Jörgensen’s Dilemma

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by Jaap Hage

I. the dilemma and its purported relevance

In 1937/8, Jörgen Jörgensen published a brief paper in Erkenntnis, called Imperatives and Logic. This paper turned out to be very influential in the fields of normative reasoning and deontic logic. In it, Jörgensen presented what has become known as Jörgensen’s dilemma. Briefly restated in modern terminology, this dilemma boils down to the following: On the one hand: The relation of logical implication only holds between sentences with truth values. Imperatives do not have truth values. Therefore there can be no logical implication or derivation between imperatives. On the other hand: It seems evident that it is possible to derive imperatives from two premises if at least one of them is an imperative too, as is illustrated by the following two examples:

Keep your promises Love your neighbour as you love yourself
This is one of your promises Love yourself
Therefore: Keep this promise Therefore: Love your neighbour

Jörgensen’s dilemma seems at first sight to be a rather innocent logical puzzle, without much practical interest. However, in combination with the view that ought judgements are a kind of imperatives, the appearance of innocence disappears, because then Jörgensen’s dilemma seems to threaten the possibility of valid arguments that deal with ought judgements and maybe even the rationality of normative discourse. Does it follow from the prohibition to steal in general that it is forbidden to steal during night time? Or is it impossible to derive anything from the prohibition to steal, because this prohibition is essentially a command not to steal and commands have no truth values and can therefore not figure in valid logical inferences? Several ‘solutions’ have been proposed for Jörgensen’s dilemma, all of which try to rescue the rationality of normative discourse and the existence of logical relations between norms. They argue that:
1. imperatives do have truth values.
2. sentences without truth values can figure in logically valid arguments.
3. Jörgensen’s dilemma is only of limited importance.
These three approaches will be discussed in the following three sections. However, before continuing with these solutions, I want to mention a paper by Von Wright (Von Wright 1985), in which he argues that there cannot be a logic of norms proper, because norms have no truth values, but that it is possible to formulate guidelines for a rational legislator who does not want to create norms that ‘contradict’ each other in the sense that they prescribe incompatible things. These guidelines come very close to a traditional logic for norms.

II. imperatives with ‘truth’ values

One way to solve the dilemma is to deny the second of its presuppositions, namely that imperatives have no truth values. (The denial of the first presupposition - that logic only deals with sentences which have truth values - is discussed in section III.) It is possible to distinguish at least two variants of this approach. The first variant reads that the logic of imperatives is ‘really’ the logic of normative propositions. The second variant reads that the ‘value’ with which logic deals need not be a ‘truth’ value.
A normative proposition describes some normative situation, for instance that in Belgium it is normally forbidden to drive on the left. Such normative propositions are true or false, just like other propositions. Arguably, the examples which Jörgensen gave of valid arguments containing imperatives are convincing because they are really examples of arguments containing normative propositions. The example about the promises should on this interpretation be read as:
It is the case that you ought to keep your promises
This is one of your promises
Therefore it is the case that you ought to keep this promise
Because both premises and the conclusion of this argument have truth values, its validity can be judged by means of ordinary deductive logic. There is no problem at all. This solution of Jörgensen’s dilemma is bought by detaching the logic of normative propositions from imperatives. Since the dilemma specifically deals with imperatives, however, this seeming solution does not touch the essence of Jörgensen’s dilemma. Maybe this is not a problem to worry about (see section IV), but nevertheless reinterpreting the examples as dealing with normative propositions is not a real solution for Jörgensen’s dilemma.
Imperatives prescribe behaviour; they do not describe anything. Therefore they have no truth values. This is problematic for logic in its fashionable interpretation, because logic deals with the transmission of truth from the premises of an argument to the conclusion. Arguably, however, the transmitted value is not necessarily atruth value; it may be some related other value, such as ‘being justified’ or ‘being acceptable’. If logic is concerned with such a broadened value, it may be able to deal with both descriptive sentences which have truth values in the narrow sense and with imperatives which have only a value in the broad sense. An argument would on this view be valid, if and only if its conclusion must be justified (assuming that ‘being justified’ is the broadened value) if all of its premises are justified. (see Brouwer 1982 and Soeteman 1989, p. 51f.)
This way to deal with Jörgensen’s dilemma has its attractions, but there are also some problems connected to it. One is that it is not very clear what the broadened ‘truth’ value would stand for. It seems a rather artificial construction, with as its sole function to rescue the role of logic in the area of deontic reasoning. What does it mean that an imperative has this broadened truth value? A second problem is that it is not clear on beforehand what the ‘logic’ of entities with this broadened truth value would be. One thing would be certain, namely that this logic is not anymore the classic logic that deals with the transmission of truth from premises to conclusion. Somehow, broadening the notion of a truth value changes the nature of the logical enterprise.

III. broadening the conception of logic

Our discussion of broadening the notion of a ‘truth’ value, which dealt with the horn of Jörgensen’s dilemma about the truth value of imperatives, has lead us to the other horn of the dilemma, namely the capability of logic to deal with entities without truth values. This subject touches upon the very nature of logic, and its discussion must therefore be a discussion of what logic is.
To begin, we must distinguish between logica utens, the informal standards by means of which arguments in daily life are evaluated, and logica docens, precisely specified and usually formalised theories about valid inference. Let us call a specific theory that belongs to logica docens a logical system. It has become customary to specify such logical systems syntactically, that is by means of a set of axioms and a set of inference rules by means of which theorems can be derived from the axioms.
Such a ‘logical’ system has no meaning; the symbols used in it are merely defined by the role they fulfil in the system. As such, it is not even a logical system in the sense that it has to do with logica utens. To make it into a real logical system, the system must be interpreted. The usual way to do this is by adding semantics to it. Some symbols are interpreted as sentences, others as logical operators. Moreover, it is assumed that the sentences have a value, which is usually taken to stand for either true or false. The inference rules are really taken as rules of inference, in the sense that the theorems are taken as following logically from the axioms. And finally the idea of following logically must be made precise. This is usually done by saying that a conclusion follows logically from a set of premises, if and only if, given the semantics of the logical system, the conclusion is true on all interpretations of the premises. This is just a technical way of saying that the conclusion of the argument must be true if all the premises are true.
It turns out that it is well possible to give a logical system a slightly different interpretation, which makes it suitable for dealing with norms without truth values. One characteristic of this interpretation is that the values it assigns to sentences are not interpreted as truth values. For instance, if the values of the system are represented by the symbols ‘0’ and ‘1’, and if these values are assigned to the symbols ‘p’, ‘q’, and ‘r’, it is possible to interpret ‘p’, ‘q’, and ‘r’ not as descriptive sentences, but as imperatives, and the symbols ‘1’ and ‘0’ as being issued, respectively not being issued. Under this interpretation, the logical system in question might be said to be logic of imperatives and is a counter example to Jörgensen’s presupposition that logic can only deal with entities having a truth value. An approach in which some symbols of a logical system are interpreted as standing for imperative-like entities is taken in several publications of Alchourrón (e.g. Alchourrón and Martino 1990).
Broadening the conception of logic may solve the problem how logic can deal with imperatives and norms, but at the cost that the nature of logic becomes unclear. If logic deals with the transmission between entities (what kinds of entities?) of some value (what value precisely?), has this still something to do with the assessment of arguments? And if so, what kind of assessment? These are questions that need to be answered if logic is interpreted different from what is usual.

IV. is Jörgensen’s dilemma important?

It has probably become clear from the foregoing that the dilemma formulated by Jörgensen with regard to imperatives has gained much of its importance from the issue whether it is possible to have a logic of norms. Apparently the presupposition that norms have something to do with imperatives is widely shared. It becomes time to take a closer look at this presupposition.
Let us take the presupposition that norms are a kind of imperatives for granted for a while. Clearly Jörgensen’s dilemma is very relevant for legal theory then. However, legal norms are clearly not ordinary imperatives, if ordinary imperatives are taken to be commands. Otherwise than commands, norms persist in time. Apparently, the notion of an imperative is changed by the presupposition that norms are a kind of imperatives. The issue at stake becomes whether it is possible that logic deals with norms that prescribe behaviour, rather than describe (normative) facts. We have seen that it is possible to devise such a logic, but only at the cost of redefining the nature of logic.
Suppose, then, that we do not take the presupposition that norms are a kind of imperatives for granted. Suppose, on the contrary, that norms have little to do with imperatives (see Hage 2005, p. 173f.). Then Jörgensen’s dilemma has only limited relevance, because it does not touch upon the possibility of a logic of norms. Moreover, if Jörgensen’s examples are interpreted strictly as dealing with imperatives in the sense of commands, not as norms, it is questionable whether they illustrate the possibility to reason with these imperatives. If somebody gives the command to keep promises, does he also give the command to keep this particular promise? Or is it merely so that his command also applies to this particular promise? In the latter case, the logic at stake would be the logic of normative propositions, not the logic of imperatives.

Related Entries

Deontic Logic.

Annotated bibliography

Alchourrón, C.E. and A.A. Martino, Logic Without Truth, Ratio Juris 3 (1990), no. 1, p. 46-67.

A paper in which Alchourrón and Martino develop the idea that logical systems can be taken primarily in their syntactic rendering, which avoids the problem of truth values.

Brouwer, P.W., Over de toepassing van de tweewaardige logica in het rechtsdenken, in A. Soeteman and P.W. Brouwer (eds.), Logica en recht, Tjeenk Willink: Zwolle 1982, p. 17-30.

In this paper, Brouwer links the use of a broader notion of truth value with the reinterpretation of syntactically characterised logical systems. In this way he combines the approaches of Soeteman and Alchourrón.

Hage, J.C., Studies in Legal Logic, Springer: Berlin 2005, chapter 6: What is a Norm?

In this chapter, I argue amongst others that theories which link norms to imperatives are seriously flawed. If this argument is correct, the relevance of Jörgensen’s dilemma for legal theory and deontic logic is very limited.

Jörgensen, J., Imperatives and Logic, Erkenntnis 7 (1937/8), p. 288-296.

The paper in which Jörgensen’s dilemma was originally formulated.

Kelsen, H., Allgemeine Theorie der Normen, Manzsche: Wien 1979.

This book contains Hans Kelsen’s final view about the logic of norms. In chapter 50, Kelsen explicitly addresses the problem posed by Jörgensen’s dilemma and denies the possibility of logical relations between norms.

Soeteman, A., Logic in Law, Kluwer: Dordrecht 1989, chapter III: The Possibility of Deontic Logic.

In this chapter, Soeteman develops the idea that ‘valid’ and ‘invalid’ can be used as the deontic counterparts of the truth values in alethic logic. This illustrates the approach to Jörgensen’s dilemma based on broadening the notion of a truth value.

Weinberger, O., The Logic of Norms Founded on Descriptive Language, Ratio Juris 4 (1991), no. 3, 284-307.

This paper contains an overview of the techniques that have been used to deal with the logic of norm sentences and as a consequence, also with one of the major ways to handle Jörgensen’s dilemma.

Wright, G.H. von, A Pilgrim’s ProgressPhilosophers on Their Own Work vol. 12, Peter Lang: Bern 1985, p. 269f.

The work in which Von Wright argues that a logic of norms in the strict sense is not possible, and provides a different interpretation of a logic of norms in the form of guidelines for a rational legislator. In his paper IsThere a Logic of Norms? (Ratio Juris 4 (1991), p. 265-283), came back on this position and reinterpreted the guidelines as logic proper again.

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