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Formal analysis of responsibility attribution in a
multimodal framework
Daniela Glavaniˇcov´a1and Matteo Pascucci2,3
1Department of Logic and Methodology of Sciences, Comenius University in
Bratislava
2Institute of Logic and Computation, TU Wien
3Institute of Philosophy, Slovak Academy of Sciences
daniela.glavanicova@gmail.com, matteo.pascucci@tuwien.ac.at
Abstract. The present article is devoted to a logical treatment of some
fundamental concepts involved in responsibility attribution. We specify a
theoretical framework based on a language of temporal deontic logic with
agent-relative operators for deliberate causal contribution. The frame-
work is endowed with a procedure to solve normative conflicts which
arise from the assessment of different normative sources. We provide a
characterization result for a basic system within this framework and il-
lustrate how the concepts formalized can be put at work in the analysis
of examples of legal reasoning.
Keywords. Multi-agent Deontic Logic – Legal Reasoning – Norm Inter-
pretation – Responsibility
1 Introduction
The philosophical literature on responsibility is so rich that is almost intractable,
while considerably less attention has been devoted to this notion in the logi-
cal literature until recent decades; nowadays formal accounts of responsibility
are being developed in various frameworks, such as action logic [8, 12], STIT
logic [15,16], game theory [4], lambda calculus [17] or precedence logic [2]. Each
account focuses on certain aspects of responsibility that are relevant in the un-
derlying framework, without aiming at an exhaustive picture, since responsibility
attribution involves an impressive variety of levels of analysis.
To give an idea of this variety we start by pointing out, as in [12], that
an individual (or a group) may be held responsible either for an action or for
some consequences of an action. Responsibility for an action does not entail
responsibility for its consequences, given that an individual cannot foresee all
consequences of what he/she does. However, an individual may deliberately act
in order to obtain a given outcome. In the latter case responsibility can be
attributed with respect to both the action performed and the state-of-affairs
achieved. If we restrict our attention to responsibility for consequences of actions,
then, as observed in [8], an individual may be taken to be responsible either
2 D. Glavaniˇcov´a and M. Pascucci
for some state-of-affairs that should obtain in the future or for some state-of-
affairs that occurred in the past. Future-oriented responsibility can be sometimes
thought of as allocation of duties, which is especially important in scenarios
involving many agents that need to coordinate their behaviour in order to achieve
a common goal. Past-oriented responsibility can be sometimes thought of as
culpability for something that has happened.
Focusing on past-oriented responsibility, one can further distinguish, along
the lines of [15], between causal and agentive responsibility —where the for-
mer encompasses also cases of accidental contribution to the attainment of a
relevant state-of-affairs, while the latter makes explicit reference to voluntary
contribution— or between active and passive responsibility. Active responsibil-
ity means that an agent did something to produce a certain outcome, while
passive responsibility means that an agent could have prevented something from
being the case but did not.
In the present article, we take a novel perspective on the analysis of respon-
sibility and add another piece to the puzzle by looking at the role played by
normative sources and their interpretation. We introduce a very simple logical
framework where it is possible to make explicit reference to normative sources
from which obligations, permissions and prohibitions arise and whose content
may vary with time and be interpreted in different ways. We will see that this
framework allows one to capture many aspects of the debate around responsibil-
ity that are directly relevant in the legal domain and that have not been formally
addressed so far. For instance, it allows for a treatment of cases of responsibility
alleviation related to normative conflicts (e.g., when a more important normative
source permits or prescribes something forbidden by a subordinated normative
source), as well as of cases of retrospective attribution of responsibility (i.e., when
some law now in effect is used to evaluate something that occurred in the past,
where the relevant laws were possibly different).
The structure of the paper is as follows: section 2 provides some theoretical
background for the notions of responsibility we will be dealing with; section 3
outlines the formal framework, which is based on a multimodal language; section
4 is devoted to the formal rendering of various concepts involved in responsibility
attribution in the legal domain; the applicability of the proposed framework will
be illustrated with examples; section 5 concludes the paper, pointing to some
interesting applications and directions for future research.
2 The theoretical framework
In this section we will illustrate some core aspects of the theoretical framework
for responsibility attribution we want to formalize: causal contribution, context
of evaluation and interpretation of norms. To this aim, we will employ useful
distinctions taken from the philosophical analysis of responsibility.
Causal contribution. The first distinction is that between responsibility for ac-
tions and responsibility for outcomes. An outcome of an action can be identified
Formal analysis of responsibility attribution in a multimodal framework 3
with a state-of-affairs. Our attention will be restricted to cases of responsibil-
ity for outcomes. An agent may causally determine an outcome or just causally
contribute to an outcome. As [3,4] acknowledge, a general definition of responsi-
bility cannot rely on an agent’s causal determination of the outcome, since this
would not apply to cases in which the realization of the outcome depends on the
behaviour of several agents. For instance, Alfred attempted to poison Carl, but
his attempt failed due to Barbara’s intervention: in this case we still want Alfred
to be responsible for “creating the possibility” of Carl’s being poisoned. There-
fore, the causal contribution of an agent is intended to represent a triggering
condition for a certain outcome, even if the outcome is in the end avoided due
to the behaviour of other agents. An analogous argument can be used in cases
of causal overdetermination (see [5] for a detailed discussion): both Alfred and
Barbara attempted to poison Carl and in the end it was Barbara who managed
to achieve the goal. Responsibility is not restricted to Barbara: Alfred is indeed
responsible for an attempted crime.
Furthermore, sometimes responsibility for an outcome is attributed to a
group of agents even if only some of the members of the group causally con-
tributed to the outcome (see the notion of collective responsibility in [7]); there-
fore, the role played by causal contribution is different in the case of individual
and group responsibility. For instance, before the introduction of video surveil-
lance systems in stadiums it was sometimes the case that a football team was
punished with some sanction due to the behaviour of a restricted number of ul-
tras. In similar scenarios it was common to say that the supporters of the team
(as a group) were responsible for the sanction, since it was not always possible
to identify the specific individuals who misbehaved.
In judicial reasoning it is important to assess whether the causal contribution
of an agent to an outcome was deliberate or not. Our analysis will be focused
on deliberate causal contribution which will be represented in terms of hyperin-
tensional operators for causal contribution: an agent can deliberately contribute
to realizing the proposition expressed by a formula φwithout being aware that
he/she is contributing to realizing the proposition expressed by a formula ψlog-
ically equivalent to φ.
Context of evaluation. Responsibility attribution will be here treated as a relative
issue, depending on a certain context of evaluation. First, a person might be held
responsible with reference to a certain normative source (set of norms) and not
responsible with reference to another. The ultimate decision on whether a person
or a group of people is to be blamed for something often depends on a hierarchy
of normative sources [1]. Different norms can disagree, giving rise to normative
conflicts; in this case, a norm can be derogated due to its incompatibility with
a more important one. To capture this aspect, which is fundamental both for
responsibility attribution and for responsibility alleviation, we will employ a
mechanism for conflict resolution which produces all-things-considered norms
relative to a specified ordering of normative sources.
4 D. Glavaniˇcov´a and M. Pascucci
Second, normative sources can change with time and thus norms valid at
the time of an agent’s conduct need not be valid at the time of a responsibility
ascription (and the other way around). For instance, a law currently in effect
may be used to evaluate the conduct of an individual if it is more favourable to
him/her than the law in effect when the relevant conduct occurred. Changes in
normative systems often occur due to the growth of our knowledge. For instance,
if it is not known that a certain compound is toxic, that compound will not
appear on the list of chemicals to be avoided, and the relevant regulation on
compounds will not prohibit its usage in commercial products. As soon as a
certain part of our society acknowledges that the compound is toxic, it is added
to the list of toxic chemicals and its usage becomes prohibited. Thus, the relevant
regulation changes.
In our framework the interaction between normative sources and time will
be central to define three types of responsibility: prospective,historic and ret-
rospective. Prospective responsibility is responsibility for something that should
obtain either now or in the future, according to some norm currently in effect.
Historic responsibility is responsibility for something that obtained in the past
but was at that time prohibited by some norm. Retrospective responsibility is
responsibility for something that obtained in the past but is prohibited by some
norm currently in effect. We will illustrate how prospective responsibility can be
used to define both historic and retrospective responsibility.
Interpretation of norms. Another important aspect of norms is their interpre-
tation. Since norms are written in a natural language, they often have an am-
biguous meaning and different readings can give rise to controversies in courts.
The interpretation of norms is especially challenging when a new regulation is
released or when a regulation written in one language has to be translated into
another language. In legal reasoning it is therefore convenient to keep track of the
different interpretations of a norm in order to see which are their consequences.
In our account we will focus on the role played by propositional synonymy in
norm interpretation. For instance, consider the following sentences, which are
parts of the police caution used, respectively, in the UK and in the US [6]:
A. “You do not have to say anything unless you wish to do so, but what you
say may be given in evidence.”
B. “You have the right to remain silent. If you give up the right to remain silent,
anything you say can and will be used against you in a court of law.”
One may interpret B as a way of paraphrasing A (hence, as expressing a logically
equivalent proposition) or as a sentence with a more specific meaning (hence, as
expressing a proposition strictly entailing the one expressed by A), arguing that
the fact that what one says can and will be used against this person in a court
entails that it may be given in evidence, but not vice versa.
The problem of establishing when two sentences express logically equivalent
propositions recently raised a certain interest also in the area of deontic logic.
Borrowing an example from [10], the proposition expressed by the sentence “You
Formal analysis of responsibility attribution in a multimodal framework 5
ought to drive” is logically equivalent to the proposition expressed by the sen-
tence “You ought to drive or to drive and drink” in many traditional systems of
deontic logic, such as SDL (Standard Deontic Logic). In our framework, we will
employ hyperintensional deontic operators to avoid problems of this kind.
3 The formal framework
Consider a countable set of propositional variables V ar ={p1, p2, p3, ...}and let
Agt ={a1, ..., an}and Src ={s1, ..., sm}be a finite set of agents and a finite
set of normative sources, respectively. A normative source represents a set of
norms. Here we restrict ourselves to the case in which sets of norms are pairwise
disjoint; that is, each pair of normative sources s, s0∈Src does not include any
shared norm.
Definition 1 The language Lis defined by the following grammar:
φ::= p|¬φ|φ→φ|φ∼φ|Hφ|Gφ|Caiφ|Osj
aiφ
The modal operators used in Lcan be interpreted as follows: Hmeans “in all
possible past states”, G“in all possible future states”, Cai“agent aideliberately
contributed to” and Osj
ai“according to normative source sj, it is obligatory for
agent aithat”. The operator ∼is used for the relation of propositional synonymy:
an expression like φ∼ψmeans that the formulas φand ψhave the same semantic
content (i.e., they denote synonymous propositions).4The boolean operators for
conjunction (∧), disjunction (∨) and material equivalence (≡), as well as the
temporal operators for “in some possible past state” (P) and “in some possible
future state” (F) can be defined in the usual way, in particular: P φ =def ¬H¬φ
and F φ =def ¬G¬φ. Furthermore, we provide also a straightforward definition
for two source-relative operators of obligation concerning a group of agents X⊆
Agt:
Osj
∀Xφ=def ^
ai∈X
Osj
aiφ Osj
∃Xφ=def _
ai∈X
Osj
aiφ
In other words, an expression like Osj
∀Xφmeans that all agents belonging to
Xhave a duty with respect to the realization of φ, while an expression like
Osj
∃Xφmeans that some agent belonging to Xhas a duty with respect to the
realization of φ. Notice that in both cases we can speak of a duty of the group
Xwith respect to the realization of φ. A duty of all agents can therefore be
expressed via Osj
∀Agtφ.
Definition 2 The system RN S (‘Responsibility and Normative Sources’) is
specified by the following axiomatic basis (for every ai∈Agt):
4We here adopt a Fregean notion of proposition [11]: a proposition is the thought
(or semantic content) expressed by a sentence, rather than a function from possible
states to truth-values. Therefore, we can say that two logically equivalent formulas
denote different propositions. In the models used for our logical system a particular
interpretation of the norms establishes whether two propositions are synonymous.
6 D. Glavaniˇcov´a and M. Pascucci
A0 All substitution instances of tautologies of the Propositional Calculus;
A1 φ∼φ;
A2 (φ∼ψ)→(ψ∼φ);
A3 (φ∼ψ)→((ψ∼χ)→(φ∼χ));
A4 (φ∧ψ)∼(ψ∧φ);
A5 H(φ→ψ)→(Hφ →Hψ);
A6 G(φ→ψ)→(Gφ →Gψ);
A7 φ→H F φ;
A8 φ→GP φ;
A9 (Hφ →HHφ);
A10 (Gφ →GGφ);
A11 (Hφ ∧Gφ ∧φ)→GHφ;
A12 Caiφ→(φ∨F φ);
A13 (φ∼ψ)→(H(φ∼ψ)∧G(φ∼ψ)∧HG(φ∼ψ));
A14 (φ∼ψ)→(φ≡ψ);
A15 (φ∼ψ)→(χ∼χ0), where χ0results from χby replacing some occurrence
of φwith ψ;
R1 if `RN S φand `RN S φ→ψ, then `RN S ψ;
R2 if `RN S φ, then `RN S Hφ ∧Gφ.
Axioms A1-A4 and A14-A15 concern properties of the relation of propositional
synonymy, axioms A5-A11 and rule R2 concern properties of temporal operators;
axioms A12 and A13 concern interactions among different operators. Even if the
axiomatic basis of RNS does not include any specific principle for operators
of obligation and only the principle A12 for operators of causal contribution, a
consequence of A14 and A15 is the following First Fundamental Theorem, that
will be discussed later:
F T 1 (φ∼ψ)→((Osj
aiφ≡Osj
aiψ)∧(Caiφ≡Caiψ)).
Note that if `φ≡ψbut φand ψare not in a relation of propositional synonymy,
then it may be the case that an agent deliberately contributed to φbut not to
ψ. In a similar vein, if φand ψare equivalent but not synonymous, it may be
the case that sjprescribes φbut not ψ. In this way our framework captures the
hyperintensional flavour of deliberate causal contribution and of deontic modals.
Definition 3 The language Lis interpreted on relational models of kind M=
hW, Cnt, ≺, f , c, o, V iwhere:
–Wis a set of states denoted by w,w0,w00, etc.;
–Cnt is a set of semantic contents (propositions) denoted by k,k0,k00, etc.;
–≺ ⊆ W×Wis a relation that can be called temporal precedence;
–f:L × W−→ Cnt is a function that can be called content assignment;
–c:Agt ×W−→ ℘(Cnt)is a function that can be called causal contribution
assignment;
–o:Agt ×Src ×W−→ ℘(Cnt)is a function that can be called obligation
assignment;
Formal analysis of responsibility attribution in a multimodal framework 7
–V:V ar −→ ℘(W)is a valuation function.
For every w∈W,p∈V ar,φ∈ L,ai∈Agt and sj∈Src,f(φ, w) is the semantic
content of (i.e., the proposition expressed by) formula φat state w,c(ai, w) is
the set of propositions towards whose realization agent aiprovides a deliberate
causal contribution at state w,o(ai, sj, w) is the set of propositions that are
obligatory for agent ai(i.e., the duties of ai) with reference to normative source
sjat state wand V(p) the set of states at which the propositional variable p
holds. Since in our framework a normative source can vary with time, there are
cases in which o(ai, sj, w)6=o(ai, sj, v) for w6=v; hence, an agent may have
different duties with respect to the same normative source at different states.
Furthermore, notice that the semantic content of a formula may vary with states
as well; however, we will see that it does not vary in an arbitrary way. Finally,
given two states w, v ∈Ws.t. w≺v, we will say that vis a successor of w.
Definition 4 Truth-conditions for formulas of Lwith reference to a state win
a model Mare specified below:
–M, w piff w∈V(p), for any p∈V ar;
–M, w ¬φiff M, w 2φ;
–M, w φ→ψiff either M, w 2φor M, w ψ;
–M, w φ∼ψiff f(φ, w) = f(ψ, w);
–M, w H φ iff M, v φfor all v∈Ws.t. v≺w;
–M, w Gφ iff M, v φfor all v∈Ws.t. w≺v;
–M, w Caiφiff f(φ, w)∈c(ai, w);
–M, w Osj
aiφiff f(φ, w)∈o(ai, sj, w).
The notions of validity of a formula in a model (denoted by Mφ) and in
a class of models are defined in the usual way.
Definition 5 We say that two states wand vin a model Mare related by a
temporal path (in symbols, w v)iff there is a sequence of states (w0, ..., wn)
s.t. w0=w,wn=vand, for every is.t. 0≤i≤n−1, either (I) wi≺wi+1 or
wi+1 ≺wi.
Definition 6 We denote by Cmthe class of all models Msatisfying the following
properties (for every w, w0, w00 , v ∈Wand ai∈Agt):
P1 (w≺w0and w0≺w00)implies w≺w00 ;
P2 (w0≺wand w00≺w)implies (w0≺w00 or w00 ≺w0or w00 =w0);
P3 f(φ∧ψ, w) = f(ψ∧φ, w);
P4 f(φ, w)∈c(ai, w)implies (M, w φor ∃u(w≺uand M, u φ));
P5 w v implies (f(φ, w) = f(ψ, w)implies f(φ, v ) = f(ψ, v));
P6 f(φ, w) = f(ψ, w)implies M, w φ≡ψ;
P7 f(φ, w) = f(ψ, w)implies f(χ, w) = f(χ0, w), whenever χ0is obtained from
χby replacing some occurrence of φwith ψ.
8 D. Glavaniˇcov´a and M. Pascucci
P1 and P2 describe two fundamental features of (possibly) indeterministic repre-
sentations of time: transitivity and no branching towards the past. P3 says that
the semantic content of a conjunction does not depend on the order of the con-
juncts. P4 says that if an agent aideliberately contributed to φ, then aicreated
the possibility for φ: either φholds now or it holds in some possible future state
(consider the examples involving Alfred, Barbara and Carl discussed in section
2). P5 says that the relation of propositional synonymy is invariant across states
related by a temporal path. P6 means that two formulas have the same semantic
content at a state only if they have the same truth-value. P7 represents the idea
that the semantic content is at least weakly compositional (see, e.g., [19]), in
the sense that the semantic content of a formula is determined by the semantic
content of its subformulas. For instance, if the sentences “Alan is drunk” and
“Alan is inebriated” are taken to have the same semantic content (i.e., to express
synonymous propositions), then “It ought to be that Alan is not drunk while
driving” and “It ought to be that Alan is not inebriated while driving” have the
same semantic content as well.
Let us have a closer look at the shape of models for Sand at some intuitions
they can represent. Due to the properties P1 and P2, a model Mis the union of
a family of disjoint trees T,T0,T00,... which possibly branch towards the future.5
Given a state win a tree T, a branch bstemming from wis a maximal chain of
successors of w. Every tree Tis a maximal set of states that are pairwise related
by a temporal path; hence, due to P5, the relation of propositional synonymy
is invariant across all states of T. A tree can be used to represent the temporal
evolution of an indeterministic scenario according to a certain interpretation
of the norms; such an interpretation is given by the relations of propositional
synonymy holding in the tree. Therefore, a model can be used to compare the
temporal evolution of a scenario according to different interpretations of the
norms (one for each tree).
We will now prove that the system RNS is characterized by the semantics
provided. Let [φ]RNS ={ψ:`RN S φ≡ψ}and EqRN S ={[φ]RN S :φ∈ L}.
Proposition 1 The class of models Cmis non-empty.
Proof. Immediate, by taking a model M=hW, Cnt, ≺, f , c, o, V is.t. W={w1},
≺=∅,Cnt =E qRN S and for all φ∈ L,ai∈Agt and sj∈Src,f(φ, w1) =
[φ]RN S ,c(ai, w1) = o(ai, sj, w1) = ∅. It can be easily verified that Msatisfies
properties P1-P7.
Proposition 2 RNS is sound with respect to the class of models Cm.
Proof. The validity of A0, A5-A11 and R1-R2 is straightforward, in the light of
well-known results in correspondence theory for tense logic [20]. The validity of
A1-A3 easily follows from the truth-conditions of φ∼ψand the validity of A4
5In these models the past of a state is deterministic, given that there is no branching
towards the past; hence, we can simplify the reading of Hand Pas “in all past
states” and “in some past state”, respectively.
Formal analysis of responsibility attribution in a multimodal framework 9
from P3. Concerning A12, suppose that M, w Caiφ; then f(φ, w)∈c(ai, w)
and, by P4, one can infer that either (I) M, w φor (II) there is some u∈W
s.t. w≺uand M, u φ, so M, w Caiφ→(φ∨F φ). Concerning A13,
assume that M, w (φ∼ψ)∧ ¬(H(φ∼ψ)∧G(φ∼ψ)∧HG(φ∼ψ)); then
M, w ¬H(φ∼ψ)∨ ¬G(φ∼ψ)∨P F ¬(φ∼ψ). This means that there is some
v∈Ws.t. M, v ¬(φ∼ψ) and either (I) v≺wor (II) w≺vor (III) there is
some us.t. u≺wand u≺v. In all cases (by Definition 5) w v, so, by P5, we
must have that M, v (φ∼ψ): contradiction. Finally, the validity of A14 and
A15 follows immediately from P6 and P7.
Proposition 3 RNS is complete with respect to the class of models Cm.
Proof. For any formula φwhich is not provable in RN S, there is a maximally
RNS-consistent set of formulas Γincluding ¬φ. The canonical model for RNS
will be denoted by M+=hW+, Cnt+,≺+, f +, c+, o+, V +i.W+is the set of all
maximally RNS-consistent sets of formulas. Cnt+is a set of semantic contents
having the cardinality of L. The relation ≺+is such that for every w, v ∈W+,
w≺+viff {φ:Gφ ∈w} ⊆ viff {φ:Hφ ∈v} ⊆ w.6The function f+is s.t.
for every w∈W+and φ, ψ ∈ L, we have f+(φ, w) = f+(ψ, w) iff φ∼ψ∈w.
The relations c+and o+are s.t. for every w∈W+,ai∈Agt,sj∈Src, and
φ∈ L, we have f+(φ, w)∈c+(ai, w) iff Caiφ∈wand f+(φ, w)∈o+(ai, sj, w) iff
Osj
aiφ∈w. The valuation function V+is defined in the standard way: V+(p) =
|p|+={w∈W+:p∈w}. Relying on the definition of M+, it can be easily
proven, using an induction on the complexity of formulas, that for every w∈W+
and φ∈ L, we have M+, w φiff w∈ |φ|+.
We now show that M+∈Cm; from this it follows that if φis not provable
in RNS, then it is not valid in Cm. The part of the proof concerning P1 and
P2 follows from well-known results concerning completeness theory of tense logic
[20]. In the case of P3, since, by A4, (φ∧ψ)∼(ψ∧φ)∈wfor every φ, ψ ∈ L and
every w∈W+, then M+, w (φ∧ψ)∼(ψ∧φ), which entails f+(φ∧ψ, w) =
f+(ψ∧φ, w). In the case of P4, suppose that there is w∈W+,ai∈Agt and
ψ∈ L s.t. f+(ψ, w )∈c+(ai, w). Then, Caiψ∈wand, by A12, φ∨Fφ ∈w; if
neither M+, w φnor M+, u φfor some us.t. w≺+u, then M+, w 2φ∨F φ,
whence φ∨F φ /∈w: contradiction. In the case of P5, suppose that w v
and that, for some formulas φ, ψ ∈ L, we have f+(φ, w) = f+(ψ, w) while
f+(φ, v)6=f+(ψ, v ). Then, M+, w (φ∼ψ) while M+, v ¬(φ∼ψ); however,
by A13, M+, w H(φ∼ψ)∧G(φ∼ψ)∧H G(φ∼ψ), so, as a consequence
of the definition of the relations and ≺+, we also have (φ∼ψ)∈vand
M+, v (φ∼ψ): contradiction. In the case of P6, we have that f+(φ, w) =
f+(ψ, w) entails φ∼ψ∈w, so, by A14, φ≡ψ∈wand M+, w φ≡ψ. In
the case of P7, suppose that for some state wwe have f+(φ, w) = f+(ψ, w) but
f+(χ, w)6=f+(χ0, w) for some χ0obtained from χby replacing some occurrence
6Here we take for granted the fact that in canonical models for systems of tense
logic the derivability of A7 and A8 makes it possible to have only one accessibility
relation for temporal reference, rather than two (one for Hand one for G). See [20]
for details.
10 D. Glavaniˇcov´a and M. Pascucci
of φwith ψ. Then, M+, w φ∼ψand, by A15, M+, w χ∼χ0, which means
f+(χ, w) = f+(χ0, w): contradiction.
Notice that, as a consequence of the characterization result obtained and
of the principle FT1, in every model in the class Cmthe set of propositions
which are obligatory for an agent aiat a state wwith respect to a normative
source sj(i.e. o(ai, sj, w)) and the set of propositions towards whose realization
aideliberately contributed at w(i.e. c(ai, w)) are closed under propositional
synonymy.
We will now show that obligations and deliberate causal contributions also
preserve the commutative property of binary boolean operators.
Proposition 4 Let ♣ ∈ {∧,∨,≡}; then `RN S Osj
ai(φ♣ψ)→Osj
ai(ψ♣φ)and
`RNS Cai(φ♣ψ)→Cai(ψ♣φ)for any ai∈Agt.
Proof. Axioms A4 and A15 tell us that the result holds for ♣=∧; hence, we
need to show that `RNS (φ∨ψ)∼(ψ∨φ) and `RN S (φ≡ψ)∼(ψ≡φ) in order
to apply A15 also in the cases ♣=∨and ♣=≡. We can rely on the semantic
characterization of RNS with respect to Cm. Assume that f(φ∨ψ, w ) = kfor
some win a model Mbelonging to Cm. Since (φ∨ψ) =def ¬(¬φ∧ ¬ψ), then
f(¬(¬φ∧ ¬ψ), w) = k. By A4, we know that f(¬φ∧ ¬ψ, w) = f(¬ψ∧ ¬φ, w);
hence, by A15, f(¬(¬ψ∧¬φ), w) = f(ψ∨φ, w) = k. Assume that f(φ≡ψ , w) =
k0. Since φ≡ψ=def ¬(φ∧¬ψ)∧¬(ψ∧¬φ), then f(¬(φ∧¬ψ)∧¬(ψ∧¬φ), w ) = k0.
By A4, we know that f(¬(φ∧¬ψ)∧¬(ψ∧¬φ), w) = f(¬(ψ∧ ¬φ)∧ ¬(φ∧ ¬ψ), w );
hence, f(¬(ψ∧ ¬φ)∧ ¬(φ∧ ¬ψ), w) = f(ψ≡φ, w) = k0.
The system RNS is supported by a mechanism for conflict resolution. First,
we introduce a relation over the set Src so that ss0means that normative
source sis overridden by normative source s0; we take to be a strict partial
order, namely (for all s, s0, s00 ∈Src):7
–¬(ss);
–(ss0∧s0s00)→ss00 .
A normative source that is not overridden by any other can be called maximal.
Due to the properties of ,Src always includes at least one maximal normative
source. Second, we define an operator O∗
aifor agent-relative all-things-considered
obligation, as follows:
O∗
aiφ=def Osj
aiφfor some maximal sj∈Src.
Notice that the set of all-things-considered obligations for an agent is not re-
quired here to be consistent, differently from what usually is the case in the
literature (see, e.g., [13]). Indeed, in real-life scenarios there are sometimes con-
flicts among two or more normative sources that neither override each other
nor are overridden by other normative sources. Such conflicts cannot be solved,
unless one revises some of the normative sources involved or rearranges their
7For a more elaborated formulation of a hierarchy among normative sources, see [1].
Formal analysis of responsibility attribution in a multimodal framework 11
hierarchy. We omit the analogous truth-conditions for the two operators of all-
things-considered obligation making reference to groups of agents, that is O∗
∀X
and O∗
∃X.
We would like to point out that a normative source need not correspond to
the set of norms found in a specific legal text, it may also be a proper subset of
all norms in a text or a collection of norms taken from different texts, provided
that they have a common status. For instance, a maximal normative source
may be also thought of as a collection of peremptory norms. A peremptory
norm (jus cogens), such as the prohibition of torture in international law, is a
principle which, by definition, does not admit any derogation (namely, it cannot
be overridden by any other). Something prescribed by a peremptory norm can
be represented in this framework via an expression of kind O∗
∀Agtφ.
4 Formal analysis of responsibility attribution
The most basic notion used in the present section is that of prospective respon-
sibility. We will adopt the following very general definitions for this notion:
Prospective Responsibility (single agent)
PRaiφ=def O∗
aiφ, provided that φdoes not include any operator for
past reference (Hor P).
Source-specific Prospective Responsibility (single agent)
PRsj
aiφ=def Osj
aiφ, provided that φdoes not include any operator for
past reference (Hor P).
Prospective responsibility in this sense means that an agent has a certain obli-
gation (or prohibition, when φis a negative formula) towards the present or the
future —either an all-things-considered obligation, or an obligation with refer-
ence to a specific normative source. An attribution of prospective responsibility
may also concern a sequence of states to be achieved and duties of other agents.
For instance, the expression Osj
ai(p∧F(q∧F Osj
akr)) means that, according to
normative source sj, agent aiis responsible for the sequential achievement of p
and qand for successively ensuring the duty of agent aktowards the achievement
of r. Variations of the definitions of prospective responsibility involving groups
of agents are easily obtained by replacing Osj
aiand O∗
aiwith Osj
∀X,Osj
∃X, etc.
Other two core concepts are those of historic responsibility and retrospective
responsibility. In order to define these we will employ the notion of prospective
responsibility and the notion of deliberate causal contribution; furthermore, we
will employ a notion of historic avoidability of causal contribution, which re-
quires some preliminary remark. Our attention is here focused on responsibility
ascription for a state-of-affairs that obtained at some point in the past due to
an agent’s (or a group of agents’) causal contribution, while it could have never
obtained (neither in the past nor in the future). For instance, Mark and Emma
stole a car two days ago, but three days ago it was still possible for both Mark
and Emma to conduct their entire life without stealing any car. More generally,
12 D. Glavaniˇcov´a and M. Pascucci
Definition 7 The causal contribution on φof a group of agents X⊆Agt is
historically avoidable at a state wof a model Miff:
(I) there is a state w0≺ws.t., for all ai∈X, we have M, w0¬Caiφ;
(II) for all w00 ≺w0and for all ai∈X, we have M, w00 ¬Caiφ;
(III) there is a branch bstemming from w0s.t. for all v∈band for al l ai∈X,
we have M, v ¬Caiφ.
This notion of historic avoidability can be formally represented in Lby the
following expression, whose truth at a state wof a model in Cmguarantees
properties (I)-(III) of Definition 7:
Historic Avoidability (every agent in a group)
HA∀Xφ=def P F Vai∈X(¬Caiφ∧H¬Caiφ∧G¬Caiφ)
In the case of a single agent, the definition at issue boils down to:
Historic Avoidability (single agent)
HAaiφ=def P F (¬Caiφ∧H¬Caiφ∧G¬Caiφ)
Normative sources affect responsibility attribution across time. A group of
agents Xcan be held responsible for a certain state-of-affairs φthat is prohibited
either with reference to a normative source that was in effect at the time in which
some (or every) agent ai∈Xdeliberately contributed to φor with reference to
a normative source presently in effect, but intended to have also a retrospective
validity. In representing the two cases, we restrict our attention to maximal
normative sources (i.e., all-things-considered norms). In the first case, one has
the following formal definitions of historic responsibility:
Historic Responsibility (some agent in a group)
HR∃Xφ=def P(Wai∈XCaiφ∧HA∀Xφ∧PR∀X¬φ)
Historic Responsibility (every agent in a group)
HR∀Xφ=def P(Vai∈XCaiφ∧HA∀Xφ∧PR∀X¬φ)
In the second case, instead, one has the following definitions of retrospective
responsibility:
Retrospective Responsibility (some agent in a group)
RR∃Xφ=def P(Wai∈XCaiφ∧HA∀Xφ)∧PR∀X¬φ
Retrospective Responsibility (every agent in a group)
RR∀Xφ=def P(Vai∈XCaiφ∧HA∀Xφ)∧PR∀X¬φ
Historic and retrospective responsibility do not exclude each other: indeed, it
can be the case that the relevant normative sources remain unmodified across
time and thus both definitions can be applied to describe a certain scenario.
We provide also the simplified definition of historic responsibility in the case
of a single agent; retrospective responsibility for a single agent can be obtained
in an analogous way:
Formal analysis of responsibility attribution in a multimodal framework 13
Historic Responsibility (single agent)
HRaiφ=def P(Caiφ∧HAaiφ∧PRai¬φ)
Furthermore, we can introduce corresponding notions of responsibility with ref-
erence to a specific normative source, such as:
Source-specific Historic Responsibility (single agent)
HRsj
aiφ=def P(Caiφ∧HAaiφ∧PRsj
ai¬φ)
Simultaneous and posterior alleviation with respect to what is prescribed by a
normative source that is derogated (sj) can be defined as follows:
Simultaneous Alleviation (single agent)
SAsj
aiφ=def P(Caiφ∧HAaiφ∧PRsj
ai¬φ∧ ¬PRai¬φ)
Posterior Alleviation (single agent)
PAsj
aiφ=def HRsj
aiφ∧ ¬PRai¬φ
Notice that in the system RNS, we have the following Second Fundamental
Theorem, due to A14 and A15:
F T 2 (φ∼ψ)→(Nφ ≡N ψ), where Nis any notion of responsibility/alleviation
defined in the present section.
Therefore, responsibility/alleviation attribution is invariant under propositional
synonymy. Let us now show how the framework works in terms of some examples.
Example 1: the special militia. Alan and Bill, the only two members of
a special militia, simultaneously shot at a single victim, Colin, since they sus-
pected that he was a spy. However, the military code of the special militia has
always prohibited to kill spies. Neither of the two bullets which were fired by Alan
and Bill was sufficient for killing, but the two bullets together led Colin to lose
enough blood to die. Let kbe the proposition that Colin is killed and M={a, b}
be the special militia, where ais Alan and bis Bill; let wbe the state of evalua-
tion, in which Colin is already dead, and ube the state in which Alan and Bill
shot Colin (hence u≺w). We want to formally express the fact that the special
militia is responsible for the death of Colin. At uit is true that Cakand Cbk
and that their causal contribution is historically avoidable, since we can imagine
that nothing forced Alan and Bill to act in such a way and that Colin’s murder
could have never taken place. Hence, we have that Vx∈M(Cxk∧HAxk∧PRx¬k)
holds at uand that PVx∈M(Cxk∧HAxk∧PRx¬k) holds at w. Therefore, at w
the special militia is historically responsible for the death of Colin. In this case,
both Alan and Bill are individually responsible as well, even if none of the two
causally determined Colin’s death.
Example 2: the toxic compound. A toy company Tconsists of two fac-
tories, fand g(so T={f, g}), and gused to produce toys with a compound
that was recognized as toxic only few years ago, such as lead paint. Companies
14 D. Glavaniˇcov´a and M. Pascucci
who have produced toys with lead paint are required to withdraw their products
from the market, since they are responsible for the distribution. We want to
claim that Tis retrospectively responsible for the distribution of dangerous toys
by virtue of the new legislation and so that Thas to take action. Let wbe the
state at which we are evaluating things and lthe proposition that lead paint is
used in toys. We know that there is some state u≺ws.t. Cglholds at u; further-
more, in an indeterministic world the causal contribution of gon lis historically
avoidable at u, so HAglholds there as well. Finally, at uthe legislation on toxic
compounds (s) is such that the use of lead paint is not prohibited, though it is
prohibited at w, due to successive scientific discoveries. Therefore, ¬PRs
g¬lholds
at uand PRs
g¬lholds at w. In this scenario, at w, we can conclude that not only
the factory g, but the company Titself is retrospectively responsible for the use
of lead paint, and so has to take action. Indeed, the formula RR∃Tlholds at w.
Example 3: the food thief. In 2011, a homeless (a) attempted to steal a
small amount of food (t) in Italy, which counts as an offence according to the
legal source regulating offences of this kind (s1), such as small thefts, whether
completed or attempted. However, in the Italian legal system, there is a norm
(s2) of so-called state of necessity which allows for exceptions to generally valid
norms. As a matter of fact, awas judged innocent by the Supreme Court of Cas-
sation in virtue of s2: stealing a small amount of food when in extreme need does
not constitute a crime.8Let wbe the state at which the action of ais evaluated
by the Supreme Court of Cassation. We can say that there is some previous state
us.t. Catholds at u. The causal contribution of aon tis historically avoidable
at uunder the assumption of an indeterministic world and tis prohibited by
s1at w, so PRa¬tholds at w. Therefore, even if at wone can attribute to a
retrospective (as well as historic, since the relevant regulations have not changed
from uto w) responsibility on the theft on the basis of s1, the Supreme Court of
Cassation sentences that a’s responsibility is alleviated by the higher-normative
source s2(i.e., s1s2). Hence, from the perspective of the Supreme Court,
we have a case of simultaneous alleviation due to the interaction between two
normative sources. This fact is represented by the truth of SAatat w.
Example 4: the two ships. This example is also known as “Raffles v. Wichel-
haus” and is taken from [18]. In 1864 a buyer purchased bales of cotton that
were to be sent from Bombay to Liverpool on a ship named the “Peerless”.
When the contract regulating the transaction was made there were two ships
called the Peerless (though, the two parties were not aware of this fact): one
of them was supposed to leave India in October, the other in December. While
the buyer expected the goods to be on the October ship, the seller placed them
on the December ship. The two parties interpreted the contract in two different
ways. Technically speaking, while the buyer took the statements “the bales of
cotton are placed on the Peerless” (p) and “the bales of cotton are placed on the
ship which leaves India in October” (q) as bearing a relation of propositional
8This case is discussed here: https://www.bbc.com/news/world-europe-36190557
Formal analysis of responsibility attribution in a multimodal framework 15
synonymy, the seller did not acknowledge such a relation. In order to model the
controversy here, we need to take two states of evaluation, wseller and wbuyer,
which represent the alternative interpretations of the contract (s) followed by
the two parties. In the sketched model the states wseller and wbuy er are not re-
lated by any temporal path (i.e., we do not have wseller wbuyer ); they rather
belong to two disjoint trees of the model, Tseller and Tbuyer, which are associated
with the two different interpretations of the contract. If we represent the legal
divergence in terms of historic responsibility, then we have that p∼qholds at
wbuyer and that, in the light of the principle FT2, RRs
seller¬qentails RRs
seller¬p
at such state. However, since p∼qdoes not hold at wseller , then RRs
seller¬q
does not entail RRs
seller¬pat such state.
5 Concluding remarks
We developed a theoretical framework for the analysis of responsibility based on
three main ingredients: causal contribution, context of evaluation (provided by
several normative sources which may vary with time) and norm interpretation.
We represented this framework within a simple system of temporal and multi-
agent deontic logic, called RNS, where it is possible to define many fine-grained
notions of responsibility attribution and alleviation. We supported RNS with
a mechanism for conflict resolution based on a hierarchy of normative sources
and showed that RNS can be characterized by a certain class of models. Fi-
nally, we illustrated how the formal definitions provided can be used to analyse
heterogeneous examples of legal reasoning.
As far as future directions of research are concerned, we have not discussed a
mechanism to handle defeasible norms that is provided, in nuce, by the present
framework and that requires further investigation. Consider the following norms:
(I) “aought to bring about φgiven ψ” and (II) “aought not to bring about φ
given ψand χ.” Since normative sources are just sets of norms, we can take a
normative source s1which includes exactly (I) and a normative source s2which
includes exactly (II). We can then formalize the two norms as ψ→Os1
aφand
(ψ∧χ)→Os2
a¬φ. Then, if s1and s2are the only relevant normative sources,
by taking s1s2, we get (ψ∧χ)→(O∗
a¬φ∧ ¬O∗
aφ).
Another direction is to examine a richer framework of temporal logic, such
as a STIT-based or a CTL-based one, which would allow us to provide more
refined definitions of responsibility. From a philosophical perspective it would be
relevant to examine what kind of indeterminism is needed for the very possibility
of responsibility, taking the notion of avoidability as the starting point. Other
directions would include examining various specifically legal concepts, which are
the stock-in-trade of lawyers, and considering applications of our framework for
responsibility attribution in the areas of multi-agent systems and autonomous
vehicles (see, e.g., [9] and [14]).
Acknowledgements. We are grateful to Olivier Roy, his group, and Timo
Lang. Daniela Glavaniˇcov´a was supported by the Slovak Research and Devel-
16 D. Glavaniˇcov´a and M. Pascucci
opment Agency under the contract no. APVV-17-0057 and by the grant no.
UK/414/2018. Matteo Pascucci was funded by the WWTF project MA16-028.
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