily either open this window or close it. Therefore he will necessarily disobey one of the pair of orders Od( ~pTp) and Of( ~pTp) but not necessarily disobey one of the pair of orders Od( ~pTp) and Od(pT ~p). The last order he can neither obey nor disobey on the occasion in question.

Let it be assumed that the two orders Od( ~pTp) and Od(pT ~p) are given for one single occasion only. Then they mean in terms of our window-illustration that the agent to whom the orders are given should close the window if it is open and open it if it is closed (on that occasion). In practice an authority would give both orders only if he does not himself know what the state of the world is or will be on the occasion in question. There is nothing uncommon or odd about such cases.

Let it be assumed that the two orders are general with regard to the occasion (see Ch. V Sect. 11). Then they mean in terms of our illustration that the agent to whom the orders are given should close the window whenever he finds it open and open it whenever he finds it closed. Now assume that the first order applies to the situation at hand and that the agent obeys and closes the window. Thereby he creates a situation to which the second order becomes applicable. He ought now to open the window. If he obeys he creates a situation to which the first order applies. And so forth ad infinitum. The case is noteworthy—also from a logical point of view.

I shall say that the two general orders jointly constitute a pair of Sisyphos-orders. Generally speaking: a set of orders which are general with regard to the occasion will be said to constitute a set of Sisyphos-orders if and only if obedience to all the orders which apply under given conditions of application necessarily creates new conditions of application (of some or all of the orders).

One could introduce a notion of deontic equilibrium. The world we shall say can be brought to deontic equilibrium with a (consistent) set of orders if it is possible to obey all the orders which apply to any given state of the world without creating ad infinitum a new state of the world to which some of the orders apply. The two orders to open a certain window whenever possible and to close it whenever possible form a consistent set—but the world cannot be brought to deontic equilibrium with it.

To issue Sisyphos-orders such as ‘Open the window whenever it is closed and close it whenever it is open’ may be cruel. But it is not nonsensical in the same sense in which to issue inconsistent orders such as ‘Open the window but leave it closed’ is nonsensical.

7. In order to see the ontological significance of the conditions of consistency (and compatibility) we shall consider in some detail the case of two commands the content of one of which is the internal negation of the content of the other. Why is it let us ask that a command to open a window and a prohibition to do this i.e. a command to leave it closed contradict each other are incompatible?

It is here pertinent to note that the two commands (the command and the prohibition) can be reasonably said to contradict each other only if they refer to the same window are addressed to the same agent and are for the same occasion. If on an occasion when a certain window is closed I ask a person to open it and on another occasion when this same window is again closed I ask the same or another person to leave the same window closed there is no contradiction between my orders. But if I command a person to open a window and command the same person to leave the same window closed on the same occasion then it would seem I can rightly be accused of contradicting myself logically. The two commands annihilate one another they cannot exist together ‘in logical space’ as one might put it.

But on the other hand: if x orders z to open a window and y prohibits z to open the same window on the same occasion is there then contradiction? It is true that it is logically impossible for z to obey both orders. But is it logically impossible for the two orders to coexist? Is there not room for them both in logical space? It seems off-hand reasonable to think that they can coexist. On the view which I have here taken of the nature of commands and prescriptions generally this seems plausible too. On this view the coexistence of the two commands which we just mentioned (normally) means that x wants z to open the window and y wants z to leave it closed on the same occasion. This is no logical contradiction; but it can truly be called a ‘conflict’. It is an instance of what I shall call a conflict of wills.

Now then: Why is it logically possible for x to command z to open the window and for y to command z to leave it closed but not logically possible for x to command z to open the window and at the same time to prohibit him to do this? Or is this last after all possible too? Can commands or norms in general ever contradict one another?

I wish I could make my readers see the serious nature of this problem. (It is much more serious than any of the technicalities of deontic logic.) It is serious because if no two norms can logically contradict one another then there can be no logic of norms either. There is no logic we might say in a field in which everything is possible. So therefore if norms are to have a logic we must be able to point to something which is impossible in the realm of norms. But that we can do this is by no means obvious.

It is important to realize that it will not do to answer the question why it should be called logically impossible to command and prohibit the same thing by saying that this is impossible because it is logically impossible for one and the same man both to do and forbear one and the same thing at the same time. For if I order a man to do something and you prohibit him to do the same it is also logically impossible that the man should obey both of us but nevertheless perfectly possible that there should be this command and this prohibition.

Commands as we have said earlier manifest efforts to make people do or forbear things. It is clear that one cannot on the same occasion make the same man do and forbear the same thing since it is logically impossible for a man to do and forbear the same thing at the same time. It is also clear that I can try to make him do the thing and you try on the same occasion to make him forbear the thing—although it is logically impossible that we should both succeed. So why could it not be that one man on the same occasion should try to make another agent both do and forbear the same thing? Well how does a norm-authority try to make people do or forbear things? By threats of punishment before the act and by punitive measures when disobedience has taken place and in other ways (cf. Ch. VII Sect. 14). If someone were to punish a child in one way if it does a certain thing and also to punish it though perhaps in a somewhat different way if the child abstains from doing this same thing—can he then be said both to try to make the child do this thing and to try to make him abstain from doing it? In the absence of criteria we can say nothing at all. The concept of trying has still to be moulded to fit this case. But we should certainly feel inclined to say that such behaviour as that which we just described looks queer and purposeless. And if the agent described to us his own action by saying that he tries to make the punished child do and also tries to make it forbear the same act we should say that we do not understand him or that he behaves irrationally or perhaps even that he is mad.

We can illustrate the problem in pictures. A man a is walking along with another man s. a has a cane or whip in one hand and holds a rope with the other hand. The rope is tied round the waist of s. (It may be more attractive to the imagination to think of s as a dog rather than as a man.) They pass by various objects. Sometimes when they come to an object a drives s towards the object with the whip. Sometimes he pulls him back with the rope. Sometimes he lets s go towards the object if s wants to. Sometimes he lets s turn away from the object. These four cases answer to the four basic norm-situations of positive and negative command and positive and negative permission respectively.

	push on	Od-case
	pull back	Of-case
	let go	Pd-case
	let retreat	Pf-case

Now comes another man b. He also has a whip in one hand and a rope in the other. He ties the rope round s's waist a and b both walk along with s. Sometimes when a threatens s with the cane and urges him towards an object b pulls him back. Then a and b try to make s do opposite things s cannot please both his masters; it is logically impossible for him to do so. But this does not make it impossible for a to go on hitting s with the whip or for b to pull in the rope. There is nothing illogical or even irrational in this.

Remove b from the picture a is alone with s. When they pass by a certain object a drives s on to it with the whip and holds him back with the rope. Can s do this? I just described it in intelligible terms. We as it were see it happen in the imagination. The question is very much like this: Can a man both push and pull in opposite directions one and the same object at the same time? He can pull it with one hand and push it with the other and the object will move in the direction of whichever hand is the stronger. He could do this to test which hand of his is the stronger. But if he said that he does this because he wants to make the object move in the one direction and also wants to make it move in the other direction we should think that he was joking with us or was mad. A psychologist would perhaps speak of him as a ‘split personality’. He acts as two men would act who contested about the object.

The upshot of this argument is as follows:

That norms can contradict each other logically is not anything which logic ‘by itself’ can show. It can be shown if at all only from considerations pertaining to the nature of norms; and it is far from obvious whether it can be shown even then. The only possibility which I can see of showing that norms which are prescriptions can contradict one another is to relate the notion of a prescription to some idea about the unity and coherence of a will.

Of the will which does not make incompatibilities its objects it is natural to use such attributes as a rational or reasonable or coherent or consistent will.

The ontological significance of the formal notion of compatibility of norms is possibility of coexistence we said at the beginning of Section 5. We now realize that at least so far as prescriptions are concerned the identification of compatibility with possibility of co-existence is subject to an important qualification. The prescriptions must have the same authority. (This was the qualification to which we alluded at the end of Section 4.)

I shall here introduce the notion of a corpus of norms. By this I understand a set of prescriptions which all have the same authority.

Thus for prescriptions the ontological significance of compatibility is the possibility of coexistence within a corpus. The consistency of a set of prescriptions means the possibility that the set constitutes a corpus. Incompatibility of prescriptions means the impossibility of their coexistence within a corpus. The inconsistency of a set of prescriptions finally means the impossibility i.e. necessary non-existence of a certain corpus.

Contradiction between prescriptions can be said to reflect an inconsistency (irrationality) in the will of a norm-authority. One and the same will cannot ‘rationally’ aim at incompatible objects. But one will may perfectly well ‘rationally’ want an object which is incompatible with the object of another ‘rational’ will. Because of the first impossibility prescriptions which do not satisfy our formal criteria of compatibility cannot coexist with a corpus of norms. Because of the second possibility prescriptions which do not satisfy these criteria can yet exist within different corpora and in this sense coexist.

In terms of the will-theory of norms the inconsistency of a set of commands means that one and the same norm-authority wants one or several norm-subjects to do or forbear several things which at least in some circumstances it is logically impossible conjunctively to do or forbear.

In terms of the will-theory the inconsistency of a set of commands and permissions means this: one and the same norm-authority wants one or several norm-subjects to do or forbear several things and also lets them do or forbear several things. Something which the authority lets the subject(s) do or forbear is however at least in some circumstances logically impossible to do or forbear together with everything which he wants them to do or forbear. This too we count as irrational willing.

That permissions never contradict each other means that it is not irrational to let people do or forbear several things which it is not logically possible to do or forbear conjunctively on one and the same occasion. To let them do this is to let them freely choose their mode of action.

8. Self-inconsistent norms we have said cannot exist. They thus have what might also be called necessary non-existence. The question may be raised: are there norms which must exist or which have necessary existence?

The question can conveniently be divided into three:

(a) Are there norms which necessarily exist simpliciter?

(b) Are there norms which necessarily exist if certain other norms (as a matter of fact) exist?

(c) Are there norms which necessarily exist if certain other norms (as a matter of fact) do not exist?

The second of the three questions is the most important. It is virtually the same as the question of entailment between norms. We shall discuss it in the next section.

The first question may present interesting aspects e.g. in connexion with a theonomous view of morality. If God is a being endowed with necessary existence and if he has given a moral law to man must we then not also think that the moral commands exist ‘of necessity’? This kind of question we do not discuss in the present work at all. I have no idea how to answer or even how to tackle the question. But I do not think it can be dismissed as pure nonsense.

The only comment on question (a) which I shall make here concerns the notion of what I propose to call a tautologous norm. A norm of O- or of P-character will be called tautologous if and only if its content satisfies the following requirement: the positive normal form of the df-expression for the content contains as disjuncts all the act-descriptions which answer to some or several of the conditions of applications of the norm.

An example of a tautologous norm is the command expressed by O(d( ~pTp) ∨ f( ~pTp)). The symbolic expression for its content is in normal form. The normal form enumerates all the modes of action which are possible under the conditions expressed by ~pT~p. Another example is the command expressed by O(d( ~pTp) ∨ d ~pT ~p) ∨ f( ~pTp) ∨ f( ~pT ~p)). The normal form of its content enumerates all modes of action which are logically possible under the conditions expressed by ~pTpand by ~pT ~p. A third example of a tautologous norm is the permission expressed by P(d(pTp) & d(qTq) ∨ d(pTp) & f(qTq) ∨ f(pTp) & d(qTq) ∨ f(pTp) & f(qTq)). The modes of action which its content covers are all the modes which are possible under the conditions expressed by (pT ~p) & (qT ~q).

What does the command expressed by O(d( ~pTp) ∨ f( ~pTp)) require of the subject to whom it is addressed? Let p stand for ‘The window is open’. The demand then is to open or leave closed a window which is closed (and does not open ‘of itself’). Whatever the agent does in the situation in question he necessarily either opens the window or leaves it closed. (Assuming that this is an act which he can do.) The command therefore does not properly speaking ‘demand’ anything at all. This is why we call it tautological.

What does the above tautologous permission permit? It applies to a situation when both of two given states of affairs obtain but vanish unless prevented from vanishing. The permission is to prevent both from vanishing or to prevent the one but not the other or to let both vanish. Since this is what the agent will do anyway the permission properly speaking does not ‘permit’ anything at all.

Tautologous prescriptions are thus commands which do not demand anything or permissions which do not permit anything in particular. It is easily seen that the negation-norms of tautologous prescriptions are self-inconsistent prescriptions. Since these latter necessarily do not exist shall we say that the former do necessarily exist?

We could say this and no harm would follow. But we need not say this. The logically most appropriate reaction to the case seems to me to be to deny tautologous prescriptions the status of (‘real’) prescriptions. We exclude them from the range of the concept. The justification for this may be sought in our ontology of prescriptions. There is no such thing as making or (‘actively’) letting people do things which they will necessarily do in any case. Therefore it makes no sense to say that people are commanded or permitted to do such things either.

The only comment on question (c) which we shall make here concerns the relation of a norm to its negation-norm.

We have so far left open the question whether a norm and its negation-norm form an exhaustive alternative (cf. the discussion in Section 4). If they do then we could conclude from the factual non-existence of a norm to the existence of its negation-norm. The existence and non-existence of either norm may be in itself contingent. What would be necessary is that either the one or the other exists.

Let the norm be e.g. the command expressed by O(d(pTp) & d(qTq) ∨ f(pTp) & f(qTq)). Its negation-norm is then the permission expressed by P(d(pTp) & f(qTq) ∨ f(pTp) & d(qTq)). Must it necessarily be the case that an agent is either commanded to continue both of two states or to let them both vanish or permitted to continue one of them and to let the other vanish?

It is easily recognized that the problem whether a norm and its negation-norm form an exhaustive disjunction is a generalization of the problem which we discussed in Ch. V Sects. 13–16 of the mutual relations of the norm-characters of command and permission.

If we accept the view that a norm and its negation-norm form an exhaustive disjunction then we are forced to accept the inter-definability of the two norm-characters also. Permission would then be mere absence of a command (prohibition) ‘to the contrary’ but also command would be absence of permission ‘to the contrary.’ The exact meaning of the phrase ‘to the contrary’ is explained in terms of the relation between a norm and its negation-norm.

Since we have decided not to accept the view of permission as absence of prohibition (cf. Ch. V Sect. 16) we are therefore also forced to reject the idea that a norm and its negation-norm form an exhaustive disjunction. A norm and its negation-norm cannot both exist i.e. coexist within a corpus. But they can both be absent from a corpus.

9. We shall now define the notion of entailment between norms. Consider a consistent set of self-consistent norms and a self-consistent norm. We want to determine the conditions under which this single norm shall be said to be entailed by the set of norms.

We consider the negation-norm of the single norm. We add it to the set. We test the enlarged set of norms for consistency under each one of the conditions when the negation-norm applies. There are three possibilities as regards the results of the test. Either they are all positive or some are negative or they are all negative. In the third case we say that the negation-norm is absolutely incompatible with the original set of norms (cf. above Section 5). This is the possibility which is of relevance to entailment. For we define:

A consistent set of self-consistent norms entails a given self-consistent norm if and only if the negation-norm of the given norm is absolutely incompatible with the set.

Consider the command expressed by Od( ~pTp). We want to know whether an order to produce the state of affairs described by p is entailed by a set of prescriptions which have already been given. We consider the negation-norm expressed by Pf( ~pTp). We test whether a permission to leave this state unproduced is absolutely incompatible with the prescriptions of the set. That there is absolute incompatibility means that under no circumstances (conditions of application) could one avail oneself of a permission to leave the state in question unproduced without disobeying some of the commands (prohibitions) in the set of prescriptions. In other words: Only by producing the state in question can one obey the commands (prohibitions) which have already been given. In this sense the original set of prescriptions will be said to entail a command to produce this state.

Consider the prohibition expressed by Of( ~pTp). Is a prohibition to produce the state of affairs described by p entailed by a set of given prescriptions? Test the permission expressed by Pd( ~pTp) for compatibility with the set. Assume that it is absolutely incompatible with the set. This means that under no circumstances could one (avail oneself of a permission to) produce the state in question without disobeying some of the commands (prohibitions) in the set of prescriptions. Only by observing the prohibition to produce this state can one under all circumstances obey the commands (prohibitions) which have already been given. In this sense the original set of prescriptions entails the new prohibition.

Consider the permission expressed by Pd( ~pTp). Is a permission to produce the state of affairs described by p entailed by a set of prescriptions? We test the prohibition expressed by Of( ~pTp) for compatibility with the set. Assume that there is absolute incompatibility. This means that in no circumstances could one observe the prohibition to produce the state described by p without either disobeying some command or not being able to avail oneself of some permission among those which have already been given. Only by actually producing the state described by p can one under all circumstances obey all the commands and avail oneself of any one of the permissions which have already been given. In this sense the original set of prescriptions entails a permission to produce this state.

To the case of the negative permission expressed by Pf( ~pTp) applies mutatis mutandis what was said of the positive permission expressed by Pd( ~pTp).

When a set of norms entails a further norm we shall also say that the norms of the set jointly (‘conjunctively’) entail this further norm.

10. Prescriptions which are entailed by a given set of norms I shall call derived commands prohibitions and permissions.

One could speak of derived prescriptions as the commitments of a norm-authority or lawgiver. If it turns out that a lawgiver cannot under any conditions consistently with the prescriptions which he has already given order a certain act to be done then he has in fact permitted its forbearance. If he cannot consistently prohibit an act then he has in fact permitted it. If he cannot permit it he has forbidden it. If he cannot permit its forbearance he has commanded its doing.

The ontological significance of the word ‘cannot’ should be plain from our discussion (in Section 7) of compatibility. That an authority ‘cannot’ give a certain prescription consistently with other prescriptions which he has already given means that an attempt to give this prescription would signalize an inconsistency in his will. He would then want or allow things to be done which for reasons of logic cannot be done.

When we call the derived prescriptions ‘commitments’ this should be understood in a factual and not in a normative sense. We did not say that if a lawgiver cannot consistently prohibit an act then he ought to permit it etc. But we said that if he cannot forbid it he has permitted it etc.

That an authority has prohibited something entails that he can and is prepared to see to it that this thing is not done. He threatens prospective trespassers with punishment and takes steps to punish those who in fact disobey. In what sense if any can the authority be said to do this also with regard to the entailed prohibitions? Is it not logically possible that the authority shows great anxiety to make his will effective as far as his manifest prohibitions and orders are concerned but is completely indifferent towards the conduct of the norm-subjects as far as the derived prescriptions are concerned?

The answer to the last question is that such an attitude on the part of the norm-authority is not logically possible. For let us recall what on our definition it means to say that a certain prohibition is entailed by a given set of prescriptions. It means that it is not logically possible in any circumstances to do the prohibited thing without disobeying some orders or breaking some prohibitions which have already been given (are in the set). If therefore the authority manifests anxiety to make the norm-subjects obey these latter commands and prohibitions e.g. by punishing the disobedient he ipso facto also manifests anxiety to make the subjects observe the entailed prohibition.

The derived commands prohibitions and permissions of a corpus of prescriptions we could say are as much ‘willed’ by the norm-authority as the original commands prohibitions and permissions in this corpus. The derived norms are necessarily in the corpus with the original ones. They are there although they have not been expressly promulgated. Their promulgation is concealed in the promulgation of other prescriptions.

11. We shall now use the proposed definition of entailment for the purpose of proving some important entailment-relations between norms.

First we show that a O-norm of a given content entails a P-norm of the same content. For short: Ought entails May or Obligation entails Permission.

We conduct the proof in terms of an example. Its general significance should be immediately clear.

Consider the command expressed by Od(pTp) and the ‘corresponding’ permission expressed by Pd(pTp). The negation of the permissive norm is the command (prohibition) expressed by Of(pTp). We have to show that the first and the third norms are absolutely incompatible. This we have already done in Section 6. Thus the first norm entails the second.

Consider why it is not the case that May entails Ought e.g. that the permission expressed by Pd(pTp) entails the command expressed by Od(pTp). The negation of the command is the permission Pf(pTp). It is true that one cannot on one and the same occasion avail oneself both of a permission to do and of a permission to forbear one and the same thing. But this impossibility does not on our definition mean that the permissions were incompatible. Hence the proposed entailment does not follow either.

12. Consider an order to do one (or both) of two things each one of which can be done under the same conditions of application. An order to a person to stop smoking or to leave the room would be an example. I shall call this a disjunctive order (disjunctive obligation). A disjunctive order does not mean that the subject ought to do one thing or ought to do another thing. It means that the subject ought to do one thing or do another thing. The subject is normally free to choose between the two modes of conduct.

Compare this with an order to do one of two things which cannot be done under the same conditions of application. An order to a person to open a door or keep it open would be an example. This should be called neither a ‘disjunctive order’ nor a ‘disjunction of orders’. The order amounts in fact to two orders. The one is an order to do a certain thing should certain conditions be satisfied e.g. a certain door be closed. The other is an order to do a certain other thing should certain other conditions be satisfied e.g. a certain door be open but would close unless prevented. The two conditions are incompatible. The two orders can therefore never be both executed on the same occasion. The obedient subject will execute one or the other if there is an opportunity of executing either. The subject is here never free to choose between two modes of conduct. Instead of the disjunctive form of the order ‘Open the door or keep it open’ one could use the conjunctive form ‘Open the door if it is closed and keep it open if it is (already) open’. The conjunctive form makes it more plain that in fact two orders are being given.

The form of words ‘Open the door and keep it open’ would normally be used to enunciate an order first to open a door which is (now) closed and then keep it open i.e. not let it close (again). This is an order of different logical structure from either of the two orders which we have just compared viz the disjunctive order and the ‘conjunction’ of two orders. It commands two things to be done in a certain order of time. It cannot be resolved into two orders which are given for the same occasion. In this it differs from the order enunciated with the words ‘Open the door or keep it open’. But it may become resolved into two orders one for an earlier occasion and another for a later occasion. In the theory of norm-kernels which we are now studying it is assumed that the norms under consideration are given for the same occasion. The theory of norm-kernels is therefore not in its present form adequate to deal with orders of the type of ‘Open the door and keep it open’.

Consider now an order expressed in symbols by O(d( ~pTp) ∨ d(pTp)). An instantiation would be the above example of an order to open a door or keep it open. It may easily be shown that on our definition of entailment this order entails the order expressed by Od( ~pTp). We form the negation of this last order. It is the permission expressed by Pf( ~pTp) to leave the state of affairs described by p unproduced. Its sole condition of application is given by the change-description ~pT ~p. This is one of the two conditions of application of the disjunctive order. The disjunctive order requires that under this condition the state described by p be produced. The permission leaves the subject free to leave the state unproduced. Obviously it is logically impossible to obey the order and avail oneself of the permission. Hence the order expressed by Od( ~pTp) is entailed.

By exactly similar argument it is shown that the order expressed by O(d( ~pTp) ∨ d(pTp)) entails the order expressed by Od(pTp).

We can also show that on our definition of entailment the two orders expressed by Od( ~pTp) and Od(pTp) jointly entail the order expressed by O(d( ~pTp) ∨ d(pTp)). The negation of the last is the permission expressed by P(f( ~pTp) ∨ f(pTp)). It has two conditions of application. The one is also the condition of application of the first of the two orders. The other is also the condition of application of the second of the two orders. It is logically impossible both to obey an order to produce a certain state and to avail oneself of a permission to leave it unproduced. It is also logically impossible both to obey an order to continue a certain state and to avail oneself of a permission to let it vanish. Hence the permission expressed by P(f( ~pTp) ∨ f(pTp)) is absolutely incompatible with the set of two orders. Hence finally the two orders jointly entail the order expressed by O(d( ~pTp) ∨ d(pTp)).

The generalization of the example should be clear to the reader. Let an order have n conditions of application. It is then equivalent to a set of n orders each of which has only one condition of application. The contents of these n orders are those ‘parts’ of the contents of the first order which answer to its several conditions of application. (That the order and the set of orders are ‘equivalent’ means that the order entails each one of the orders of the set individually and that it is entailed by all the orders of the set jointly.)

I shall call this the Rule of O-distribution.

13. A permission to do at least one of two things which can both be done under the same conditions I shall call a disjunctive permission. Often when there is a disjunctive permission to do one of two things there is also a permission to do the one and a permission to do the other. But this is not necessarily the case. A disjunctive permission is not equivalent to a set of several permissions.

A permission to do one of several things no two of which can be done under the same conditions is however equivalent to a set of permissions. A permission for example to open a window or to close it is tantamount to a permission to open the window if closed and a permission to close it if open.

It is easily seen that the order expressed by Of( ~pTp) is absolutely incompatible with the permission expressed by P(d( ~pTp) ∨ d(pT ~p)). Hence the permission expressed by Pd( ~pTp) is entailed by the first permission. By similar argument it is shown that the permission expressed by Pd(pT ~p) is entailed by it.

It is also the case that the order expressed by O(f( ~pTp) ∨ f(pT ~p)) is absolutely incompatible with the set of two permissions expressed by Pd( ~pTp) and by Pd(pT ~p). Hence the permission expressed by P(d( ~pTp) ∨ d(pT ~p)) is entailed by the set.

I shall call the generalization of these findings the Rule of P-distribution.

14. Thanks to the rules of distribution every prescription with several conditions of application may become ‘resolved’ into a set of prescriptions each one of which has only one condition of application. The members of the set of prescriptions we call the constituents of the original prescription. Depending upon the character of the original prescription we distinguish between O- and P-constituents.

We can make a systematic list of all the possible O- and P-constituents which can be expressed in terms of a given number n of (atomic) states of affairs. We start from the systematic list of all possible state- change- and act-descriptions which can be thus expressed (see Ch. IV Sect. 5). We consider the set of act-descriptions which answer to a given change-description. There are 2ⁿ such act-descriptions. We then consider the set of disjunctions of act-descriptions which can be formed of these 2ⁿ act-descriptions. Counting the act-descriptions themselves as one-membered disjunctions there are in all 2⁽²ⁿ⁾−1 such disjunctions. Each of them is the content of one possible O-constituent and one possible P-constituent. Thus we get in all 2(2⁽²ⁿ⁾−1) constituents which answer to a given change-description. Since there are in all 2²ⁿ change-descriptions the total number of constituents which answer to n states of affairs is 2²ⁿ×2(2⁽²ⁿ⁾−1) or 2^{2n + 2n + 1}−2^{2n + 1}.

For n = 1 the formula yields the value 24. For n = 2 it yields 480.

One single state of affairs thus determines 24 possible norm-constituents. Their formation is a simple matter. We begin with the two act-descriptions d(pTp) and f(pTp) which answer to the change-description pTp. Of them only one disjunction can be formed viz. d(pTp) ∨ f(pTp). We thus get three O-constituents Od(pTp) and Of(pTp) and O(d(pTp) ∨ f(pTp)) and three P-constituents Pd(pTp) and Pf(pTp) and P(d(pTp) ∨ f(pTp)) answering to the change-description pTp. In a similar manner we form the six constituents answering to pT ~p the six answering to ~pTp and the six answering to ~pT ~p.

Of these 24 constituents however the 8 which have a disjunctive content express what we have called (Section 8) tautologous norms. We may not wish to count them as genuine prescriptions at all. If we omit them the number of constituents is reduced to 16.

Generally speaking if the tautologous constituents are excluded the total number of constituents which answer to n states of affairs is reduced by 2^{2n + 1} and becomes 2^{2n + 2n + 1}−2^{2n + 2}. For n = 1 the formula yields the value 16 and for n = 2 it yields 448.

Two single states of affairs determine 2²ⁿ change-descriptions. If the states are described by p and q the first change-description in the list is (pTp) & (qTq). To it answer four act-descriptions viz. d(pTp) & d(qTq) and d(pTp) & f(qTq) and f(pTp) d(qTq) and f(pTp) & f(qTq). Of these six two-termed disjunctions can be formed four three-termed disjunctions and one four-termed disjunction. Counting the four act-descriptions themselves as one-termed disjunctions we thus get in all 15 disjunctions. They are the contents of 15 O- and 15 P-constituents. Not counting the two tautologous constituents the contents of which is the four-termed disjunction d(pTp) & d(qTq) ∨ d(pTp) & f(qTq) ∨ f(pTp) & d(qTq) ∨ f(pTp) & f(qTq) we have in all 28 constituents. Since there are 16 change-descriptions in the list the total number of not-trivial constituents will be 16 times 28 which is 448.

15. If the content of a norm is an internal consequence of the content of another norm then the first norm is entailed by the second. This is true independently of the character of the norm.

For example: the mode of action described by d(pTp) & d(qTq) ∨ d(pTp) & f(qTq) is an internal consequence of the mode of action described by d(pTp) & d(qTq). The internal negation of the first is described by f(pTp) & d(qTq) ∨ f(pTp) & f(qTq). This last is internally incompatible with the mode of action described by d(pTp) & d(qTq). From this incompatibility (and our definition of entailment) it follows both that the command expressed by O(d(pTp) & d(qTq)) entails the command expressed by O(d(pTp) & d(qTq) ∨ d(pTp) & f(qTq)) and that the permission expressed by P(d(pTp) & d(qTq)) entails the permission expressed by P(d(pTp) & d(qTq) ∨ d(pTp) & f(qTq)).

If the content of a command or permission is an internal consequence of the conjunction of the contents of two or more commands then the first command (permission) is entailed by the set of commands.

Consider for example the two commands expressed by O(d(pTp) & d(qTq) ∨ d(pTp) & f(qTq)) and by O(d(pTp) & f(qTq) ∨ f(pTp) & d(qTq)). The conjunction of their content is the act described by d(pTp) & f(qTq). One can obey both commands only by doing this act. Its internal negation is the act described by d(pTp) & d(qTq) ∨ f(pTp) & d(qTq) ∨ f(pTp) & f(qTq). One could avail oneself of a permission to do this last act only by disobeying at least one or possibly both of the commands in question. Hence the command expressed by O(d(pTp) & f(qTq)) is entailed by the first two commands jointly.

If the content of a permission is an internal consequence of the conjunction of the contents of one or several commands and of one permission then the first permission is entailed by the set of one or several commands and one permission.

Consider for example the command expressed by O(d(pTp) & d(qTq) ∨ d(pTp) & f(qTq)) and the permission expressed by P(d(pTp) & f(qTq) ∨ f(pTp) & d(qTq)). The internal negation of the conjunction of their contents is the act described by d(pTp) & d(qTq) ∨ f(pTp) & d(qTq) ∨ f(pTp) & f(qTq). One can obey the first command and a command to do this last act only by doing the act described by d(pTp) & d(qTq). But this would make it impossible to avail oneself of the above permission. Hence the three-termed disjunctive action cannot be commanded. Its internal negation must be a permitted action. This means that the permission expressed by P(d(pTp) & f(qTq)) is entailed by the first command and the first permission jointly. It is possible to merge the three theorems concerning entailment which have been mentioned in this section into two. Let us adopt the convention that the phrase ‘the conjunction of the content of a command (permission) with the contents of none or one or several command(s)’ shall mean ‘the content of a command (permission) or the conjunction of the content of this command (permission) with the content(s) of one or several command(s)’. Then we have the following two entailment-theorems:

• (i) If the content of a command (or permission) is an internal consequence of the conjunction of the content of a command with the contents of none or one or several other commands then the first command (permission) is entailed by the second command or by the set of it and the other commands.

• (ii) If the content of a permission is an internal consequence of the conjunction of the content of a permission with the contents of none or one or several commands then the first permission is entailed by the second permission or by the set of it and the commands.

The rule that Ought entails May is easily seen to be a special case of the first of these two theorems.

It is essential to the entailment-theorems which we have been discussing in this section that the consequence-relations between norm-contents should be of the kind which we have called internal. For external consequences the theorems are not valid.

Thus for example d(pTp) ∨ d(pT ~p) is an external consequence of d(pTp). But from say Od(pTp) does not follow O(d(pTp) ∨ d(pT ~p)). That it must be thus is easily understood. For by virtue of the Rule of O-distribution Od(pT ~p) follows from O(d(pTp) ∨ d(pT ~p)). If therefore O(d(pTp) ∨ d(pT ~p)) followed from Od(pTp) then one could conclude by transitivity that Od(pT ~p) follows from Od(pTp). This means that one could deduce an order to destroy a state from an order to continue it.

It is intuitively obvious that no norm can entail another norm to the effect that something ought to or may or must not be done under conditions when the first norm does not apply. A norm can only have consequences for the circumstances in which it applies itself. This is reflected in the formal theory by the fact that only internal relationships of consequence between norm-contents have repercussions in the form of relationships of entailment between norms.

16. We have distinguished between a descriptive and a prescriptive interpretation of the (atomic) O- and P-expressions. The metalogical notions of (self—) consistency compatibility and entailment which we have defined in this chapter are in the first place relevant to the prescriptive interpretation. They concern the logical properties of the norms themselves. The ontological significance of those properties however has to be explained in terms of the (possible) existence of norms. Hence this significance will be reflected in the descriptive interpretation too. For on the descriptive interpretation the O- and P-expressions express norm-propositions. And norm-propositions are to the effect that such and such norms exist.

O- and P-expressions descriptively interpreted and their molecular complexes we have called OP-expressions.

Every OP-expression expresses a truth-function of the propositions expressed by the atomic O- and/or P-expressions which are the constituents (descriptively interpreted) of the atomic O- and/or P-expressions of which the given OP-expression is a molecular complex. We shall call these constituents of its atomic components the constituents of the OP-expression itself.

Which truth-function of its constituents a given OP-expression is can be investigated and decided in a truth-table. If this truth-function is the tautology we shall call the given expression an OP-tautology or deontic tautology.

It may be of particular interest to know whether the proposition expressed by one OP-expression entails the proposition expressed by another OP-expression. In order to find out this we form a third OP-expression which is the material implication of the first and the second. We test it in a truth-table. If and only if it is a deontic tautology the (proposition expressed by the) first OP-expression entails the (proposition expressed by the) second OP-expression. The truth-tables of deontic logic differ from ordinary truth-tables (of propositional logic) in that certain combinations of truth-values in the constituents of the tables are excluded as being impossible. Which the excluded combinations are is determined by the definitions of consistency compatibility and entailment for norms (and theorems derived from these definitions).

This is the rule for the construction of truth-tables in Deontic Logic:

Given an OP-expression. We replace the atomic O- and/or P-expressions in it by the conjunctions of their constituents. The constituents are made uniform with regard to the atomics-expressions (variables p q…) which occur in the entire OP-expression.

The distribution of truth-values over all possible constituents which are determined by the atomic p-expressions in the whole OP-expression is subject to the following restrictions:

• (i) If the content of an O- or P-constituent is inconsistent the constituent must be assigned the value ‘false’ (cf. above Section 3). (ii) If the contents of two or more O-constituents or of one or several O- and one P-constituent are internally incompatible all constituents cannot be assigned the value ‘true’ (cf. above Section 6).
• (iii) If an O- and a P-constituent have the same content then if the first constituent is assigned the value ‘true’ the second constituent must also be assigned the value ‘true’ (cf. above Section 11).
• (iv) If the content of an O-constituent is an internal consequence of the content of another O-constituent or of the conjunction of the contents of several O-constituents then if the latter are all assigned the value ‘true’ the former must also be assigned the value ‘true’ (cf. above Section 15).
• (v) If the content of a P-constituent is an internal consequence of the content of another P-constituent or of the conjunction of the contents of one P-constituent and one or several O-constituents then if the latter are all assigned the value ‘true’ the former must also be assigned the value ‘true’ (cf. above Section 15).

When the truth-table of the given OP-expression is being constructed care must be taken that only such distributions of truth-values occur in the table as are allowed by the rules for the distribution of truth-values over all possible constituents which can be formed in terms of the atomic p-expressions in the entire OP-expression. (These constituents may all occur in the truth-table but some may also be missing.)

These are examples of deontic tautologies:

Od(pTp) →Pd(pTp)

Od(pTp) → ~Of(pTp)

Od(pTp) → ~Pf(pTp)

O(d(pTp) ∨ d(pT ~p)) → Od(pTp)

Od(pTp) & Od(pT ~p) → O(d(pTp) ∨ d(pT ~p))

P(d(pTp) ∨ d(pT ~p)) → Pd(pTp)

Pd(pTp) & Pd(pT ~p) → P(d(pTp) ∨ d(pT~p))

O(d(pTp) ∨ d(pT ~p)) & O(d(pTp) ∨ d( ~pTp)) → Od(pTp)

O(d(pTp) ∨ d(pT ~p)) & P(d(pTp) ∨ d( ~pTp)) → Pd(pTp)

The reader will immediately recognize how these formulae may be said to ‘reflect’ the very rules for the construction of truth-tables in Deontic Logic. Their proof in a truth-table is therefore completely trivial. The non-trivial aspect of the proof of those tautologies is an application to the particular formulae in question of the definitions of consistency compatibility and entailment for norms.

• 1. Cf. my paper “On the Logic of Negation” in Soc. Sci. Fenn. Comm. Phys.-Math. XXII 4 (1959).