## 1. Informal Preliminaries and Background



Deontic logic has been regularly influenced by reflection on the logic
of modal notions, such as *necessity* (in varying senses of the
term). In particular, analogies between alethic (truth-implicating)
modal notions and deontic notions were noticed before the fourteenth
century in Europe, where we might say that the rudiments of formal
(though not symbolic) deontic logic experienced its initial European
stirrings. In Islamic thought, such analogies go back at least as far
as the tenth century.



Although interest in what can be arguably called formal aspects of
deontic logic continued off and on, the trend toward studying logic
using the *symbolic and exact* techniques of mathematics began
primarily in the nineteenth century, and became dominant in the
twentieth century. Work in twentieth century symbolic modal logic
provided the explicit impetus for von Wright (1951a, 1951b), the
central early figure in the emergence of deontic logic as a
full-fledged branch of *symbolic* logic in the twentieth
century. However, we note that prior to von Wright 1951a, there was
one significant earlier episode in symbolic deontic logic, namely
Mally 1926. See supplement A:
 Mally and Symbolic Deontic Logic.[6]



In the remainder of this section, we first spell out a few
folk-logical features of alethic modal notions, and then give an
impressionistic sense of how natural it was for early developments of
deontic logic to explore deontic analogs to those features.


### 1.1 Some Informal Rudiments of Alethic Modal Logic



Alethic modal logic is the logic of necessary truth and related
notions. Consider six basic alethic modal notions, expressed as
sentential operators--constructions that, when applied to a
sentence, yield a sentence (as does "it is not the case
that"):



> 
> * it is necessary (necessarily true) that
> * it is possible that
> * it is impossible that
> * it is non-necessary that
> * it is contingent that
> * it is non-contingent
>  that[7]
> 
> 
> 



Although all of the above operators are generally deemed definable in
terms of any one of the first four, the necessity operator, usually
symbolized as a box, \(\Box\), is typically taken as primitive and the
rest defined accordingly. Where "\(\neg\)",
"&" , and "\(\vee\)" denote classical
negation, conjunction, and disjunction, the definitions are as
follows:



> 
> * It is possible that \(p\) \((\Diamond p)\)
>   
> 
> \( \eqdf \neg \Box \neg p\)
> * It is impossible that \(p\)
>   
> 
> \( \eqdf \Box \neg p\)
> * It is non-necessary that \(p\)
>   
> 
> \( \eqdf \neg \Box p\)
> * It is contingent that \(p\)
>   
> 
> \( \eqdf \neg \Box p \amp \neg \Box \neg p\)
> * It is non-contingent that \(p\)
>   
> 
> \( \eqdf \Box p \vee \Box \neg p\)
> 
> 
> 



It was also routinely assumed that the following threefold partition
of propositions holds:



![three boxes containg respectively the words Necessary, Contingent, and Impossible. Necessary and Contingent are braced as Possible; Necessary and Impossible are braced as Non-Contingent, and Contingent and Impossible are braced as Non-Necessary](fig-three-a.svg)

Fig. 1




The three rectangular cells are intended to be jointly exhaustive and
mutually exclusive: every proposition is either necessary, contingent
(possibly true but also possibly false), or impossible, but no
proposition is more than one of these. The possible propositions are
those that are either necessary or contingent, the non-necessary
propositions are those that are either impossible or contingent, and
the non-contingent ones are those either necessary or impossible.



Thirdly, dating back at least to the twelfth and thirteenth century
(Knuuttila 2008), the following modal square of opposition is often
 noted:[8]



![first square diagram: link to extended description below](fig-square-a.svg)

Fig. 2
 [An extended description of figure 2.]




Furthermore it is generally assumed that the following hold:



> 
> 
> If \(\Box p\) then \(p\) (if it is necessary that \(p\), then \(p\) is
> true).
> 
> 
> 
> If \(p\), then \(\Diamond p\) (if \(p\) is true, then it is
> possible).
> 
> 
> 



These indicate that the idea of necessity here is alethic or
truth-implicating. Many of the ideas above were noted in medieval
times. (See, e.g., the entry on
 medieval theories of modality.)


### 1.2 The Traditional Scheme and the Modal Analogies



We now turn to some of the analogies involved in what is a
corresponding bit of deontic folk logic, following the exposition in
McNamara 1996a,b. This is a minor elaboration of elements that can be
found explicitly in von Wright 1951b and Prior 1955 [1962]. However,
much of it has roots in medieval Islamic and late medieval European
thought, where possible analogies between deontic modals and alethic
modals were being explored and formal deontic schemes were
 proposed.[9]



Expressed as sentential operators, the six normative statuses of what
we will call "the Traditional Scheme" are:



> 
> * it is obligatory that \((\OB)\)
> * it is permissible that \((\PE)\)
> * it is impermissible that \((\IM)\)
> * it is omissible that \((\OM)\)
> * it is optional that \((\OP)\)
> * it is non-optional that
>  \((\NO)\).[10]
> 
> 
> 



The first three are the most often cited, the fourth is often not
labeled, and the fifth has regularly been mislabeled as "it is a
matter of *indifference* that", and so likewise for the
sixth. Typically, one of the first two is taken as primitive, and the
others defined in terms of it, but any of the first four can play the
same defining role. The most prevalent approach is to take \(\OB\) as
primitive, and define the rest as
 follows:[11]


 

\[ \begin{align} \PE p & \eqdf \neg \OB \neg p\\ \IM p & \eqdf \OB \neg p \\ \OM p & \eqdf \neg \OB p \\ \OP p & \eqdf (\neg \OB p \amp \neg \OB \neg p).\\ \NO p & \eqdf (\OB p \vee \OB \neg p).\\ \end{align} \]


These definitions imply that something is permissible iff (if and only
if) its negation is not obligatory, impermissible iff its negation is
obligatory, omissible iff it is not obligatory, optional iff neither
it nor its negation is obligatory, and non-optional iff it is either
obligatory or impermissible. Call this "The Traditional
*Definitional* Scheme (TDS)". It, or minor variants, are
found as far back as Ockham in European thought, and quite explicitly
in Leibniz in the modern
 period.[12]
 If one began with \(\OB\) alone and considered the formulas on the
right above, one could easily be led to consider them as at least
candidate defining conditions for those on the left. Although not
uncontestable, they are natural, and this scheme has been very widely
employed. Now if the reader looks back at our use of the necessity
operator in defining the remaining five alethic modal operators, it
will be clear that these are perfectly analogous to the five deontic
definitions above. From the formal standpoint, the one is merely a
syntactic variant of the other: just replace \(\OB\) with \(\Box\),
\(\PE\) with \(\Diamond\), etc.



In addition to the TDS, it was traditionally assumed that the
following, call it "The Traditional Threefold Classification
(TTC)" holds:



![three boxes containg respectively the words Obligatory, Optional, and Impermissible. Obligatory and Optional are braced as Permissible; Obligatory and Impermissible are braced as Non-Optional, and Optional and Impermissilbe are braced as Omissible.](fig-three-b.svg)

Fig. 3




As in the modal case, the thought is that all propositions are divided
into three jointly exhaustive and mutually exclusive classes: every
proposition is obligatory, impermissible or neither (that is,
optional) and no proposition falls into more than one of these three
categories. Furthermore, the permissible propositions are those that
are obligatory or optional, the omissible propositions are those that
are impermissible or optional, and the non-optional propositions are
those that are either obligatory or impermissible. This classification
too has roots centuries
 ago.[13]
 The reader can easily confirm that this natural scheme is also
perfectly analogous to the threefold classification we gave above for
necessary, contingent and impossible propositions.



Furthermore, as indicated earlier, "The Deontic Square"
(DS) is also often cited in the historical literature:



![second square diagram: link to extended description below](fig-square-b.svg)

Fig. 4
 [An extended description of figure 4.]




The relationships between the logical operators at the corners are to
be interpreted as in the modal square of opposition. The two squares
are perfectly analogous. If we weave in nodes for optionality and
non-optionality, we get a deontic
 hexagon:[14]



![a hexagon diagram: link to extended description below](fig-hex-a.svg)

Fig. 5
 [An extended description of figure 5.]




Given these analogies, it is unsurprising that our primitive operator,
read here as "it is obligatory that", is often referred to
as "deontic *necessity*". However, there are also
obvious disanalogies with alethic necessity. Before, we cited two
principles for alethic necessity whose deontic analogs are clearly
false:



> 
> * If \(\OB p\) then \(p\) (if it is obligatory that \(p\), then
> \(p\) is true).
> * If \(p\), then \(\PE p\) (if \(p\) is true, then it is
> permissible).
> 
> 
> 



Obligations can be violated, and impermissible things do
 happen.[15]
 However, as researchers turned to generalizations of alethic modal
logic, they began to consider wider classes of modal logics, including
ones where the necessity operator was *not* truth-implicating.
This too encouraged seeing deontic logic as falling within modal logic
so-generalized. In fact, recognizing possibilities like this helped to
fuel the generalizations of what began with a focus on alethic modal
logic to *normal modal logics* (Lemmon & Scott 1977).



We will revisit the Traditional Scheme in a more precise and unifying
way in the next section. There is a sense in which it is an earlier
informal and widely endorsed fragment of the most well-known deontic
logic, a logic that makes the connection to alethic modal logic even
tighter. We now turn to that logic.


## 2. Standard Deontic Logic


### 2.1 SDL Syntax



*Standard Deontic Logic* (SDL) is the most cited and studied
system of deontic logic, and one of the first deontic logics
axiomatically specified. SDL includes classical propositional logic,
and is essentially a distinguished member of the most studied class of
propositional modal logics, "normal modal logics". SDL is
a "monadic" deontic logic, since its primitive deontic
operator is a one-place operator (like \(\neg\), and unlike &): it
applies to a *single* sentence to yield a compound
 sentence.[16]
 This operator, \(\OB\), is taken as primitive within SDL; as
explained in
 Section 1.2,
 other operators such as \(\PE\) and \(\IM\) can be defined from it
following the Traditional Definitional Scheme.



Assume again that we have a language of classical propositional logic
(PC) with an infinite set of propositional variables, the truth
functional operators \(\neg\) and \(\rightarrow\) (and perhaps any of
&, \(\vee\) and \(\leftrightarrow\) as well), and the operator,
\(\OB\). SDL is then often axiomatized as
 follows:[17]




|  |  |  |
| --- | --- | --- |
| SDL: | All tautologous formulas of the language | (TAUT) |
|  | \(\OB (p \rightarrow q) \rightarrow(\OB p \rightarrow \OB
q)\) | (\(\OB\)-K) |
|  | \(\OB p \rightarrow \neg \OB \neg p\) | (NC) |
|  | If \(\vdash p\) and \(\vdash p \rightarrow q\) then \(\vdash
q\) | (MP) |
|  | If \(\vdash p\) then \(\vdash \OB p\) | (\(\OB\)-NEC) |



TAUT combined with modus ponens (MP) gives us the full inferential
power of PC. \(\OB\)-K, which is the K-axiom common to all normal
modal logics, tells us that if a material conditional is obligatory,
and its antecedent is too, then so is its
 consequent.[18]
 NC tells us that \(p\) is obligatory only if its negation
isn't. The rule \(\OB\)-NEC tells us that if anything is a
theorem, then the claim that that thing is obligatory is also a
theorem. Note that this guarantees that something is always obligatory
(even if only logical
 truths).[19]



Each of the distinctively deontic principles, \(\OB\)-K, NC, and
\(\OB\)-NEC are contestable, and we will consider criticisms of them
in
 Section 6.
 However, to avoid immediate confusion for those new to deontic logic,
it is perhaps worth noting that \(\OB\)-NEC is generally deemed a
convenience that, among other things, assures that SDL is in fact one
of the well-studied normal modal
 logics.[20]
 Few have spilled blood to defend its cogency substantively, and often
these practical compromises are wise, especially in early stages of
research.



SDL can be seen as a normalized version of von Wright's original
proposal in von Wright 1951a, though there are some significant
differences in terms of the formal language, validities, and its
interpretation. A quick comparison of both systems can be found in the
supplement B on
 von Wright's 1951a System and SDL.



Below we list some theorems and two important derived rules of
 SDL.[21]




|  |  |
| --- | --- |
| \(\neg \OB \bot\) | (\(\OB\)-OD) |
| \(\OB (p \amp q) \rightarrow(\OB p \amp \OB q)\) | (\(\OB\)-M) |
| \(\OB p \vee \OP p \vee \IM p\) | (\(\OB\)-Exhaustion) |
| \(\OB \top\) | (\(\OB\)-N) |
| If \(\vdash p \rightarrow q\) then \(\vdash \OB p\rightarrow \OB
q\) | (\(\OB\)-RM) |
| If \(\vdash p \leftrightarrow q\) then \(\vdash \OB
p\leftrightarrow \OB q\) | (\(\OB\)-RE) |



We will be discussing virtually all of these subsequently. For now,
let's indicate a few derivations. We show that \(\OB\)-RM is a
derived rule of SDL, and note three
 corollaries.[22]


* **Show:** If \(\vdash p \rightarrow q\), then
\(\vdash \OB p \rightarrow \OB q\). (\(\OB\)-RM)
* **Proof:** Suppose \(\vdash p \rightarrow q\). Then
by OB-NEC, \(\vdash \OB (p \rightarrow q)\), and then by \(\OB\)-K,
\(\vdash \OB p \rightarrow \OB q\).
* **Corollary 1:** \(\vdash \OB p \rightarrow \OB (p
\vee q)\). (Weakening)
* **Corollary 2:** If \(\vdash p \leftrightarrow q\)
then \(\vdash \OB p \leftrightarrow \OB q\) (\(\OB\)-RE)
* **Corollary 3:** \(\vdash \OB (p \amp q)
\rightarrow(\OB p \amp \OB q)\). (\(\OB\)-M)



An alternative formulation of SDL can be found in supplement C on
 Alternative Axiomatization of SDL.



SDL can be strengthened by adding additional axioms; in particular, we
might consider adding axioms with nested deontic operators. For
example, suppose we added the following formula as an axiom to
SDL:

 

\[ \tag{\(\OB\)-U} \OB (\OB p \rightarrow p) \]


Call the resulting system "SDL+" for easy reference in
this section. This says (roughly) that it is required that obligations
be fulfilled. This is not a theorem of SDL (as we will see in
 Section 2.3),
 so SDL+ is a genuine strengthening of SDL. Furthermore, it makes a
logically *contingent* proposition (i.e., that \(\OB p
\rightarrow p)\) obligatory *as a matter of deontic logic*. SDL
does not have this feature. With this addition to SDL, it is easy to
prove \(\OB \OB p \rightarrow \OB p\), a formula involving an iterated
occurrence of our main
 operator.[23]
 This formula asserts that if it is obligatory that \(p\) be
obligatory, then \(p\) *is* obligatory. (Cf. "the only
things that are required to be obligatory are those that actually
 are").[24]
 It should be noted that these are often given an impersonal ought
reading, so that \(\OB\)-U would be read as "It ought to be the
case that if it ought to be the case that \(p\) then it
 is)".[25]


### 2.2 The Traditional Scheme Revisited



In identifying the Traditional Definitional Scheme (TDS) in
 Section 1.2,
 we noted that it was widely recognized that we could have taken
another notion than *obligation* from our first four primary
normative statuses and defined the rest in terms of that one. So it is
expected that we should be able to generate the corresponding
equivalences derivatively from the scheme we did settle on, where
\(\OB\) is taken as primitive. For example, it is obviously desirable
to have \(\OB p \leftrightarrow \neg \PE \neg p\) as part of the
Traditional Scheme, as this expresses the equivalence between what
would have been definiens and definiendum had "\(\PE\)"
been taken as primitive. However, this equivalence is not thus far
derivable. For \(\OB p \leftrightarrow \neg \PE \neg p\) is
definitionally equivalent to \(\OB p \leftrightarrow \neg \neg \OB
\neg \neg p\), that is, \(\OB p \leftrightarrow \OB \neg \neg p\), but
the latter formula is not a tautology, so we cannot in turn derive
\(\OB p \leftrightarrow \neg \PE \neg p\). So more is presupposed in
the Traditional Scheme than what has been noted above. A natural
thought is a rule of inference needs to be added, one allowing for the
replacement of provably equivalent formulas in the context of
"\(\OB\)", namely \(\OB\)-RE in Corollary 2 above
Let's take this to be tacitly part of the Traditional
Scheme.



Suppose now that we have the same language as that of SDL, along with
the rule RE governing \(\OB\). We saw in
 Section 1.2
 that in addition to TDS, the Traditional Threefold Classification
(TTC) and/or the Deontic Square (DS) of Opposition were often
explicitly endorsed in pre-symbolic investigations of deontic logic.
Symbolically formulated, these are:

 

\[\begin{align} \tag\*{DS} (\OB p &\leftrightarrow \neg \OM p) \amp (\IM p \leftrightarrow \neg \PE p) \amp \neg(\OB p \amp \IM p) \\ & {} \amp \neg(\neg \PE p \amp \neg \OM p) \amp (\OB p \rightarrow \PE p) \amp (\IM p \rightarrow \OM p).\\ \end{align} \] \[\begin{align} \tag\*{TTC} (\OB p \vee \OP p &\vee \IM p) \amp \neg(\OB p \amp \IM p) \\ &{} \amp \neg(\OB p \amp \OP p) \amp \neg(\OP p \amp \IM p). \end{align} \]


Given TDS and RE, it turns out that DS and TTC are each
*tautologically equivalent* to the principle that obligations
cannot conflict (and thus to one another):

 

\[\tag\*{NC:} \neg(\OB p \amp \OB \neg p). \]


For in primitive notation, DS becomes

 

\[\begin{align} 
(\OB p &\leftrightarrow \neg \neg \OB p) \amp (\OB \neg p \leftrightarrow \neg \neg \OB \neg p) \amp \neg(\OB p \amp \OB \neg p) \\ 
 & {} \amp \neg(\neg \neg \OB \neg p \amp \neg \neg \OB p) \amp (\OB p \rightarrow \neg \OB \neg p) \\ 
 & {} \amp (\OB \neg p \rightarrow \neg \OB p), 
\end{align}\]


and although the first two conjuncts are tautologies, the remaining
four are each tautologically equivalent to NC above. Similarly, TTC
becomes

 

\[\begin{align} (\OB p &\vee(\neg \OB p \amp \neg \OB \neg p) \vee \OB \neg p) \\ &{} \amp [\neg(\OB p \amp \OB \neg p) \amp \neg(\OB p \amp (\neg \OB p \amp \neg \OB \neg p)) \\ &{} \quad\: {} \amp \neg((\neg \OB p \amp \neg \OB \neg p) \amp \OB \neg p)], \end{align} \]


and although the exhaustiveness clause is tautological, as are the
last two conjuncts of the exclusiveness clause, the first conjunct of
that clause is just NC
 again.[26]



So the Traditional Scheme rests squarely on the soundness of NC, along
with \(\OB\)-RE and the TDS. In fact, the Traditional Scheme can be
seen as a widely endorsed, informal counterpart of a deontic logic
composed of just PC, \(\OB\)-RE and NC. Since that logic is a (proper)
fragment of SDL, it follows that the Traditional Scheme (symbolized)
is entailed by
 SDL.[27]


### 2.3 SDL Semantics



The reader familiar with elementary textbook logic will have perhaps
noticed that the deontic square and the modal square both have even
better-known analogs for the quantifiers as interpreted in classical
predicate logic:



![square diagram; link to extended description below](fig-square-c.svg)

Fig. 6
 [An extended description of figure 6.]




Though less widely noted in textbooks, there is also a threefold
partition for classical quantifiers regarding any condition,
\(p\):



![three boxes diagram: link to extended description below](fig-three-c.svg)

Fig. 7
 [An extended description of figure 7.]




These quantificational analogies reflect possible inspiration behind
what is most often called "possible worlds semantics" for
such logics, to which we now
 turn.[28]
 Once the analogies are noticed, spelling out this sort of semantics
seems all but
 inevitable.[29]



We choose a standard "Kripke-style" possible world
semantics for SDL (Kripke 1959,
 1963).[30]
 We assume that we have a set of possible worlds, \(W\), and a
relation, \(A\), relating worlds to worlds, with the intention that
\(Aij\) iff \(j\) is a world where everything that holds at \(j\) is
acceptable from the standpoint of \(i\), so that everything obligatory
at \(i\) holds in \(j\). For brevity, we will call all worlds so
related to \(i\), "\(i\)-acceptable" worlds and denote
them by
 \(A^i\).[31]
 We then add that the acceptability relation is "serial":
for every world, \(i\), there is at least one \(i\)-acceptable world.
Finally, propositions are either true or false at a world, never both,
and when a proposition, \(p\), is true at a world, we will often
indicate this by referring to that world as a
"\(p\)-world". The truth-functional operators have their
usual behavior at each world.



The fundamental idea here is that the normative status of a
proposition from the standpoint of a world \(i\) can be assessed by
looking at how that proposition fairs at the \(i\)-acceptable worlds.
For any given world, \(i\), we can easily picture the \(i\)-accessible
worlds as all corralled together in logical space, where seriality is
reflected by a small dot representing the presence of at least one
\(i\)-acceptable world. The intended truth-conditions, relative to
\(i\), for our five deontic operators can now be pictured as
follows:



![a diagram of 6 boxes: link to extended description below](fig-box-a.svg)

Fig. 8
 [An extended description of figure 8.]




Thus, \(p\) is *obligatory* iff it holds in all the
\(i\)-acceptable worlds, *permissible* iff it holds in some
such world, *impermissible* iff it holds in no such world,
*omissible* iff its negation holds in some such world,
*optional* iff \(p\) holds in some such world, and so does
\(\neg p\), *and non-optional when* \(p\) *either holds at
all such worlds or at none.* If a formula is true at every world
in any such model of serially-related worlds, then the formula is
 *valid*.[32]



To illustrate the workings of this framework, consider NC, \(\OB p
\rightarrow \neg \OB \neg p\). This is valid in this framework because
of seriality. For suppose that \(\OB p\) holds at any world \(i\) in
any model. Then each \(i\)-accessible world is one where \(p\) holds,
and by the seriality of accessibility, there must be at least one such
world. Call it \(j\). Now we can see that \(\neg \OB \neg p\) must
hold at \(i\) as well, for otherwise, \(\OB \neg p\) would hold at
\(i\), in which case, \(\neg p\) would have to hold at all the
\(i\)-accessible worlds, including \(j\). But then p as well as \(\neg
p\) would hold at \(j\) itself, which is impossible (by the semantics
for "\(\neg\)"). The other axioms and rules of SDL can be
similarly shown to be valid, as can all the principles listed above as
derivable in SDL.



In contrast, \(\OB\)-U, the axiom \(\OB (\OB p \rightarrow p)\) that
we added to SDL to get SDL+, is not valid in the class of serial
models. In order to validate \(\OB\)-U, we need the further
requirement of "secondary seriality": that any
\(i\)-acceptable world, \(j\), must be in turn *acceptable to
itself*. We can illustrate such an \(i\) and \(j\) as follows:



![two dots labelled i and j; an arrowed line points from i to j; another arrowed line points from j back to j](fig-graph-a.svg)

Fig. 9




Here we imagine that the arrow connectors indicate relative
acceptability, thus here, \(j\) (and only \(j)\) is acceptable to
\(i\), and \(j\) (and only \(j)\) is acceptable to \(j\). If all
worlds that are acceptable to any given world have this property of
self-acceptability, then our axiom is valid. For assume that this
property holds throughout our models, and for reductio, suppose that
for some arbitrary world \(i, \OB (\OB p \rightarrow p)\) is false at
\(i\). Then not all \(i\)-acceptable worlds are worlds where \(\OB p
\rightarrow p\) is true. So, there must be an \(i\)-acceptable world,
say \(j\), where \(\OB p\) is true, but \(p\) is false. Since \(\OB
p\) is true at \(j\), then \(p\) must be true at all \(j\)-acceptable
worlds. But by stipulation, \(j\) is acceptable to itself, so \(p\)
must be true at \(j\), but this contradicts our assumption that \(p\)
was false at \(j\). Thus \(\OB (\OB p \rightarrow p)\) must be true at
all worlds, after all. Two counter-models showing that \(\OB\)-U is
not derivable in SDL, and that SDL \(+ \OB \OB p \rightarrow \OB p\)
does not imply \(\OB\)-U can be found in supplement D:
 Two Counter-Models Regarding Additions to SDL.



We should also note that one alternative semantic picture for SDL is
where we have a set of world-relative ordering relations, one for each
world \(i\) in W, where \(j \ge\_i k\) iff \(j\) is as good as \(k\)
(and perhaps better) relative to \(i\). We might then assume that from
the standpoint of any world \(i\), a) each world is as good as itself,
b) if one is as good as a second, and the second is as good as a
third, then the first is as good as the third, c) and for any two
worlds, either the first is as good as the second or vice versa (i.e.,
respectively, each such \(\ge\_i\) is reflexive, transitive, and
connected in W). As a matter of fact, this approach also determines
SDL. (See Goble 2003 for one of the few proofs of this claim.) If we
then add "The Limit Assumption", that for each world
\(i\), there is always at least one world as good as all worlds (i.e.,
one \(i\)-best world) we can easily generate our earlier semantics for
SDL derivatively. We need only add the natural analogue here to our
prior truth-conditions for \(\OB\): \(\OB p\) is true at a world \(i\)
iff \(p\) is true at all the \(i\)-best worlds:



![a vertical bar: link to extended description below](fig-bar-a.svg)

Fig. 10
 [An extended description of figure 10.]




Essentially, the ordering relation coupled with the limit assumption
just gives us a way to *generate* the set of \(i\)-acceptable
worlds instead of taking them as primitive in the semantics: \(j\) is
\(i\)-acceptable iff \(j\) is \(i\)-best. Once generated, we look only
at what is going on in the \(i\)-acceptable (as \(i\)-best) worlds to
interpret the truth-conditions for the deontic operators, just as with
our simpler Kripke-Style semantics. The analogue to the seriality of
our earlier \(i\)-acceptability relation is also assured by the Limit
Assumption, since it entails that for each world \(i\), there is
always some \(i\)-best (so \(i\)-acceptable) world. Although this
ordering semantics approach appears to be a bit of overkill here, as
we shall see in
 Section 4
 and
 Section 5,
 it became quite important later on in the endeavor to develop
expressively richer deontic logics. For now, we turn to the
second-most well-known approach to monadic deontic logic, one in which
SDL emerges *derivatively*.


## 3. The Andersonian-Kangerian-Leibnizian Reduction



The *Andersonian-Kangerian* reduction is dually-named in
acknowledgment of Kanger's and Anderson's independent
formulation of it around the same
 time.[33]
 As Hilpinen 2001a points out, the approach is adumbrated much earlier
in Leibniz. We follow Kanger's development here, noting
Anderson's toward the end.


### 3.1 Syntax



Assume that we have a language of classical modal propositional logic,
with a distinguished (deontic) propositional constant:




"\(d\)" for "all (relevant) normative demands are
met".




Now consider the following axiom system, "K*d*":




|  |  |  |
| --- | --- | --- |
| K*d*: | All Tautologies | (TAUT) |
|  | \(\Box(p \rightarrow q) \rightarrow(\Box p \rightarrow \Box
q)\) | (K) |
|  | \(\Diamond d\) | \((\Diamond d)\) |
|  | If \(\vdash p\) and \(\vdash p \rightarrow q\) then \(\vdash
q\) | (MP) |
|  | If \(\vdash p\) then \(\vdash \Box p\) | (NEC) |



K*d* is just the normal modal logic K with \(\Diamond d\)
 added.[34]
 \(\Diamond d\) is interpreted as telling us that it is possible that
all normative demands are met. In import when added to system K, it is
similar to (though stronger than) the "No Conflicts" axiom
of SDL. *All* of the Traditional Scheme's deontic
operators are easily defined derivatively once we provide "the
reductive clause" for \(\OB\):

 

\[\OB p \eqdf \Box(d \rightarrow p)\]


So in K*d*, \(p\) is *obligatory* if \(p\) is necessitated
by all normative demands being met, *permissible* if \(p\) is
compatible with all normative demands being met,
*impermissible* if \(p\) is incompatible with all normative
demands, etc. Since none of the operators of the Traditional Scheme
are taken as primitive, and the basic logic is a modal logic with
necessity and possibility as the primitive modal operators, this is
referred to as "a reduction" (of deontic logic to modal
logic)



Proofs of SDL-ish formulas are then just K-proofs of the corresponding
modal formulae involving
 "\(d\)".[35]
 It is well-known that all theorems of SDL are derivable in K*d*
and conversely, that if some formula in the language of SDL is not a
theorem of SDL, then its translation following the Kangerian reduction
is also not a theorem of
 K*d*.[36]
 In addition to containing all theorems of SDL, we note a few theorems
that are specific to K*d* because of the non-overlapping
syntactic ingredients, \(d, \Box\), and \(\Diamond\):




|  |  |
| --- | --- |
| \(\vdash \OB d \) | \((\OB d)\) |
| \(\vdash \Box(p \rightarrow q) \rightarrow(\OB p \rightarrow \OB
q)\) | (RM') |
| \(\vdash \Box p \rightarrow \OB p\) | (Nec') |
| \(\vdash \OB p \rightarrow \Diamond p\) | ("Kant's
 Law")[37] |
| \(\vdash \neg \Diamond(\OB p \amp \OB \neg p)\) | (NC') |



These are easily
 derived.[38]



Although our underlying modal system is just K, adding further
non-deontic axiom schema (i.e., those that cannot be abbreviated via
SDL formulas, nor involving \(d\) specifically) can nonetheless have a
deontic impact. To illustrate, suppose we added a fourth axiom, one to
the effect that necessity is truth-implicating, standardly called
axiom "T":

 

\[ \Box p \rightarrow p \tag{T} \]


Call the system that results from adding this formula to our current
system "KT*d*". The addition of T makes derivable our
previously mentioned axiom \(\OB\)-U of SDL+, which we have shown is
not derivable in SDL
 itself:[39]


 

\[ \vdash \OB (\OB p \rightarrow p) \]


So, reflecting on the fact that SDL+ is derivable in KT*d*, we
see that the Andersonian-Kangerian reduction must either rely on a
non-truth-implicating conception of necessity in order for its pure
deontic fragment to match SDL, or SDL itself is not susceptible to the
Andersonian-Kangerian reduction. Put another way, the most plausible
version of the Andersonian-Kangerian reduction can't help but
view "Standard Deontic Logic" as too weak.



Anderson's approach is practically equivalent to Kanger's.
First, consider the fact that we can easily define another constant in
K*d*, as follows:

 

\[ s \eqdf \neg d. \]


where this new constant would now be derivatively read as follows:



> 
> 
> "some (relevant) normative demand has been violated".
> 
> 
> 



Clearly our current axiom, \(\Diamond d\), could be replaced with
\(\neg \Box s\), asserting that it is not necessary that some
normative demand is violated. We could then define \(\OB\) as:



\[ \OB p \eqdf \Box(\neg p \rightarrow s). \]


Essentially, Anderson took this equivalent course with
"\(s\)" being his primitive (initially standing for
something like "the sanction has been invoked" or
"there is a liability to sanction"), and \(\neg \Box s\),
as the axiom added to some modal system (e.g., at least as strong as
modal system KT).



We should also note that Anderson was famous as a founding figure in
 relevance logic,
 and along with using strict implication, \(\Box(p \rightarrow q)\),
he also explored the use of a relevant (and thus neither material nor
strict) form of implication, \(\Rightarrow\), to express the reduction
as: \(\OB p \eqdf \neg p \Rightarrow
 s\).[40]
 (A bit more on this can be found within the entry on
 Mally's deontic logic.
 See further references there.) This alternative reflects the fact
that there is an issue in both Kanger's and Anderson's
strict necessitation (or entailment) approaches of just what notion of
"necessity" is involved in claiming that meetings all
normative demands (or avoiding the sanction) in some sense
*necessitates* \(p\).



As a substantive matter, how should we think of these
"reductions"? For example, should we view them as giving
us an *analysis* of what it is for something to be obligatory?
Taking Kanger's approach first, it would seem that \(d\) must be read
as a distinctive deontic ingredient, if we are to get the derivative
deontic reading for the "reduced" deontic operators. Also,
as our reading suggests, it is not clear that \(d\) does not, at least
by intention, express a complex quantificational notion involving the
very concept of obligation (demand) as a proper part, namely that all
obligations have been fulfilled, so that the "reduction",
*presented as an analysis*, would appear to be circular. If we
read \(d\) instead as "ideal circumstances obtain", the
claim of a substantive reduction or analysis appears more promising,
until we ask: Are the circumstances ideal only with respect to meeting
normative *demands or obligations*, or are they ideal in other
(for example supererogatory) ways that go beyond merely satisfying
normative *demands*? Anderson's "liability to
sanction" approach may appear more promising, since the idea
that something is obligatory if (and only if) and because
non-compliance necessitates (in some sense) liability to (or perhaps
desert of) punishment does not appear to be circular (unless the
notion of "liability" itself ultimately involves the idea
of permissibility of punishment), but is it plausible? Alternatively,
perhaps a norm that is merely an *ideal* cannot be
*violated*, in which case perhaps norms that have been
*violated* can be distinguished (as a subset) from norms that
have not been *complied* with, and then the notion of an
obligation as something that must obtain unless some norm is violated
will not be obviously circular. There is a substantive philosophical
question lingering here that the language of a "reduction"
brings naturally to the surface, but is sloughed over. The formal
utility of the reduction does not hinge on this, but its philosophical
significance does.


### 3.2 Semantics



The semantic elements here are in large part analogous to those for
SDL. We have a binary relation again, but this time instead of a
relation interpreted as relating worlds *acceptable* to a given
world, here we will have a relation, \(R\), relating worlds
"accessible" to a given world (e.g., possible relative to
the given world). The only novelties are two: (1) we add a simple
semantic element to match our syntactic constant "\(d\)",
and (2) we add a slightly more complex analog to seriality, one that
links the accessibility relation to the semantic element added in
order to model \(d\).



Once again, assume that we have a set of possible worlds, \(W\), and
we assume that we have a relation, \(R\), relating worlds to worlds,
with the intention that \(Rij\) iff \(j\) is accessible to \(i\)
(e.g., \(j\) is a world where everything true in \(j\) is possible
relative to
 \(i\).[41]
 For brevity, we will call all worlds possible relative to \(i\),
"\(i\)-*accessible* worlds" and denote them by
\(R^i\). For the moment, no restrictions are placed on the relation
\(R\). We can illustrate the truth-conditions for necessity, and
possibility with obvious abbreviations, as follows:



![two boxes: link to extended description below](fig-box-b.svg)

Fig. 11
 [An extended description of figure 11.]




Here we imagine that for any given world, \(i\), we have corralled all
the \(i\)-accessible worlds together. We then simply look at the
quantificational status of \(p\) (and/or \(\neg p)\) in these
\(i\)-accessible worlds to determine \(p\)'s modal status back
at \(i\). For example\(,\) at a given world \(i, p\) is
*necessary* if \(p\) holds throughout \(R^i\) and
*possible* if \(p\) holds somewhere in \(R^i\).



The only *deontic* element in the syntax of K*d* is our
distinguished constant, \(d\), intended to express the fact that all
normative demands are met\(.\) To model that feature, we simply assume
that the worlds are divided into those where all normative demands are
met and those that are not. We denote the former subset of worlds by
"DEM" in a model. Then \(d\) is true at a world \(j\) iff
\(j\) belongs to DEM. Here is a picture where \(d\) is true at an
arbitrary world, \(j\):



![a box diagram: link to extended description below](fig-vbox-a.svg)

Fig. 12
 [An extended description of figure 12.]




Since \(j\) is contained in DEM, that means all normative demands are
met at
 \(j\).[42]



Corresponding to simple seriality for SDL (that there is always an
\(i\)-acceptable world), we assume what I will call "strong
seriality" for K*d*: for every world \(i\), there is an
\(i\)-accessible world that is among those where all normative demands
are met. In other words, for every world \(i\), the intersection of
the \(i\)-accessible worlds with DEM is non-empty. Given the truth
conditions for \(d\), strong seriality validates \(\Diamond d\),
ensuring that for any world \(i\), there is always some
\(i\)-*accessible* world where \(d\) is true:



![two overlapping blue boxes: link to extended description below](fig-vbox-b.svg)

Fig. 13
 [An extended description of figure 13.]




Given these semantic elements, if you apply them carefully to the
definitions of the deontic operators of K*d*, you will see that
in each case, the normative status of \(p\) at \(i\) depends on
\(p\)'s relationship to this intersection of the i-accessible
worlds and the worlds where all normative demands are met:



![six pairs of two overlapping blue boxes: link to extended description below](fig-vbox-c.svg)

Fig. 14
 [An extended description of figure 14.]




If that intersection is permeated by \(p\)-worlds, \(p\) is
obligatory; if it contains some \(p\)-world, \(p\) is permissible,
 etc.[43]



If we wish to validate T, \(\Box p \rightarrow p\) (and derivatively,
\(\OB\)-U, \((\OB (\OB p \rightarrow p))\), we need only stipulate
that the accessibility relation, R, is reflexive: that each world
\(i\) is \(i\)-accessible (possible relative to itself):



![a blue dot labelled i with an arrowed line going from the dot and back to it.](fig-graph-d.svg)

Fig. 15




For then \(\Box p \rightarrow p\) must be true at any world \(i\), for
if \(\Box p\) is true at \(i\), then \(p\) is true at each
\(i\)-accessible world, which includes \(i\), since \(i\) is
self-accessible. This will indirectly yield the result that \(\OB (\OB
p \rightarrow p)\) is true in all such models as well.



Having introduced deontic logic's classical points of departure,
we now turn to a number of problems, paradoxes, and related logics
that are studied in the field today. Here, our aim is mainly to map
out the various topics that are central stage in deontic logic; for
reasons of space, we cannot present theories or logics in much
detail.


## 4. Conditional Obligations and Chisholm's Puzzle


### 4.1 Chisholm's Puzzle and SDL



Consider the following four premises (Chisholm 1963a) and their most
straightforward symbolization in SDL:



(1)
It ought to be that Jones goes to assist his neighbors.
(1')
\(\OB g\).


(2)
It ought to be that if Jones goes, then he tells them he is
coming.
(2')
\(\OB (g \rightarrow t)\).


(3)
If Jones doesn't go, then he ought not tell them he is
coming.
(3')
\(\neg g \rightarrow \OB \neg t\).


(4)
Jones doesn't go.
(4')
\(\neg g\).


(1)-(4) certainly appear to describe a possible situation. They
appear to constitute both a *mutually consistent* and
*logically independent* set of sentences. Note that (1) is a
*primary obligation*, saying what Jones ought to do
unconditionally. (2) is a *compatible-with-duty obligation*,
telling us what else Jones ought to do on the condition that Jones
fulfills his primary obligation. In contrast, (3) is a
*contrary-to-duty obligation* or "imperative" (a
"*CTD*") expressing what Jones ought to do
conditional on his violating his primary obligation. (4) is a factual
claim, which conjoined with (1), implies that Jones *violates*
his primary obligation. Thus this puzzle also places not only deontic
*conditional* constructions, but the *violability* of
obligations, at center stage. It raises the challenging question: what
constitutes proper reasoning about what to do in the face of
violations of obligations?



One obvious answer would be: just apply SDL to
(1')-(4') above. The following table displays the
difficulties in trying in to interpret the reasoning in SDL:





|  |  |  |
| --- | --- | --- |
| **First Path** | **Second Path** | **Third Path** |
| (1') \(\OB g\) | (1') \(\OB g \) | (1') \(\OB g\) |
| (2') \(\OB(g \rightarrow t)\) | (2') \(\OB(g \rightarrow t)\) | (2'') \(g \rightarrow \OB t\) |
| (3') \(\neg g \rightarrow \OB \neg t \) | (3'' ) \(\OB (\neg g \rightarrow \neg t) \) | (3') \(\neg g \rightarrow \OB \neg t\) |
| (4') \(\neg
g\) | (4') \(\neg
g\) | (4') \(\neg
g\) |
| 
From (1'), (2'), \(\OB t.\)

From (3'), (4'), \(\OB \neg t.\)

By NC, consistency is lost. | 
(1') implies (3'' ).

Independence is lost. | 
(4') implies (2'').

Independence is lost. |




The first path, as Chisholm notes, leads to contradiction via
\(\OB\)-K, MP and NC. The second and third paths take a different
route, interpreting the conditionals in (2) and (3) uniformly.
However, for the second path, the symbolization of (3) follows from
that of (1), and for the third path, the symbolization of (2) follows
from that of (4). Each SDL symbolization of the original violates one
of our desiderata: either mutual consistency or joint
independence.


### 4.2 Two Types of Detachment and Two Ways of Enriching SDL



If von Wright launched deontic logic as an academic area of research,
Chisholm's paradox, and similar cases, were the booster rocket
that provided the escape velocity deontic logic needed from
 subsumption
 under normal modal logics, thus solidifying deontic logic's
status as a distinct branch of
 logic.[44]
 It is now virtually universally acknowledged that Chisholm was right:
the sort of conditional deontic claim expressed in (3) can't be
faithfully represented in SDL, nor more generally by a composite of
some sort of unary deontic operator and *material* conditional.
The views diverge however where it comes to exactly what should be
added to SDL in order to represent conditional obligations. This
divergence in turn relates to a distinction between two types of
detachment in deontic
 reasoning.[45]



Let "\(\OB (q\mid p)\)" be shorthand for a conditional
obligation or ought statement like that in the natural language
sentence, (3), above. So we will read \(\OB (q\mid p)\) as "if
\(p\), then it ought to be (or it is obligatory) that \(q\)".
Suppose we also assume that monadic obligations are disguised dyadic
obligations, per the following analysis:

 

\[ \OB p \eqdf \OB (p\mid \top). \]


This analysis has been widely
 employed.[46]



With this in mind, we distinguish between two relevant types
of"detachment principles" (Greenspan 1975) that we might
ascribe to these iffy-'ought's:



* *Factual Detachment* (FD): \(p \amp \OB (q\mid p) \vdash
\OB q\)
* *Deontic Detachment* (DD): \(\OB p \amp \OB (q\mid p)
\vdash \OB
 q\).[47]




*Factual detachment* tells us that from \(p\), and a deontic
conditional to the effect that if \(p\) then it ought to be that
\(q\), we can conclude that it ought to be that \(q\). *Deontic
Detachment* in contrast tells us that from it ought to be that
\(p\) and if \(p\), then it ought to be that \(q\), we can conclude
that it ought to be that \(q\). If we interpret a deontic conditional
as a material conditional with an obligatory consequent (as in
(3') above), FD, but not DD is supported. Conversely, if we
interpret deontic conditionals as obligatory material conditionals (as
in (2') above), DD, but not FD is supported. Although we have
shown earlier that neither of these interpretations is acceptable, the
contrast reveals a general problem. Unrestricted endorsement of both
types of detachment is not tenable, since it leads implausibly to the
conclusion that Jones is both obligated to tell (the neighbor he is
coming) and obligated to not tell. Thus researchers tended to part
company over which principle of the two to endorse (Loewer and Belzer
1983).



The Factual Detachment camp typically endorses the view that the
conditional in (3) in the Chisholm Quartet needs to be interpreted as
a non-material conditional, but otherwise things are as they seem in
(3): we have a conditional obligation that is a simple composite of a
non-material conditional and a pure unary deontic operator in the
consequent:




\(\OB (q\mid p) \eqdf p \Rightarrow \OB q\), for some independent
 conditional.[48]




Typically, the conditional \(\Rightarrow\) is of the sort made famous
by Stalnaker and Lewis (Stalnaker 1968, Lewis
 1973).[49]
 It is then generally maintained that deontic detachment is flawed,
since the conditional obligations like those in (2) tell us only what
to do in ideal circumstances, but they do not necessarily provide
 "cues"[50]
 for action *in the actual world*, where things are often
typically quite sub-ideal, as (4) combined with (1) indicate. Thus
from the fact that Jones ought to go and he ought to tell if he goes,
it doesn't follow that what he ought to *actually* do is
tell--that would be so only if it was also a fact that he goes to
their aid. At best, we can only say that he ought *ideally* to
go.



This suggestion seems a bit more difficult when we change the
conditional to something like "If Doe does kill his mother, then
it is obligatory that Doe kills her gently" (Forrester 1984).
The idea that my obligation to not kill my mother gently (say for an
inheritance) merely expresses an "ideal" obligation, but
not an actual obligation, given that I will kill her, seems hard to
swallow. So this case makes matters a bit harder for those favoring a
factual detachment approach for generating actual obligations.
Similarly, it would seem that if it is impermissible for me to kill my
mother, then it is impermissible for me to do so gently, or to do so
while
 dancing.[51]
 So *carte blanche* factual detachment seems to allow the mere
fact that I will take an action in the future (killing my mother) that
is horribly wrong and completely avoidable now to render obligatory
another horrible (but slightly less horrible) action in the future
(killing my mother gently). The latter action must be completely
avoidable if the former is, and the latter action is one that I would
*seem* to be equally obligated to not perform intuitively.



The deontic detachment camp represented conditional obligations via
dyadic non-composite obligation operators modeled syntactically on
conditional probability, following Danielsson 1968; B. Hansson 1969;
von Wright 1956, 1964; Lewis 1973, 1974; and Feldman
 1986.[52]
 On this view, deontic conditionals are viewed as idioms: the meaning
of the compound is not a straightforward function of the meaning of
the parts. The underlying intuition regarding the Chisholm example is
that even if it might be true that we will violate some obligation,
that doesn't get us off the hook from obligations that derive
from the original one that we will violate. If I must go help and I
must inform my neighbors that I'm coming, if I do go help, then
I must inform them, and the fact that I will in fact violate the
primary obligation does not block the derivative obligation any more
than it blocks the primary obligation.



One early semantic picture for the latter camp was that a sentence of
the form \(\OB (q\mid p)\) is true at a world \(i\) iff the \(i\)-best
\(p\)-worlds are all \(q\)-worlds. \(\OB q\) is then true iff \(\OB
(q\mid \top),\) and so iff all the unqualifiedly \(i\)-best worlds are
\(q\)-worlds (B. Hansson 1969). Note that this weds preference-based
semantic orderings with dyadic conditional
 obligations.[53]
 This reflects a widespread trend. Factual detachment does not work in
this case, since even if our world is an I-don't-go-help-world,
and the best among the I-don't-go-help-worlds are
I-don't-call-worlds, it does not follow that the
*unqualifiedly* best worlds are I-don't-call-worlds. In
this example, these authors would maintain that the unqualifiedly best
worlds are both I-go worlds and I-call worlds, and the fact that I
won't do what I'm supposed to do won't change
that.



But one is compelled to ask those in the Deontic Detachment camp: what
then is the point of such apparent conditionals if we can't ever
detach them from their apparent antecedents, and how are these
conditionals related to regular ones? This seems to be the central
challenge for this camp. Thus they often endorse a restricted form of
factual detachment, of which the following is a representative
instance:



* *Restricted Factual Detachment*: \(\Box p \amp \OB (q\mid
p) \rightarrow \OB q\).




Here \(\Box p\) might mean various things, for example that \(p\) is
physically unalterable or necessary as of this moment in
 history.[54]
 Only if \(p\) is settled true in some sense, can we conclude from
\(\OB (q\mid p)\) that \(\OB q\). This certainly helps, but it still
leaves us with a bit of a puzzle about why this apparent composite of
a conditional and a deontic operator is actually some sort of
primitive
 idiom.[55]


## 5. Other Enrichments of SDL



While conditional obligations and the puzzles they generate call for
extensions of SDL, other problems and distinctions in normative
discourse motivate further, mostly orthogonal enrichments. In this
section we point to a few of these.


### 5.1. Supererogation and Allied Normative Concepts



Even among monadic deontic notions, and even if we stay within the
realm of all-things-considered, unconflicted obligations (cf.
 Section 6.3),
 SDL displays various expressive weaknesses compared to our everyday
and more theorized ethical argumentation.


#### 5.1.1 Optionality vs. Indifference



Let us start with the distinction between indifference and
 optionality.[56]
 In general, optionality of a given state of affairs or
action need not imply that it is morally indifferent. For instance, it
may be optional that you attend a faculty meeting, but not a matter of
indifference that you do so. However, when deontic logicians and
ethicists gave an operator label for the condition \((\neg \OB p \amp
\neg \OB \neg p)\), it was almost invariably "It is indifferent
that \(p\)", "\(\IN p\)". It would seem to follow
from the theorem

 

\[\OB p \vee(\neg \OB p \amp \neg \OB \neg p) \vee \IM \neg p,\]


that

 

\[(\neg \OB p \amp \neg \IM \neg p) \rightarrow \IN p,\]


that is, everything that is neither obligatory nor prohibited is a
matter of indifference. But many actions are neither obligatory nor
prohibited, yet they are hardly matters of indifference. A
particularly intriguing class of such actions are those that are
deemed supererogatory (e.g., volunteering for a costly or risky good
endeavor where others are equally qualified and no one person is
obligated). SDL can represent optionality, but not indifference,
despite the fact that the latter concept has been a purported target
for representation since nearly its beginning.


#### 5.1.2 Must vs. Ought



The distinction between optionality and indifference is moreover
strongly linked to that between "must" (and its dual
"can") and 
 "ought".[57]
 Consider the following utterance:



(1)
Although you can skip the meeting, you ought to attend.


This seems to be a perfectly sensible piece of advice. (1) appears to
imply that it is optional that you attend--that you *can*
attend and that you *can* fail to attend. The latter two uses
of "can" express permissibility. Yet "ought"
is routinely the reading given for deontic necessity in deontic logic
(and in ethical theory), and then "permissibility" is
routinely presented as its dual. However, if we symbolize (1) above
accordingly, we get,

 

\[\tag{\(1'\)} \PE \neg p \amp \OB p\]


which is just \(\neg \OB p \amp \OB p\) in disguise (given \(\OB\)-RE
and the Traditional Definitional Scheme). So (1'), given NC,
yields a contradiction. Another way to put this is that the
"can" of permissibility seems to be the dual of
"must" not the dual of "ought". This yields a
dilemma: either deontic necessity represents "ought", in
which case, its dual does not represent permissibility (and neither
does any other construction in SDL), or permissibility is represented,
but "ought" is inexpressible, despite the ubiquitous
assumption otherwise. That "ought" is the dual of
permissibility is really a largely overlooked dubious yet pervasive
bipartisan presupposition in both ethical theory and deontic
 logic.[58]


#### 5.1.3 The Least One Can Do



A widely neglected normative notion is exemplified by the following
 complaint:[59]



(LYCD)
You ought to have come home on time; *the least you could have
done* was called, and you didn't do even that.


The expression in the second clause of (LYCD), i.e., "the least
you could do", has been largely ignored in the literature on
deontic logic and ethical theory both. (LYCD) appears to express the
idea that there is some minimal but acceptable alternative. This
notion of what is minimally acceptable among the permissible options
is not expressible in SDL.


#### 5.1.4 Supererogation and Permissible Suboptimality



To go beyond the call of duty or to supererogate is to permissibly do
more than you would do if you did the least you could 
 do.[60]
 It has a
mirror image, *permissible suboptimality*, or that which you
can but ought not do. These are plainly not representable in SDL,
given the preceding remarks.



McNamara 1996c and Mares & McNamara 1997 provides a semantic and
logical framework for distinguishing "must" from
"ought", indifference from optionality, as well as
distinctly representing "the least you can do" idiom and
analyzing one central sense of "supererogation" via that
otherwise unstudied idiom. As a measure of the increased expressive
power, the three-fold partition of normative positions for SDL is
extended to a twelve-fold partition; with an extension to accommodate
praiseworthiness and blameworthiness (McNamara 2011a,b), an
eighty-four fold partition results. See McNamara forthcoming for a
thorough introduction to this area of research.


### 5.2 Deontic Logic and Agency



We routinely talk about both what *ought to be* and what people
*ought to do*. These hardly look like the same things (for
example, the latter notion calls for an agent, the former does not).
This issue, and the general issue of representing agency in deontic
logic has been much discussed, and continues to be an area of active
concern. In the present section we provide an overview of the
literature on deontic logics that allow for the expression of agency;
in
 Section 6.2
 we consider a more fundamental revision of SDL in order to
accommodate deontic claims about actions.


#### 5.2.1 Agency as a Modal Operator



Let us start by introducing some basics from the logical study of
 agency.[61]
 Here, the idea is that we treat expressions of the type "x
brings it about that" or "x sees to it that" as
agent-indexed modal operators. Focusing, as we will do here, on the
case of a single agent called Jane Doe, let us introduce the following
notation:



* \(\BA\):
Jane Doe brings it about
 that.[62]




Then the following claims expressing Jane Doe's agency with
respect to a proposition p are to be distinguished:



* \(\BA
p\): Jane Doe brings it about that \(p\)
* \(\BA
\neg p\): Jane Doe brings it about that \(\neg
p\)
* \(\neg
\BA p\): Jane Doe does not bring it about that
\(p\)
* \(\neg
\BA \neg p\): Jane Doe does not bring it about
that \(\neg p\).




Plainly, if neither of the first two hold, then the conjunction of the
last two holds. In such a case we might say that Jane Doe is
*passive* with respect to \(p\), or more adequately, passive
with respect to herself bringing about p or its
 negation.[63]
 Let's introduce \(\PV\) as an abbreviation for
 this:[64]


 

\[\PV p \eqdf \neg \BA p \amp \neg \BA \neg p.\]


See also supplement E:
 Non-Performance versus Refraining/Forbearing.



Having thus enriched our propositional language, we can now consider
some logical principles for the new operator. Virtually all accounts
take it to satisfy the rule for replacement of equivalents:




If \(p \leftrightarrow q\) is a theorem, so is \(\BA p \leftrightarrow
\BA q\) (\(\BA\)-RE),




as well as the truth scheme,




\(\BA p \rightarrow p\)
 (\(\BA\)-T).[65]


Taken together, these principles entail that, relative to a fixed
proposition \(p\) and an agent, a threefold partition emerges:





|  |  |  |
| --- | --- | --- |
| \(\BA p\) | \(\PV p\) | \(\BA {\neg p}\) |



Fig. 16



#### 5.2.2 The Meinong-Chisholm Reduction



Returning to deontic matters, we can now ask: how do we represent
agential obligation? With an agency operator in hand, we might invoke
the famous *Meinong-Chisholm Reduction*: the idea that what
Jane ought to do is what it ought to be that she
 does.[66]
 If we regiment this a bit using our operator for agency, we get the
following versions of the "reduction":



> 
> 
> *Meinong-Chisholm Reduction*: Jane Doe is obligated to bring it
> about that \(p\) iff it is obligatory that Jane Doe brings it about
> that \(p\).
> 
> 
> 



This is sometimes taken to be a reduction of personal, agentive
obligation to impersonal obligation on the one hand, and (mere) agency
on the other. In terms of modal operators, it is a reduction of
"ought to do" to "ought to be" and
"brings it about
 that".[67]
 Although not uncontested (cf. Section 5.2.4 below), by relying on
this analysis we can make many fine distinctions. Suppose, for
instance, that "Jane Doe lights the candle" is equivalent
to "Jane Doe sees to it that the candle is lit", and let
"the candle is lit" be represented by p. Then we can
distinguish the following claims, while keeping track of their formal
interrelations:



(1)
Jane Doe ought to light the candle
(1')
\(\OB \BA p\)


(2)
Jane Doe ought not to light the candle
(2')
\(\OB \neg \BA p\)


(3)
Jane Doe ought to bring it about that the candle is not lit
(3')
\(\OB \BA \neg p\)


(4)
Jane Doe may (is permitted to) light the candle
(4')
\(\neg \OB \neg \BA p\)


(5)
Jane Doe may not (is permitted not to) light the candle
(5')
\(\neg \OB \BA p\)


Here, (2') would, e.g., express that Jane Doe is positively
obligated to not bring it about that \(p\); note that this need not
preclude that some other agent can still bring it about that \(p\), or
that \(p\) is in itself obligatory. Note also that (2') is
distinct from (5'), which merely expresses that it is not
required of Jane Doe that she lights the candle. That is, Jane's
lighting the candle would be a violation of (2), but not of (5). (3)
is, on most accounts of agency and obligation, strictly stronger than
(2): (3) implies not only that Jane should not light the candle, but
that she should bring it about that no one else lights the candle
either (e.g., by keeping it in some locked box).



Logics that combine operators for (impersonal) obligation and agency
have been very widely
 employed.[68]
 Krogh and Herrestad 1996 reinterpret the Meinong-Chisholm analysis so
that the deontic operator is *personal*, yet not agential. This
is arguably a more plausible way to preserve a componential analysis
of agential obligation. McNamara 2004a also makes the case that a
person's being obligated to be such that a certain condition
holds (e.g., being obligated to be at home at noon, as promised) is
the more basic idiom, and being obligated to bring about something is
just being obligated to be such that you do bring it about.


#### 5.2.3 Normative Positions



One way in which agency operators have been fruitfully employed is in
the study of what are called *normative
 positions*.[69]
 A set of normative positions is intended to describe the set of
mutually exclusive and jointly exhaustive positions that a person or
set of persons may be in regarding a proposition and with respect to a
logic governing a set of selected primitive normative statuses and/or
agency operators. Recall the partition regarding how Jane Doe may be
positioned agentially with respect to \(p\): she may either bring
about \(p\), bring about its negation, or be passive with respect to
\(p\). Now also recall our partition with respect obligation from
 Section 1:





|  |  |  |
| --- | --- | --- |
| \(\OB p\) | \(\PE p \amp \PE {\neg p}\) | \(\IM {p}\) |



Fig. 17 




We might consider "merging" these two partitions, as it
were, and try to get a representation of the possible ways Jane Doe
may be positioned normatively with respect to her agency regarding
\(p\). Given certain choices of logic for \(\BA\) and for \(\OB\), we
obtain a set of mutually exclusive and exhaustive "normative
positions":




|  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- |
| \(\OB \BA p\) | \(\OB \BA \neg p\) | \(\OB \PV p\) | \(\PE \BA p\)

\(\PE \BA \neg p\)

\(\PE \PV p\) | \(\PE \BA p\)

\(\PE \BA \neg p\)

\(\OB \neg \PV p\) | \(\PE \BA p\)

\(\OB \neg \BA \neg p\)

\(\PE \PV p\) | \(\OB \neg \BA p\)

\(\PE \BA \neg p\)

\(\PE \PV p\) |


#### 5.2.4 Challenges to the Meinong-Chisholm Analysis



The Meinong-Chisholm analysis that lies at the core of the theory of
normative positions is, however, not uncontested. Following an earlier
criticism by Geach (1982), in the third chapter of the influential monograph,
Horty (2001), Horty has attacked one specific form of the
Meinong-Chisholm reduction from a semantic viewpoint. Recall that in
the standard semantics for SDL, for any formula p, \(\OB\)p is true
whenever p holds at all the deontically best alternatives. Horty shows
that, when combined with the STIT semantics for agency championed in
Belnap et al. 2001, this gives rise to counterintuitive
 cases.[70]
 Suppose, e.g., that an agent can choose between two things: to gamble
or not to gamble. Simplifying somewhat, say all possible states of
affairs are ones where either the agent gambles, or he doesn't.
If the agent gambles, there is a possibility that he wins $5, but
there is also a possibility that he loses the $5 he currently has. If
he doesn't gamble, then he keeps his money but cannot make any
profit. Now, granting that worlds where one has more money are better
than worlds where one has less, the best worlds are those where the
agent wins $5. Consequently, the best worlds are worlds in which the
agent does gamble. Hence, where g denotes that the agent gambles,



\[\OB \BA g\]


is true in this case. Yet intuitively, it is not true that the agent
ought to gamble. Absent further information, gambling and not gambling
both have their pros and cons in this case, and hence one would not
make any recommendation for
 either.[71]
 More generally, Horty takes this example to show that from "it
ought to be that the agent brings about \(p\)", it does not
follow that "the agent ought to bring about
 \(p\)".[72]



Horty's argument has been criticized from various angles: one
may, e.g., question whether the same applies to non-standard semantics
for \(\OB\) (Broersen & van der Torre 2003; McNamara 2004b), or
one may argue that very similar issues also arise in context where no
agency is involved (Danielsson 2002). Finally, it should be noted that
at the syntactic level, relatively little changes to the basic
properties of the (now agent-relative) operator \(\OB\): it will still
be a KD-type normal modal operator. However, its interaction with
other such modal operators, and the relation between obligations of
different agents and groups can now be studied in much more
fine-grained
 terms.[73]
 Moreover, Horty's incorporation of game-theoretic ideas into
the semantics of deontic logic opens up a whole range of new
possibilities for logical investigations, a suggestion from Bartha
2002 that was taken up in Tamminga 2013.


#### 5.2.5 Deontic Complements



Another issue that has been disputed in this context is whether or not
deontic operators necessarily call for agential complements or 
 not.[74]
 Consider the




*Strict Deontic Complement Thesis* (SDCT): Each fundamental
deontic status must be followed immediately by an operator ascribing
agency to an agent (here, by "\(\BA\)") to be
well-formed.




A *strict omission* (refraining) is now a formula of the form
\(\RF p\) (i.e., \(\BA \neg \BA p)\). "\(\neg \BA p\)" is
just a *non-action*. *Strict deontic omissions* are
deontic operators immediately followed by strict omissions. Note that
if we substitute "\(\BA p\)" for "\(p\)" in
the equivalences associated with the Traditional Definitional Scheme,
we obtain formulas that violate SDCT. For instance, the first schema
would yield

 

\[\IM \BA p \leftrightarrow \OB \neg \BA p\]


Essentially, non-action statements would have to be replaced by strict
omissions.



Belnap et al. 2001 provisionally defends SDCT. McNamara 2004a raises
doubts about this thesis. He notes that we are sometimes obligated to
be a certain way (e.g., to be in our office), and furthermore, it is
plausible to think that agential obligations reduce to these stative
obligations-- obligations to be the agents of states of affairs,
so that obligations to be a certain way are analytically prior to
agential obligations. A related issue is whether there can be cases
where a certain person \(i\) faces a given obligation, but these
obligations can (in principle or even in practice) be fulfilled by
some other person \(j\)--and hence, they are not, strictly
speaking, agential obligations for \(i\) (Krogh & Herrestad 1996;
McNamara 2004a).


### 5.3 Other Modalities



Aside from the preceding operators in this section, there are a number
of other types of modal operators one could sensibly add to a given
deontic logic. Far from being exhaustive, let us just give a few
examples and mention well-known issues that arise from such
enrichments.



In some normative systems (e.g., legal systems), explicit permissions,
prohibitions and obligations are given. So it would seem to be
possible for there to be normative systems with gaps: systems where
something is neither explicitly obligatory, impermissible, nor
permissible because, say, it is neither explicitly commanded,
prohibited or permitted by the relevant authority. For instance, it
may well have been undecided, legally speaking, whether storing
someone's search data on a website was permissible or not prior
to any new law being cast to regulate this practice. Yet \(\OB p
\vee(\PE p \amp \PE \neg p) \vee \IM p\) is a thesis
("exhaustion") of SDL (given the Traditional Definitional
Scheme), which makes any such gaps impossible. There is no place for
this fourth category in SDL, as it does not allow for such
*normative gaps* (von Wright
 1968).[75]
 SDL is not designed to represent a notion of explicitly stated
obligations, permissions, etc.



One way to accommodate gaps, without revising the Traditional
Definitional Scheme or the underlying logic of the classical
connectives, is by enriching SDL with a modal operator \(\NS\) that
refers explicitly to the normative system at hand: \(\NS\OB p\) then
means something like "according to the normative system, \(p\)
is obligatory". Normative gaps can be expressed by such
sentences as

 

\[\neg \NS\OB p \amp \neg \NS (\PE p \amp \PE \neg p) \amp \neg \NS\IM p.\]


Note that this is perfectly consistent with the claim

 

\[\NS (\OB p \vee(\PE p \amp \PE \neg p) \vee \IM p),\]


which just says that according to the normative system, the
proposition \(p\) is either obligatory, optional, or
 impermissible.[76]



Other types of modalities that display interesting interaction with
deontic notions are temporal modalities. For instance, it is plausible
to think that one is obligated *to do* something only if that
thing is in the future. This suggests that the deontic necessity
operator quantifies over states that form a subset of the set of all
future states; and hence a principle of the type



(FO)
 If in all future states, \(p\) is the case, then \(\OB p\)


should hold. Similar observations can be made in terms of the
Andersonian-Kangerian reduction of SDL. If we read the deontic
constant \(d\) atemporally as "all obligations past, present,
and future are met", then the only relevant worlds are those so
ideal that in them there has never been a single violation of a
mandatory norm. But as a parent, I may be obligated to lock the front
door at night even though this would not be a norm unless there had
been past violations of other norms (e.g., against theft and murder).
People also acquire obligations over time, create them for themselves
and for others, discharge them,
 etc.[77]



Finally, one may also combine deontic and epistemic or doxastic
logics. Such combinations may serve two distinct aims: to capture
distinctively epistemic obligations such as "Jane ought to know
that there is no cure to this illness (as she is a doctor)", or
to study subjective, knowledge-dependent obligations as in
"Given what you know about their production, you ought not to
buy these clothes". For a famous puzzle concerning the first
type of extension, see supplement F on
 the Paradox of Epistemic Obligation.



For the second type of extension, a recent puzzle that has received
much attention is the so-called Miners Paradox (Kolodny &
MacFarlane 2010; Parfit ms. [see
 Other Internet Resources]).
 A proposal for dealing with the latter in the context of deontic STIT
logic can be found in Horty 2019; earlier proposals for dealing with
knowledge-based obligations are Pacuit, Parikh and Cogan 2006 and
Broersen 2008. Clearly, however fundamental the monadic operators
represented in SDL, they are but a fragment of those employed in
normative discourse.


## 6. Revising the Foundations of Deontic Logic



When one meets a problem in formalizing certain aspects of reasoning
via a given formal system (e.g., SDL), one may diagnose the situation
in two different ways: as caused by a lack of expressive power in the
formalism, one that can be remedied by augmentation, or as a
fundamental problem with the initial formalism itself calling for
modification or replacement of the initial formalism. In the present
section, we survey some attempts to revise the foundations of deontic
logic, in view of specific puzzles related to SDL and its
extensions.


### 6.1. Jorgensen's Dilemma and Procedural Semantics for Deontic Logic



A view still held by some researchers within deontic logic and
meta-ethics, and particularly popular in the first few decades
following the emergence of positivism, was that evaluative sentences
are not the sort of sentences that can be either true or false. If
that view is correct, we seem to be facing a dilemma. On the one hand,
it seems there can be no logic of normative sentences, since logic is
the study of what follows from what, and one thing can follow from
another only if the things in question are of the sort that can be
either true or false. So there can be no deontic logic. On the other
hand, some normative sentences do seem to follow from others, and
there appear to be logical relations between distinct (possibly
logically complex) deontic statements, so there can be deontic logic
after all. What to do? This is Jorgensen's dilemma.



A widespread distinction that was put forward in an attempt to
overcome the dilemma is that between a *norm* and a
*normative
 proposition*.[78]
 Consider a normative sentence such as "You may park here for
one hour". This sentence may be used by an authority to provide
permission on the spot or it may be used by a passerby to report on an
already existing norm (e.g., a standing municipal regulation). The
activity of using a normative sentence as in the first case is
sometimes referred to as "norming"--it creates a norm
by granting permission by the very use of the sentence. The second use
is often said to be descriptive, since the sentence is then not used
to grant permission, but to report that permission to do so is a
standing state. It is often maintained that the two uses are mutually
exclusive, and only the latter use allows for truth or falsity. Some
have however challenged the exclusiveness of the division, by blending
semantics and speech-act theory (especially regarding performatives).
They thereby suggest that one who is in authority to grant a
permission can not only grant it by performing the speech act of
uttering the relevant sentence (as in the first example), but also
thereby makes what it said true-- that the person *is*
permitted to park (as in the second example with the
 passerby)).[79]



If one agrees that norm-propositions are distinct from norms, one may
insist that the realm of deontic logic is limited to
norm-propositions, perhaps interpreted with a standard semantics in
terms of truth at possible worlds. Where some normative system, \(n\),
is the basis in some state, \(w\), for a set of normative
propositions, one could just say that \(\OB p\) is true at a state,
\(w\), in virtue of the fact that according to the norms in \(n\) that
apply in \(w\), \(p\) is required. However, this still leaves a
Jorgensenian gap: it remains unclear what it means that a set of
norms requires a certain state of affairs (given the way the world
currently is). Isn't there still a need for a framework to model
systems of norms, their possible structural features, and how they
generate normative propositions? If a normative system \(n\) requires
\(p\) and \(n\) also requires \(q\), *how* does it do so, and
does it then also require \((p \amp q)\), and if so, *in virtue of
what* features of
 \(n\)?[80]



An alternative route is to give up on truth-functional (possible
world) semantics. One may, e.g., assume a primitive notion of
"norms" (sometimes also called "imperatives"),
which generate "obligations" in a purely syntactic
fashion. Originating in the work on conflict-tolerant deontic logics
(van Fraassen 1973, cf. Hansen 2008 and Hansen 2013), this route
became most influential under the banner of Input/Output logic
(Makinson & van der Torre 2000, 2001; Parent & van der Torre
2013). The "semantics" of I/O-logics is purely
operational: it specifies a set of obligations (propositional
formulas) that result from the application of factual detachment to
(a) a set of factual statements (closed under PC) and (b) a set of
conditional norms, i.e., pairs of propositional formulas, closed under
a given set of logical rules, R. Variations regarding R give rise to
various "input/output relations" (hence, I/O-logics). See
Parent and van der Torre 2018 for a short primer.



The I/O-logic framework has proven useful in a number of respects: it
puts factual detachment where, for many, it belongs: at the very core
of deontic reasoning. It allows for many fine-grained distinctions and
nuanced accounts of the interplay between detachment and logical
reasoning about facts and norms that seem hard to characterize in
terms of classical possible worlds-semantics. Finally, it can
accommodate various notions of permission and their relation to
obligation (Makinson & van der Torre 2003). On the downside, how
to integrate a procedural semantics from I/O-logic with semantics for
epistemic logics, logics of agency, and the like is far from obvious.
More generally, since in I/O-logic the notion of obligation
is--by definition--one of an "output", it is
unclear how, e.g., truth-functional connectives or other modal
operators can be applied to statements concerning obligation (e.g., it
is believed that murder is wrong and sometimes done), or what a nested
statement like "it is permissible that it is obligatory that
p" would mean in this
 framework.[81]


### 6.2 Deontic Logic and Actions



We often apply terms such as "must", "ought",
and "can" to actions rather than to states of affairs. For
instance, we would say that "Jimmy must tie his
shoelaces", rather than "it is obligatory (for Jimmy) that
Jimmy's shoelaces are tied". Likewise, we make claims such
as "Jack must get the groceries" and "you can turn
left here". In each of these three examples, it seems we signal
the deontic status of a certain action-- *an agent-initiated
way of changing* the current world, or bringing it into a
different state. However, the by now common reading of
SDL--deviating from von Wright's original interpretation
(recall
 supplement B, von Wright's 1951a System and SDL),
 has it that we can always apply the deontic modals to propositions
instead.



Various semantics have been developed in which deontic terms are
applied directly to actions, conceived as transitions between states
in a Kripke-model. For instance, following the reduction of SDL to
alethic modal logic (cf.
 Section 3),
 Meyer 1988 adds an Andersonian "violation" constant V to
 propositional dynamic logic
 (PDL). In this framework, [\(\alpha\)]V expresses that executing the
action \(\alpha\) necessarily leads to a violation, hence,
"\(\alpha\) is forbidden". Where \(-\alpha\) denotes the
negation of action \(\alpha\), [\(-\alpha\)]V then represents that
\(\alpha\) is
 obligatory.[82]
 Note that intuitively, this formula can be read as: "whenever
\(\alpha\) is not carried out, a violation occurs".
Alternatively, one may define a notion of *strong*
permissibility of an action \(\alpha\), by introducing a Kangerian
constant "OK" and expressing that \(\alpha\) is strongly
permissible by [\(\alpha\)]OK. This means, intuitively, that whenever
one executes the action \(\alpha\), then one is guaranteed to end up
in a world that is deontically acceptable (no violation
 occurs).[83]



Meyer's analysis reduces the deontic status of a given action
(at a given state w) to the occurrence of violations in its terminal
states (reachable from w through that action), giving rise to
so-called "goal norms". In contrast, one may also consider
"process norms" that concern all the intermediate states
that are "passed by" when carrying out a given action at a
state. Deontic logics of the latter type have been proposed in van der
Meyden 1996; see Broersen 2004, Kulicki and Trypuz 2017, Canavotto and
Giordani 2019 for discussions of this distinction.



A different approach to actions in deontic logic originates in
Segerberg 1982, and further developed *inter alia* in Kulicki
and Trypuz 2017. Here, a *Boolean algebra of actions* is
proposed--analogous to the Boolean algebra of PC--and one
defines functions that specify which (basic and complex) actions are
permissible, forbidden, or required. An expression like "\(P
\alpha\)" then just means that \(\alpha\) is a member of the set
of permitted actions, and such expressions are used to build more
complex ones using classical propositional connectives. Whereas in
dynamic deontic logic, the deontic behavior of complex actions is
governed by the underlying logic of actions (i.e., PDL), in deontic
logics over Boolean action algebras, specific frame conditions are
imposed to ensure that, e.g., "action \(\alpha\) is
forbidden" entails that "doing \(\alpha\) and \(\beta\)
(say, simultaneously) is
 forbidden".[84]


### 6.3 Rejecting Core Principles of SDL: OB-NEC and OB-RM



Rather than questioning its semantic foundations on their own terms,
one may also argue against SDL in view of the logical principles it
validates. Here we outline a number of distinct arguments and examples
that have been raised against specific principles of SDL, after which
we show how some of these can be overcome by moving to weaker
systems.



Let us start with \(\OB\)-NEC. This rule requires that any
tautological statement is obligatory. Now consider the following
statement:



(1)
Nothing is obligatory.


It seems that what (1) expresses, an absence of obligations, is
possible. For example, consider a time when no rational agents existed
in the universe. Why should we think that any obligations--even
those pertaining to tautological states of affairs--existed
 then?[85]



If we are reading \(\OB\) as simply "it ought to be the case
that", it is not clear that there is anything counterintuitive
about \(\OB \top\) (now read as, essentially, "it ought to be
that contradictions are false"), but there is also no longer any
obvious connection to what is obligatory or permissible for that
reading, or to what people ought to do. A different way of avoiding
the counterintuitive consequences of \(\OB\)-NEC is by distinguishing
between "vacuous" obligations such as \(\OB (p \vee \neg
p)\) and those that require the satisfaction of some logically
contingent statement such as \(\OB (p \amp \neg q)\). This can, e.g.,
be done by enriching one's language with a modal operator
\(\Box\) for "settled true" (cf. Belnap 2001 et al., Horty
2001).



Next on the list of problematic principles is \(\OB\)-RM, also known
as the principle of Inheritance. This principle states that whenever
something is obligatory, then everything that is a logical consequence
is also obligatory ("inherits" that status). There are
various well-known examples highlighting problems with \(\OB\)-RM. The
earliest and most well-known challenge to RM is *Ross's
Paradox* (Ross 1941). From



(1)
"It is obligatory that the letter is mailed"


formalized as \(\OB m\), we can derive:



(2)
"It is obligatory that the letter is mailed (m) or the
letter is burned (b)."


Formalized as \(\OB(m \vee b)\). It seems rather odd to say that an
obligation to mail the letter entails an obligation to mail the letter
or burn it (where burning the letter is presumably forbidden), one
that can be fulfilled by burning the letter, which invites from the
offender: "Well at least I did one thing right by burning
it".



While weakening from \(m\) to \(m \vee b\) gives problems, similar
issues arise when we weaken a conjunction to one of its conjuncts.
This is illustrated by the famous *Good Samaritan Paradox*
(Prior
 1958):[86]



(1)
It is obligatory that Jones help Smith who has been robbed.
(2)
It is obligatory that Smith has been robbed.


Supposing that Jones helps Smith who has been robbed if and only if
Jones helps Smith and Smith has been robbed, it would appear that a
correct way to symbolize (1) and (2) in SDLs is:

 

\[\tag{\(1'\)} \OB (h \amp r) \] \[\tag{\(2'\)} \OB r\]


By \(\OB\)-RM, \(\OB(h \amp r)\) entails \(\OB r\). However, it hardly
seems that if helping the robbed man is obligatory it follows that his
being robbed is likewise
 obligatory.[87]
 A much-discussed, epistemic variant of the Good Samaritan is
discussed in supplement F,
 the Paradox of Epistemic Obligation.



There have been various responses to these RM-related paradoxes. Some
defend RM and blame the Ross paradox on an elementary confusion
(Follesdal & Hilpinen 1971) or cite pragmatic features to
explain away the puzzle (Castaneda 1981): no one would, e.g.,
merely utter the statement that John ought to mail or burn the letter,
knowing that in fact John ought to just mail the letter. Similarly, it
has been argued that the Good Samaritan paradox is really a
conditional obligation paradox, and so RM is not the real source of
the paradox (Castaneda 1981; Tomberlin 1981). However, since
these paradoxes all at least *appear* to depend on \(\OB\)-RM,
a natural solution to the problems is to undercut the paradoxes by
rejecting \(\OB\)-RM itself. Two accessible and closely related
examples of approaches to deontic logic that reject \(\OB\)-RM from a
principled philosophical perspective are Jackson 1985 and Goble
 1990a.[88]
 A third, somewhat different principled strategy is proposed in
Cariani 2013 and studied in formal detail in Van De Putte 2019.
Finally, in S. O. Hansson 1990, and more elaborately in Hansson 2001,
S.O. Hansson develops systems of deontic logic where he analyzes
prohibitive and prescriptive deontic notions in terms of abstract
properties of various preference orderings (e.g., a normative status
is prohibitive whenever anything worse than something that has that
status also has it). He also sees \(\OB\)-RM as the main culprit in
the paradoxes of standard deontic logic, and thus he methodically
explores non-standard frameworks where \(\OB\)-RM is not
 sound.[89]



One very general semantic framework for logics that can invalidate
necessitation, inheritance, or other principles of normal modal logics
is the neighborhood semantic
 framework.[90]
 , A neighborhood semantics works with a set of possible worlds (just
like Kripke-semantics) but replaces the accessibility relation with a
neighborhood function \(\rN\) that maps every world \(w\) onto a set
of sets \(X, Y,\ldots\) of possible worlds \(\rN(w)\), often thought
of as a set of propositions. On such a semantics, \(\OB p\) is true at
a world \(w\) iff the set of all \(p\)-worlds is in the neighborhood
of \(w\), i.e.

 

\[ M, w \vDash \OB p \textrm{ iff } |p| \in \rN(w) \]


On the weakest logic characterized by such neighborhood models (the
minimal "classical modal logic", E), only \(\OB\)-RE is
valid. Additional validities and rules can be obtained by imposing
specific conditions on the neighborhood functions. Furthermore,
neighborhood semantics can be naturally generalized to deal with
conditional obligations (Chellas 1974, 1980), and combined with
semantic accounts of related concepts such as agency and ability
(McNamara 2019). It can also accommodate conflict-tolerant logics, as
we will see next.


### 6.4 Deontic Dilemmas and Conflict-Tolerant Deontic Logics: Rejecting NC



*A conflict of obligations* is a situation where there are two
or more obligations and it is not possible for all of them to be
fulfilled. A specific, paradigmatic case is the one where we have two
obligations that are logically incompatible, such as:



(1)
It is obligatory that I now meet Jones (say, as promised to Jones,
my friend).
(2)
It is obligatory that I now do not meet Jones (say, as promised to
Smith, another
 friend).[91]


Here it would seem that I have a conflict of obligations, in fact a
quite direct and explicit one, since what I promised one person would
happen, I promised another would not happen. People do (e.g., under
pressure or distraction) make conflicting promises (cf. conflicting
appointments), and it appears that they incur conflicting obligations
as a
 result.[92]
 But consider the natural representation of these in SDLs:



\[\tag{\(1'\)} \OB j\] \[\tag{\(2'\)} \OB \neg j\]


These premises are inconsistent in SDL. First, they are simply
excluded by NC \((\OB p \rightarrow \neg \OB \neg
 p)\).[93]
 But merely giving up NC is not sufficient. In fact, any normal modal
logic (hence, any logic that contains K and Necessitation, or
equivalently, \(\OB\)-RM, \(\OB\)-C, and RE based on PC) will generate
deontic explosion:



(DEX)
For every \(p, q: \OB p, \OB \neg p \vdash \OB q\).


In effect this means that once one has conflicting obligations,
everything is obligatory. Apart from this trivializing aspect, the
validity of \(\OB\)-RM and \(\OB\)-C also means that having two
incompatible obligations implausibly reduces to having a single
self-contradictory obligation, cf.
 supplement G, Collapse of conflicts into impossible obligations.



As for \(\OB\)-N and \(\OB\)-RM, this problem can be tackled in
various ways. Let us start with those who develop conflict-tolerant
deontic logics, by giving up one or more of the aforementioned
principles. This gives rise to three distinct classes of such
logics:


* *Non-aggregative deontic logics*, which invalidate
\(\OB\)-C (aka the Aggregation rule for OB). Here, the classical
example is the logic \(\bP\) (see, e.g., Goble 2003), which is
characterized by various semantics that weaken specific features of
SDL's possible worlds-semantics.
* *Inheritance-free deontic logics*, or logics that only
satisfy a weakened form of \(\OB\)-RM. Here, Goble's DPM-systems
(Goble 2005) are particularly well-studied.
* *Paraconsistent (and possibly relevant) deontic logics*,
which refute the *Ex Falso Quodlibet* principle that underlies
deontic explosion. These logics preserve the full "modal
part" of SDL, at the cost of giving up on classicality for
negation. Semantically, they require non-classical (or so-called
"impossible") worlds in which, e.g., both \(p\) and \(\neg
p\) can be true.



However, giving up on any of these three principles may be too
drastic. Suppose, e.g., that, apart from the obligations (1) and (2)
mentioned above, I also face the obligation to visit Tom (\(t\)), and
in case I visit Tom, to hand him over a letter from Jane (\(l\)). So
we have as premises \(\OB j\), \(\OB \neg j\), \(\OB t\), \(\OB (t
\rightarrow l)\). Suppose moreover that \(t\) and \(l\) are perfectly
compatible with either of \(j\) or \(\neg j\). Then any of the three
above-mentioned strategies would still invalidate the inference to
\(\OB l\): we can no longer infer that I ought to hand over the
letter.



More recently, it has been argued that by using a non-monotonic
framework such as adaptive logics, one can accommodate deontic
conflicts and yet retain most of the inferential power of SDL. The
rough idea behind this account is that one only rejects certain
principles of SDL when they concern obligations that are in fact
conflicted (in view of the premises). See Goble 2013 and Van de Putte,
Beirlaen, and Meheus forthcoming for an overview of such adaptive
deontic logics.



Another shortcoming of most existing conflict-tolerant deontic logics
is that they only focus on the above-mentioned paradigmatic case of
binary conflicts between logically contradictory obligations. In
contrast, conflicts often arise between obligations that are just
incompatible in view of the circumstances one finds oneself in, and
hence have a contingent
 basis.[94]
 In principle, SDL would be perfectly fine with such
"contingent" conflicts, and we could express them in an
enriched language by formulas of the type

 

\[\tag{DC} \OB p \amp \OB q \amp \neg \Diamond(p \amp q)\]


However, (DC) in combination with (\(\OB\)-C) yields triviality as
soon as we endorse a minimalistic version of Kant's law that
*ought implies can*:

 

\[\tag{OiC} \OB p \rightarrow \Diamond p\]


Similar problems arise if we try to formalize contingent deontic
conflicts following the Andersonian-Kangerian reduction, as shown in
 supplement H, A puzzle surrounding Kant's law.



Let us note that a long-ignored and challenging further puzzle for
conflicting obligations, sometimes called "van Fraassen's
Puzzle" and inspired by van Fraassen 1973 has deservedly
received increased attention: Horty 1994, 2003; van der Torre and Tan
2000; McNamara 2004a; Hansen 2004; and Goble 2005. For a brief general
overview, see Section 8.4 of Hilpinen and McNamara 2013.



An altogether different approach towards conflicting obligations
consists in distinguishing between *prima facie* (or *pro
tanto*) obligations, and all-things considered (*pro toto*)
obligations, and insisting that SDL and its siblings are logics of the
latter notion. This distinction has been famous at least since (Ross
1939). It can be illustrated by *Plato's Dilemma* (Lemmon
 1962a):[95]


1. I'm obligated to meet you now for a light lunch meeting at
the restaurant.
2. I'm obligated to rush my choking child to the hospital.



Here we seem to have an indirect non-explicit conflict of obligations,
if we assume that satisfying both obligations is practically
impossible. Yet, unlike in our prior example, where the two promises
might naturally have been on a par, we would all agree that the
obligation to help my child *overrides* my obligation to meet
you for lunch. In other words, the first obligation is
*defeated* by the second obligation, which *takes
precedence*. Ordinarily, we would also assume that no other
obligation overrides my obligation to rush my child to the hospital,
so that this obligation is an *all things considered non-overridden
obligation*, but not so for the obligation to meet you for lunch.
Furthermore, we are also prone to say that the situation is one where
the general obligation we have to keep our appointments (or to keep
our promises, still more generally) has an
*exception*--the circumstances are extenuating.



If we define an "obligations dilemma" as a conflict of
obligations where neither of the conflicting obligations is overridden
(cf. Sinnott-Armstrong 1988), then the case above involves a conflict
of obligations but not an obligations dilemma. Once we acknowledge
conflicts of obligation, there is the further issue of representing
the logic of reasoning about conflicting obligations where some do or
don't override others, some are or are not defeated, some are,
some are not all things considered non-overridden obligations, where
obligation dilemmas do or don't exist, where some obligations
hold generally, but not unexceptionally, some are absolute, etc. So a
central issue here is that of conflicting obligations *of different
weight* and the *defeasibility* of obligations. Clearly,
there is no mechanism in SDL for this, since SDL does not allow for
conflicts to begin with, yet this is an issue that goes well beyond
that of merely having a logic that allows for conflicts. There have
been a variety of approaches to this dilemma, and to defeasibility
among conflicting
 obligations.[96]


### 6.5 Hyperintensionality of Deontic Modals and Truthmaker Semantics



Recently, the puzzles related to \(\OB\)-NEC, Inheritance, and Deontic
Explosion have been taken to reflect a more general (alleged) feature
of deontic modals, i.e., their *hyperintensionality*. In the
syntactic sense, a modal operator is hyperintensional if it is not
closed under the replacement of classical, truth-functionally
equivalent formulas. On this view, \(\OB p\) is, e.g., not equivalent
to \(\OB\) \(((p \amp q) \vee(p \amp \neg q))\), even if the truth of
\(p\) always coincides with the truth of \((p \amp q) \vee(p \amp \neg
q)\). Semantically, this means that such an operator cannot be
interpreted as an intensional modal operator, using (classical)
possible-worlds semantics.



A promising recent semantic framework for hyperintensional logics is
that of truthmaker semantics (Fine 2017). On such a semantics,
propositions are interpreted by "truthmakers", resp.
"falsemakers" rather than sets of worlds, where these
entities are related to one another in terms of a mereological
(parthood) relation. For instance, a truthmaker for "it
rains" would be simply a state of it's raining, nothing
less and nothing more. A truthmaker for "(it rains and the sun
shines) or (it rains and the sun does not shine)" can be either
of two things: a fusion of a state of it's raining with a state
of the sun's shining, or a fusion of a state of it's
raining with a state of the sun's not shining (and hence, the
latter, a falsemaker of "the sun shines"). Formally, the
set of truthmakers of \(p\) may, e.g., be \(\{X, X'\}\), while the set
of truthmakers of \((p \amp q) \vee(p \amp \neg q)\) may be \(\{X\*Y,\)
\(X\*Z,\) \(X'\*Z,\) \(X'\*Z\}\), with the "\*" symbol
denoting a fusion
 operation.[97]
 Fine applied this framework to the logic of imperatives and to
deontic logic proper in a series of papers (Fine 2018a,b) where
"actions" play the role usually played by truthmakers in
the semantics of indicative sentences, and analogous to the
truthmaking and falsemaking relations between truthmakers and
propositions, such actions can be either in compliance with or
contravening a given command. For instance, the imperative "Sing
or dance!" is interpreted in terms of (a) the actions that are
in (exact) compliance with it, and (b) the actions that are in (exact)
contravention of it. For the case of conditional obligations,
additional complications arise, as one may also *avoid* a
conditional by making its antecedent
 false.[98]
 Similar ideas, though worked out in different ways, can be found in
Anglberger and Korbmacher 2020, which also relate truthmaker semantics
to Goble's logic BDL from Goble 2013. For a full-fledged
discussion of the hyperintensionality of normativity and formal
approaches to it, see Faroldi 2019.


## 7. Conclusion



Plainly, there are a number of outstanding problems for deontic logic.
Some see this as a serious defect; others see it merely as a serious
challenge, even an attractive one. There is some antecedent reason to
expect that the challenges will be great in this area. Normativity is
challenging generally, not just in deontic logic. Normative notions
appear to have strong pragmatic features. Normative notions must
combine with notions for agency and with temporal notions to be of
maximal interest--which introduces considerable logical
complexity. There is also reason to think that there are hidden
complexities in the interaction of normative notions and conditionals.
Finally, there appears to be a wide array of normative notions with
interesting interactions, some easily conflated with others (by
ethicists as much as deontic logicians). Clearly, there is a lot of
work to be done.