1 The paradox
Let \(W\) be a finite set of possible worlds of a given model, and let \(\mathscr P\) denote the power set operation.
Combining the systems from several papers, the full list of Carmo and Jones’ conditions on a function \(ob : \mathscr P(W) \to \mathscr P(\mathscr P(W))\) is as follows.
Axiom 5(a): \(\emptyset \not\in \operatorname{ob}(X)\). (Obligations cannot be impossible to fulfil.)
Axiom 5(b): If \(Y\cap X=Z\cap X\) then \(Y\in \operatorname{ob}(X)\) iff \(Z\in \operatorname{ob}(X)\). (Whether \(Z\) is obligatory in context \(X\) depends only on \(Z\) through \(Z\cap X\).)
Axiom 5(c\(^-\)) (2002, page 319; finite version of 5(c\(^*\)), 2013, which is called 5(c) in 2022): If \(Y\in \operatorname{ob}(X)\) and \(Z\in \operatorname{ob}(X)\) and \(X\cap Y\cap Z\ne \emptyset \), then \(Y\cap Z\in \operatorname{ob}(X)\). (Closure under intersection.)
Axiom 5(c) (1997 and 2002, page 287): If \(Y\in \operatorname{ob}(X)\) and \(Z\in \operatorname{ob}(X)\) then \(Y\cap Z\in \operatorname{ob}(X)\). (Closure under intersection, even for disjoint sets.)
Axiom 5(d): If \(Y\subseteq X\) and \(Y\in \operatorname{ob}(X)\) and \(X\subseteq Z\), then \((Z\setminus X)\cup Y\in \operatorname{ob}(Z)\). (If \(Y\) is obligatory in context \(X\) then a form of the material conditional “\(X \to Y\)” holds in a wider context.)
Axiom 5(e): If \(Y\subseteq X\) and \(Z\in \operatorname{ob}(X)\) and \(Y\cap Z\ne \emptyset \), then \(Z\in \operatorname{ob}(Y)\). (If \(Z\) is obligatory in the context \(X\) then it remains obligatory in any subcontext in which it is still possible.)
Axiom 5(f): If \(X\in \operatorname{ob}(Y)\) and \(X\in \operatorname{ob}(Z)\) then \(X\in \operatorname{ob}(Y\cup Z)\). (If \(X\) is obligatory in both of the contexts \(Y\) and \(Z\), then it is obligatory in their union.)
Axiom 5(g): If \(Y\in \operatorname{ob}(X)\) and \(Z\in \operatorname{ob}(Y)\) and \(X\cap Y\cap Z\ne \emptyset \), then \(Y\cap Z\in \operatorname{ob}(X)\). (A form of transitivity for obligations.)
Carmo and Jones gave several axioms on their function \(\mathrm{ob}\).
Axiom 5(a) for a function \(\operatorname{ob}: \mathcal P(U) \to \mathcal P(\mathcal P(U))\) says that \(\emptyset \notin \operatorname{ob}(X)\) for all \(X\in \mathcal P(U)\).
Axiom 5(b) for a function \(\operatorname{ob}: \mathcal P(U) \to \mathcal P(\mathcal P(U))\) says that
The weak, but potentially infinite, version of Axiom 5(c) for a function \(\operatorname{ob}: \mathcal P(U) \to \mathcal P(\mathcal P(U))\) says that
The strong version of Axiom 5(c) for a function \(\operatorname{ob}: \mathcal P(U) \to \mathcal P(\mathcal P(U))\) says that If \(Y\in \operatorname{ob}(X)\) and \(Z\in \operatorname{ob}(X)\) then \(Y\cap Z\in \operatorname{ob}(X)\).
Axiom 5(d) for a function \(\operatorname{ob}: \mathcal P(U) \to \mathcal P(\mathcal P(U))\) says that
If \(A\in \operatorname{ob}(X)\) whenever \(A\subseteq X\) and \(a_1\ne a_2\) then \(A\cup \{ a_1\} \in \operatorname{ob}(A\cup \{ a_1,a_2\} )\).
Axiom 5(e) for a function \(\operatorname{ob}: \mathcal P(U) \to \mathcal P(\mathcal P(U))\) says that If \(Y\subseteq X\) and \(Z\in \operatorname{ob}(X)\) and \(Y\cap Z\ne \emptyset \), then \(Z\in \operatorname{ob}(Y)\).
The argument.
Axioms 5(b)(e)(f) will now be used to derive a paradox.
Suppose that James is taking an exam on which the possible grades, in decreasing order of quality, are A, B, C, D and F.
If seems reasonable to assume that given that James’ grade is A or B, it ought to be A.
Moreover, given that the grade is C or D, it ought to be C.
Then by axiom 5(b), given that the grade is C or D, it ought to be A or C. Moreover, also by 5(b), given that the grade is A or B, it ought to be A or C. In other words, James ought to make sure the proposition “the grade is A or C” is true (of course, ideally by getting an A).
It also follows that the grade ought to be A or C given that it is A, B, C, or D, this time using axiom 5(f) applied to the two previous statements. (If this is getting hard to follow, the reader may rest assured that a formalization is available, see 2.1.)
So far this is not entirely unintuitive. But now we use 5(e), and conclude that since A or C was obligatory in the context A, B, C, or D it must remain obligatory in a more restrictive context in which it is still possible, namely B, C, or D.
Finally, using 5(b) again, we conclude that given that the grade is B, C, or D, it ought to be C. While there is such a concept as “gentleman’s C”, surely this is a paradox. 1
A literary perspective.
Let us consider this paradox above in another context, beside students and their grades.
Suppose James has the following options.
Marriage to Alice, his favorite.
Marriage to Alice’s sister Beatrice, or one of several other women he also quite likes.
Remaining a lonely bachelor.
Marriage to Deirdre, whom he despises.
Then our conclusion is like that of the Bee Gees in 1977:
If I can’t have you, I don’t want nobody.
From despondency to mathematics.
Hage [ 5 ] thinks the task Carmo and Jones and others have set themselves is impossible.
From Hage’s book review, in the context of discussing a man who (i) should visit his neighbor, (ii) if he does visit, should call to say that he is coming, and (iii) is not in fact coming:
[...] Actually, A does not assist his neighbours and therefore should not call them. The two intuitions presuppose a different role for deontic logic, namely, reasoning about what is ideally the case, and reasoning about what ought to be done in the real world. These two roles are hard to reconcile in one logic, unless the logic combines the two kinds of reasoning in different parts. The paper by Carmo and Jones in the present volume provides such a combination logic. Nevertheless, many have attempted to do what is, in my opinion, undoable 2 and this has lead to many modifications of the Standard System of Deontic Logic (Hilpinen, 1971, p. 13f.; Chellas, 1980, p. 190f.). These new systems have lead to new paradoxes that deal with CTD obligations.
Given Hage’s sentiment and the paradox we have presented, one might temporarily become despondent. If deontic systems always have flaws, why pursue them? However, in the course of uncovering this paradox I also discovered interesting mathematical structure in the axioms 5(a)–5(g).
Given a set \(W\) and a family \(\mathcal F\) of functions \(f :\mathscr P(W)\to \mathscr P(\mathscr P(W))\), we say that \(f_0\in \mathcal F\) is the least element of \(\mathcal F\) if for all \(f\in \mathcal F\) and all \(X\subseteq W\), \(f_0(X)\subseteq f(X)\).
The least model of a set of axioms \(\mathscr A\) concerning a variable function \(\mathrm{ob} :\mathscr P(W)\to \mathscr P(\mathscr P(W))\) is the least element of the collection \(\mathcal F\) of all functions \(\mathrm{ob} :\mathscr P(W)\to \mathscr P(\mathscr P(W))\) satisfying all axioms in \(\mathscr A\).
“Axiom” here is used to mean simply a condition on \(\mathrm{ob}\), although we may note that all the axioms 5(a)–(g) may be formulated in first-order set theory.
The least model of 5(b) given certain “oughts” has a very natural characterization, as does the least model of 5(b), 5(d), and 5(f).
To be specific, let us write \(\mathrm{Ought}(A\mid B)\) to mean that for each \(X\subseteq B\), if \(A\cap X\ne \emptyset \) then \(A\in \operatorname{ob}X\). This is the semantic condition used by Carmo and Jones for the conditional obligation operator \(O(A\mid B)\). A pair of oughts \((\mathrm{Ought}(A\mid W)\), \(\mathrm{Ought}(B\mid W \setminus A))\) forms a basic contrary-to-duty obligation of \(B\) given that our duty \(A\) has failed to be observed.
Definitions 9 and 10 introduce canon\(_2\) and canon\(_2\)II, “canonical” models of some of Carmo and Jones’ axioms with two parameters: a set of most desirable worlds, A, and a set of worlds that are desirable when A is not available, B.
The set canon\(_2\) A B X consists of all contexts that are obligatory in the context X in the model canon\(_2\) A B.
It is defined to be: if \(X \cap B = \emptyset \) then \(\emptyset \), else: if \(X \cap A = \emptyset \) then \(\{ T \mid X \cap B \subseteq T\} \), else \(\{ T \mid X \cap A \subseteq T\} \).
The set canon\(_2\)II A B X is defined by: if \(X \cap B = \emptyset \) then \(\emptyset \), else: if \(X \cap A = \emptyset \) then \(\{ Y \mid X \cap B = X \cap Y\} \) else \(\{ Y \mid X \cap A = X \cap Y\} \).
In other words, if \(Y \in \operatorname{ob}(X)\) under a canon₂_II model then generically \(X\cap Y\) consists exactly of the most desirable worlds in \(X\).
The least model \(\operatorname{ob}\) (under inclusion) of \(\mathrm{Ought}(A\mid W)\), \(\mathrm{Ought}(B\mid W \setminus A)\), and axiom 5(b) is canon₂_II A B.
Even though the model arises from assuming 5(b) only, it also satisfies axioms 5(a), 5(c), 5(e) and 5(g). This indicates a certain robustness of our definitions.
In other words, \(X\cap Y\) contains at least all the most desirable worlds in \(X\). Since axiom 5(f) follows from 5(a)(b)(c)(d), as shown by Carmo and Jones, the model can alternatively be characterized as the least one satisfying the latter four axioms and the two specified oughts.
The two families of models in 11 and 12 represent two alternative approaches to contrary-to-duty obligations as discussed in [ 7 ] (called I and II there). They do not exhaust the interesting models by any means: for instance, we may have conflicting obligations. This may lead us to prefer 5(c) to 5(c)*, in particular. For a concrete example, suppose that James has received acceptances on separate marriage proposals to both Alice and Beatrice, but cannot marry them both.
Classification.
In mathematics, classification theorems are fairly common. For example, the finite abelian groups have a straightforward characterization and the finite simple groups a complicated one. Vector spaces over a fixed field \(\mathbb F\) of finite dimension are characterized by their dimension \(d\), hence the single parameter \(d\) determines the space up to isomorphism.
In deontic logic, where axioms and rules are added based on moral intuitions, we should perhaps not expect structures that are mathematically natural enough to be classifiable.
However, for the full theory of Carmo and Jones’s axioms 5(a)–5(e) with the strong version 5(c)* in which we do not impose nondisjointness, we can characterize its models completely. The only nontrivial ones basically say that there is just one bad world and the only obligation is to avoid it:
Let \(W\) be a finite set of possible worlds and let \(\operatorname{ob}: \mathscr P (W)\to \mathscr P(\mathscr P(W))\). The following are equivalent:
\(\operatorname{ob}\) satisfies the full system suggested in [ 2 ] : 5(a), (b), (c)*, (d) and (e).
One of the following three holds:
\(\operatorname{ob}(X)=\emptyset \) for all \(X\).
\(\operatorname{ob}(X)=\{ Y \mid \emptyset \ne X \subseteq Y\} \) for all \(X\).
There is a distinguished possible world \(a\) (the “forbidden” world) such that for all \(X\), \(\operatorname{ob}(X)=\{ Y \mid X \cap Y \ne \emptyset \text{ and } X\setminus \{ a\} \subseteq Y\} \).
Details on the proof of Theorem 13 can be found in the next section.