1 Introduction
Modern research into the use of deontic logic to represent and reason with normative statements began with the introduction in [10] of what has become known as standard deontic logic (SDL). SDL is based on a propositional language and uses modal operators for obligation and permission where, intuitively, means that it ought to be the case (or, it is obligatory) that the formula be true, and as the dual, means it is permitted that the formula be true.
Dyadic deontic logic (DDL) was introduced in [11] to cater for paradoxes in SDL resulting from socalled contrarytoduty obligations as described in [3]
. In DDL, dyadic modal operators (adapted from the conventions used to express conditional probabilities) enable modal formulae to express the conditions in which obligations and permissions apply so that, for example,
means that it is obligatory that the formula be true, provided or given that is true.The standard semantics used for DDL were introduced by Hansson in [5], who imposes restrictions on the language such that modal operators cannot be nested and mixed formulae (connecting modal and propositional formulae) are not permitted.
In [1], Åqvist proposes system G, a system of dyadic deontic logic which extends Hansson’s logic with a universal necessity operator . Intuitively means that the formula is necessarily true (and the dual means the formula is possibly true).^{1}^{1}1[1] uses the operators (necessarily) and (maybe), but we have opted for the standard ‘box’ and ‘diamond’ normally used with alethic logics and consistent with other recent authors such as Parent (for example, in [8]).
We present an interesting property of System G, namely that every formula with nested and operators is equivalent to another formula without any nesting. It has already been established that this is the case for logics in [7]. In System G, the operator is an operator and it therefore follows from [7] that nestings of the are superfluous. The operator however presents additional challenges as the nesting may occur within either argument of the operator. We can still prove that despite this, nestings of the operator and of combinations of the and operator are superfluous. These results provide an a posteriori justification to Hansson’s original syntactic restrictions.
The rest of the paper is structured as follows. In Section 2 we introduce some preliminaries including the syntax, semantics and proof theory of System G. In Section 3 we show our main result that every System G formula with nested and operators is equivalent to another formula without any nesting. We do this using both a semantic and a syntactic argument. As part of the proof, we also show that in System G, all formulae can be expressed using the modality only. We conclude in In Section 4. Longer proofs of the lemmas and theorems are provided in the appendix.
2 Preliminaries
2.1 Some background on nesting deontic operators
The issue of whether deontic logic formalisms should allow operators to be nested within the scope of others has been considered as early as in [6]. Sentences in which deontic operators are nested are sentences like (taken from [6]):

It is obligatory that it is obligatory that everyone keeps his promises.

It ought to be the case that what ought to be the case is the case.
The issue, from a deontic logic point of view, is whether statements such as the above ones should be objects in a logic or should rather be ignored (on different bases such as being nonmeaningful, or trivial or uninteresting).
As pointed out in [4] (quoted in [12]), little attention was initially paid to nested modalities. In [11], von Wright defines the language of both his original and dyadic systems to exclude nesting with no explicit discussion of why this should be the case. Hansson in [5] follows this tradition and also explicitly forbids nesting in his language, commenting however that despite this restriction “almost all philosophical problems discussed in connection with deontic logic are expressible”. In [1], Åqvist identifies nesting as one of the differences between Hansson’s language and his proposed System G but does not discuss any implications of this difference.
Some work has been done on the representation and meaning of nested modalities in some normative systems, such as in socalled stit logics in [2]. However, despite the fact that it is considered a powerful logic for handling contrarytoduty obligations, to the best of our knowledge, no research has been done on nested modalities in System G. In this paper we aim to address this and provide a technical answer to the issue of nesting (within System G).
Results of this type are wellknown for normal monadic modal logics (e.g., for [7]). In System G, the operator is an operator and it therefore follows from [7] that nestings of the are superfluous. The operator however presents additional challenges as the nesting may occur within either argument of the operator. We prove that despite this, nestings of the operator and of combinations of the and operator are superfluous.
2.2 Language, Syntax and Semantics of System G
2.2.1 Language
Formally the language of system G is defined thus
Definition 1 (Language of System G)
Let be a set of propositional atoms. The language of system G, is defined by the following inductive syntax, where and
Conjunction, material implication and biconditional are defined from negation and disjunction in the expected way. We treat as a propositional atom in , and define as . Parentheses may be used to clarify the order of operators.
is read as ‘ is obligatory, given ’. is read as ‘Everywhere ’. It follows that the duals of these operators (‘Permitted’) and (‘Somewhere’) are defined as follows:
2.2.2 Semantics
These formulae are evaluated using Kripkestyle possible world semantics based on the socalled class of models defined by Åqvist in [1] as follows:
Definition 2 ( model)
Given a set of propositional atoms , a model is a structure , where

is a nonempty set of possible worlds

is a reflexive, transitive, fully connected and limited (see below) binary relation over that satisfies the limit assumption condition described below.

is a valuation function that assigns a truth set to every proposition in .
The weak preference relation captures the idea that for any two worlds and , if then is at least as good as . The relation is fully connected if and only if any two worlds in are comparable, or formally that:
(1) 
The limitedness property can be characterised in terms of the function, defined as follows.
Definition 3 ( function)
(2) 
Intuitively, are the elements of that are at least as good as any other element in . We say that is the optimal set or the set of optimal elements of X.
The preference relation is then limited if and only if for every nonempty subset of there is at least one optimal element, or formally:
(3) 
Definition 4 (Truth at a point)
The notion of a formula being true at a world s in a model , denoted , is defined inductively as follows:
(4)  
(5)  
(6)  
(7)  
(8)  
(9) 
( denotes the truth set of , i.e. the set of worlds in which is true.)
The truth definitions for (the dual of ) and (the dual of ), derived from (8) and (9) are as follows:
(10)  
(11) 
We say that a formula is valid in models if it is true in all worlds of all models, and denote this . (We sometimes drop the subscript if the class of models being referred to is clear.) If is valid with respect to a class of models, we say that is a validity of that class of models. We say that two formulae and are semantically equivalent if is a validity.
Observation 1
The modalities and , as well as their duals, are global modalities in the following sense:
Note that, as a consequence, for a formula of the form in any model , we either have or . Likewise for the other modalities.
2.2.3 Axiomatics
Åqvist proposes the following axioms and rules for system G (names are from [8]).
All propositional tautologies  (PL)  
(S5)  
(DfP)  
(COK)  
(Abs)  
(CON)  
(Ext)  
(Id)  
(C)  
(D*)  
(S)  
(MP)  
(N) 
We refer to this axiom system as system G, and so we say that a formula is derivable in system G if it can be derived using this calculus. We denote this by . (For readability, we sometimes drop the subscript if the system referred to is clear). If is derivable in system G, we say that is a theorem of system G. We say that two formulae and are provably equivalent if is a theorem of System G.
3 Nested formulae
In this section we present our main finding that any system G formula which contains nested modal operators is equivalent to some other formula without nesting. We present a semantic and a syntactic argument for this theorem.
Both arguments use the following definitions and lemma.
Definition 5 (Modal depth)
The modal depth of a formula is defined inductively as
where represents the arithmetic maximum, i.e. for any , if then , otherwise .
Definition 6 (Unnested disjunctive normal form (UDNF))
Let be the sublanguage of without formulae containing the operator. We say that a formula is in Unnested Disjunctive Normal Form (UDNF) if it is a disjunction of conjunctions of the form
where and all of the formulae () are propositional formulae ( and are considered propositional formulae). The formula is called a canonical conjunction and the formulae are called prenex formulae.
Observe that by the above definition, formulae in UDNF have a maximum modal depth of 1.
The following lemma from [7] guarantees that a prenex formula within a formula in UDNF can always be moved to the outermost level.
Lemma 1
(Meyer and van der Hoek [7] Lemma 1.7.6.2) If is in UDNF and contains a prenex formula , then is equivalent to a formula of the form where , and are all in UDNF.
Proof
is in UDNF so where all the ’s are canonical conjunctions. Suppose occurs in . As is one of the conjuncts of , can be written as , where collects the remaining conjuncts in (or if ). Taking to be gives the desired result .
3.1 Semantic argument
First we state the following proposition which provides a means to obtain for every formula, a semantically equivalent formula.
Proposition 1
(12) 
The following lemma guarantees that an or formula within the scope of another can be brought out of that scope.
Lemma 2
Let stand for an arbitrary but fixed formula.
(13)  
(14)  
(15)  
(16) 
We can now state our main theorem as follows.
Theorem 3.1
For every formula , there exists a formula such that is in UDNF and
3.2 Syntactic argument
The syntactic argument has the same structure as the semantic argument with Proposition 1 and Lemma 2 replaced by syntactic counterparts.
The following is a syntactic counterpart to Proposition 1.
Proposition 2
(17) 
The following is a syntactic counterpart to Lemma 2.
Lemma 3
Let stand for an arbitrary but fixed formula.
(18)  
(19)  
(20)  
(21) 
Theorem 3.2
Every formula is provably equivalent to a formula in UDNF.
4 Conclusion
When he introduced restrictions on nesting and in [5],Hansson commented that in SDL, despite these restrictions, “almost all philosophical problems discussed in connection with deontic logic are expressible in his language.” He, however, offers no justification for this conjecture and, in moving from SDL to the dyadic language, a formal definition of the dyadic language used in not given, and it is not explicit whether these restrictions are to be kept. It is generally considered that it was his intention that the dyadic language be so restricted, and this is stated explicitly, for example, in [9]. We note also that the list of example valid and invalid formulae given in [5] contain no nested formulae.
From a technical point of view, it seems unnatural to forbid certain operators to appear in the scope of others. The main goal of our work has been to see whether such a restriction is necessary, and whether it actually limits the ability of the system to express deontic concepts. We have demonstrated that the answer to both questions is negative: forbidding nested modalities is not technically needed, and, on the other hand, not allowing them does not restrict the expressive power of the language either. This in some sense settles Hansson’s conjecture, at least with respect to system G: if any interesting problems can be expressed in the full dyadic language, they also can in a restricted version where nesting is not permitted.
References
 Åqvist [1986] L Åqvist. Some results on dyadic deontic logic and the logic of preference. Synthese, 66:95–110, 1986.
 Belnap and Bartha [1995] N Belnap and P Bartha. Marcus and the problem of nested deontic modalities. In Modality, morality, and belief: essays in Honor of Ruth Barcan Marcus, pages 174–197. 1995.
 Chisholm [1963] R M Chisholm. Contrarytoduty imperatives and deontic logic. Analysis, 24(2):33–36, 1963.
 Goble [1966] L F Goble. The iteration of deontic modalities. Logique Et Analyse, 9:197–209, 1966.
 Hansson [1969] B Hansson. An Analysis of some Deontic Logics. Noûs, 3(4):373–398, 1969.
 Marcus [1966] R B Marcus. Iterated deontic modalities. Mind, 75(300):580–582, 1966.
 Meyer and van der Hoek [1995] JJ Ch. Meyer and W van der Hoek. Epistemic Logic for AI and Computer Science (Cambridge Tracts in Theoretical Computer Science). Cambridge University Press, 1995. ISBN 0511569858.
 Parent [2008] X Parent. On the Strong Completeness of Åqvist’s Dyadic Deontic Logic G. In DEON, pages 189–202, 2008.
 Spohn [1975] W Spohn. An analysis of Hansson’s dyadic deontic logic. Journal of Philosophical Logic, 4:237–252, 1975.
 von Wright [1951] G H von Wright. Deontic Logic. Mind, 60(237):1–15, 1951.
 von Wright [1964] G H von Wright. A New System of Deontic Logic. In R Hilpinen, editor, Deontic Logic: Introductory and Systematic Readings, pages 105–120. D. Reidel, 1964.
 Wansing [1998] H Wansing. Nested deontic modalities: Another view of parking on highways. Erkenntnis, 46:185–199, 1998.
Appendix 0.A Proof of Proposition 1
Proof
From left to right:
Assume that for an arbitrary model and world , .
Therefore, by the truth definition of (9), .
Therefore, there are no worlds, so .
Therefore, by the definition of opt (2) .
Therefore , by the truth definition of (8), .
From right to left:
Assume .
Therefore, by the truth definition of (8), .
Therefore, as , .
Therefore, given the limit assumption (3), .
Therefore .
Therefore, by the truth definition of (9), .
Appendix 0.B Proof of Lemma 2
Proof
From left to right:

Let be a model, and let .

Assume

Therefore, by (8), . Call this (A).

Given Observation 1, there are two cases: Either or .

If

Then

Therefore, given (A),

Therefore, by equation (8),

Therefore, given that ,

Therefore,


If :

Then

Therefore, given (A),

Therefore, by equation (8),

Therefore, given that ,

Therefore,

From right to left:

Let be a model, and let
item[] Assume .

There are two cases. Either or .
From left to right:

We assume that for some arbitrary model and arbitrary world ,

Therefore, by equation (8), . Call this (D).

By Observation 1, there are two cases. Either or .

If :

[]

Then

Therefore,

Therefore, given (D),

Therefore, by equation (8),

Therefore, given that ,

Therefore,


If :

Then

Therefore,

Therefore, given (D),

Therefore, by equation (8),

Therefore, given that ,

Therefore,

From right to left:

We assume that for some arbitrary model and arbitrary world , .

There are two cases. Either or
Appendix 0.C Proof of Theorem 3.1
Proof
The theorem can be proved by induction on the syntax of (Definition 1).
There are 5 cases to consider.

For the base case, if is a propositional atom, then by definition it is in UDNF.

If then we show that if has an equivalent formula in UDNF, so does . Suppose = for some . By Definition 6 this is in UDNF. Then , where each is of the form
The result then follows by applying the distributive law:

If then we show that if and have an equivalent formula in UDNF, so does . Suppose and where and are in UDNF (i.e. disjunctions of canonical conjunctions). Then which is also a disjunction of canonical conjunctions and thus in UDNF.

If then we show that if and have an equivalent formula in UDNF, so does . In the case where both and are propositional formulae, is already in UDNF. If contains a prenex formula, we can, using Lemma 1, assume that or for some prenex formula . In the first case, item 15 of Lemma 2 tells us how to remove the prenex outside the scope of the outer , in the second case we need item 16 to do this. Likewise, if contains a prenex, we can write either as , or else as . In the first case, we use item 14 of Lemma 2 to move the prenex outside the scope of , and otherwise we use item 13. We can repeat this process of removing a prenex from within the scope of until none are left, and finally, use the distributive law above to bring the result in normal form.

If then we show that if has an equivalent formula in UDNF, so does . By Proposition 1 is equivalent to . An equivalent formula can be obtained by applying cases 1 to 4 above to this formula.
Appendix 0.D Proof of Proposition 2
Proof
First we note the following rules of inference that are known to be sound in propositional logic namely the rules of hypothetical syllogism (HS), biconditional introduction (BI), subconditional elimination (SCE), contraposition(Contra) and substitution of provable equivalences (Subst).
(HS)  
(BI)  
(SCE)  
(Contra)  
(Subst) 
In Subst, the notation means the formula exactly like except that instances of the formula within are replaced by the formula . SCE is a special case of conditional elimination, where the conditional being eliminated is part of another conditional.
Appendix 0.E Proof of Lemma 3
Proof
This proof uses the propositional rules of inference HS, BI and Subst given in Appendix 0.D.
We first derive the formula gExt (a generalisation of ) and and (two special cases of ):
The derivation is as follows:
Going back to the main lemma, to derive (18) we show that the following sequence of formulae are provably equivalent. (Note that (19) can be derived in a similar way, by replacing with .):
by substituting for and for in the propositional tautology
, given that (Abs) and (T) by applying BI to obtain , and then Subst.
, given the propositional tautology and , by successive applications of Subst.
, in a similar way, given and , by successive applications of Subst.
, by given that and from Abs, T and BI, by Subst.
Next we also derive the formula gCOK (a generalisation of ) and and (two special cases of ):
The derivation is as follows:
Going back to the main lemma again, to derive (20) we show that the following sequence of formulae are provably equivalent. (Note that (21) can be derived in a similar way, by replacing with .):
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