IX: Deontic Logic: Hypothetical Norms

1. Hypothetical prescriptions order or permit or prohibit a certain mode of action to some subject(s) on some occasion(s) assuming that the occasion(s) satisfy certain conditions—in addition to providing an opportunity for performing the action.

Formally hypothetical prescriptions differ from categorical ones in the statement of their conditions of application. The conditions of application of categorical prescriptions can be ‘read off’ from a statement of their contents. The conditions are that the occasion(s) for which the prescription is given should provide an opportunity for performing the commanded permitted or prohibited action. The conditions of application of hypothetical prescriptions require special statement. The requirement is usually fulfilled by appending an ‘if-then’—clause to the norm-formulation. For example: ‘If it starts raining shut the window’ If you have finished your homework before dinner you may see your friends in the evening’ ‘If the dog barks don't run’.

Our first problem concerns the nature of the conditionality which is characteristic of hypothetical prescriptions (and other hypothetical norms). We could also say that it concerns the ‘formalization’ of the ‘if-then’—clause which normally occurs in their formulation.

2. Compare the following two types of schematic sentence: ‘One ought to (may must not) should such and such contingencies arise do……’ and ‘Should such and such contingencies arise then one ought to (may must not) do…#8230;’. Ordinary usage does not maintain a sharp distinction between two meanings here. But the two different orders of words may be said to hint at a distinction which the logician has to note.

The second schema contains a deontic sentence as a part. The deontic sentence appears in the consequent of a conditional sentence whose antecedent contains the sentence ‘Such and such circumstances arise’. Shall we say that the schema is that of a descriptve sentence which conditions a prescriptive sentence? I do not think we should say this. Deontic sentences it will be remembered have a typical ambiguity. They can be understood prescripively or descriptively. In the kind of context now under consideration they should I think be interpreted descriptively (or else we shall become involved in logical difficulties). The schema will then be that of a descriptive sentence which conditions another descriptive sentence. The whole thing says ‘If such and such is (will be) the case then such and such is (will be) the case too’. The antecedent speaks of things which happen. The consequent speaks of norms (prescriptions) which there are (will be). I shall say that the schema is of a sentence which expresses a hypothetical norm-proposition. The proposition is true or false depending upon whether there are (will be) such and such norms should such and such things happen. It is clear that hypothetical norm-propositions are quite different from hypothetical norms (prescriptions).

The first schema may be said to be itself a deontic sentence. As such it admits of a descriptive and a prescriptive interpretation. Descriptively interpreted it expresses a norm-proposition. This proposition is to the effect that a certain norm (prescription) exists. The norm-proposition is categorical and not hypothetical. It says that there is such and such a norm—not that should such and such be the case there is a norm. Prescriptively interpreted however the schema is of a norm-formulation. The norms (prescriptions) which sentences of that form are used for enunciating are hypothetical norms.

The scope of the deontic operator in the formulation of a hypotheical norm includes or stretches over the conditional clause in the formulation. What is subject to condition in the norm is the content i.e. a certain action. The character is not conditioned. We could also say that a hypothetical norm does not contain a categorical norm as a part.

3. We have distinguished between hypothetical and technical norms (cf. Ch. I Sect. 7). Technical norms are concerned with that which ought to or may or must not be done for the sake of attaining some end. They too are normally formulated by means of a conditional clause. ‘If you want to escape from becoming attacked by the barking dog don't run.’ Here not-running is thought of as a means to escaping attack by the dog.

The reason why a hypothetical prescription is given is often or perhaps normally that the prescribed action is thought of as a means to some end. In the ‘background’ of a hypothetical prescription there is thus often a technical norm. The reason for example why the order ‘If the dog barks don't run’ is given to somebody may be anxiety that the subject of the order should not be bitten by the dog.

Wanting something as an end can be regarded as a contingency which may arise (in the life of a person). A hypothetical prescription can be given for such cases too. ‘Should you want to climb that peak consult him first.’ This could express a genuine hypothetical order. There may but need not be any means-end connexion between consulting that person and success in the projected enterprise of climbing the hill. That is: the order can ‘exist’ independently of the existence of any such causal ties. This observation should make it clear that a technical norm is not the same as a hypothetical norm for the special case when the conditioning circumstances happen to be the pursuit of something as an end.

It seems to me that a difference between hypothetical and technical norms is that the answer to the question what is subject to a condition is different for the two types of norm. In the case of a hypothetical norm it is the content of the norm which is subject to a condition. In the case of a technical norm it is the existence of the norm which is subject to condition. The ‘if-then’—sentence says: Should you want that as an end (but not otherwise) then you ought to (may must not) do thus and thus. The ‘if-then’—sentence is of the second rather than of the first of the two schematic types which we mentioned and discussed in Section 2. It is a descriptive sentence. The proposition expressed by it is a hypothetical norm-proposition.

If this is a correct view of the matter the technical norm itself is categorical and not hypothetical. The existence of the norm however is hypothetical. The ‘if-then’—sentence is not a norm-formulation but a statement of the conditions under which something will become imperative (permissible) for an agent.

4. For our theory of hypothetical norms we need an extension of our previous Logic of Action. We need a logical theory of conditioned action i.e. action performed on occasions which satisfy certain conditions (in addition to affording opportunities for performing the actions themselves).

We introduce a new symbol /.

By an elementary /-expression we understand an expression which is formed of an elementary d- or f-expression to the left and an elementary T-expression to the right of the stroke /. For example: d(pTp) / qTq is an elementary /-expression.

By an atomic /-expression we understand an expression which is formed of a (atomic or molecular) df-expression to the left and a (atomic or molecular) T-expression to the right of /. For example: (d(pT ~p) ∨ f( ~qTq)) / rTs is an atomic /-expression.

By /-expressions finally we understand atomic /-expressions and molecular complexes of atomic /-expressions. For example: d(pTp) / qTq & ~(f((p&r) T( ~p & ~r)) / ~ sT ~s) is a /-expression.

A /-expression describes a generic action which is performed by an unspecified agent on an unspecified occasion when a certain generic change takes place (independently of the action). The generic change be it observed can also be a non-change. For example: The elementary /-expression d(pTp) / qTq describes that which an (unspecified) agent does who on some (unspecified) occasion when the state described by q obtains and remains independently of action prevents the state described by p from vanishing.

/-expressions belong to (formalized) descriptive discourse. They are schematic representations of sentences which express propositions. Their combination by means of truth-connectives to form molecular complexes is therefore entirely uncontroversial.

5. df-expressions may be regarded as degenerate or limiting cases of /-expressions.

Thus for example d(pTp) and d(pTp) / (qTqqT ~q ∨ ~qTq ∨ ~qT ~q) obviously describe the same action. The first expression says that the state of affairs described by p is continued (prevented from vanishing). The second says that the state described by p is continued on some occasion when the state described by q either is and remains or is but vanishes or is not but comes into being or is not and remains absent. Since what is being said about the state described by q is trivially true the end part of the second description can be omitted as vacuous. The df-expression and the /-expression say in fact the same.

Generally speaking: any df-expression may be regarded as a degenerate form of a /-expression in which the df-expression in question stands to the left of / and an arbitrary T-tautology stands to its right.

But may not d(pTp) be regarded as a degenerate form of d(pTp) / pT ~p also? Generally speaking: May not any df-expression be regarded as a degenerate form of a /-expression in which the df-expression in question stands to the left of / and a description of the conditions for performing the action described by the df-expression to the right of /?

As will be seen presently the answer to these questions is affirmative. There are thus two senses or ways in which df-expressions may be said to represent limiting cases of /-expressions. The second of these two conceptions of df-expressions as limiting cases is however provable on the basis of the first with the aid of other principles of our logical theory of /-expressions.

6. As will be remembered there are four types of elementary T-expressions and eight types of elementary df-expressions. Since thus the expression to the left of / in an elementary /-expression may be any one of eight types and the expression to the right any one of four types it follows at once that there are 32 types of elementary /-expressions. We could list them beginning with d(pTp) / qTq and ending with f( ~pT ~p) / ~qT ~q.

Since elementary T-expressions and also elementary df-expressions of the same variable but of different types are mutually exclusive it is obvious that any two elementary /-expressions of different types but containing the same variable to the left of/ and the same variable to the right of / are mutually exclusive also.

The 32 elementary types of /-expression do not ipso facto form an exhaustive disjunction. They do it only on condition that the eight elementary types of df-expression in them form an exhaustive disjunction. This condition is fulfilled for an arbitrary agent and state of affairs provided it is within the ability of the agent in question to continue and produce and destroy and suppress the state in question when there is an opportunity (cf. above Ch. IV Sect. 2). We shall here assume that this condition is actually satisfied for any agent and state that may enter our consideration.

7. Consider an atomic /-expression. Assume that the df-expression to the left is self-inconsistent i.e. expresses a df-contradiction. This means that it describes a logically impossible mode of action. It is clear that on this assumption the /-expression too is inconsistent. An action which it is logically impossible to perform in any case cannot be performed under certain conditions either. Assume next that the T-expression to the right is self-inconsistent i.e. expresses a T-contradiction. This means that it describes a logically impossible transformation of the world. It is clear that on this assumption the /-expression too is inconsistent. Under logically impossible conditions no action is possible either.

An atomic /-expression is thus inconsistent if the df-expression to the left or the T-expression to the right of the sign / (or both) is inconsistent. This however is not the sole condition of inconsistency.

The occasion on which the action described by an atomic /-expression is done has (i) to satisfy the conditions stated by the T-expression to the right of / and (ii) to afford an opportunity for doing the action described by the df-expression to the left of /. It can happen that the conditions stated by the T-expression and the conditions for doing the action described by the df-expression are consistent in themselves but mutually incompatible.

Consider for example the expression d(pTp) / pTp. The df-expression to the left is consistent so far as the laws of the Logic of Action are concerned. The T-expression to the right is consistent so far as the laws of the Logic of Change are concerned. But the /-expression itself is obviously inconsistent. It says that somebody prevents the state described by p from vanishing in a situation when this state obtains and does not vanish unless destroyed. But under such circumstances it is not (logically) possible to ‘prevent’ the state in question from vanishing. This can be done only in a situation when the state in question obtains and does vanish unless prevented.

Formally the inconsistency of d(pTp) / pTp is reflected in the incompatibility in the Logic of Change of the expressions pTp and pT ~p. The first states the condition which the occasion for doing the action in question has to satisfy in addition to affording an opportunity of doing the action. The second states the condition which the occasion has to satisfy in order to afford an opportunity of doing the action. The two conditions are incompatible. (pTp) & (pT ~p) expresses a T-contradiction.

These observations relating to the self-inconsistency of the expression d(pTp) / pTp can easily be generalized. An atomic /-expression is inconsistent if the T-expression to the right of / is in the Logic of Change incompatible with the T-expression which states the conditions for doing the action described by the df-expression to the left of /. An atomic /-expression is inconsistent we could also say if the conjunction of the two T-expressions in question expresses a T-contradiction.

From the meaning of /-expressions as we have explained it the validity of the following principle is obvious:

If we replace the T-expression to the right of / in a given /-expression by the conjunction of itself and the T-expression which states the conditions for doing the action described by the df-expression to the left of / then the new /-expression is logically equivalent with the original /-expression.

For example: (d( ~pTp) & f( ~qTq)) /rTr is logically equivalent with (d( ~pTp) & f( ~qTq)) / (( ~pT ~p) & ( ~qT ~q) & (rTr)).

We can accordingly speak of a ‘shorter’ and ‘longer’ form of any given atomic /-expression. In the longer form the T-expression to the right states both the conditions which the occasion has to satisfy in order to afford an opportunity of action and the conditions which the occasion has to satisfy in addition to affording an opportunity of action. The variables which appear in the df-expression to the left all appear in the T-expression to the right. But the T-expression may contain additional variables.

When an atomic /-expression is in the longer form it is in fact consistent if and only if the T-expression to the right of the symbol / is consistent.

8. Consider two atomic /-expressions. The two df-expressions to the left may or may not contain the same variables. The same holds good for the two T-expressions to the right.

Let the variable p occur in one of the df-expressions but not in the other. Then can be made to appear in this latter by conjoining to it the eight-termed disjunction d(pTp) ∨…#8230; ∨ f( ~pT ~p). By this procedure one can procure that the two df-expressions contain exactly the same variables.

Let the variable p occur in one of the T-expressions but not in the other. Then can be made to appear in this latter by conjoining to it the four-termed disjunction (pTp) ∨… ∨ ( ~pT ~p). By this procedure one can achieve that the two T-expressions contain exactly the same variables.

Atomic /-expressions which contain the same variables in the df-expressions to the left of / and the same variables in the T-expressions to the right of / will be called uniform (with regard to the variables).

Uniform atomic /-expressions which are in the ‘longer’ form will satisfy the additional condition that the variables which occur in the df-expressions to the left of / form a sub-set of the variables which occur in the T-expressions to the right of /.

9. Every /-expression expresses a truth-function of elementary /-expressions. That this must be the case is intuitively obvious from considerations about distributability.

We consider an atomic /-expression. We assume that the df-expression to the left and the T-expression to the right are both in the positive normal form.

Let for example the expression be d(pTp) / (qTqqT ~q). A state of affairs is prevented from vanishing on an occasion when another state obtains and either remains or vanishes independently of action. Obviously this means the same as (d(pTp) / qTq) ∨ (d(Tp) / qT ~q).

Let the expression be d(pTp) / (qTq & rTr). This means the same as (d(pTp) / qTq) & (d(pTp) / rTr).

Let the expression be (d(pTp) ∨ d(pT ~p)) / qTq. This means the same as (d(pTp) / qTq) ∨ (d(pT ~p) / qTq).

Let finally the expression be (d(pTp) & d(qTq)) / rTr. This means the same as (d(pTp) / rTr) & (d(qTq) / rTr).

It is important to remember that the whole /-expression refers to one and the same agent and occasion. Assume for example that (d(pTp) / qTq) ∨ (d(pTp) / qT ~q) meant that either some agent on some occasion when the state described by q is and remains prevents the state described by p from vanishing or some agent on some occasion when the state described by q is but vanishes prevents the state described by p from vanishing. Then the expression would not be identical in meaning with d(pTp) / (qTqqT ~q).

If atomic /-expressions are truth-functions of elementary /-expressions then all /-expressions must be truth-functions of elementary /-expressions.

10. Consider an arbitrary /-expression. It is a molecular complex of atomic /-expressions. We make its atomic constituents uniform (with regard to the variables) according to the procedure described in Section 8. We replace the df-expressions to the left and the T-expressions to the right of / in the atomic /-expressions by their positive normal forms. Thereupon we carry out the four types of distribution mentioned in Section 9. The original /-expression has then become transformed into a molecular complex of elementary /-expressions. The elementary /-expressions we call the /-constituents of the original /-expression.

Which truth-function of its /-constituents a given /-expression is can be investigated and decided in a truth-table. The distribution of truth-values over the constituents is subject to the following two restrictions:

• (i) Uniform elementary /-expressions are mutually exclusive and jointly (all 32 of them) exhaustive.
• (ii) Inconsistent elementary /-expressions must be assigned the value ‘false’. An elementary /-expression is inconsistent if and only if the T-expression to the right contradicts the condition of doing the action described by the elementary d- or f-expression to the left.

If a /-expression is the tautology of its /-constituents we call it a /-tautology. If it is the contradiction of its constituents we call it a /-contradiction.

11. Let there be an arbitrary /-expression. We replace it according to the transformations described in Section 10 by a complex of its /-constituents. This complex we transform into its perfect disjunctive normal form. This is a disjunction of conjunctions of elementary /-expressions and/or their negations. We replace every negation of an elementary /-expression by a 31-termed disjunction of elementary /-expressions which are uniform with the first and form with it an exhaustive disjunction. We transform the expression obtained after these replacements into its perfect disjunctive normal form. This will be a disjunction of conjunctions of (unnegated) elementary /-expressions. We call it the positive normal form of the original /-expression.

12. In Section 5 we showed that df-expressions may be regarded as degenerate or limiting cases of /-expressions. Unconditioned action we could also say is a limiting case of conditioned action. It is the limiting case when the condition of action is tautological. Similarly categorical norms may be regarded as degenerate or limiting cases of hypothetical norms.

Consider the expression d(pTp). According to what was said in Section 5 it may become ‘translated’ into the /-expression d(pTp) / (qTqqT ~q ∨ ~qTq ∨ ~qT ~q). If the letter O or P is prefixed to the first expression we obtain the symbol for the norm-kernel of a categorical command and permission respectively. If the letter O or P is prefixed to the second expression we obtain the symbol for the norm-kernel of a hypothetical command and permission respectively. ‘Axiomatically’ we shall regard the two symbols for norm-kernels as ‘intertranslatable’. A command or permission to do something unconditionally may be regarded as a command or permission to do something under conditions which are tautologously satisfied.

Since there is no restriction on the choice of a variable for the tautologous T-expression to the right of / the expression d(pTp) may also become ‘translated’ by d(pTp) / (pTppT ~p ∨ ~pTp ∨ ~pT ~p). By virtue of the distribution-principles mentioned in Section 9 the last expression is equivalent to d(pTp) / pTpd(pTp) / pT ~pd(pTp) / ~pTpd(pTp) / ~pT ~p. According to the criteria of consistency given in Section 7 the first the third and the fourth disjunct in this four-termed disjunction of elementary /-expressions is inconsistent. The whole expression is thus tautologically equivalent to d(pTp) / pT ~p. Generally speaking: any df-expression may become ‘translated’ into a /-expression in which the given df-expression stands to the left of / and a statement of the condition of doing the action described by it stands to the right of / (cf. Section 5).

Corresponding to the two ways in which df-expressions may be regarded as limiting cases of /-expressions there are two ways in which categorical norms may be regarded as limiting cases of hypothetical norms. Od(pTp) may be regarded as an ‘abbreviation’ of an expression of the form O(d(pTp) / (qTqqT ~q ∨ ~qTq ∨ ~qT ~q)) or of the form O(d(pTp) / pT ~p). And the corresponding is true of P(pTp).

We can now generalize the notion of an OP-expression which we introduced in Ch. V Sect. 4.

By an atomic O-expression (P-expression) we understand an expression formed of the letter O (P) followed by a df- or by a /-expression. The atomic O- and P-expressions are thus symbols of norm-kernels of categorical or of hypothetical norms.

By an OP-expression we understand any atomic O- or atomic P-expression or molecular complex of atomic O- and/or P-expressions.

An OP-expression in the general sense of the term can thus be a molecular compound containing symbols both of categorical and of hypothetical norm-kernels. The symbolic statement of many of the theorems which we are going to prove will be such ‘mixed’ OP-expressions. When ‘mixed’ expressions are handled for the purposes of proofs it is often convenient to replace expressions of categorical norm-kernels in them by such expressions of hypothetical norm-kernels of which the first may be regarded as degenerate or limiting cases.

13. The principles of the logic of categorical norms (norm-kernels) which we discussed in the last chapter are with minor modifications also the principles of the logic of hypothetical norms. The logic of hypothetical norms (norm-kernels) has no new independent principles of its own.

The ‘minor modifications’ to which we referred concern the notions of the content the conditions of application and of the negation-norm of a given norm. They have to be redefined so as to become applicable also to hypothetical norms.

Consider an atomic OP-expression in which the /-expression following after the letter O or P is atomic. By the content of the hypothetical norm in question we understand the action described by the df-expression to the left of / in the /-expression.

By the condition of application of the norm we understand the conjunction of the change which is the condition of doing the action described by the df-expression to the left of / and the change described by the T-expression to the right of /.

By the negation-norm of the given norm finally we understand a norm of opposite character whose content is the internal negation of the content of the original norm and the conditions of application of which are the same as those of the original norm.

For example: the content of the hypothetical command expressed by O(d(pTp) /qTq) is the action described by d(pTp). Its condition of application is the change described by pT ~p & qTq. Its negation-norm finally is the norm whose kernel is expressed by P(f(pTp) / qTq).

These definitions will have to be generalized for the case when the /-expression following after the letter O or P in the OP-expression is not atomic. In this case we have to think of the /-expression as being in the positive normal form. It is then a disjunction of conjunctions of elementary /-expressions. Consider such a conjunction in the normal form. We form the conjunction of the elementary d- and/or f-expressions to the left of the symbols / in it. Thereupon we form the conjunction of the T-expressions stating the conditions of doing the acts described by these and/or /-expressions and the T-expressions to the right of the symbols /. These two operations are performed on each one of the conjunctions in the normal form. The operations give us two conjunctions for each conjunction in the normal form. The one is a conjunction of elementary d- and/or f-expressions; the other is a conjunction of elementary T-expressions. The disjunction of all the conjunctions of the first kind states the content of the hypothetical norm in question; the disjunction of all the conjunctions of the second kind states its conditions of application. The negation-norm of the given hypothetical norm finally is a norm of opposite character whose content is the internal negation of the content of the original norm and the conditions of application of which are the same as those of the original norm.

For example: The content of the hypothetical norm with the norm-kernel O(d(pTp) / qTqd(pT ~p) / qT ~q) is the action described by d(pTp) ∨ d(pT ~p). Its condition of application is the change described by (pT #8764;p & qTq) ∨ (pTp & qT ~q). The symbol for the norm-kernel of its negation-norm finally is P(f(pTp) / qTqf(pT ~p) / qT ~q).

Having redefined the notions of content conditions of application and negation-norm the definitions of the notions of compatibility and entailment can without further modification be transferred to the theory of hypothetical norms. The notion of consistency we define as follows: The norm-kernel of a hypothetical norm is consistent if and only if the /-expression after the letter O or P in the symbol of this norm-kernel is consistent.

14. We easily prove that the following formula is a deontic tautology: Od(pTp) →O(d(pTp) /qTq). The proof is as follows: We replace in accordance with the principles of ‘translation’ given in Section 12 the antecedent of the implication-formula by O(d(pTp) / qTqqT ~q ∨ ~qTq ∨ ~qT ~q). The /-expression after the letter O can be replaced by d(pTp) / qTqd(pTp) / qTq ~qd(pTp) / ~qTqd(pTp) / ~qT ~q. If we apply the Rule of O-Distribution (Ch. VIII Sect. 12) to the above implication-formula we get the formula (O(d(pTp) / qTq) & O(d(pTp) / qT ~q) & O(d(pTp) / ~qTq) & O(d(pTp) / ~qT ~q)) →O(d(pTp) / qTq). This is easily recognized as a tautology of the Logic of Propositions.

In the above proof we assumed the validity of the Rule of O-Distribution for hypothetical norms. We could however have proved the same formula without this assumption directly on the basis of our definition of entailment. We would then have to show that the negation-norm of O(d(pTp) / qTq) which is P(f(pTp) / qTq) is absolutely incompatible with Od(pTp). The two norms have only one condition of application in common viz. pT ~p & qTq. The conjunction of their contents under this condition is d(pTp) & f(pTp). This conjunction is inconsistent. P(f(pTp) / qTq) has no condition of application which is not also a condition of application of Od(pTp). Hence P(f(pTp) / qTq) is not only incompatible but absolutely incompatible with Od(pTp). It follows that O(d(pTp) / qTq) is entailed by Od(pTp).

Similarly we establish the tautological character of the formula Pd(pTp) →P(d(pTp) / qTq) either directly on the basis of our definition of entailment or with the aid of the Rule of P-distribution and principles of the Logic of Propositions.

Generalizingly we can state the two theorems which we have proved in this section as follows:

If something is unconditionally obligatory then it is also obligatory under any particular circumstances and if something is unconditionally permitted then it is also permitted under any particular circumstances.

15. In this and the next few sections I shall take up for discussion some principles of deontic logic which I have acknowledged as true in previous publications and which other writers in the field seem on the whole to have accepted. It will be seen that the principles in question will either have to be rejected altogether or reformulated so as to avoid some error which was implicit in their original formulation. I shall refer to my previous system of deontic logic with the name ‘the old system’.

In the old system the O-operator was conjunctively distributive. In the symbolism of that system the formula O(A & B) ↔OA & OB expressed a deontic tautology. The idea was that one ought to do two things jointly if and only if one ought to do each one of the things individually. For example: One ought to open the window and shut the door if and only if one ought to open the window and ought to shut the door.

Is this a logical truth? Doubts are raised by the following considerations: A command to open a window and shut a door applies to a situation when a certain window is closed and a certain door is open. A command to open a window applies to a situation when a certain window is closed—irrespective of whether a certain door is open or not. The two commands have different conditions of application. How could then the one entail the other?

The nearest formal analogue in the new system to the above formula of the old system would be O(d( ~pTp) & d( ~qTq)) ↔ Od( ~pTp) & Od( ~qTq). It may easily be shown that this formula does not express a deontic tautology. To this end we need only show that O(d( ~pTp) & d( ~qTq)) does not entail Od( ~pTp). This is done as follows:

The negation-norm of Od( ~pTp) is Pf( ~pTp). It has four conditions of application in terms of the two states described by p and by q respectively. These conditions are the changes described by ~pT ~p & qTq and ~pT ~p & qT ~q and ~pT ~p & ~qTq and ~pT ~p & ~qT ~q. The norm expressed by O(d( ~pTp) & d( ~qTq)) has only one condition of application viz. the change described by ~pT ~p & ~qT ~q. Under this one condition of application the two norms are incompatible as shown by the fact that f( ~pTp) & d( ~pTp) & d( ~qTq) is inconsistent. But the mere fact that the first of the two norms applies under conditions in which the second does not apply is enough to warrant that their incompatibility is not absolute. Hence on our definition of entailment O(d( ~pTp) & d( ~qTq)) does not entail Od( ~pTp).

An order to produce both of two states does not entail an order to produce the first of them unconditionally. But it obviously entails an order to produce the first of them on condition that the occasion in question affords an opportunity for producing the second as well. O(d( ~pTp) & d( ~qTq)) in other words entails O(d( ~pTp) / ~qT ~q). This is easily proved as follows:

The negation-norm of O(d( ~pTp) / ~qT ~q) is P(f( ~pTp) / ~qT ~q). The sole condition of application of the norms O(d( ~pTp) & d( ~qTq)) and P(f( ~pTp) / ~qT ~q) is the change described by ~pT ~p & ~qT ~q. Under this condition the two norms are incompatible. Their incompatibility moreover is absolute. Hence the categorical norm expressed by O(d( ~pTp) & d( ~qTq)) entails the hypothetical norm expressed by O(d( ~pTp) / ~qT ~q). By similar argument we show that it entails the hypothetical norm O(d( ~qTq) / ~pT ~p). Very easily too it is shown that the two hypothetical norms jointly entail the categorical norm. The following formula is a deontic tautology: O(d( ~pTp) & d( ~qTq)) ↔ O(d( ~pTp) / ~qT ~q) & O(d( ~qTq) / ~pT ~p).

A conjunctive categorical obligation may thus become resolved into a conjunction of hypothetical obligations. The tendency to think that it may become resolved into a conjunction of categorical obligations probably arises from the fact that we think of the norms as having the same conditions of application and ignore that there may be conditions under which some of them apply and others not.

16. In the old system the P-operator was disjunctively distributive. In the symbolism of this system the formula P(AB) ↔ PAPB expressed a deontic tautology. The idea was that one may do at least one of two things if and only if one may do the one or may do the other. This principle was the very cornerstone on which the old system of deontic logic rested.

The principle however has to be rejected. From the fact that one is unconditionally permitted to do one or the other of two things it does not follow that one is unconditionally permitted to do the one or unconditionally permitted to do the other. (The converse entailment however is valid.)

As was shown in Section 14 if something is unconditionally permitted it is permitted also under any particular conditions. Now it may happen that whatever the conditions are one is permitted to do one or the other of two things but that under some conditions doing one thing is forbidden and under some other conditions doing the other thing is forbidden. One may for example be permitted always to leave either the door or the window of a certain room open but not permitted to leave the door open at night and not permitted to leave the window open in the morning. These considerations should convince us that the principle of the disjunctive distributivity of the P-operator is not a logical truth.

One can sustain this insight by formal considerations. Let there be an unconditional permission expressed by P(d(pTp) & f(qTq) ∨ f(pTp) & d(qTq)). Some agent is on some occasion unconditionally allowed either to continue one state and let another vanish or to let the first vanish and continue the second. Let there further be a hypothetical order to that same agent for that same occasion expressed by O((d(pTp) & d(qTq) ∨ f(pTp) & d(qTq) ∨ f(pTp) & f(qTq)) / rTr). This is a prohibition to continue the first state and let the second vanish should a third state (r) obtain on the occasion in question and remain unless destroyed through action. Let there finally be a hypothetical order to that same agent for that same occasion expressed by O((d(pTp) & d(qTq) ∨ d(pTp) & f(qTq) ∨ f(pTp) & f(qTq)) / rT ~r). This is a prohibition to let the first state vanish and continue the second should a third state (r) obtain on the occasion in question but vanish unless continued through action. The three norms viz. the categorical permission and the two hypothetical prohibitions are compatible. The reader can easily convince himself of this by constructing a table in which are listed the conditions of application and the parts of the contents of the various norms which apply under the respective conditions.

It may easily be shown that if there is a categorical disjunctive permission then it is impossible that both the disjunct modes of action should be categorically prohibited. It is also impossible that both should be hypothetically prohibited under the same conditions. But it is possible that one of the modes of action is prohibited under some conditions and the other under some other conditions. From the fact that it is impossible that both modes of action are categorically prohibited it does not follow that at least one of them must be categorically permitted.

17. Sometimes when an agent does something he thereby becomes committed to doing something else. If he does the first he ought to do the second. Promising might be given as an example. By giving a promise an agent commits himself to doing the act which fulfils the promise.

In the old system of deontic logic the symbol O(AB) was proposed as a formalization of the notion of commitment. It was suggested that the symbol might be read as follows: ‘It is obligatory to do B if one does A’ or alternatively ‘It is forbidden to do A without also doing B’.

Some theorems on commitment were proved in the system. One of them was the formula PA & O(AB) → PB. Another was the formula O(AB) & O ~ BO ~ A. The first was read: ‘Doing something permitted can commit one only to doing something else which is also permitted.’ And the second: ‘An act the doing of which commits one to a forbidden act is itself forbidden.’

The suggested formalization of commitment is highly problematic and the reading of formulae is very free indeed. It is obvious that a much more refined symbolism is needed for coping adequately with the notion of commitment and for expressing the ideas aimed at in the above theorems.

How then should the notion of commitment be formalized? I do not think the question has a unique answer. For by ‘commitment’ one can mean several things of rather different logical character.

One sense of ‘commitment’ has to do with the very notion of a hypothetical norm. Consider for example the command expressed by O(d( ~pTp) / qTq). It orders the production of the state of affairs described by p if the state described by q obtains and remains unless destroyed through action. Assume now that this second state can be produced through action. If then an agent produces the change described by ~qTq and the state thus produced does not vanish ‘of itself’ unless prevented he thereby commits the agent who is the subject of the hypothetical command to produce the change described by ~pTp. If the agent who produces the first change is the same as the subject of the hypothetical command we can speak of auto-commitment. If the agents are different we can speak of alio-commitment. Both cases are of obvious importance in many legal and moral contexts. Agreement contract and promise may be regarded as instances of auto-commitment.

A satisfactory account of this notion of commitment is not possible within our theory of norm-kernels. For commitment in this sense involves action on at least two distinct though related occasions for acting. First one state is transformed and then another state which exists simultaneously with the result of the transformation ought in its turn to be transformed. This can be formalized only within a symbolism which has signs for occasions. Thus it cannot be formalized within the theory of norm-kernels.

There is however another notion of commitment which concerns action on one occasion only. Its study falls within the theory of norm-kernels.

The definition of commitment in the old system was based on the notion of an ‘implication-act’. Commitment was defined as the obligatoriness of an act of this kind. The act was symbolized by a material implication formula which obeyed the laws of the Logic of Propositions and no special rules of its own. This symbolism is inadequate. The question therefore is urgent how the notion of an ‘implication-act’ shall be formalized in the notation of our Logic of Action.

The formula pq is a schematic description of a compound state of affairs. Does the ‘implication-act’ consist in the production through action of a state of this kind? In that case its symbolic expression would be d(p & ~qTpq). The ‘implication-act’ would consist in the transformation through action of a p & ~q-world into either a p & q-world or a ~p & q-world or a ~p & ~q-world. Commitment would be the obligatoriness of such action.

It may be of some interest to study acts of the schematic description d(p & ~qTpq). It seems to me excluded however that their study would be of relevance to the notion of commitment. The reading of Od(p & ~qTpq) as ‘one ought to do q if one does p’ does not appear at all natural.

The idea of producing (or having to produce) one state if one produces another obviously applies to an initial situation in which neither of these two states obtains. The notion which we are trying to ‘formalize’ concerns action in a world described by ~p & ~q. The mode of action in question consists in this that this world is not transformed into a p-world unless it is also transformed into a q-world. Or conversely if it is transformed into a p-world it is also transformed into a q-world. It is not unnatural to call this mode of action an ‘implication-act’. Obligatoriness of this mode of action means that it is forbidden to produce the first of two states and forbear to produce the second.

The symbolic expression of the prohibition to produce the state of affairs described by p and forbear to produce the state described by q is O(d( ~pTp) & d( ~qTq) ∨ f( ~pTp) & d( ~qTq) ∨ f( ~pTp) & f( ~qTq)). This is the nearest formal equivalent in the new deontic logic to the symbol O(AB) of the old system.

Is it a logical necessity that if one is categorically permitted to produce the state of affairs described by p and categorically prohibited to produce p and forbear to produce q then one is also categorically permitted to produce q? The answer obviously is negative. A categorical permission to produce q is a permission to produce it also on an occasion which does not afford an opportunity for producing p. And it is clear that a permission to produce q on such an occasion cannot be deduced from norms which do not apply to this occasion at all. These considerations show—as may also be intuitively felt—that there is a logical flaw involved in the entailment theorem of the old system which was given the wording ‘Doing something permitted can commit one only to doing something else which is also permitted’.

This however is a valid formula of deontic logic: Pd( ~pTp) & O(d( ~pTp) & d( ~qTq) ∨ f( ~pTp) & d( ~qTq) ∨ f( ~pTp) & f( ~qTq)) → P(d( ~qTq) / ~pT ~p). In words: If one is unconditionally permitted to produce a certain state of affairs but unconditionally forbidden to produce this state and forbear to produce a certain other state then one is also permitted to produce this second state under circumstances which constitute an opportunity for producing the first state. The proof which is easy is left as an exercise to the reader.

This too is a valid formula of deontic logic: O(d( ~pTp) & d( ~qTq) ∨ f( ~pTp) & d( ~qTq) ∨ f( ~pTp) & f( ~qTq)) & Of( ~qTq) → O(f( ~pTp) / ~qT ~q). In words: If one is unconditionally forbidden to produce (the state of affairs described by) p and forbear to produce q whose production is itself unconditionally forbidden then one is also forbidden to produce p under circumstances which constitute an opportunity for producing q.

The last two formulae are what correspond in the new system to the formulae PA & O(AB) → PB and O(AB) & O ~ BO ~ A of the old system.

18. In the old system this was a valid formula: O ~ AO(AB). It was read: ‘Doing the forbidden commits one to doing anything.’ This was an analogue in deontic logic to one of the well-known Paradoxes of Implication. Another analogue was OBO(AB). It was read: ‘Doing anything commits one to doing one's duty.’ We could call these two formulae Paradoxes of Commitment.

The impact of the paradoxes is that they make debatable the attempt to formalize the notion of commitment by means of O(AB). As we know there are independently of the ‘paradoxes’ conclusive reasons for regarding this formalization as inadequate.

It is an observation of some interest that corresponding ‘paradoxes’ arise for the suggested formalization of commitment through O(d( ~pTp) & d( ~qTq) ∨ f( ~pTp) & d( ~qTq) ∨ f( ~pTp) & f( ~qTq)). For it may easily be shown that this expression is entailed both by Of( ~pTp) and by Od( ~qTq).

These findings are not ‘paradoxical’ if we render them in words as follows: If it is categorically forbidden to do a certain thing then it is also forbidden to do this thing in conjunction with any other thing; and if it is categorically obligatory to do a certain thing then it is also obligatory to do this thing irrespective of whether one does or forbears to do a certain other thing. The air of paradox comes in when we speak of the conjunctive prohibition and obligation as a ‘commitment’.

The proper conclusion to be drawn from these ‘paradoxes’ is in my opinion that the suggested formalization of the notion of commitment is not (entirely) satisfactory. The way out of these ‘paradoxes’ is not however to abandon the notion of commitment which we are trying to formalize in favour of that notion of commitment which concerns action on different occasions. My suggestion is that we should replace the suggested formalization by the following amplified form of it: O(d( ~pTp) & d( ~qTq) ∨ f( ~pTp) & d( ~qTq) ∨ f( ~pTp) & f( ~qTq)) & P(d( ~pTp) / ~qT ~q) & P(f( ~qTq) / ~pT ~p).

It may be shown that this expression entails P(f( ~pTp) / ~qT ~q) and also P(d( ~qTq) / ~pT ~p).

The amended definition of commitment amounts to the following: The fact that it is prohibited to do a certain thing and forbear a certain other thing on some occasion gives rise to a commitment to do the second thing if one does the first then and then only when the agent is normatively free i.e. permitted to do or forbear the first thing and also normatively free i.e. permitted to do or forbear the second thing on the occasion in question.

In the notion of commitment there is thus involved not only the notion of obligation but also the notion of permission. This is not surprising. To commit oneself normatively is to ‘bind oneself’ normatively to give up a freedom. Therefore one cannot commit oneself to an action which one is already normatively bound to do. Nor can one commit oneself by action from which one is normatively bound to abstain.

The two theorems of deontic logic which we discussed in the last section retain their validity as theorems. But they cease to be theorems on commitment. If in the two formulae under discussion we replace the originally suggested formalization of commitment by the amended formalization the formulae reduce to tautologies of the Logic of Propositions.

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