ContrarytoDuty Paradoxes and Counterfactual Deontic Logic
Abstract
In this paper, I will discuss some examples of the socalled contrarytoduty (obligation) paradox, a wellknown puzzle in deontic logic. A contrarytoduty obligation is an obligation telling us what ought to be the case if something forbidden is true, for example: ‘If she is guilty, she should confess’. Contrarytoduty obligations are important in our moral and legal thinking. Therefore, we want to be able to find an adequate symbolisation of such obligations in some logical system, a task that has turned out to be difficult. This is shown by the socalled contrarytoduty (obligation) paradox. I will investigate and evaluate one kind of solution to this problem that has been suggested in the literature, which I will call the ‘counterfactual solution’. I will use some recent systems that combine not only counterfactual logic and deontic logic, but also temporal logic, in my analysis of the paradox. I will argue that the counterfactual solution has many attractive features and that it can give a fairly satisfactory answer to some examples of the contrarytoduty paradox, but that it nevertheless has some serious problems. The conclusion is that, notwithstanding the many attractive features of the solution, there seem to be other approaches to the paradox that are more promising.
Keywords
Contrarytoduty paradox Chisholm’s paradox Deontic logic Contrarytoduty obligation Counterfactual logic1 Introduction
A contrarytoduty obligation is an obligation telling us what ought to be the case if something forbidden is true. Here are some examples of sentences that often express such obligations: ‘If she is guilty, she should confess’, ‘If he has hurt his friend, he should apologise to her’, ‘If you are not going to keep your promise to him, you ought to call him’, ‘If the books are not returned by the due date, you must pay a fine’. We might also say that a contrarytoduty obligation is a conditional obligation where the condition (in the obligation) is forbidden, or where the condition is fulfilled only if a primary, unconditional obligation is violated. She should not be guilty; but if she is, she should confess. He should not have hurt his friend; but if he did, he should apologise. You should keep your promise to him; but if you are not going to do so, you ought to call him. The books ought to be returned by the due date; but if they are not, you must pay a fine.
Contrarytoduty obligations turn up in discussions concerning guilt, blame, restoration, reparation, confession, punishment, repentance, retributive justice, compensation, apologies, damage control, etc. Therefore, they are important in our moral and legal thinking. The rationale of a contrarytoduty obligation is the fact that most of us do neglect our primary duties from time to time and yet it is reasonable to believe that we should make the best of a bad situation, or at least that it matters what we do when this is the case. We want to be able to find an adequate formal representation of such obligations; it has, however, turned out to be difficult to do that. The socalled contrarytoduty (obligation) paradox, sometimes called the contrarytoduty imperative paradox, which has been discussed by deontic logicians for more than fifty years now, has clearly shown this. Roderick Chisholm was one of the first philosophers to address this puzzle (Chisholm 1963). Since then many different versions of this problem have been mentioned in the literature (see, e.g., Powers 1967; Åqvist 1967, 2002; Forrester 1984; Prakken and Sergot 1996; and Carmo and Jones 2002, for some examples).
The contrarytoduty paradox arises when we try to symbolise certain intuitively consistent sets of ordinary language sentences, sets that include at least one contrarytoduty obligation sentence, by means of ordinary counterparts available in various monadic deontic logics, such as socalled Standard Deontic Logic and similar systems. The formal representations often turn out to be inconsistent, in the sense that it is possible to deduce contradictions from them, or else they might violate some other intuitively plausible condition, for example that the sentences in a representation should be independent of each other.
In this paper, I will discuss some examples of the contrarytoduty paradox, and I will investigate and evaluate one kind of solution to this puzzle that has been suggested in the literature. I will call this proposal ‘the counterfactual solution’. The idea behind this kind of solution is that we can solve the paradox if we combine deontic logic with counterfactual logic. A contrarytoduty obligation is interpreted as a kind of conditional obligation that involves a counterfactual conditional. Mott (1973) and Niles (1997), for example, are sympathetic to this kind of approach, while Tomberlin (1981) and DeCew (1981) criticise it.
I will not discuss which, if any, contrarytoduty obligations are true. In this paper, I am primarily interested in the logical form of such obligations and the paradoxes they seem to generate. Before we can answer the question whether a particular contrarytoduty obligation is true or not, it seems that we must know what we mean by a ‘contrarytoduty obligation’.
In my analysis of the paradox, I will use some recent systems that combine not only counterfactual logic and deontic logic, but also temporal logic (Rönnedal 2016; see the appendix). In those systems, we can avoid some problems with earlier similar counterfactual solutions, and we can make an important distinction between ‘afterward’ (‘backward’), ‘beforehand’ (‘forward’) and ‘timeless’ or ‘parallel’ contrarytoduty obligations. In an ‘afterward’ contrarytoduty obligation, the content of the obligation comes after the time of the condition. ‘If you do not keep your promise, you ought to apologise (afterwards)’ can be used to express such an obligation. The content of the obligation, that you apologise, is supposed to take place after you violate your primary, unconditional obligation to keep your promise, if, in fact, you do not keep it. In a ‘beforehand’ contrarytoduty obligation, the content of the obligation comes before the time of the condition. ‘If you are not going to keep your promise, you ought to call your friend (beforehand)’ can be used to express such an obligation. The content of the obligation, that you call your friend, is supposed to take place before you violate your primary obligation to keep your promise. If, in fact, you will not keep your promise, you ought to call your friend beforehand. In a ‘parallel’ contrarytoduty obligation, the content of the obligation is supposed to take place at the same time as the condition. ‘If there is a dog, there ought to be a warning sign’ can be used to express such an obligation. The content of the obligation, that there is a warning sign, is supposed to be fulfilled at the same time as the primary obligation, that there is no dog, is violated.
I will argue that the counterfactual solution presented in this paper has many attractive features and that it can give a fairly satisfactory answer to some examples of the contrarytoduty paradox, primarily examples that involve afterward contrarytoduty obligations, but that it nevertheless has some serious problems, especially with parallel and beforehand contrarytoduty paradoxes. The conclusion is that, notwithstanding the many attractive features of the solution, there seem to be other approaches to the paradox that are more promising.
The plan of the paper is as follows. In Section 2, I will describe one example of a contrarytoduty paradox, and in Section 3, I will show how this version can be solved in counterfactual temporal alethicdeontic logic. Section 4 includes several arguments for this solution, and in Section 5, I will consider some problems with the approach. Section 6 contains a summary of the paper and a conclusion.
2 The ContrarytoDuty (Obligation) Paradox
In this section, I will begin by discussing a particular version of a contrarytoduty paradox that involves an afterward contrarytoduty obligation. I will call this version ‘the (afterward) promise (contrarytoduty) paradox’. Then, in later sections, I will mention some other versions.
Scenario I: The (Afterward) Promise (ContrarytoDuty) Paradox (after Prakken and Sergot 1996)
Consider the following scenario. It is Monday and you promise a friend to meet her on Friday to help her with some task. Suppose, further, that you always meet your friend on Saturdays. In this example, the following sentences all seem to be true:
NCTD
 N1

(On Monday it is true that) You ought to keep your promise (and see your friend on Friday).
 N2

(On Monday it is true that) It ought to be that if you keep your promise, you do not apologise (when you meet your friend on Saturday).
 N3

(On Monday it is true that) If you do not keep your promise (i.e. if you do not see your friend on Friday and help her out), you ought to apologise (when you meet her on Saturday).
 N4

(On Monday it is true that) You do not keep your promise (on Friday).
Let NCTD = {N1, N2, N3, N4}. N3 is a contrarytoduty obligation (or expresses a contrarytoduty obligation). If the condition is true, the primary obligation that you should keep your promise (expressed by N1) is violated. Furthermore, it is an afterward contrarytoduty obligation. If you do not keep your promise on Friday, you ought to apologise afterwards, when you meet your friend on Saturday. NCTD appears to be consistent, it does not seem possible to deduce any contradiction from this set. Nonetheless, if we try to formalise NCTD in socalled Standard Deontic Logic, for instance, we immediately encounter some problems. Standard Deontic Logic is a wellknown logical system described in most introductions to deontic logic (see e.g. Gabbay, Horty, Parent, van der Meyden and van der Torre (eds.), Gabbay et al. 2013, pp. 36–39). It is basically a normal modal system of the kind KD (Chellas 1980). In Åqvist (2002) this system is called OK+. For introductions to deontic logic, see e.g. Hilpinen (1971), (Hilpinen 1981) and Gabbay et al. (2013). Consider the following symbolisation:
SDLCTD
 SDL1

Ok
 SDL2

O(k → ¬a)
 SDL3

¬k → Oa
 SDL4

¬k
O is a sentential operator that takes a sentence as argument and gives a sentence as value. Informally, ‘Op’ says ‘It ought to be (or it should be) the case that (or it is obligatory that) p’. ‘¬’ is the standard negation sign, and ‘→’ the standard material implication sign, wellknown from ordinary propositional logic. In SDLCTD, ‘k’ stands for ‘You keep your promise (meet your friend on Friday and help her with her task)’ and ‘a’ abbreviates ‘You apologise (to your friend for not keeping your promise)’. In this symbolisation, SDL1 is supposed to express a primary obligation and SDL3 a contrarytoduty obligation telling us what ought to be the case if the primary obligation is violated. However, the set SDLCTD = {SDL1, SDL2, SDL3, SDL4} is not consistent in Standard Deontic Logic. O¬a is entailed by SDL1 and SDL2, and from SDL3 and SDL4 we can derive Oa. Hence, we can deduce the following formula from SDLCTD Oa ∧ O¬a (‘It is obligatory that you apologise and it is obligatory that you do not apologise’), which directly contradicts the socalled axiom D, that is, the schema ¬(OA ∧ O¬A). (‘∧’ is the ordinary symbol for conjunction.) ¬(OA ∧ O¬A) is included in Standard Deontic Logic (usually as an axiom). Clearly this sentence rules out explicit moral dilemmas. Since NCTD seems to be consistent, while SDLCTD is inconsistent, something must be wrong with our formalisation, with Standard Deontic Logic or with our intuitions. In a nutshell, this puzzle is the contrarytoduty (obligation) paradox.
3 Solutions to the ContrarytoDuty (Obligation) Paradox: The Counterfactual Solution
There are several possible responses to the contrarytoduty paradox. The various solutions that have been suggested in the literature can be divided into five categories: quick solutions, e.g. reject some axioms or rules of inference that are necessary to derive the contradiction, try to find some alternative formalisation of NCTD or reject some of our intuitions about this set; operator solutions that introduce several deontic operators that can be used to symbolise different kinds of unconditional obligations (e.g. Åqvist 1967; Jones and Pörn 1985; Carmo and Jones 2002); connective solutions that interpret the expression ‘if, then’ in a nonstandard way (e.g. Mott 1973; Niles 1997; Bonevac 1998); action or agent solutions that argue that deontic logic should be combined with some kind of action logic (e.g. Castañeda 1977, 1981, 1989; Meyer 1988; Bartha 1993); and temporal solutions, which use some combination of deontic logic and temporal logic to solve the puzzle (e.g. van Eck 1982; Loewer and Belzer 1983; Feldman 1986, 1990; Bartha 1993; Åqvist 2003).
In this section, I will consider the counterfactual solution to the contrarytoduty paradox and in sections 4 and 5, I discuss some arguments for and against this approach. The idea behind this kind of solution is that we can solve the paradox if we combine deontic logic with counterfactual logic. A contrarytoduty obligation is interpreted as a kind of conditional obligation that involves a counterfactual conditional. Mott (1973) and Niles (1997), for example, are sympathetic to a view of this kind, while Tomberlin (1981) and DeCew (1981), for instance, criticise it. I will, however, use some recent systems that combine not only counterfactual logic and deontic logic, but also temporal logic, in my analysis of the paradox (Rönnedal 2016; see the appendix). In those systems, we can avoid some problems with earlier similar counterfactual solutions and we can make an important distinction between ‘afterward’, ‘beforehand’ and ‘parallel’ contrarytoduty obligations. (For some early attempts to combine deontic logic and temporal logic, see e.g. Montague 1968, and Chellas 1969.)
In a counterfactual temporal alethicdeontic logic of the kind introduced by Rönnedal (2016), it is possible to symbolise the concept of a conditional obligation in at least four interesting ways: (A □ → OB), O(A □ → B), (A □ ⇒ OB) and O(A □ ⇒ B). □ → (and □⇒) is a twoplace, sentential operator that takes two sentences as arguments and gives one sentence as value. ‘A □→ B’ (and ‘A □⇒ B’) is often read ‘If A were the case, then B would be the case’. (The differences between □ → and □ ⇒ are unimportant in this context and I will focus on □ → .) According to the counterfactual solution, contrarytoduty obligations are conditional obligations of this kind where the condition is forbidden or violates some primary obligation. I will show how counterfactual temporal alethicdeontic logic can be used to avoid the contrarytoduty paradox.
In this section, I will consider one possible formalisation of NCTD that seems to be among the most plausible in counterfactual temporal alethicdeontic logic. In sections 4 and 5, I will discuss several other attempts to symbolise some other examples of contrarytoduty paradoxes. First, however, I will explain some symbols: t_{1} (in the formalisation below) refers to the moment on Monday when you make your promise, t_{2} refers to the moment on Friday when you should keep your promise and t_{3} refers to the moment on Saturday when you should apologise if you do not keep your promise on Friday. R is a temporal operator; ‘Rt_{1}A’ says that it is realised at time t_{1} (it is true at t_{1}) that A, etc. NCTD can now be symbolised in the following way:
CFCTD
 CF1

Rt _{1} ORt _{2} k
 CF2

Rt_{1}O(Rt_{2}k □ → Rt_{3}¬a)
 CF3

Rt_{1}(Rt_{2}¬k □ → Rt_{2}ORt_{3}a)
 CF4

Rt_{1}Rt_{2}¬k
Note that Rt_{1}Rt_{2}¬k is equivalent to Rt_{2}¬k. This means that it is true on Monday that it is true on Friday that you do not keep your promise if and only if it is true on Friday that you do not keep your promise. Accordingly, we can use Rt_{2}¬k as an alternative symbolisation of N4. Furthermore, note that it might be true on Monday that you will not keep your promise on Friday (in some possible world) even though this is not a settled fact, that is, even though it is not historically necessary. In some possible worlds you will keep your promise on Friday and in some possible worlds you will not. F4 is true at t_{1} (i.e. on Monday) in the possible worlds where you do not keep your promise at t_{2} (i.e. on Friday).
In counterfactual temporal alethicdeontic logic truth is relativised to worldmoment pairs. In other words, a sentence can be true in one possible world ω at a particular time τ even though it is false in some other possible world, say ω′, at this time, that is, at τ, or false in this world, that is, in ω, at another time, say τ′. A temporally settled sentence satisfies the following condition: if it is true (in a possible world), it is true at every moment of time (in this possible world); and if it is false (in a possible world), it is false at every moment of time (in this possible world). Some (but not all) sentences are temporally settled. All the sentences in CFCTD are temporally settled; ORt_{2}k, O(Rt_{2}k □ → Rt_{3}¬a) and O(Rt_{2}¬k □ → Rt_{3}a) are examples of sentences that are not, as their truth values may vary from one moment of time to another (in one and the same possible world).
Let us call the counterfactual temporal alethicdeontic system that includes all basic alethic, deontic and temporal rules, all rules in Tables 3, 4, 5 and 6 and all rules in Table 7 except T – c7 in Rönnedal (2016), all alethic rules that correspond to the fact that the alethic accessibility relation is ‘reflexive’, ‘symmetric’ and ‘transitive’, all deontic rules that correspond to the fact that the deontic accessibility relation is ‘serial’, ‘transitive’ and ‘Euclidean’, and all temporal rules that corresponds to the fact that the temporal accessibility relation is ‘transitive’, ‘comparable’ and does not branch towards the future or the past, Strong Counterfactual Temporal AlethicDeontic Logic (SCTADL) (see the appendix). SCTADL is a very strong counterfactual temporal alethicdeontic system. Basically, all philosophically interesting systems seem to be a part of this logic. Moreover, let CFCTD = {CF1, CF2, CF3, CF4}. I will show that CFCTD is consistent in all systems weaker than or deductively equivalent to the system SCTADL. To establish this, it is enough to show that CFCTD is consistent in SCTADL. Then it follows that the result holds in all weaker systems too. To show that CFCTD is consistent in SCTADL, it is enough to come up with a model of the right kind where all elements in CFCTD are true in some possible world at some time. This establishes that the set is satisfiable, and hence, by the soundness results in Rönnedal (2016), that the set is consistent in SCTADL and in every weaker system.
Model I
Consider the following supplemented counterfactual temporal alethicdeontic model. W = {ω_{1}, ω_{2}, ω_{3}, ω_{4}}; W is a set of possible worlds, ω_{1}, etc. are possible worlds in W. The expression ‘ω_{2} ≥_{ω1}ω_{3}’ says that ω_{2} is at least as similar to ω_{1} as is ω_{3}. We use ‘ω_{2} >_{ω1}ω_{3}’ to say that ω_{2} is more similar to ω_{1} than ω_{3} is. ω_{2} > _{ω1}ω_{3} if and only if ω_{2} ≥ _{ω1}ω_{3} and not ω_{3} ≥ _{ω1}ω_{2}. We have the following similarity relations between the possible worlds in W: ω_{1} > _{ω1}ω_{2}, ω_{2} > _{ω1}ω_{4}, ω_{3} > _{ω1}ω_{4}, ω_{2} > _{ω2}ω_{1}, ω_{1} > _{ω2}ω_{3}, ω_{3} > _{ω2}ω_{4}, ω_{3} > _{ω3}ω_{4}, ω_{4} > _{ω3}ω_{2}, ω_{2} > _{ω3}ω_{1}, ω_{4} > _{ω4}ω_{3}, ω_{3} > _{ω4}ω_{1}, ω_{1} > _{ω4}ω_{2}. The model satisfies all conditions in Table 2 and all conditions except C – c7 in Table 1 in Rönnedal (2016) (see the appendix). The alethic accessibility relation is an ‘equivalence’ relation (i.e. it is ‘reflexive’, ‘symmetric’, and ‘transitive’ (at every moment in time), etc.). The deontic accessibility relation is ‘serial’, ‘transitive’ and ‘Euclidean’ (and even ‘functional’) (at every moment in time). T = {τ_{1}, τ_{2}, τ_{3}}, where T is a set of moments of time, and τ_{1}, etc., are particular moments of time in T. τ_{1} < τ_{2} < τ_{3}. ‘τ_{1} < τ_{2}’ says that τ_{1} is before τ_{2}. The temporal relation is ‘transitive’, ‘comparable’, and it does not branch towards the future or the past. At τ_{1}, ω_{1} is deontically accessible from every possible world (and no other world is deontically accessible from any world). ω_{1} is deontically accessible from ω_{1} and ω_{2}, and ω_{3} is deontically accessible from ω_{3} and ω_{4} at τ_{2}. No other possible world is deontically accessible from any world at τ_{2}. At τ_{3}, every possible world is deontically accessible from itself. No possible world is deontically accessible from any other possible world at τ_{3}. At τ_{1}, all possible worlds are alethically accessible from all possible worlds. At τ_{2}, ω_{2} is alethically accessible from ω_{1} and ω_{1} is alethically accessible from ω_{2}. At τ_{2}, ω_{3} can see ω_{4} alethically and ω_{4} can see ω_{3} alethically. At all times every possible world is alethically accessible to itself. v(t_{1}) = τ_{1}, v(t_{2}) = τ_{2}, and v(t_{3}) = τ_{3}. k is true in ω_{1} and ω_{2} at τ_{2} and k is false in ω_{3} and ω_{4} at τ_{2}. a is false in ω_{1} and ω_{4} at τ_{3} and a is true in ω_{2} and ω_{3} at τ_{3}. For our purposes, we do not need any further information about this model. Model I is a model of the right kind.
In ω_{3} at τ_{1}, all sentences in CFCTD are true. Hence, by the soundness results in Rönnedal (2016), the set is consistent in SCTADL and in every weaker system. Therefore, it appears to be the case that we can solve the contrarytoduty paradox in counterfactual temporal alethicdeontic logic, at least the promise paradox. CFCTD is not inconsistent in any philosophically interesting counterfactual temporal alethicdeontic system.
4 Arguments for the Counterfactual Solution
In this section, I will discuss some reasons why the counterfactual solution is attractive. Together these reasons provide a strong, although not conclusive, case for this approach.^{1}
Reason (1): CFCTD is consistent
We have shown above that CFCTD is consistent (in SCTADL and in every weaker system). It does not seem to be possible to deduce any contradiction from NCTD. Hence, we want our symbolisation of this set to be consistent. In this respect, CFCTD is an intuitively plausible formalisation of NCTD.
Reason (2): CFCTD is nonredundant
A set of sentences is nonredundant if and only if no sentence in this set follows from the rest. NCTD appears to be nonredundant. Consequently, we want our symbolisation of NCTD to be nonredundant. In certain monomodal deontic systems, for instance Standard Deontic Logic, we can try to solve the contrarytoduty paradox by finding some other formalisation than SDLCTD of the sentences in NCTD. Instead of SDL2 we can use k → O¬a and instead of SDL3 we can use O(¬k → a). If we employ these formulas, we obtain three consistent alternative symbolisations of NCTD. Anyhow, O(¬k → a) follows from Ok in every socalled normal deontic system, including Standard Deontic Logic, and k → O¬a follows from ¬k by propositional logic. So, these alternatives are not nonredundant. Intuitively, N3 does not seem to be deducible from N1, and N2 does not seem to follow from N4. In counterfactual temporal alethicdeontic logic, we can avoid this problem. All sentences in CFCTD are independent of each other in SCTADL and in every weaker system. To establish this claim, it is enough to show that the elements of CFCTD are independent of each other in SCTADL. Then it follows that the result also holds for all weaker systems. To prove this, I will establish the following four propositions.
Proposition 1
CF1 is not deducible from {CF2, CF3, CF4} in SCTADL. To prove this proposition, it is sufficient to come up with a model M of the right kind, a world ω in M and a time τ in M such that all members of {CF2, CF3, CF4} are true in ω at τ and CF1 false in this world at this time. Then the proposition follows by the soundness results in Rönnedal (2016). Consider the following model.
Model II
Model II is exactly like Model I except that the deontic accessibility relation is modified in the following way. At τ_{1}, ω_{4} is deontically accessible from every possible world. No other possible world is deontically accessible from any possible world at τ_{1}. All members of {CF2, CF3, CF4} are true and CF1 false in world ω_{4} at time τ_{1}. Consequently, CF1 is not deducible from {CF2, CF3, CF4} in SCTADL (or any weaker system).
Proposition 2
CF2 is not deducible from {CF1, CF3, CF4} in SCTADL. The following model proves this proposition.
Model III
Model III is exactly like Model I except that a is true in ω_{1} at τ_{3}. All members of {CF1, CF3, CF4} are true and CF2 false in world ω_{3} at time τ_{1}. Therefore, CF2 is not deducible from {CF1, CF3, CF4} in SCTADL (or any weaker system).
Proposition 3
CF3 is not derivable from {CF1, CF2, CF4} in SCTADL. To prove this claim we use the following model.
Model IV
Model IV is exactly like Model I except that a is false in ω_{3} at time τ_{3}. All members of {CF1, CF2, CF4} are true and CF3 false in world ω_{1} at time τ_{1}. Hence, CF3 is not deducible from {CF1, CF2, CF4} in SCTADL (or any weaker system).
Proposition 4
CF4 is not derivable from {CF1, CF2, CF3} in SCTADL. Consider Model I: in world ω_{2} at time τ_{1} all members of {CF1, CF2, CF3} are true and CF4 false. Hence, CF4 is not derivable from {CF1, CF2, CF3} in SCTADL (or any weaker system), by the soundness of SCTADL.^{2}
Reason (3): CFCTD is dilemma free
Standard Deontic Logic minus the axiom D, ¬(OA ∧ O¬A), is the minimal socalled normal deontic logic K. SDLCTD does not entail any contradiction in K. Hence, it is possible to avoid the contrarytoduty paradox in ordinary monomodal deontic logic if we give up this axiom. There might be independent reasons for rejecting D. If there are, or could be, genuine moral dilemmas, D cannot in general be true. But even if there are or could be genuine moral dilemmas (which is debatable), rejecting D does not seem to be a good solution to the contrarytoduty paradox for several reasons. Firstly, even if we reject axiom D, it is counterintuitive to claim that a dilemma follows from NCTD. We can still deduce the sentence Oa ∧ O¬a from SDLCTD in every normal deontic system, which says that it is obligatory that you apologise and it is obligatory that you do not apologise. And this proposition does not seem to follow from NCTD. Secondly, if there are any moral dilemmas of this kind, we can derive the claim that everything is both obligatory and forbidden in every normal deontic system, which is absurd (see Reason 4 below). Thirdly, such a solution might still have problems with the socalled pragmatic oddity (see Reason 5 below).
The solution in counterfactual temporal alethicdeontic logic is dilemma free. The sentence Rt_{1}ORt_{3}¬a is derivable from CF1 and CF2 (see Reason 10 below) and from CF3 and CF4 we can deduce the formula Rt_{2}ORt_{3}a (see Reason 11 below) (in some systems). Thus, it is possible to derive the following sentence from CFCTD: Rt_{1}ORt_{3}¬a ∧ Rt_{2}ORt_{3}a (in certain systems). ‘Rt_{1}ORt_{3}¬a’ says ‘On Monday [when you have not yet broken your promise] it ought to be the case that you do not apologise on Saturday’, and ‘Rt_{2}ORt_{3}a’ says ‘On Friday [when you have broken your promise] it ought to be the case that you apologise on Saturday’. Nonetheless, ORt_{3}a and ORt_{3}¬a are not true at the same time. Neither Rt_{1}ORt_{3}¬a ∧ Rt_{1}ORt_{3}a nor Rt_{2}ORt_{3}¬a ∧ Rt_{2}ORt_{3}a is deducible from CFCTD in any interesting counterfactual temporal alethicdeontic system. So, this is not a moral dilemma. Since NCTD seems to be dilemma free, we want our formalisation of NCTD to be dilemma free too; and CFCTD is, as we have seen, dilemma free. This is one good reason to be attracted to the counterfactual solution.
Reason (4): It is not possible to derive the proposition that everything is both obligatory and forbidden from CFCTD
If there is at least one moral dilemma, we can derive the conclusion that everything is both obligatory and forbidden in every normal deontic logic (even without the axiom D). Since FA (‘It is forbidden that A’) is equivalent to O¬A, this follows from the fact that Oa ∧ O¬a entails Or for any r in every normal deontic logic. This is clearly absurd. NCTD does not appear to entail that everything is both obligatory and forbidden. Accordingly, we do not want our symbolisation to entail this. In counterfactual temporal alethicdeontic logic we can avoid this problem. Consider Model I above. Every sentence in CFCTD is true in ω_{3} at time τ_{1} in this model. For all that, Rt_{1}ORt_{3}¬a is true while Rt_{1}O¬Rt_{3}¬a and Rt_{1}FRt_{3}¬a are false in this world at this time.
Reason (5): CFCTD does not entail the socalled pragmatic oddity
It does not seem to follow from NCTD that you should keep your promise and apologise (for not keeping your promise). Accordingly, we do not want our symbolisation of NCTD to entail this counterintuitive conclusion or anything similar to it. Despite that, O(k ∧ a) is derivable from SDLCTD in every socalled normal deontic logic (with or without the axiom D), and ‘O(k ∧ a)’ says that it is obligatory that you keep your promise and apologise (for not keeping your promise). Even if this claim is not inconsistent, it is certainly very odd. This is the socalled pragmatic oddity, which is a problem for many possible solutions to the contrarytoduty paradox (Prakken and Sergot 1996). In counterfactual temporal alethicdeontic logic, we can avoid this puzzle. Neither O(Rt_{2}k ∧ Rt_{3}a), Rt_{1}O(Rt_{2}k ∧ Rt_{3}a) nor Rt_{2}O(Rt_{2}k ∧ Rt_{3}a) is deducible from CFCTD in any interesting counterfactual temporal alethicdeontic system. Consider Model I above. In ω_{3} at τ_{1} all sentences in CFCTD are true, but neither O(Rt_{2}k ∧ Rt_{3}a), Rt_{1}O(Rt_{2}k ∧ Rt_{3}a) nor Rt_{2}O(Rt_{2}k ∧ Rt_{3}a) holds in this world at this time. This proves our claim.
Reason (6): The counterfactual solution is applicable to (at least apparently) actionless and agentless contrarytoduty examples
If we combine deontic logic with some kind of action logic, for example some kind of Stit (‘Seeing to it’) logic (Bartha 1993), or dynamic logic (Meyer 1988), it might be possible to solve some contrarytoduty paradoxes. However, there also seem to be examples of contrarytoduty paradoxes that involve action and agentless contrarytoduty obligations. It appears to be difficult to solve these paradoxes in such systems. (For more on Stit logic, see Horty 2001, and Belnap et al. 2001).
Scenario II: ContrarytoDuty Paradoxes Involving (apparently) action and/or agentless contrarytoduty obligations
Consider the following scenario. At t_{1} (some arbitrary point in time when there is no fire in the house), it ought to be the case that there is no fire in the house at t_{2} (some time later than t_{1}). (At t_{1}) It ought to be that if there is no fire in the house (at t_{2}), then it is not the case that the sprinkler goes off (at t_{3}, a point in time as soon as possible after t_{2}). (At t_{1}) If there is a fire in the house (at t_{2}), then (at t_{2}) the sprinkler ought to go off (at t_{3}). Moreover, there is a fire in the house (at t_{2}). In this example, all of the following claims seem to be true:
NACTD
 AN1

(At t_{1}) There ought to be no fire in the house (at t_{2}).
 AN2

(At t_{1}) It ought to be that if there is no fire in the house (at t_{2}), then it is not the case that the sprinkler goes off (at t_{3}).
 AN3

(At t_{1}) If there is a fire in the house (at t_{2}) then (at t_{2}) the sprinkler ought to go off (at t_{3}).
 AN4

(At t_{1} it is the case that at t_{2}) There is a fire in the house.
AN1 tells us what ought to be the case unconditionally, and AN3 expresses a contrarytoduty obligation. The condition in AN3 is fulfilled only if the primary obligation expressed by AN1 is violated. But AN3 does not seem to involve any action or particular agent in any ordinary sense. The scenario is an example of an action and agentless contrarytoduty paradox. AN3 says something about what ought to be the case if the world is not as it ought to be according to AN1. It is difficult to find a plausible symbolisation of NACTD and similar paradoxes in dynamic deontic logic or in Stit logic.
But for all that, in counterfactual temporal alethicdeontic logic, we have no trouble symbolising (apparently) action and agentless contrarytoduty obligations. The logical form of the sentences in NACTD exactly parallels the logical form of the sentences in NCTD. Hence, we can solve contrarytoduty paradoxes of this kind in exactly the same way as we solved our original paradox.^{3}
Reason (7): We do not have to postulate any primitive conditional obligations
In counterfactual temporal alethicdeontic logic, a conditional obligation can be expressed by a combination of a counterfactual conditional and an ordinary (unconditional) obligation. We do not have to introduce any new primitive dyadic deontic operators. According to various dyadic and temporal dyadic deontic solutions, we need some new primitive dyadic deontic operator (e.g. van Eck 1982; Loewer and Belzer 1983; Feldman 1986, 1990; Åqvist 2003). (For more on dyadic deontic logic, see e.g. Rescher 1958; von Wright 1964; Danielsson 1968; Hansson 1969; van Fraassen 1972, 1973; Lewis 1974; von Kutschera 1974; Spohn 1975; Cox 1978; Åqvist 1984, 1987, 2002; Åqvist and Hoepelman 1981, Goble 2003, 2004; Parent 2008, 2010.) Some philosophers have been critical of this approach (e.g. Bonevac 1998). They think a proper theory of conditional obligations, including contrarytoduty obligations, will be the product of two separate components: a theory of the conditional and a theory of obligation. The counterfactual solution satisfies this ‘requirement’. (For some other criticisms, see e.g. Tomberlin 1989, 1989b.)
Reason (8): We can express the idea that an obligation has been violated
It has been suggested that the contrarytoduty paradox can be solved by using some kind of nonmonotonic logic (see e.g. Bonevac 1998). However, it is not obvious that we can explain the difference between violation and defeat in such a logic. If you will not see your friend and help her, the primary, unconditional obligation to keep your promise will be violated. It is not the case that this obligation is defeated, overridden or cancelled. The conditional norms in NCTD do not cancel each other out. Nor is it the case that one of the norms defeat or override the other. Smith (1994) and Prakken and Sergot (1996) stress the difference between violation and defeat.
In counterfactual temporal alethicdeontic logic, it is easy to express the idea that an obligation will be violated, or has been violated. At t_{1}, CF4 [Rt_{1}Rt_{2}¬k]—for example—expresses the idea that the unconditional norm CF1 [Rt_{1}ORt_{2}k] will be violated. The fact that a norm will be violated does not entail that this is necessary, settled or determined. At t_{2}, on the other hand, we can say that it is settled or historically necessary that you do not keep your promise (in the worlds where you in fact do not keep it). Then, we might want to say that it is no longer obligatory that you keep your promise. In spite of that, the sentence Rt_{1}ORt_{2}k is still true at t_{2} (in the worlds where you do not keep your promise), that is, it is still true that you should have kept your promise. So, when you do not keep your promise, you violate this ‘earlier’ duty.
Reason (9): We can symbolise higher order contrarytoduty obligations
There are contrarytoduty obligations of a ‘higher order’ or ‘degree’. This is illustrated by the following scenario:
Scenario III: The Higher Order ContrarytoDuty Paradox
Scenario III is similar to Scenario I. Although, we now suppose that you do not meet your friend on Saturday and apologise. So, you do not only violate the primary obligation to keep your promise, but also the contrarytoduty obligation to apologise on Saturday given that you do not keep your promise. In this case, we can assume that there is a ‘higher order’ or ‘degree’ contrarytocontrarytoduty obligation to apologise (on Sunday when you do in fact meet your friend) given that you neither keep your promise nor apologise on Saturday. In this scenario all of the following sentences seem to be true:
NHCTD
 HN1

(On Monday it is true that) You ought to keep your promise (and see your friend on Friday).
 HN2

(On Monday it is true that) It ought to be that if you keep your promise, you do not apologise (on Saturday).
 HN3

(On Monday it is true that) It ought to be that if you keep your promise, you do not apologise (on Sunday).
 HN4

(On Monday it is true that) If you do not keep your promise (i.e. if you will not see your friend on Friday and help her out), then (on Friday) you ought to apologise (on Saturday).
 HN5

(On Monday it is true that) You do not keep your promise (on Friday).
 HN6

(On Monday it is true that) If you do not keep your promise (on Friday) and you do not apologise (on Saturday), then (on Saturday) you ought to apologise (on Sunday).
 HN7

(On Monday it is true that) You do not apologise (on Saturday).
Let NHCTD = {HN1,. .., HN7}. HN4 is an ordinary, firstorder or firstdegree contrarytoduty obligation that tells us what ought to be the case if the primary obligation expressed by HN1 is violated. HN6 expresses a contrarytocontrary to duty obligation, a secondorder or seconddegree contrarytoduty obligation. The condition in this obligation is satisfied only if the primary obligation expressed by HN1 is violated and the firstorder contrarytoduty obligation to apologise (on Saturday) is violated. A plausible solution to the contrarytoduty paradox should be able to deal with higherorder contrarytoduty obligations as well as ordinary firstdegree contrarytoduty obligations. In our counterfactual temporal alethicdeontic systems, we can symbolise such higherorder contrarytoduty obligations. NHCTD can, for example, be formalised in the following way in counterfactual temporal alethicdeontic logic:
FHCTD
 HF1

Rt _{1} ORt _{2} k
 HF2

Rt_{1}O(Rt_{2}k □ → Rt_{3}¬a)
 HF3

Rt_{1}O(Rt_{2}k □ → Rt_{4}¬a)
 HF4

Rt_{1}(Rt_{2}¬k □ → Rt_{2}ORt_{3}a)
 HF5

Rt_{1}Rt_{2}¬k [Rt_{2}¬k]
 HF6

Rt_{1}((Rt_{2}¬k ∧ Rt_{3}¬a) □ → Rt_{3}ORt_{4}a)
 HF7

Rt_{1}Rt_{3}¬a [Rt_{3}¬a]
Let FHCTD = {HF1,..., HF7}. NHCTD seems to be consistent, nonredundant, etc. Hence, we want our symbolisation of this set to have these properties too. Not all solutions to the contrarytoduty paradox can solve such ‘higher order’ contrarytoduty paradoxes. Yet, FHCTD is consistent, nonredundant, etc. This is a good reason to accept the counterfactual solution.
Reason (10): We can derive ‘ideal’ obligations from CFCTD
N1 and N2 in NCTD seem to entail that you ought not to apologise. You ought to keep your promise, and it ought to be that if you keep your promise, then you do not apologise (for not keeping your promise). So, ideally you ought not to apologise. We want the symbolisation of NCTD to reflect this intuition. In many counterfactual temporal alethicdeontic systems, for instance SCTADL, Rt_{1}ORt_{3}¬a is deducible from CF1 [Rt_{1}ORt_{2}k] and CF2 [Rt_{1}O(Rt_{2}k□ → Rt_{3}¬a)]. The tableau below proves this.
 (1)
Rt_{1}ORt_{2}k, w_{0}t_{0}
 (2)
Rt_{1}O(Rt_{2}k □ → Rt_{3}¬a), w_{0}t_{0}
 (3)
¬Rt_{1}ORt_{3}¬a, w_{0}t_{0}
 (4)
¬ORt_{3}¬a, w_{0}t_{1} [3, DR1]
 (5)
P¬Rt_{3}¬a, w_{0}t_{1} [4, ¬O]
 (6)
sw_{0}w_{1}t_{1} [5, P]
 (7)
¬Rt_{3}¬a, w_{1}t_{1} [5, P]
 (8)
¬¬a, w_{1}t_{3} [7, DR1]
 (9)
ORt_{2}k, w_{0}t_{1} [1, Rt]
 (10)
O(Rt_{2}k □ → Rt_{3}¬a), w_{0}t_{1} [2, Rt]
 (11)
Rt_{2}k, w_{1}t_{1} [6, 9, O]
 (12)
Rt_{2}k □ → Rt_{3}¬a, w_{1}t_{1} [6, 10, O]
 (13)
Rt_{2}k → Rt_{3}¬a, w_{1}t_{1} [12, DR2]
 (14)
Rt_{3}¬a, w_{1}t_{1} [11, 13, MP]
 (15)
¬a, w_{1}t_{3} [14, Rt]
 (16)
* [8, 15]
Informally, ‘Rt_{1}ORt_{3}¬a’ says that it is true at t_{1} (i.e. on Monday) that it ought to be the case that you will not apologise on Saturday when you meet your friend. Even so, Rt_{2}ORt_{3}¬a is not derivable from CF1 and CF2 (see Reason 3 above). On Friday, when you have broken your promise, and when it is no longer historically possible for you to keep your promise, then it is not obligatory that you do not apologise on Saturday. On the contrary, on Friday, it is obligatory that you apologise when you meet your friend on Saturday (see Reason 11). Nevertheless, it is reasonable to claim that it is true on Monday that it ought to be the case that you do not apologise on Saturday. For on Monday it is not a settled fact that you will not keep your promise; on Monday, it is still possible for you to keep your promise, which you ought to do. These conclusions correspond well with our intuitions about Scenario I.
Reason (11): We can derive ‘actual’ obligations from CFCTD
 (1)
Rt_{1}(Rt_{2}¬k □ → Rt_{2}ORt_{3}a), w_{0}t_{0}
 (2)
Rt_{1}Rt_{2}¬k, w_{0}t_{0}
 (3)
¬Rt_{2}ORt_{3}a, w_{0}t_{0}
 (4)
¬ORt_{3}a, w_{0}t_{2} [3, DR1]
 (5)
P¬Rt_{3}a, w_{0}t_{2} [4, ¬O]
 (6)
sw_{0}w_{1}t_{2} [5, P]
 (7)
¬Rt_{3}a, w_{1}t_{2} [5, P]
 (8)
¬a, w_{1}t_{3} [7, DR1]
 (9)
Rt_{2}¬k □ → Rt_{2}ORt_{3}a, w_{0}t_{1} [1, Rt]
 (10)
Rt_{2}¬k, w_{0}t_{1} [2, Rt]
 (11)
Rt_{2}¬k → Rt_{2}ORt_{3}a, w_{0}t_{1} [9, DR2]
 (12)
Rt_{2}ORt_{3}a, w_{0}t_{1} [10, 11, MP]
 (13)
ORt_{3}a, w_{0}t_{2} [12, Rt]
 (14)
Rt_{3}a, w_{1}t_{2} [6, 13, O]
 (15)
a, w_{1}t_{3} [14, Rt]
 (16)
* [8, 15]
Note that Rt_{1}ORt_{3}a is not derivable from CF3 and CF4 (see Reason 3). According to Rt_{1}ORt_{3}a, it is true at t_{1} (i.e. on Monday) that you should apologise to you friend on Saturday when you meet her. But on Monday it is not yet a settled fact that you will not keep your promise to your friend, on Monday it is still open to you to keep your promise. Accordingly, it is not true on Monday that you should apologise on Saturday. Since it is true on Monday that you ought to keep your promise, and it ought to be that if you keep your promise then you do not apologise, it follows that it is true on Monday that it ought to be the case that you do not apologise on Saturday (see Reason 10). These facts correspond well with our intuitions about Scenario I.
Reason (12): We can avoid the socalled dilemma of commitment and detachment
(Factual) Detachment is an inference pattern that allows us to infer or detach an unconditional obligation from a conditional obligation, given that the condition obtains. Thus, if detachment holds for the conditional (contrarytoduty) obligation that you should apologise if you do not keep your promise (if detachment is possible), then if it is in fact true that you do not keep your promise, we can derive the unconditional obligation that you should apologise.
According to the socalled dilemma of commitment and detachment (van Eck 1982, p. 263; see also Greenspan, 1975): (1) Detachment should be possible, for we cannot take seriously a conditional obligation if we cannot detach an unconditional obligation from it; and (2) Detachment should not be possible, for if detachment is possible, the following kind of situation would be inconsistent—A, it ought to be the case that B given that A; and C, it ought to be the case that notB given C. But, such a situation is not necessarily inconsistent.^{4}
This dilemma appears to be a problem for many different solutions to the contrarytoduty paradox, for example solutions that use some kind of pure dyadic deontic logic. In pure dyadic deontic logic, we cannot derive the unconditional obligation that it is obligatory that A (OA) from the dyadic obligation that it is obligatory that A given B (O(A/B), O[B]A) and B. But if this is true, how can we take such conditional obligations seriously? In counterfactual temporal alethicdeontic logic we can avoid this dilemma. From A □ → OB and A we can detach OB (in most interesting systems, for example SCTADL). However, the following set of formulas is not necessarily inconsistent: {A, O(A □ → B), C, O(C □ → ¬B)}. We have seen above that Rt_{2}ORt_{3}a, but not Rt_{1}ORt_{3}a, is detachable from Rt_{1}Rt_{2}¬k and Rt_{1}(Rt_{2}¬k □ → Rt_{2}ORt_{3}a) (in certain systems). These conclusions correspond well with our intuitions.
Reason (13): The structure of the sentences in CFCTD reflects the structure of the sentences in NCTD
In N2 (It ought to be that if you keep your promise, you do not apologise.), ‘ought’ has ‘wide scope’ and in N3 (If you do not keep your promise, you ought to apologise.) ‘ought’ has narrow scope. This is reflected in the formal sentences CF2 and CF3. In CF2 [Rt_{1}O(Rt_{2}k □ → Rt_{3}¬a)], ‘O’ has wide scope and in CF3 [Rt_{1}(Rt_{2}¬k □ → Rt_{2}ORt_{3}a)] ‘O’ has narrow scope. Some solutions to the paradox do not respect this difference in structure between N2 and N3.
Reason (14): We do not have to postulate several different unconditional obligation operators
Some philosophers have suggested that we should solve the contrarytoduty paradox by introducing different kinds of obligations, symbolised by distinct obligation operators (e.g. Åqvist 1967; Jones and Pörn 1985; and Carmo and Jones 2002). Assume, for example, that there are two obligation operators O_{1} and O_{2} that represent ideal and actual obligations, respectively. Then it might be possible to derive the formula O_{1}¬a ∧ O_{2}a from CFCTD instead of Oa ∧ O¬a. But O_{1}¬a ∧ O_{2}a is not inconsistent with the axiom D; ‘O_{1}¬a ∧ O_{2}a’ says that it is ‘ideallyobligatory’ that you do not apologise and it is ‘actuallyobligatory’ that you apologise. Still, this solution is problematic, since the derived ‘ideal’ obligation not to apologise does not seem to be of another kind compared to the derived ‘actual’ obligation to apologise.
In counterfactual temporal alethicdeontic logic, we do not have to postulate several different kinds of unconditional obligations. The unconditional obligation to not apologise is the same kind of obligation as the derived unconditional obligation to apologise. The only difference is that they hold at different times; the ‘ideal’ obligation not to apologise holds at τ_{1} (on Monday) and the ‘actual’ obligation to apologise holds at τ_{2} (on Friday).
Reason (15): The assignment of logical form to each of the norms in the representation of a contrarytoduty scenario is independent of the other norms in it
According to Carmo and Jones (2002), the assignment of logical form to each of the norms in the representation of a contrarytoduty scenario should be independent of the other norms in it. The counterfactual solution meets this requirement. Not all solutions do that (see Carmo and Jones 2002).
This ends our discussion of reasons for the counterfactual solution. In light of these arguments, the counterfactual solution seems to be among the most plausible so far suggested in the literature. It appears to be the case that it can solve at least contrarytoduty paradoxes that involve afterward contrarytoduty obligations. Despite that, it also has some serious problems. In the next section, I will consider some arguments against this solution. (For more on some problems, see DeCew 1981, and Tomberlin 1981, and for some responses, see Niles 1997.)
5 Problems with the Counterfactual Solution
In this section, I will consider three arguments against the counterfactual solution. According to the first one, the solution introduced in this paper does not add anything new to the discussion. According to the second argument, the counterfactual solution cannot handle timeless (or parallel) contrarytoduty paradoxes, and according to the third one, it cannot solve beforehand contrarytoduty paradoxes. It seems to me that the first argument can be answered. The solution in this paper does add something new to the discussion, and this is a good reason why it is interesting to investigate it. The second argument is more problematic, even though I do not think it is conclusive (see the reply in Section 5.2.5). The third is probably the most disturbing. Moreover, it suggests that there are other solutions to the paradox that are better than the counterfactual solution, all things considered, notwithstanding the fact that the counterfactual solution is very attractive and that it appears to be the case that it can solve afterward contrarytoduty paradoxes.
5.1 Argument 1: The Solution in this Paper Does Not Add Anything New to the Discussion
According to the first argument, the counterfactual solution in this paper does not add anything new to the discussion. The argument consists of two propositions. According to the first, the solution does not add anything because we can already solve the contrarytoduty paradox in counterfactual deontic logic without a temporal part. According to the second, it does not add anything since we can solve the contrarytoduty puzzle in a temporal deontic logic without any counterfactual part. I will now argue that both of these claims are problematic.
In pure counterfactual deontic logic, we can symbolise NCTD in the following way: {Ok, O(k □ → ¬a), ¬k □ → Oa, ¬k}. Nevertheless, in most systems, this set is inconsistent, whereas CFCTD is consistent. This problem can be avoided by using the following formalisation instead: {Ok, k □ → O¬a, ¬k □ → Oa, ¬k}. But there are other problems with this symbolisation. We can, for example, not derive the ‘ideal’ obligation that you should not apologise from this set. Furthermore, both of these solutions generate the pragmatic oddity. In other words, both sets entail the following formula, O(k ∧ a), which says that it is obligatory that you keep your promise and apologise (for not keeping your promise). This is surely counterintuitive. Moreover, it is difficult to find any other plausible symbolisation of NCTD in pure counterfactual deontic logic without any temporal part. (See DeCew 1981, for some further critique.) The counterfactual temporal solution does not have any of these problems. So, it is clearly a step forward. It is better than similar solutions in pure counterfactual deontic logic without a temporal part, at least in some important respects. In the light of all the arguments for the solution, it is clearly worth investigating it.
According to the second proposition in this argument, we do not really need the counterfactual in our symbolisation. We can solve afterward contrarytoduty paradoxes already in temporal deontic logic. We can use the following symbolisation instead: {Rt_{1}ORt_{2}k, Rt_{1}O(Rt_{2}k → Rt_{3}¬a), Rt_{1}(Rt_{2}¬k → Rt_{2}ORt_{3}a), Rt_{1}Rt_{2}¬k}. It seems that this solution has all the advantages that the counterfactual solution has. The set is consistent, nonredundant, etc. (in most, perhaps all, interesting logics). So, we do not really need to include any counterfactual part in our systems. However, even if it is true that this set is consistent, nonredundant, etc., there are other arguments against this kind of solution. Suppose that instead of Rt_{1}Rt_{2}¬k we have Rt_{1}Rt_{2}k. Then it follows that Rt_{1}(Rt_{2}¬k → Rt_{2}ORt_{3}¬a) and, in fact, that Rt_{1}(Rt_{2}¬k → Rt_{2}ORt_{3}A) for any A. Rt_{1}(Rt_{2}¬k → Rt_{2}ORt_{3}a) is supposed to express a contrarytoduty obligation. But if this is true, it seems that Rt_{1}(Rt_{2}¬k → Rt_{2}ORt_{3}¬a) and Rt_{1}(Rt_{2}¬k → Rt_{2}ORt_{3}A) also express contrarytoduty obligations. This is counterintuitive. It is implausible to claim that the proposition that you keep your promise on Friday entails the following contrarytoduty obligation: if you do not keep your promise (on Friday), then you ought not to apologise (on Saturday). In fact, if Rt_{1}(Rt_{2}¬k → Rt_{2}ORt_{3}A) expresses a contrarytoduty obligation, we can derive a whole set of problematic conclusions, for example, that if you do not keep your promise (on Friday), then you ought to lie, steal, cheat, rape, etc. (on Saturday). This is surely counterintuitive. The counterfactual solution does not have this problem. From Rt_{1}Rt_{2}k, it does not follow that Rt_{1}(Rt_{2}¬k □ → Rt_{2}ORt_{3}A). In this respect, the counterfactual solution is better than the ‘simple’ temporal solution.
In conclusion, the first argument does not appear to succeed. The counterfactual solution in this paper does add something new and interesting to the debate. Moreover, it seems to be a fairly plausible solution to afterward contrarytoduty paradoxes. So, let us turn to the second, more problematic argument.
5.2 Argument 2: The Counterfactual Solution cannot Handle Timeless (or Parallel) ContrarytoDuty Paradoxes
There seems to be timeless (or parallel) contrarytoduty paradoxes. In a timeless (or parallel) contrarytoduty paradox, all obligations appear, in some sense, to be in force simultaneously and both the antecedent and consequent in the contrarytoduty obligation seem to ‘refer’ to the same time (if indeed they refer to any time at all). Paradoxes of this kind cannot be solved in counterfactual temporal alethicdeontic logic. Several (apparently) timeless (or parallel) contrarytoduty paradoxes are mentioned by Prakken and Sergot (1996).
Here is one example:
Scenario IV: The Dog Warning Sign Scenario (after Prakken and Sergot 1996)
Consider the following set of regulations: It ought to be that there is no dog (in a certain place). It ought to be that if there is no dog, there is no warning sign (that warns us about the dog). If there is a dog, it ought to be that there is a warning sign. Suppose, further, that there is a dog. Then all of the following sentences seem to be true:
NTCTD
 TN1

It ought to be that there is no dog.
 TN2

It ought to be that if there is no dog, there is no warning sign.
 TN3

If there is a dog, it ought to be that there is a warning sign.
 TN4

There is a dog.
TN1 expresses a primary obligation and TN3 a contrarytoduty obligation. The condition in TN3 is satisfied only if the primary obligation expressed by TN1 is violated. In this example, all obligations appear to be timeless or parallel, they appear to be in force simultaneously and the antecedent and consequent in the contrarytoduty obligation TN3 seem to refer to one and the same time (or perhaps to no particular time at all). Hence, this is an example of a timeless or parallel contrarytoduty paradox. Let NTCTD = {TN1, TN2, TN3, TN4}. NTCTD appears to be consistent, nonredundant, etc. But if this is the case, NTCTD poses a problem for the counterfactual solution in this paper. For it seems to be impossible to find a plausible formalisation of NTCTD in counterfactual temporal alethicdeontic logic.
I will now try to show this by considering some possible symbolisations of NTCTD and some problems with these.
5.2.1 Attempt 1
F1TCTD
 TF1

O¬d
 TF2

O(¬d □ → ¬w)
 TF3

d □ → Ow
 TF4

d
In F1TCTD, ‘d’ stands for ‘There is a dog’ and ‘w’ for ‘There is a warning sign’. All other symbols are interpreted as usual. Let F1TCTD = {TF1, TF2, TF3, TF4}. This seems to be the most natural interpretation of NTCTD. But there are at least two problems with this formalisation. First, on this reading, the paradox is reinstated, for F1TCTD is inconsistent in most plausible counterfactual deontic systems, for example, in SCTADL. Second, F1TCTD generates the pragmatic oddity. In this section, I will prove that this set is inconsistent. In the next, we will see that it also generates the pragmatic oddity.
 (1)
O¬d, w_{0}t_{0}
 (2)
O(¬d □ → ¬w), w_{0}t_{0}
 (3)
d □ → Ow, w_{0}t_{0}
 (4)
d, w_{0}t_{0}
 (5)
d → Ow, w_{0}t_{0} [3, DR2]
 (6)
Ow, w_{0}t_{0} [4, 5, MP]
 (7)
sw_{0}w_{1}t_{0} [T − dD]
 (8)
¬d, w_{1}t_{0} [1, 7, O]
 (9)
¬d □ → ¬w, w_{1}t_{0} [2, 7, O]
 (10)
w, w_{1}t_{0} [6, 7, O]
 (11)
¬d → ¬w, w_{1}t_{0} [9, DR2]
 (12)
¬w, w_{1}t_{0} [8, 11, MP]
 (13)
* [10, 12]
So, this symbolisation does not seem to succeed. Let us turn to a second attempt.
5.2.2 Attempt 2
F2TCTD
 TF1

O¬d
 TF2b

¬d □ → O¬w
 TF3

d □ → Ow
 TF4

d
Let F2TCTD = {TF1, TF2b, TF3, TF4}. From TF3 and TF4, we can deduce Ow, but it is not possible to derive O¬w from TF1 and TF2b, at least not in most reasonable counterfactual deontic systems. Hence, we cannot derive a contradiction from this set. However, this symbolisation is problematic for at least three different reasons. First, the structure of TF2b does not reflect the structure of the English sentence, TN2. In TN2, the expression ‘ought’ has wide scope, but in TF2b, the Ooperator has narrow scope. Of course, this is not a conclusive argument. But it is nice if our formal sentences mirror our English sentences as close as possible. Similar remarks hold for similar arguments in this section. Second, we cannot derive the ideal obligation that there is no warning sign from TF1 and TF2b. Nevertheless, the proposition that there ought to be no warning sign seems to follow from TN1 and TN2. Third, this solution also generates the pragmatic oddity.
In conclusion, this attempt does not seem to succeed either. Our third attempt below avoids the pragmatic oddity. So, maybe this symbolisation can be used to solve timeless or parallel contrarytoduty paradoxes.
5.2.3 Attempt 3
F3TCTD
 TF1

O¬d
 TF2

O(¬d □ → ¬w)
 TF3b

O(d □ → w)
 TF4

d
Let F3TCTD = {TF1, TF2, TF3b, TF4}. In this set, not only TN2 but also TN3 is represented by a sentence where the Ooperator has wide scope. From this set, we can derive O¬w from TF1 and TF2, but Ow does not follow from TF3b and TF4. The set is not inconsistent. Moreover, the solution does not generate the pragmatic oddity. O(¬d ∧ w) does not follow from F3TCTD. Nevertheless, there are at least three problems with this symbolisation. First, O(d □ → w) does not reflect the structure of the natural language sentence, TN3. In TN3, the word ‘ought’ has narrow scope, but the Ooperator in O(d □ → w) has wide scope. Second, we cannot derive the ‘actual’ obligation that there ought to be a warning sign from TF3b and TF4. But it seems to follow that there ought to be a warning sign from TN3 and TN4. Third, O(d □ → w) does not seem to be true. This is problematic since we have stipulated that TN3 is true in our scenario. So, the symbolisation of this sentence should also be true. We will now consider the argument for this claim.
All of the following sentences seem to be true: O(¬d □ → ¬w), ¬d □ → O¬w and d □ → Ow. But O(d □ → w) appears to be false. According to the standard truthconditions for counterfactuals, A □ → B is true in a possible world ω if and only if B is true in every possible world that is as close as (as similar as) possible to ω in which A is true; and OA is true in a possible world ω if and only if A is true in every possible world that is deontically accessible from ω. If we think of the truthconditions in this way, O(d □ → w) is true in ω_{@} (our world) just in case d □ → w is true in all ideal worlds (i.e. in all possible worlds that are deontically accessible from ω_{@}), that is, if and only if, in every ideal world, ω′ deontically accessible from ω_{@}, w is true in all the worlds that are as close to ω′ as possible in which d is true. But in all ideal worlds, there is no dog (in the specified place), and in all ideal worlds, if there is no dog, there is no warning sign (warning us about the dog). From this, it follows that in all ideal worlds, there is no warning sign. Accordingly, in all ideal worlds, there is no dog and there is no warning sign. Take an ideal world, say ω′. In the closest d world(s) to ω′, ¬w seems to be true (since ¬w is true in ω′). If this is correct, d and ¬w are true in one of the closest d worlds to ω′. So, d □ → w is not true in ω′. Hence, O(d □ → w) is not true in ω_{@} (in our world). In conclusion, if this argument is sound, we cannot avoid the contrarytoduty paradox by using the symbolisation, F3TCTD.
5.2.4 Attempt 4
I will mention one more attempt to symbolise NTCTD. So far, we have not used any temporal operators in our formalisations. This reflects the idea that we are dealing with a timeless contrarytoduty paradox. Might it perhaps be possible to solve the paradox if we introduce some temporal operator(s) in the symbolisation? The antecedent and the consequent in the two conditional norms expressed by TN2 and TN3 are supposed to ‘refer’ to the same moment in time. So, the only plausible formalisation that includes some temporal operators seems to be the following:
F4TCTD
 TCF1

Rt_{1}ORt_{1}¬d
 TCF2

Rt_{1}O(Rt_{1}¬d □ → Rt_{1}¬w)
 TCF3

Rt_{1}(Rt_{1}d □ → Rt_{1}ORt_{1}w)
 TCF4

Rt _{1} d
Where ‘t_{1}’ refers to some arbitrary point in time. Unfortunately, this does not help. For F4TCTD = {TCF1, TCF2, TCF3, TCF4} is also inconsistent. From TCF1 and TCF2, we can derive the sentence Rt_{1}ORt_{1}¬w, and TCF3 and TCF4 entail the formula Rt_{1}ORt_{1}w. Together these imply Rt_{1}ORt_{1}¬w ∧ Rt_{1}ORt_{1}w, which is a kind of moral dilemma. Moreover, F4TCTD entails Rt_{1}ORt_{1}(¬d ∧ w), which is a version of the pragmatic oddity. So, this strategy does not work either.
Therefore, the only reasonable conclusion seems to be that the counterfactual solution discussed in this paper cannot be used to solve timeless or parallel contrarytoduty paradoxes. Nonetheless, maybe the arguments discussed so far are not decisive. I will now consider one possible counterargument.
5.2.5 Possible Reply
So far we have assumed that NTCTD is consistent, and that it does not generate any (moral) dilemma. But this is not obviously correct. Is it really true that this set does not entail any contradiction or any (moral) dilemma? One way to respond to the arguments above is to deny this. It is possible that NTCTD is inconsistent and/or entails a (moral) dilemma. And if this is the case, it is not particularly surprising that our symbolisation, F1TCTD, which is the most natural rendering of NTCTD, is inconsistent. In fact, if this is the case, our formalisation of NTCTD should be inconsistent. The intuition that NTCTD does not entail any contradiction can be explained by the fact that NTCTD is similar to afterward contrarytoduty paradoxes, such as NCTD, which are not inconsistent. This fools us into believing that NTCTD is also consistent. But, in fact, it is not. Consequently, it is not necessarily the case that the timeless or parallel contrarytoduty paradoxes refute the counterfactual solution presented in this paper. At the very least, we should not dismiss this possibility without very good arguments. The fact that a certain timeless contrarytoduty scenario appears to be consistent is not enough to conclude that it is consistent.^{5}
Hence, let us turn to our third argument, which seems to be the most problematic for the counterfactual solution.
5.3 Argument 3: The Counterfactual Solution cannot Solve Beforehand ContrarytoDuty Paradoxes
According to the third argument against the counterfactual solution, we cannot use counterfactual temporal alethicdeontic logic to solve beforehand contrarytoduty paradoxes. This seems to be the most difficult problem for this kind of solution. Consider the following scenario:
Scenario V: The (Beforehand) Promise (ContrarytoDuty) Paradox
Scenario V is similar to Scenario I. Having said that, Scenario I involves an afterward contrarytoduty obligation while Scenario V is concerned with a beforehand contrarytoduty obligation. On Monday, you promise a friend to meet her on Friday to help her with some task. Hence, you ought to keep your promise. If you will not keep your promise, you ought to call your friend on Wednesday, tell her that you will not come and apologise. In this example, the following sentences all seem to be true:
NBCTD
 BN1

(On Monday it is true that) You ought to keep your promise (and see your friend on Friday).
 BN2

(On Monday it is true that) It ought to be that if you keep your promise, you do not call your friend (on Wednesday).
 BN3

(On Monday it is true that) If you do not keep your promise (i.e. if you do not see your friend on Friday and help her out), you ought to call her (on Wednesday).
 BN4

(On Monday it is true that) You do not keep your promise (on Friday), that is, you will not see your friend.
Let NBCTD = {BN1, BN2, BN3, BN4}. BN3 is a contrarytoduty obligation (or expresses a contrarytoduty obligation). If the condition is true, the primary obligation that you should keep your promise (expressed by BN1) is not fulfilled. Moreover, it is a beforehand contrarytoduty obligation. If you will not be keeping your promise on Friday, that is, if you are not going to see your friend and help her, you ought to call her beforehand, on Wednesday. Then she will be less frustrated and she might perhaps find someone else that can help her. NBCTD seems to be consistent; it does not seem possible to deduce any contradiction from this set. However, it is difficult to find any reasonable symbolisation of NBCTD. This is an example of a beforehand contrarytoduty paradox. I will now consider some attempts to solve this paradox in counterfactual temporal alethicdeontic logic. All of these attempts are problematic for various reasons.
5.3.1 Attempt 1
F1BCTD
 FB1

Rt _{1} ORt _{3} k
 FB2

Rt_{1}O(Rt_{3}k □ → Rt_{2}¬c)
 FB3

Rt_{1}(Rt_{3}¬k □ → Rt_{1}ORt_{2}c)
 FB4

Rt_{1}Rt_{3}¬k
 (1)
Rt_{1}ORt_{3}k, w_{0}t_{0}
 (2)
Rt_{1}O(Rt_{3}k □ → Rt_{2}¬c), w_{0}t_{0}
 (3)
Rt_{1}(Rt_{3}¬k □ → Rt_{1}ORt_{2}c), w_{0}t_{0}
 (4)
Rt_{1}Rt_{3}¬k, w_{0}t_{0}
 (5)
Rt_{3}¬k □ → Rt_{1}ORt_{2}c, w_{0}t_{1} [3, Rt]
 (6)
Rt_{3}¬k → Rt_{1}ORt_{2}c, w_{0}t_{1} [5, DR2]
 (7)
Rt_{3}¬k, w_{0}t_{1} [4, Rt]
 (8)
Rt_{1}ORt_{2}c, w_{0}t_{1} [6, 7, MP]
 (9)
ORt_{3}k, w_{0}t_{1} [1, Rt]
 (10)
O(Rt_{3}k □ → Rt_{2}¬c), w_{0}t_{1} [2, Rt]
 (11)
ORt_{2}c, w_{0}t_{1} [8, Rt]
 (12)
sw_{0}w_{1}t_{1} [T – dD]
 (13)
Rt_{3}k, w_{1}t_{1} [9, 12, O]
 (14)
Rt_{3}k □ → Rt_{2}¬c, w_{1}t_{1} [10, 12, O]
 (15)
Rt_{3}k → Rt_{2}¬c, w_{1}t_{1} [14, DR2]
 (16)
Rt_{2}c, w_{1}t_{1} [11, 12, O]
 (17)
Rt_{2}¬c, w_{1}t_{1} [13, 15, MP]
 (18)
¬c, w_{1}t_{2} [17, Rt]
 (19)
c, w_{1}t_{2} [16, Rt]
 (20)
* [18, 19]
Note that it does not help to replace FB3, Rt_{1}(Rt_{3}¬k □ → Rt_{1}ORt_{2}c), by Rt_{1}(Rt_{3}¬k □ → ORt_{2}c) in the derivation above. We can still prove that the resulting set is inconsistent.
The pragmatic oddity is also a problem for our second attempt to solve the beforehand contrarytoduty paradox. The proof is the same in both cases (see below). So, let us turn to this attempt.
5.3.2 Attempt 2
In the symbolisation below, we have replaced FB2 [Rt_{1}O(Rt_{3}k □ → Rt_{2}¬c)] by FB2b [Rt_{1}(Rt_{3}k □ → Rt_{1}ORt_{2}¬c)]. According to FB2, the Ooperator has wide scope, but according to FB2b, it has narrow scope.
F2BCTD
FB1 Rt_{1}ORt_{3}k
FB2b Rt_{1}(Rt_{3}k □→ Rt_{1}ORt_{2}¬c)
FB3 Rt_{1}(Rt_{3}¬k □→ Rt_{1}ORt_{2}c)
FB4 Rt_{1}Rt_{3}¬k
Note that it does not help to replace FB3, Rt_{1}(Rt_{3}¬k □ → Rt_{1}ORt_{2}c), by Rt_{1}(Rt_{3}¬k □ → ORt_{2}c) in the proof above. We can still derive the pragmatic oddity with this sentence instead of FB3.
5.3.3 Attempt 3
Attempt 3 is similar to Attempt 1. Although, we have replaced FB3 [Rt_{1}(Rt_{3}¬k □ → Rt_{1}ORt_{2}c)] by FB3b [Rt_{1}O(Rt_{3}¬k □ → Rt_{2}c)]. In FB3b, the Ooperator has wide scope instead of narrow scope as in FB3.
F3BCTD
 FB1

Rt _{1} ORt _{3} k
 FB2

Rt_{1}O(Rt_{3}k □ → Rt_{2}¬c)
 FB3b

Rt_{1}O(Rt_{3}¬k □ → Rt_{2}c)
 FB4

Rt_{1}Rt_{3}¬k
Let F3BCTD = {FB1, FB2, FB3b, FB4}. This seems to be one of the most plausible symbolisations of NBCTD in counterfactual temporal alethicdeontic logic. The set is consistent and nonredundant, and it has several other attractive features too (in most reasonable systems). Nevertheless, there are at least four problems with this attempt. First, FB3b does not reflect the structure of BN3. In the English sentence, ‘ought’ has narrow scope; in FB3b, the deontic operator O has wide scope. Second, we cannot derive any ‘actual’ obligation from this set, that is, we cannot derive the proposition that you ought to call your friend from FB3b and FB4. But intuitively, this claim seems to follow from BN3 and BN4. A possible reply is to replace FB4 by FB4b (Rt_{1}□Rt_{3}¬k). In other words, we assume that it is true at t_{1} that it is settled at t_{3} that ¬k. Then we can derive the sentence, Rt_{1}ORt_{2}c, from FB3b and FB4b. However, the set F3bBCTD = {FB1, FB2, FB3b, FB4b} is inconsistent. We can derive both Rt_{1}ORt_{2}c and Rt_{1}ORt_{2}¬c from F3bBCTD and, from this, it follows that Rt_{1}(ORt_{2}c ∧ ORt_{2}¬c) (and that Rt_{1}O(Rt_{2}c ∧ Rt_{2}¬c)), which is inconsistent with the idea that there are no moral dilemmas. (It is also possible to use semantic tableau directly to prove that F3bBCTD is inconsistent. Though, we will not labour the details.) So, this move does not appear to succeed. Third, in our solution to the afterward contrarytoduty paradox, we used the sentence Rt_{1}(Rt_{2}¬k □ → Rt_{2}ORt_{3}a). In this formula, the Ooperator has narrow scope. It seems a little bit odd that we should use a sentence with a wide scope Ooperator to symbolise beforehand contrarytoduty obligations and a sentence with a narrow scope Ooperator to symbolise an afterward contrarytoduty obligation.
Fourth, Rt_{1}O(Rt_{3}¬k □ → Rt_{2}c) does not seem to be true. The most natural model of Scenario V appears to be the following. We have four possible worlds ω_{1} – ω_{4} and three moments of time τ_{1} – τ_{3}. At τ_{1}, every possible world is alethically accessible from every possible world. At τ_{2}, ω_{1} is alethically accessible from ω_{2} and ω_{2} from ω_{1}, and ω_{3} is alethically accessible from ω_{4} and ω_{4} from ω_{3}. Every possible world is alethically accessible to itself at every moment of time. At τ_{1}, ω_{3} is deontically accessible from every possible world. At τ_{2}, ω_{3} is deontically accessible from ω_{3} and ω_{4}, and ω_{1} is deontically accessible from ω_{1} and ω_{2}. At τ_{3}, every possible world is deontically accessible to itself. c is true in ω_{1} and ω_{2} and false in ω_{3} and ω_{4} at τ_{2}. k is true in ω_{2} and ω_{3} and false in ω_{1} and ω_{4} at τ_{3}. v(t_{1}) = τ_{1}, v(t_{2}) = τ_{2} and v(t_{3}) = τ_{3}. Furthermore, ω_{4} is more similar to ω_{3} than is ω_{1}. In this model, every sentence in F3BCTD is true (in ω_{1} and ω_{4} at τ_{1}) except FB3b. But intuitively, BN3 is true. So, we want our symbolisation of this sentence to be true also. Moreover, it seems difficult to find any other plausible model of Scenario V. This is a problem for Attempt 3.
5.3.4 Attempt 4
In F1BCTD and F2BCTD, the consequent in the contrarytoduty obligation begins with Rt_{1}. That is, we assume that the obligation to call your friend given that you will not keep your promise is in force already at t_{1}, that is, on Monday. What happens if we replace FB3 with FB3c: Rt_{1}(Rt_{3}¬k □ → Rt_{2}ORt_{2}c), which says that the obligation to call your friend is in force at t_{2} given that you will not keep your promise? Does this solve the beforehand contrarytoduty paradox? Consider the following symbolisation of NBCTD:
F4BCTD
 FB1

Rt _{1} ORt _{3} k
 FB2

Rt_{1}O(Rt_{3}k □ → Rt_{2}¬c)
 FB3c

Rt_{1}(Rt_{3}¬k □ → Rt_{2}ORt_{2}c)
 FB4

Rt_{1}Rt_{3}¬k
Let F4BCTD = {FB1, FB2, FB3c, FB4}. Rt_{1}ORt_{2}¬c follows from FB1 and FB2, and from FB4 and FB3c we can derive Rt_{2}ORt_{2}c. But Rt_{1}ORt_{2}¬c is not inconsistent with Rt_{2}ORt_{2}c. Rt_{1}ORt_{2}¬c says that it is true at time t_{1} that it is obligatory that you do not call your friend at time t_{2}; and Rt_{2}ORt_{2}c says that it is true at time t_{2} that it is obligatory that you call your friend at t_{2}, that is, on Wednesday. Obligations may change over time and something may be obligatory at one time even though it is not obligatory at a later time. F4BCTD also has some other attractive features; it is, for example, nonredundant.
Nevertheless, Attempt 4 is problematic for at least one reason. In FB3c, the consequent Rt_{2}ORt_{2}c is not an obligation that is ‘future oriented’ as genuine norms are or should be according to many deontic logicians. An obligation is future oriented if the content of the obligation involves a time that is later than the obligation itself. The consequent Rt_{2}ORt_{3}a in Rt_{1}(Rt_{2}¬k □ → Rt_{2}ORt_{3}a), for instance, is future oriented since t_{3} is later than t_{2}. Here are some other examples of futureoriented obligations: OFk, OGk, where F (‘It will at some time in the future be the case that’) and G (‘It is always going to be the case that’) are temporal operators that are future oriented.
This result is a problem for Attempt 4 if it is reasonable to assume that our counterfactual temporal alethicdeontic tableau system should include the rules that are used in the derivation. And for many applications, these rules seem plausible, for example, if we consider the past and the present as settled or historically necessary, as many deontic logicians do.
We have now considered four attempts to solve the beforehand contrarytoduty paradox. We have seen that all of these solutions are problematic. It appears to be difficult to find any other formalisation of NBCTD in counterfactual temporal alethicdeontic logic that is plausible. This is a serious problem for the counterfactual solution. In Section 5.2.5, I suggested that a possible reply to the argument from parallel contrarytoduty paradoxes against the counterfactual solution is to reject the idea that NTCTD is consistent. A similar response to the problem with beforehand contrarytoduty paradoxes does not seem to be plausible. The problem with NTCTD is that all obligations are supposed to be in force at the same time. But in beforehand contrarytoduty paradoxes, this is not the case. It is intuitively more plausible to think that NBCTD is consistent than to think that NTCTD is consistent. Therefore, Argument 3 seems to be more problematic than Argument 2.
One might argue that it is enough that the approach used in this paper can solve afterward contrarytoduty paradoxes. Perhaps one and the same solution cannot take care of all different versions of this puzzle. The fact that the counterfactual solution has problems with parallel and beforehand paradoxes is, therefore, not a decisive argument against it. But if there is some other approach that can be used to solve not only afterward contrarytoduty paradoxes but also beforehand contrarytoduty paradoxes, such a solution seems preferable. And, in fact, there might be such a solution. Several philosophers have suggested that contrarytoduty paradoxes can be solved in a system that combines dyadic deontic logic and some type of temporal logic (e.g. van Eck 1982; Loewer and Belzer 1983; Feldman 1986, 1990; Åqvist 2003). Moreover, it seems to be the case that a solution of this kind can take care of afterward and beforehand contrarytoduty paradoxes. This suggests that a solution of this type might be, all things considered, more plausible than the counterfactual solution, even though it too has problems with parallel contrarytoduty paradoxes.
6 Conclusion
In this paper, I have discussed some examples of the socalled contrarytoduty (obligation) paradox, a puzzle that has been investigated by deontic logicians for more than fifty years. A contrarytoduty obligation is an obligation telling us what ought to be the case if something forbidden is true. Here are some examples of sentences that can be used to express such obligations: ‘If he did it, he should be ashamed of himself’, ‘If the resources are unfairly distributed, they should be redistributed’, ‘If you are going to take drugs, you ought to use clean needles’. In each case, the condition is supposed to violate a primary obligation. He should not have done it, but if he did, he should be ashamed of himself. The resources should not be unfairly distributed, but if they are, they should be redistributed. You should not take drugs; but if you are going to take drugs, you should use clean needles. Contrarytoduty obligations are important in our moral and legal thinking. Therefore, we want to be able to find an adequate symbolisation of such obligations in some logical system, a task that has turned out to be difficult. This is clearly illustrated by the socalled contrarytoduty (obligation) paradox. I have investigated and evaluated one kind of solution to this problem that has been suggested in the literature, a solution that I called the ‘counterfactual solution’. I used some recent systems introduced by Rönnedal (2016) that combine not only counterfactual logic and deontic logic, but also temporal logic in my analysis of the paradox. I have argued that the counterfactual solution has many attractive features and that it can give a fairly satisfactory answer to some examples of the contrarytoduty paradox, but that it nevertheless has some serious problems. The conclusion is that, notwithstanding the many attractive features of the counterfactual solution, there seem to be other approaches to the paradox that are more promising, especially a kind of solution that uses a combination of dyadic deontic logic and temporal logic.
Footnotes
 1.
Carmo and Jones (2002) suggest that a plausible solution to the contrarytoduty paradox should meet certain conditions: (i) consistency; (ii) logical independence; (iii) applicability to (at least apparently) timeless and actionless contrarytoduty examples; (iv) analogous logical structures for conditional sentences; (v) capacity to derive actual obligations; (vi) capacity to derive ideal obligations; (vii) capacity to represent the fact that a violation of an obligation has occurred; (viii) capacity to avoid the pragmatic oddity; (ix) the assignment of logical form should be independent. In this section, we will see that the counterfactual solution meets all of these conditions except (iv) (and perhaps the first part of (iii); see Section 5.2), and that it has some other attractive features. The solution in Carmo and Jones (2002) meets all requirements they mention. But this solution has other problems that I cannot discuss in this paper. It is unlikely that there is a solution that meets all the conditions philosophers have thought a solution to the paradox should meet. We probably have to give up some of our prephilosophical intuitions. This is why the paradox is so profound.
 2.
The idea that a solution to the contrarytoduty paradoxes should be consistent and nonredundant is old, see for example Åqvist (1967).
 3.
Some might argue that there are no agentless obligations. In this paper, I do not make a sharp distinction between obligations and ‘oughts’. But we could make such a distinction, and then we could speak about agentless ‘oughts’ instead of agentless obligations. According to some deontic logicians, deontic statements must take ‘agentives’ as their complements (Belnap et al. 2001, p. 13). But this view is not shared by everyone (see e.g. McNamara 2004). To me, the sentences in NACTD clearly seem to be meaningful.
 4.
 5.
This reply can be used to meet the following possible counterargument to the topic in this paper. Suppose that pure counterfactual deontic logic can handle the contrarytoduty paradoxes satisfactorily. Then there is no need to extend it with time, as far as the paradoxes are concerned. If the pure counterfactual deontic logic cannot handle the paradoxes, then its tense version will not either. Whatever they are, all the problems the first run into will arise for parallel scenarios, where the consequent of the ‘obligation operator’ and its antecedent pertain to the same point in time. In either case, the basic framework is disqualified. Moreover, if this is true, there is no need to also consider beforehand scenarios. Nevertheless, it is quite possible that afterward, and beforehand contrarytoduty scenarios are consistent, while timeless scenarios are not. If this is true, and I am inclined to believe that it is (perhaps with some exceptions), it does not matter if the symbolisation of NTCTD is inconsistent. It is, however, problematic if the counterfactual solution cannot handle beforehand contrarytoduty paradoxes. Even if parallel contrarytoduty scenarios were consistent, we would want to know if the counterfactual solution can handle beforehand paradoxes.
Notes
Acknowledgements
I would like to thank an anonymous reviewer for his or her comments on an earlier version of this paper.
Supplementary material
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