Reason-based semantics for deontic logic
Franz Dietrich and Christian List
October 2016
short preliminary draft
Abstract
Rational choice theorists and deontic logicians both study actions, yet they
use very different models. This paper starts from two rational-choice theoretic ways to model actions – one based on a ranking of options, another
on a reasons structure – and turns each of them into a formal semantics for
deontic logic. We compare ranking-based semantics and reason-based semantics with each other, and with standard possible-worlds semantics in deontic
logic.
1
Introduction
In Dietrich and List (2016b) we have presented a unified way to represent
moral theories, be they consequentialist or non-consequentialist, universalist
or relativist, monistic or pluralistic, and so on (see also Dietrich and List
2013a, 2013b, 2016a). We have represented a moral theory in terms of a reasons structure: a formal description of ‘what matters’ and ‘how it matters’.
Such a reason-based representation can be contrasted with a ranking-based
representation: a ranking of alternatives in terms of moral value. In the
present paper we provide logical foundations for both kinds of representation. We first define a formal language of deontic logic for expressing moral
propositions. For this language we then introduce ranking-based as well as
reason-based semantics (interpretations). Further, we contrast both semantics with standard possible-worlds semantics for deontic logic based on an
accessibility relation rather than a ranking of options or a reasons structure
1
(for an introduction, see Priest 2001 and Gabbay 2013). As it will turn
out, the ranking-based and reason-based logics each strengthen the standard
possible-worlds logic: they render more sentences tautological and more inferences valid. Ranking-based semantics is limited to moral theories that
are consequentialist and universalist, whereas reason-based semantics is far
more general and can accommodate non-consequentialism and relativism.
Standard possible-worlds semantics also enjoys this sort of flexibility, but it
achieves it at the cost of being silent on the structure of the underlying moral
theory: it represents only what is permissible at a world (the theory’s deontic
content), not why it is permissible (the theory’s justifications).
Although the primary concern of a moral theory is of course the notion
of permissibility (and the dual notion of obligation), this notion is intimately
linked to another modality: that of what is feasible or (agentially) possible.
Indeed, what ought to be done depends on what can be done. Our language
will therefore include both types of modality, following frequent practice in
deontic logic.
Later versions of the paper will be enriched with further discussion, references to the literature, and formal results.
2
Moral theories and their formal representation
This section will be concise, as it merely recapitulates the key concepts in
Dietrich and List (2016b).
2.1
The deontic content of a moral theory
We construe a moral theory as a theory about what is permissible and obligatory: what someone may or ought to do. Examples of such theories are classical utilitarianism (which requires generating maximal total happiness), Kantianism (which requires obeying the categorical imperative), various virtue
ethics, moral egoism, and so forth. The deontic content of a moral theory is
the body of all its action-guiding recommendations: it includes all ‘mays’ and
‘oughts’ of the theory, not however the underlying reasons or justifications
offered by the theory. The deontic content of a theory can be captured by a
rightness function: a function which to every choice context assigns the set
of currently permissible options.
2
Formally, we fix a set X representing the universal set of all options or
alternatives which an agent may ever encounter (where X is non-empty).
A context is a choice situation: a situation in which certain options in X
are feasible or possible, while all others are infeasible. Formally, a context
is simply an object K which determines a non-empty set of (‘feasible’ or
‘possible’) options denoted [K] ⊆ X and called the choice menu. Rational
choice theory often identifies contexts K with choice menus, so that K and
[K] are one and the same object. We neither forbid, no require such a thin
notion of context. For us, a context could be a choice menu K = [K] ⊆ X,
or a richer object such as a pair K = (Y, λ) of a choice menu Y = [K] ⊆ X
and an environmental parameter λ describing the environment in which the
choice takes place. The environmental parameter λ might capture the cultural
environment, or the history preceding the choice, or the agent’s current state
of information (or awareness), or even his identity, and so on. The rationale
behind us allowing contexts to carry environmental information over and
above the choice menu is that such information may be morally relevant; for
instance, certain forms of relativism are sensitive to the cultural environment,
and moral egoism is sensitive to the agent’s identity.1
Let K be a set of contexts: those contexts which are deemed possible, or
which are being considered by the moral theory. A rightness function now is
simply a function R which to each context K in K assigns a set R(K) ⊆ [K]
of currently possible options, representing the right or permissible options
in context K. For instance, the rightness function corresponding to classical
utilitarianism maps every context K to the set R(K) = {x ∈ [K] : x generates at least as much total happiness as any other feasible option y ∈ [K]}.
The notion of a rightness function is formally the same as the notion of a
choice function in rational choice theory, except that choice functions by definition never select an empty set of options and are normally defined for ‘thin’
contexts, i.e., choice menus.
2.2
Ranking-based representations
A ranking-based representation of a moral theory, or more precisely of its
deontic content (its rightness function R), is a ranking of the options in X in
1
In principle one could write all environmental information directly into the options,
thereby using ‘thick’ options together with ‘thin’ contexts (which are simply choice menus).
This may however go against the intuitive distinction between choice options and the choice
environment, up to the point of trivialising these notions (see Dietrich and List 2016b).
3
terms of moral value such that in any context the right options are precisely
the options ranked highest among all feasible options. Formally, a rankingbased representation is a binary relation % on the set of options X (with
x % y interpretable as meaning that option x is at least as good as option y)
such that in any context K ∈ K the set of right options is
R(K) = {x ∈ [K] : x % y for all y ∈ [K]}.
For instance, the ranking-based representation of classical utilitarianism ranks
options in terms of total happiness: x % y if and only if x generates at least
as much total happiness as y.
It might at first be tempting to think that all plausible moral theories can
be given a ranking-based representation, simply by ranking the options in X
in terms of the theory’s underlying notion of ‘betterness’. This conjecture is
far from true. Let us sketch the reason. Here and in much of the discussion
we assume that options have been individuated in such a way as to (explicitly
or implicitly) capture all consequences.2 There are at least three reasons for
why a moral theory may not admit any ranking-based representation. First,
the theory need not be founded on any notion of ‘goodness’ or ‘betterness’: it
need not be built on axiological foundations. Second, the theory’s notion of
‘betterness’, if it exists, need not apply to consequences, i.e., to options in X:
the theory need not be consequentialist. Third and more subtly, even if the
theory is consequentialist, it might deem an option x better than another
y in one context while deeming y better than x in another context: the
theory might be relativistic. This was quick; we shall come back to nonconsequentialism and relativism in the next section. For now we retain that
some but not all moral theories can be given a ranking-based representation.
2.3
Reason-based representations
One may represent moral theories not just by a ranking of the options, but
alternatively by a reasons structure. A reasons structure specifies (i) which
properties are normatively relevant in each context, and (ii) how these properties matter, i.e., how bundles of such properties are ranked relative to each
other. To make this precise, we fist need the concept of a property. Properties are features that an option x may or may not have in a context K where
2
So the consequence is (or is a function of) the option alone, not of the option together
with the choice context. In other words, information relevant to the consequence should
be written into the option, not into the context.
4
it is feasible. For instance, an option may have the property that someone
dies, or that someone is made happy, or that the option is the only one the
agent could have chosen in the context. More technically, properties are features of option-context pairs. An option-context pair is simply a pair (x, K)
where K is a context in K and x is a feasible option in [K].3 A property
is some object p which determines a set of option-context pairs denoted [p].
This set is called the extension of p and contains those option-context pairs
(x, K) which have or satisfy the property, or in other worlds for which x has
or satisfies the property in context K. Under a purely extensional notion
of properties, a property p simply is a set of option-context pairs: p = [p].
But we also allow an intentional notion of properties, where a property is
endowed with or come with a set of option-context pairs without being reduced to that set. The intensional notion of a property is more general: it
allows distinct properties to be satisfied by precisely the same option-context
pairs. The property that the number of happy people is being maximized is
extensionally equivalent to, but intensionally distinct from the property that
the number of unhappy people is being minimized.
A property p is called
• a (pure) option property if its satisfaction does not depend on the context, i.e., if two option-context pairs involving the same option but
possibly distinct contexts either both have the property (belong to [p])
or both do not have the property. Examples are the property that the
option does not involve lying or that it saves a life.
• a (pure) context property if its satisfaction does not depend on the
option, i.e., if two option-context pairs involving the same context but
possibly distinct options either both have the property (belong to [p])
or both do not have it. Examples are the property that only one option
is feasible, or that there is a feasible option which saves a life, or that
the choice happens in a traditional Indian environment.
• a relational property if its satisfaction depends on both the option and
the context, i.e., if the property is neither an option property nor a
context property. An example is the property that the option is the
only currently feasible one saving a life. Whether this property holds
3
We build feasibility of the option into the notion of an option-context pair, as a slight
departure from Dietrich and List (2016b).
5
indeed depends both on the option (does the option save a life?) and
the context (does the context offer other feasible options that save a
life?).
There is an abundance of properties one might imagine. For each set of
option-context pairs one might imagine a property whose extension is that
set. Most of these properties are artificial and cannot plausibly be normatively relevant. We thus introduce a set P of properties deemed to be
candidates for normative relevance. We write
• P(x, K) for the set of all properties of the option-context pair (x, K),
• P(x) for the set of all option properties satisfied by the option x (independently of the context),
• P(K) for the set of all context properties satisfied by the context K
(independently of the option).
A reasons structure is defined to be a pair R = (N, ≥) of two objects:
• a function N , the normative relevance function, which for any context
K ∈ K specifies a set N (K) ⊆ P of properties deemed normatively relevant in context K. We place a requirement on this function. Informally,
changes in what is normatively relevant must stem from changes in context properties (to avoid arbitrariness). Formally, whenever two contexts K, K 0 ∈ K have identical context properties they induce the same
normatively relevant properties: P(K) = P(K 0 ) =⇒ N (K) = N (K 0 ).
• a binary relation ≥ on the set of property combinations (subsets of P),
the weighing relation. We interpret S ≥ T to mean that the property
combination S weighs normatively at least as much as (‘outweighs’)
the property combination T .
Given such a reasons structure, each feasible option x in a context K can
be given a moral description. The moral description of x in context K is
defined as the set, denoted N (x, K), of those properties which (i) pertain to
x in context K and (ii) are normatively relevant in context K. Formally,
N (x, K) := P(x, K) ∩ N (K).
For instance, the reasons structure representing classical utilitarianism
takes the following form. In any context K ∈ K, N (K) contains all happiness properties, i.e., all properties ph where h ranges over a fixed set of
6
possible happiness levels (say, the interval [0, ∞)) and ph means that the option generates happiness level h. A feasible option x in a context K then has
the moral description N (x, K) = {phx } where hx is the particular happiness
level generated by x. The classical utilitarian reasons struction is special in
three ways: (i) only option properties are ever relevant, i.e., belong to N (K)
for some context K; (i) there are never any changes in what is relevant, i.e.,
N (K) is the same in all contexts K; (iii) an option has a single morally relevant property, i.e., N (x, K) is singleton for all option-context pairs (x, K).
We refer to these three conditions as consequentialism, universalism, and
monism, respectively. In general, we call a reasons structure (N, ≥) or the
moral theory it represents
• context-unrelated e or (formally) consequentialist if all N (K) (K ∈ K)
contain only option properties, and context-related or (formally) nonconsequentialist otherwise;
• context-invariant or (formally) universalist if all N (K) (K ∈ K) are
the same, and context-variant or (formally) relativist otherwise;
• (formally) monistic if N (x, K) contains a single property for all optioncontext pairs (x, K), and (formally) pluralistic otherwise.
These conditions can be combined at will; an example is consequentistism
together with relativism and pluralism. Most if not all 23 = 8 combinations
are prima facie plausible and have their counterparts in moral philosophy.
Several examples are given in Dietrich and List (2016b).
A reasons structure R = (N, ≥) generates permissibility verdicts. These
are captured by a rightness function which is denoted RR and defined as follows. In any given context K ∈ K, the set of right options is the set of those
feasible options whose normatively relevant properties outweigh those of any
other feasible option: RR (K) = {x ∈ [K] : N (x, K) ≥ N (y, K) for all y ∈
[K]}. We can now formally define when a reason struction R represents a
given rightness function R: it does so if RR = R. In fact, a reasons structure represents more than permissibility verdicts, i.e., more than a rightness
function: it also represents the justifications underpinning the permissibility
verdicts. In a sense, a reasons structure aims to represent a moral theory
rather than just its deontic content. This is the key difference to rankingbased representations. The formal question of which rightness functions R
7
admit a ranking-based representation, and which admit a reason-based represenation, has a clear-cut answer; for necessary and sufficient conditions see
Dietrich and List (2016b).
3
A formal language of deontic logic
The propositional language to be defined refines the standard deontic language in three ways. First, it uses two distinct types of atomic sentences:
choice sentences interpreted as saying that a certain option is chosen, and
basic descriptive sentences which could be interpreted as describing properties of the choice and/or the context. The connection to the previous section
is obvious: choice sentences correspond to options, and basic descriptive sentences to properties. By contrast, nothing in our language will correspond
to contexts. Contexts will have no syntactic counterpart, but instead enter
the picture at the semantic stage. Second, our language includes the modality of agential possibility (in addition to that of permissibility). Third, it
includes a strict conditional, to express that a conclusion holds in all worlds
in which a premises hold. Both modalities and the strict conditional are
indeed key resources that a deontic language should ideally dispose of. To
see why, consider for instance the notorious ‘ought implies can’ principle.
It can be expressed by the schema of sentences of the form ‘if it is obligatory that φ, then it is possible that φ’ (where φ is any sentence). Such a
sentence contains a deontic modality (obligation, the dual of permissibility),
the modality of agential possibility, and the strict if-then conditional. Using
a material rather than strict conditional would have been inadequate, since
‘ought implies can’ constitutes a law-like principle rather than a contingent
feature of the actual world. One may think of many other moral laws, such
as ‘If someone helps you, you ought to thank him’; they are usually understood as necessary truths, and can be rendered in our language using strict
conditionals.
Formally, let X and P be two (disjoint) sets of atomic sentences, called
choice sentences and basic descriptive sentences. Note that we use the same
symbols as for the set of options and the set of properties in the previous
section. This is explained by the following notational convention.
Notation: Choice sentences and options are denoted by the same symbols.
So each x in X stands either for the sentence that a certain option is chosen,
or for the option itself. To anticipate, once contexts will be introduced, we
8
will also convene that each p in P stands not only for a basic descriptive
sentence, but also for a corresponding property of option-context pairs.
Our language, denoted L, contains all sentences constructible from the
atomic sentences in X and P using the connectives ¬ (not), ∧ (and ), P (it
is permissible that), (it is (agentially) possible that), and ⇒ (if-then in the
sense of a strict conditional). Formally, L is the (smallest) set which contains
all sentences in X or P and is closed under construction in that whenever L
contains φ and ψ, then L contains ¬φ, (φ ∧ ψ), Pφ, φ, and (φ ⇒ ψ). So L
contains sentences such as x (e.g., ‘you kill Mr X’), p (e.g., ‘Mr X threatens
your life’), q (e.g., ‘there is an escape route’), (p ∧ ¬q) ⇒ Px (e.g., ‘whenever
Mr X threatens your life and there is no escape route, then you are permitted
to kill Mr X’), and so on.
Other truthfunctional connectives such as ∨ (or ) and → (material ifthen), ↔ (material if-and-only-if ) are definable in the usual way from ¬ and
∧: (φ ∨ ψ) stands for ¬(¬φ ∧ ¬ψ), (φ → ψ) stands for ¬(φ ∧ ¬ψ), and so
on. Moreover the modal connectives O (it is obligatory that) and (it is
necessary or unpreventable by the agent that) are defined as the duals of P
and : Oφ stands for ¬P¬φ (it is impermissible that not φ), and φ stands
for ¬ ¬φ (it is impossible that not φ).4
4
Semantics for the language
4.1
Preliminaries
Worlds as option-context pairs: The three types of semantics to be introduced in Sections 4 – the ranking-based, reason-based, and standard deontic
semantics – have something in common. In each case an interpretation is a
triple (W, v, ∗) in which W is a set of worlds, v is a truth function on P, and
∗ is some further object, i.e., either a ranking or a reasons structure or an
accessibility relation, depending on the type of semantics. The worlds in W
will not be primitive objects, but option-context pairs. To construct a set of
worlds W , one must first choose a set K of contexts, where by Section 2 a
context is an object K which determines a choice menu [K] ⊆ X. The set
4
One could also introduce a third sort of modality: conceptual (or logical) necessity
and possibility. Conceptual necessity of φ is represented by a strict implication τ ⇒ φ
where τ is a tautology, and conceptual possibility is the dual of conceptual possibility (it
is not conceptually necessary that ¬φ).
9
of contexts then gives rise to a set of option-context pairs (worlds) given by
W = {(x, K) : K ∈ K, x ∈ [K]}.
The simplest way to construct a W is to fix a set K of choice menus K ⊆
X, thereby identifying contexts with choice menus, so that W = {(x, K) :
K ∈ K, x ∈ K}. For instance, letting K = {{x, y}, {x, y, z}} we obtain six
worlds: W = {(x, {x, y}), (y, {x, y}), (x, {x, y, z}), (y, {x, y, z}), (z, {x, y, z})}.
Another way to construct W is to define contexts as pairs K = (Y, λ) of a
choice menu Y = [K] taken from a certain set Y of possible choice menus
and an environmental parameter λ taken from a certain set Γ of possible environmental parameters. Here the set of worlds becomes W = {(x, (Y, λ)) :
Y ∈ Y, λ ∈ Γ, x ∈ Y }.
Notation: Whenever we invoke a set of worlds W , its underlying set of
contexts will be denoted K. For any world w in our sense (an option-context
pair) we denoted its option by xw and its context by Kw , so that w =
(xw , Kw ).
Agential possibility: Our notion of worlds as option-context pairs gives
us a notion of which worlds are possible at a given world. To motivate the
definition, note that a world w0 should be regarded as possible at another
w if at w the agent could have brought about w0 . Now what the agent
can control is the chosen option, not however the context: the agent cannot
choose the context. Therefore, given a set of worlds W , we define a world
w0 ∈ W to be (agentially) possible at world w ∈ W if the context is the
same as in w: Kw0 = Kw . If w0 is possible at w, then a fortiori the choice
made at w0 is feasible at w: xw0 ∈ [Kw ]. It would have been inappropriate
to define w0 to be possible at w as soon as the choice is feasible, i.e., as soon
as xw0 ∈ [Kw ], since this would have allowed changes in context, which are
beyond the agent’s control.
Interdefinability of W and K: While we have constructed a set of worlds
W from a set of contexts K, the two sets are in fact interdefinable. That
is, if we start not from a set of contexts K, but from any set W consisting
of certain pairs (x, K) of an option x in X and some object K, then we
can retrieve the implicit set of contexts as being K = {K : (x, K) ∈ W for
some x ∈ X}, and we can retrieve the choice menu offered by a context K
as being [K] = {x ∈ X : (x, K) ∈ W }. The interdefinability of W and K
implies that we could define interpretations either as triples (W, v, ∗) or as
triples (K, v, ∗). We choose the former, to maximize familiarity to readers
used to possible-worlds semantics.
10
4.2
Ranking-based semantics
We are ready to define our ranking-based semantics. A ranking-based interpretation of the language L is a triple M = (W, v, %) in which
• W is the set {(x, K) : K ∈ K, x ∈ [K[} of all option-context pairs
(worlds) given by some set of contexts K,
• v is a (truth) function from W × P to {T, F }, mapping any pair (w, p)
of a world and a basic descriptive sentence to the truth value vw (p) of
p at the world w,
• % is a binary relation (ranking) on the set of options X, where x % y
is taken to mean that option x is at least as good as option y.
The truth of an arbitrary sentence φ in L at a world w of a ranking-based
interpretation M = (W, v, %) is denoted M, w φ and defined recursively
via the following instructions:
(ato1 ) M, w x (where x ∈ X) if and only if x is chosen at w, i.e., x = xw
,
(ato2 ) M, w p (where p ∈ P) if and only if vw (p) = T ,
(¬) M, w ¬φ if and only if M, w 6 φ, i.e., not M, w φ,
(∧) M, w (φ ∧ ψ) if and only if M, w φ and M, w ψ,
() M, w φ if and only if φ holds in some agentially possible world, i.e.,
M, w0 φ for some world w0 ∈ W with same context Kw0 = Kw ,
(P) M, w Pφ if and only if φ holds in some agentially possible world in
which a best feasible option is chosen, i.e., M, w0 φ for some world
w0 ∈ W with context Kw0 = Kw such that xw0 % x for all x in [Kw ]
(= [Kw0 ]),
(⇒) M, w (φ ⇒ ψ) if and only if ψ holds in all worlds where φ holds,
i.e., M, w0 ψ for all worlds w0 in W where M, w0 φ (equivalently,
M, w0 (φ → ψ) for all worlds w0 in W ).
The first two instructions settle the truth values of atomic sentences: choice
sentences in X and basic descriptive sentences in P. The other instructions
represent the truth conditions of each connective of the language.
11
4.3
Reason-based semantics
A reason-based interpretation of the language L is a triple M = (W, v, R) in
which the set of worlds W and and truth function v are precisely the same objects as for a ranking-based interpretation, and R is a reasons structure. This
reasons structure R = (N, ≥) is defined relative to the set of options X, the
set of contexts K underlying the set of worlds W , and the set of properties P.
This relies on reinterpreting P as a set of properties (of option-context pairs)
rather than sentences, a move already announced earlier. But how exactly are
these properties defined? First of all, since worlds are option-context pairs,
properties of option-context pairs are simply properties of worlds. Now to
any sentence corresponds the property (of worlds) that the sentence is true.
To the sentence ‘Mr X’s freedom is infringed’ corresponds the property that
Mr X’s freedom is infringed. Our convention formally states as follows:
Notation: Given a set of words W and a truth function v, we consider for
each basic descriptive sentence p in P an equally-denoted property which is
satisfied by those option-context pairs (worlds) w = (x, K) ∈ W such that
vw (p) = T .
Truth of a sentence φ in L at a world w of a reason-based interpretation
M = (W, v, R) is denoted M, w φ and defined recursively via the earlier
conditions (ato1 ), (ato2 ), (¬), (∧), (), (⇒) and the following new condition
replacing (Prank ):
(Preas ) M, w Pφ if and only if φ holds in some agentially possible world
in which the choice is right according to the reasons structure, i.e.,
M, w0 φ for some world w0 ∈ W with context Kw0 = Kw and choice
xw0 ∈ RR (Kw ).
Let us restate this new truth condition (Preas ) for permissibility in a more
elementary way that avoids using the rightness function RR . Consider a
world w = (x, K). According to the reasons structure R = (N, ≥), the
set of normatively salient properties of the option x in the context K is
N (x, K) := P(x, K) ∩ N (K), i.e., the set of properties with both pertain
to (x, K) and are normatively relevant in context K. We can write N (w)
for this set since w = (x, K). With this in mind, condition (Preas ) can be
restated as follows:
(Preas ) M, w Pφ if and only if φ holds in some agentially possible world in
which the normatively relevant properties of the choice outweigh those
12
of any other feasible option, i.e., M, w0 φ for some world w0 ∈ W
with context Kw0 = Kw such that N (w0 ) ≥ N (x, Kw ) for all x ∈ [Kw ].
4.4
Standard possible-worlds semantics
While ranking-based and reason-based semantics are inspired from rational
choice theory, we now turn to more traditional possible-worlds semantics.
We shall define a variant of classic possible-worlds semantics, adapted to our
richer-than-usual language L with its additional modality of agential possibility. Roughly, the adaptation consists in (i) once again construing worlds as
option-context pairs, and (ii) ensuring the permissibility modality (captured
by an accessibility relation) respects the possibility modality (captured by
our notion of worlds). Formally, we define a standard (possible-worlds) interpretation for L as a triple M = (W, v, ρ) in which the set of worlds W and
the truth function v are the same objects as in a ranking- or reason-based
interpretation, and ρ is a binary ‘accessibility’ relation on W (wρw0 reads
‘world w0 is permissible at world w’) that respects the possibility modality
in the sense that permissible worlds are possible: whenever wρw0 then w and
w0 include the same context Kw = Kw0 . A standard interpretation M is a
standard deontic interpretation if the accessibility relation ρ is extendable: at
every world w there is at least one world w0 that is permissible, i.e., satisfies
wρw0 .
Truth of a sentence φ in L at a world w of a standard interpretation
M = (W, v, ρ) is denoted M, w φ and defined recursively via the earlier
conditions (ato1 ), (ato2 ), (¬), (∧), (), (⇒) and a new condition replacing
(Prank ) or (Preas ):
(Pstan ) M, w P(p) if and only if p holds at some accessible world, i.e.,
M, w0 p for some world w0 ∈ W such that wρw0 .
5
Relationships between the different semantics
How do the various semantics for our language L compare to each other? It is
for instance easy to see that ranking-based and reason-based semantics both
strengthen standard possible-worlds semantics, essentially because rankings
and reasons structures each give rise to certain accessibility relations. Many
13
other links exist among three semantics as well as their ‘subsemantics’ obtained by imposing additional constraints on interpretations.
In later versions of the paper these relationships will be studied in more
detail. Here we only sketch some basic points. We write RAN K, REAS
and ST AN for the ranking-based, reason-based and standard possible-worlds
semantics defined above. Generally speaking, a semantics for L is given by
a class S of admissible interpretations for L, where we limit attention to
possible-wolds semantics such as for instance RAN K, REAS, ST AN . Given
such a semantics, a set of sentences Φ ⊆ L entails φ ∈ L – written Φ φ
– if whenever all sentence in Φ are true at a world of an interpretation in
S, then also φ is true there. If ∅ φ, so that φ is true in all worlds of all
interpretations in S, we call φ a tautology and write φ. Whenever there is
ambiguity about the semantics in question we index the entailment relation by the semantics: so RAN K is the ranking-based entailment relation, REAS
the reason-based entailment relation, and so on. It may be of interest to
refine a given semantics by restraining its class of interpretations S to a
subclass S 0 ⊆ S consisting of all interpretations in S which obey a certain
property. For instance we may restrict ranking-based semantics by imposing
one or more of the following conditions:
(tran) the binary relation (ranking) % is transitive,
(comp) the binary relation (ranking) % is complete,
(top) the binary relation (ranking) % always leads to at least one topranking feasible option, i.e., for all contexts K ∈ K there is at least
one feasible option x ∈ [K] such that x % y for all y ∈ [K].
Further, we may restrict reason-based semantics by imposing one or more of
the following conditions:
(cons) reasons are context-unrelated, i.e., the reasons structure R is formally consequentialist,
(univ) reasons are context-invariant, i.e., the reasons structure R is formally
universalist,
(moni) the reasons structure is formally monistic,
(tran) the weighing relation ≥ is transitive,
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(comp) the weighting relation ≥ is complete,
(top) the weighting relation ≥ always leads to at least one top-ranking feasible option, i.e., for all contexts K ∈ K there is at least one feasible
option x ∈ [K] such that N (x, K) ≥ N (y, K) for all y ∈ [K].
Finally, one may restrict standard possible-worlds semantics to standard deontic semantics:
(ext) the accessibility relation ρ is extendable.
To denote the restriction of a semantics by certain condition(s) on the interpretation, we append the condition(s) to the semantics, separated by a
hyphen. For instance, RAN K − trans is ranking-based semantics with transitivity of the ranking, and REAS − cons − univ is consequentialist universalist reason-based semantics.
One semantics (with entailment relation ) is said to strengthen another
semantics (with entailment relation 0 ) if whenever Φ 0 φ then Φ φ. Two
semantics are equivalent if their entailment relations coincide.
Here is a preliminary and incomplete list of relationships between semantics:
• RAN K and REAS strengthen ST AN .
• RAN K − top and REAS − top strengthen ST AN − ext.
• REAS − univ is equivalent to REAS.
• REAS − cons − univ is not equivalent to (but strictly strengthens)
REAS − cons.
• REAS − cons − univ strengthensu RAN K.
References
Dietrich, F., List, C. (2013a) A reason-based theory of rational choice, Noûs
47(1): 104-134
Dietrich, F., List, C. (2013b) Where do preferences come from? International
Journal of Game Theory 42(3): 613-637
Dietrich, F., List, C. (2016a) Reason-based choice and context-dependence:
an explanatory framework, Economics and Philosophy 32(2): 175-229
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Dietrich, F., List, C. (2016b) What matters and how it matters: a choicetheoretic representation of moral theories, working paper
Gabbay, D., Horty, J., Parent, X. et al. (eds.) (2013) Handbook of Deontic
Logic and Normative Systems, London: College Publications
Priest, G. (2001) An Introduction to Non-classical Logic, Cambridge Univ.
Press
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