Deontic logic: the Moebius strip

It happens rather often in a subject area that authors’ intuitions conflict with each other. It seems to me that Makinson’s ‘Moebius strip” problem (1999) may be the occasion of conflicting intuitions about what a person’s normative situation may or can be like. Or perhaps it is not different understandings of the same thing, but rather different projects in the same subject area. At the very least, the example looks different depending on whether or not we think about the situation developing in time.

I will present the Moebius Strip example below, but will first think through the way such a developing situation can be modeled.

[I] The Moebius Strip in STIT semantics

What I want to spell out for myself, first of all, is how the Moebius Strip might look in STIT semantics, that is, in a temporal development.

I will write “!E” for the imperative “See to it that E!”. Also, I will assume that seeing to it that something be the case, if it is not already the case, takes some time. Usually, perhaps, it takes only a moment, but sometimes (consider “!There is peace on earth”) a bit longer.

Let me begin with a situation S in which a certain set of imperatives are in force DELTA = {<!A(i), if B(i)>: i = 1, 2, …, n}, and what is known, in toto, is proposition K. I will here identify S as <K, DELTA, v>, where v summarizes other features that determine truth-values of statements in S. On this basis we should be able to determine which statements of form Ought(E) are true in that situation, in ways to be spelled out.

Just to make clear what it means to be in force, in contrast to how imperatives demand action, take this example. Tim is a soldier; the captain has ordered him to stand guard, and to sound the alarm if the enemy approaches. Both these imperatives are in force, and the first ‘kicks in’ (demands action) immediately, since it has no condition. The second condition ‘kicks in’ only if (or when) Tim knows that the enemy approaches.

The main points of interpretation

I propose to understand this situation (but realize that others may understand this differently), as follows:

(1) Seeing to it that E, if E is not true already, changes the person’s situation, it puts him in a situation T different from S, in which E is true. What T is like may depend on other factors, for example, aging of the persons involved.

(2) If a person sees to it that E, then in the situation thus produced, that person knows that E. In other words, the knowledge in the new situation is a proposition that implies E.

(3) O(E) is true in that situation exactly if the person in that situation ought to see to it that E is true.

(4) Ought(E) is true in S exactly if there is a subset J of {1, .., n} such that K implies all of {A(k): k in J}, the intersection of K with all of {A(k) ∩ B(k): k in J} is not empty, and the set J is maximal in this respect.

(5) The set of imperatives that are in force, in any given situation, cannot include one that is impossible to satisfy, in and by itself, given what is known in that situation. So for example in S, if K implies A(1) then the intersection of K, A(1), and B(1) is not empty. This is the point of the principle that ought implies can.

(6) In view of (1), (2), and (5) we cannot expect that all the imperatives in force in S will also all still be in force in the succeeding situation T, in general. The dynamics of the process that occurs when agents react to imperatives must have at least some independent features that determine which imperatives are in force in the succeeding elements of that process.

The Moebius Strip example, for tenseless propositions

How does the Moebius Strip example look, if these points are all accepted? The example is of a situation S* in which three imperatives are in force:

<!A, if B>, <!C, if A>, <!~B, if C>

Although not so specified, I assume that A, B, C are mutually independent propositions.

To determine which Ought statements are true in this situation we must first ask what is known then. There are different possibilities.

[a] K = T, the tautology. In that case, assuming that A, B, C are not tautologies, there is nothing that ought to be seen to.

[b] K implies A but not B. In that case the person ought to see to it that C, and subsequently see to it that ~B. For it is only when he knows that C that the imperative <!~B, if C> kicks in. Again, no problem appears.

[c] K implies B, but not A or C. In that case Ought(A) is the case; that is all. But then, if the person obeys and sees to it that A, he lands in situation S*[B], where A is true and is part of the knowledge then.

Now we need to ask another question: are we dealing with tensed or tenseless propositions? If the counter is wet I can see to it that it is dry (a moment later). But if the counter is wet at time t, I cannot see to it that it is dry at time t. And secondly, are we dealing with an ideal agent, not subject to forgetfulness, or a less ideal one?

Let us (first) focus on the special case: the propositions are tenseless, and the agent’s knowledge does not diminish over time. Therefore in S*[B], both A and B are known to be true.

Sub-case: The same imperatives are in force in S*[B] as in S*. The first demands no action now, since A is true. But the second kicks in, and Ought(C) is the case in S[B]. If the person then sees to it that C, he lands by similar reasoning in a new situation S*[B, C] where A, B, C are all known.

In that case, if all the same imperatives are in force in S*[B, C], Ought(~B) is true there. But that contradicts point (5), since B is known to be true. Therefore the imperative <!~C, B> is not in force in situation S*[B,C].

It seems to me this is the only interesting sub-case. The suspected difficulty cannot arise, there is no possible situation sequence in which we land in a self-contradiction.

Is this counterintuitive? Do imperatives lose their force when it comes to be known that they cannot be satisfied? Imagine I have promised to give you a horse if you come home for Christmas. What happens to this promise if it turns out at Christmas that I have no horse to give you? I will have secondary obligations (I have to make this up to you), but (whatever I may have done wrong), it is not true at Christmas time that I ought to be giving you a horse. That promise has dropped out, the question of keeping it has become moot (though not without other moral consequences).

The Moebius Strip for tensed propositions

It may be objected here that I focused on too easy a case, by taking the propositions to be tenseless. Of course, if B is tenseless, and known to be true, then any imperative to see to it that ~B can’t even be in force, for it asks the impossible.

So what if we take the propositions to be tensed? Here is an example:

• If there is a fire in the garage, get the fire extinguisher
• If you have the the fire extinguisher, extinguish any fire in the garage
• If there is no fire in the garage, bring the fire extinguisher back

In our official phrasing this involves the tensed propositions expressed as follows:

• !See to it that you have the fire extinguisher, if there is a fire in the garage
• !See to it that there is no fire in the garage, if you have the fire extinguisher
• !See to it that you do not have the fire extinguisher, if there is no fire in the garage

This has the temporal sequence written on its face, so to speak, and no scenario presents any semblance of difficulty.

In one scenario, I already have the fire extinguisher, I make sure that there is no fire in the garage (by inspection or action), and then bring the fire extinguisher back to where it normally belongs. On another scenario, I learn and hence know that there is a fire in the garage. The first imperative kicks in, and I see to it that I have the fire extinguisher. At this point, again, I see to it that there is no fire in the garage, and when that is so, the third imperative kicks in and I return the fire extinguisher.

[II] Alternative Understanding of Situation: Makinson, Horty

So, what alternative understanding of a decision situation would lead one to adjust one’s logic, in response to the Moebius Strip example? Makinson answer this in section 3.2.4, with Example 4, intended to show that using a focus on maximally consistent subsets of the norms does not give the intuitively correct result.

Example 4 (Möbius strip). Let C = {(α,β), (γ,α), (~β,γ)}. Intuitively, we would like to have α, γ and not(β). But …

This intuition is geared to a non-temporal understanding of the imperatives: if it is known that β then the first and second imperatives are to be simultaneously obeyed. The reason is that the first is seen as bringing the second along with it.

This would certainly be the case if obeying it were to take no time at all (to be instantaneous), and to effect no change in the situation. It would also be appropriate if the imperatives are in a subject area where time is not a relevant concern — perhaps commands in a program designed to perform mathematical calculations?

That same intuition appears to be Horty’s in his book, when he introduces the notion that one imperative can trigger another, by making true, and known, the antecedent of the other. Again, that assumes that the situation is not changed in two steps but in one, and thus either ignores time as irrelevant (as it may indeed be in certain cases) or assumes the action to be instantaneous.

REFERENCES

Makinson, David “On a fundamental problem of deontic logic”, pp 29-53 in Norms, Logics and Information Systems. New Studies in Deontic Logic and Computer Science, edited by Paul McNamara and Henry Prakken (Amsterdam: IOS Press, Series: Frontiers in Artificial Intelligence and Applications, Volume: 49, 1999, ISBN 9051994273)

Horty, John F. Reasons as Defaults. Oxford: Oxford University Press, 2012.