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R. Trypuz
P. Kulicki
Jerzy Kalinowski’s logic of
normative sentences revisited
Abstract. The paper tackles two problems. The first one is to grasp the real meaning
of Jerzy Kalinowski’s theory of normative sentences. His formal system K1is a simple
logic formulated in a very limited language (negation is the only operator defined on ac-
tions). While presenting it Kalinowski formulated a few interesting philosophical remarks
on norms and actions. He did not, however, posses the tools to formalise them fully.
We propose a formulation of Kalinowski’s ideas with the use of a set-theoretical frame
similar to the one presented by Krister Segerberg in his A Deontic Logic of Action. At
the same time we enrich the language used by Kalinowski with more operators on actions
(parallel execution and free choice) and present an adequate axiomatisation of the result-
ing system. That allows us to disclose some unrevealed aspects of Kalinowski’s theory.
The most important one is a relation between acts which we call moral indiscernibility.
Our second problem is a proper understanding of moral indiscernibility. We show how a
repertoire of agent’s actions, defined with the use of simple observable elements of actions,
can be filtrated by the relation of moral indiscernibility. That allows us to understand the
consequences of Kalinowski’s claim that not doing something good is always bad.
Keywords: deontic action logic, moral indiscernibility of actions, Jerzy Kalinowski, Krister
Segerberg
Introduction
Jerzy Kalinowski (1916-2000) from 1934 to 1958 was affiliated with the
Catholic University of Lublin, initially as a student of law and finally as
a professor of philosophy. From 1958 he has taken several positions of a pro-
fessor of philosophy in France. His theory of norms was strongly influenced
by his moral views based on classical catholic ethics, his philosophy of law
and Lukasiewicz’s trivalent logic.
Kalinowski’s thought is not widely present in the contemporary literature
on the philosophy of law, moral philosophy and deontic logic but the main
reason is probably the fact that he published his work in French and Polish.
However, in [17] von Wright listed Kalinowski, along with O. Becker and
himself, as one of the three “founding fathers” of modern deontic logic. We
strongly believe that it is worth to study his work.
Presented by Name of Editor;Received 2014 DRAFT VERSION!
Studia Logica (2014) 0: 1–25 c
Springer 2014
2Robert Trypuz, Piotr Kulicki
This paper has two main objectives. The first one is to grasp the real
meaning of Jerzy Kalinowski’s theory of normative sentences, which is not
fully reflected in his own formal system K1. The system was one of the first
deontic logics published in the first issue of Studia Logica in 1953 [6]. It
is a simple logic formulated in a very limited language. While presenting it
Kalinowski formulated a few interesting philosophical remarks on norms and
actions. He did not, however, posses the tools to formalise them adequately.
Believing that Kalinowski’s approach towards norms and actions is worth
to be explicated and precisely understood, we have decided to build a set-
theoretical frame reflecting the main ideas of Kalinowski. We show that the
obtained frame is in fact identical to Krister Segerberg’s deontic action frame
(extended by two extra conditions) published in Studia Logica in 1982 [11].
We also compare K1with Segerberg’s deontic logic B.O.D.Both systems
share the idea of founding the meaning of normative notions (such as per-
mission, prohibition and obligation) on action theory1. They differ in their
expressive power – B.O.D.is more expressive than K1.
On the other hand, a class of adequate models for K1is a proper subset
of a class of models of B.O.D.That is because one of Kalinowski’s principles
saying that “a complement of a good action is bad”, seemingly innocent
and intuitive2, restricts the model’s universe to two elements. We show that
this outwardly paradoxical consequence has a reasonable, but still difficult
to accept, justification.
We also show the aspects in which logic K1is too weak by pointing out
the restrictions imposed on the models to which the system is insensitive.
Than we put forward SK logic, which is an extension of B.O.D..SK is sen-
sitive to all ontological restrictions in the reconstructed models. That allows
us to disclose some unrevealed aspects of Kalinowski’s theory. A relation
between acts, which we call moral indiscernibility, is the most important
one.
The second objective of the paper is to present the moral indiscernibility
relation in an intuitive and formal way. Indiscernibility can be found in
many formal theories. It is enough to mention the rough set theory (where
it is referred to objects indiscernible by attributes from a given attribute set)
or epistemic logic (where possible worlds are referred to as indiscernible, i.e.,
equally plausible, to an epistemic agent). Moral indiscernibility divides a set
1Segerberg’s approach is still present in many contemporary works in deontic logic as
shown in [13].
2Surprisingly for us, while presenting Kalinowski’s ideas to the public, we have discov-
ered that many people find the principle itself awkward.
Jerzy Kalinowski’s logic of normative sentences revisited 3
of actions into subsets having the same deontic/moral value. Intuitively, it
is founded on the scholastic synderesis rule, which directs one towards the
good and restrains him from the evil. On the basis of that rule it is clear that
from the deontic point of view the only important thing is whether what one
does is good or bad, no matter how many individual good or bad deeds can
be carried out. Formally, it appears that axioms added to B.O.D.turn its
action identity relation into moral indiscernibility. On the level of models
that is reflected by the limitation of the model’s universe to two elements.
The structure of the paper is as follows. In section 1 Kalinowski’s logic
of normative sentences is introduced. We start with presenting Kalinowski’s
philosophical view on the nature of norms and actions described in his works
(section 1.1) and express them in a set-theoretical model (section 1.2). In
section 1.3 we set out and explain when actions become morally indiscernible.
In section 1.4 we put forward Kalinowski K1logic and prove its completeness
with respect to earlier “reconstructed” models. In section 2 we first present
Krister Segerberg’s deontic action logic B.O.D.(section 2.1) and then extend
it to SK logic (section 2.2). In section 2.2.2 we prove a completeness result
for SK logic. Finally in section 2.3.1 we show how action space is filtrated
by the moral indiscernibility relation.
1. Kalinowski’s logic of normative sentences
1.1. Basic intuitions concerning norms and actions
In this section we shall put forward Kalinowski’s philosophical view on the
nature of norms and actions described in [6, 7]. According to Kalinowski,
norms like propositions are true or false3. In his approach norms of per-
mission, obligation and prohibition are specific relations between agents and
actions. In the case when a normative relation, e.g. obligation, holds be-
tween an agent awith an action α, Kalinowski says that it is true that a
ought to do α. Whether a deontic relation holds between an agent and an
action depends on the moral value of the action. In other words, according
to Kalinowski, the logical value of a norm depends on the moral values of
actions (which the norm refers to).
Kalinowski introduced two kinds of actions: actions in concreto and
actions in genere. He assumed that every action in concreto by nature is
either good (positive) or bad (negative). Thus, an agent carrying out a
particular action always does something that is either good or bad. Actions
3This is an already controversial standpoint being discussed in the logical community
at least since Jørgen Jørgensen published his famous article [5].
4Robert Trypuz, Piotr Kulicki
in genere are sets/collections of actions in concreto and as such may be
good, bad or neutral. Kalinowski explains that an action in genere is good
(bad) when it contains only good (bad) actions in concreto. Neutral actions
in genere are those which contain both good and bad actions in concreto.
Thus, the meaning of the terms “good” and “bad” will depend on whether
they are used in the context of the debate on actions in concreto or actions
in genere (the term “neutral” can be assigned only about actions in genere).
Kalinowski expressed his philosophical intuitions concerning the mean-
ing of deontic concepts of permission, obligation and prohibition in a table
similar to table 1 below (henceforward we shall omit a reference to an agent
without any loss, since in Kalinowski’s formalism only one agent appears at
once).
action α α is permitted αis obligatory αis forbidden
bad false false true
neutral true false false
good true true false
Table 1. The table depicts relations between deontic relations and moral values. One can
see that action (denoted by) αis obligatory if it is good, it is prohibited if it is bad and it
is permitted if it is good or neutral.
Many people find strange the rule saying that good actions are obligatory.
They reasonably argue (referring to their intuitions) that no one would be
able to comply with all obligations obtained by the rule since there are
(too) many good actions. But we should remember that it is known in
the deontic literature that some formulas of deontic logic when they are
taken out of their context and read in natural language appear paradoxical,
whereas understood in the spirit of a logical system are non-controversial.
Later in this article we try to explain most of the paradoxical principles of
Kalinowski’s system from the point of view of his theory.
1.2. Towards a model
Kalinowski in his works uses natural language to describe his views on de-
ontic logic. He formulated an axiomatisation for his logic of norms in [6],
but he did not provide a model for it (except for deontic tables like the one
in the previous section). In this section we shall reconstruct Kalinowski’s
philosophical view on actions and norms in a set-theoretical model step by
step. This will help us understand Kalinowski’s deontic logic better and to
Jerzy Kalinowski’s logic of normative sentences revisited 5
compare it with a similar approach of Krister Segerberg (see section 2).
Let us begin with an introduction of a nonempty set of actions in con-
creto:
Con ={e1, e2, e3, . . . }
Actions in genere are sets of actions in concreto. A power set 2Con
contains all the possible combinations of actions in concreto from Con. We
shall assume that
Gen = 2Con
There are two border cases in 2Con that require special attention, namely:
Con and ∅. Kalinowski did not take them into consideration in his works.
Thus we shall exclude them from some of our considerations as well.
Let us introduce the following two sets:
Good ⊆Con Bad ⊆Con (1)
Good is a set of all actions in concreto that are good and Bad is a set of
all actions in concreto that are bad. In the next section we shall impose
Kalinowski’s restrictions on the sets.
1.2.1. Constraints on the level of actions in concreto
Let us start with constraints on the level of actions in concreto. The following
two conditions come directly from [6, p. 151]. The first one is that no action
in concreto is at the same time good and bad. Thus, the sets Good and Bad
have to be disjoint, formally:
Good ∩Bad =∅(2)
The other condition is that actions in concreto are always either good or
bad, so we have to assume that Good and Bad cover the whole set Con4,
formally:
Good ∪Bad =Con (3)
At the same time there are good reasons to impose the following condition
on the model5:
Bad 6=Con (4)
4This restriction is known as closure principle (see [11]).
5Because of (1), condition (4) can be equivalently expressed as
Con 6⊆ B ad
6Robert Trypuz, Piotr Kulicki
If everything (i.e., every action in concreto) is bad, an agent is trapped – ev-
erything he/she does is wrong. Kalinowski seems to reject such a situation.
On the other hand, Kalinowski seems to accept as possible an optimistic
situation in which Good =Con. In these situations there are neither bad
nor neutral actions in genere and therefore all permitted actions are oblig-
atory. Certainly, the situations in which Good 6=Con (and Bad 6=Con)
are also allowed. In these situations there is a room for neutral actions (and
consequently some permitted actions are not obligatory there).
Before concluding this section, we note what follows.
Proposition 1.To satisfy conditions (1), (2), (3) and (4) the universe
Con of actions in concreto has to contain at least one element from Good.
1.2.2. Constraints on the level of actions in genere
Now let us turn to the constraints that Kalinowski imposes on actions in
genere. Let us start from the definitions of good, bad and neutral actions in
genere introduced informally in the previous section.
Definition 1.1.For any action in genere X⊆C on
1. Xis good iff X⊆Good;
2. Xis bad iff X⊆Bad;
3. Xis neutral iff X=∅or the two conditions hold: X∩Good 6=∅and
X∩Bad 6=∅.
Note that since any singleton from Gen is a subset of Good or of Bad,
it is itself good or bad. Moreover, Definition 1.1 makes ∅at the same time
good, bad and neutral. It is not quite intuitive, but we treat ∅as a special
border case of an action in genere (as we have mentioned earlier Kalinowski
did not take ∅into account). We employ such a treatment of ∅mainly for
technical reasons to be able to combine easily the model theoretic structure
for Kalinowski’s logic with the analogous system of Segerberg (see section
2), in which a similar approach is used6.
Now let us turn to Kalinowski’s principle of deontic reverse stating that
a negation of good action in genere is bad and vice versa [6, p.151]. To avoid
inconsistency we limit this principle to actions from Gen different from ∅and
Con. Thus, for every action in genere Xfrom Gen \ {∅, Con}we require
that7:
6Alternatively ∅could be excluded from good and bad actions in genere as it is only
neutral.
7Let us note that −X=Con \X.
Jerzy Kalinowski’s logic of normative sentences revisited 7
X⊆Good iff −X⊆Bad (5)
Theorem 1.2.To satisfy conditions (1), (2), (3), (4) and (5) a set of actions
in concreto Con has to contain either (i) exactly one element which is Good
or (ii) exactly two elements – one of them from Good, and the other one
from Bad.
Proof. From proposition 1 we know that Con has to have at least one good
element if conditions (1), (2), (3) and (4) are satisfied.
To exclude models of cardinality greater then 2 let us assume indirectly
that card(Con)>2. Because of (3), card(Good)≥2 or card(Bad)≥2. Let
us consider the first case (proof for the other one is analogous). Let e1and
e2be two different elements of Good. Thus both {e1}and {e2}are subsets of
Good. By (5) −{e1} ⊆ Bad. Since {e1}and {e2}are different {e2} ⊆ −{e1}
and therefore {e2} ⊆ Bad. That contradicts (2). Moreover, if a model
consists of two elements, both from Good, we have the same situation.
In case (i) from Theorem 1.2 we do not have any neutral actions other
then ∅. Thus (ii) is the only interesting case when discussing the properties
of neutral actions. For that case we can formulate the following proposi-
tion which describes the relation between a neutral action and its negation,
corresponding to condition (5) for good and bad actions.
Proposition 2.If Con 6=Good, then for every X⊆C on, if Xis a neutral
action in genere, then −Xis also a neutral action in genere.
With proposition 2 we complete the list of conditions imposed on our
deontic action model that we have found in [6]. That allows us to believe
that the model indeed formalises Kalinowski’s philosophical intuitions. As
a result we obtain a structure
DAF =hC on, Good, Badi
which we shall call a deontic action frame. In the next section we shall
prove that Kalinowski’s logic of normative sentences K1is sound and com-
plete with respect to the models based on deontic action frame DAF , which
further justifies our belief.
1.3. Morally indiscernible actions
The limitation of the size of the universe of actions in concreto given by
Theorem 1.2 makes it difficult to understand what actions in concreto and
8Robert Trypuz, Piotr Kulicki
actions in genere really are and how they can be intuitively described. First
of all, actions in concreto and actions in genere cannot be identified with
individual and generic acts respectively as understood by von Wright in [16]
and his followers. There are many more then two individual acts (understood
as individual agent’s behavior at a certain place and time). Thus, we have
to look for a different interpretation of actions in concreto and actions in
genere.
We propose to understand both of them as a special kind of abstraction
specific to moral deliberations in which we look only for a moral value of
an action. Then actions in concreto are considered within the context of a
concrete situation and actions in genere are abstracted from the situations
in which they may occur.
Example 1.Usually we refer to actions by their descriptions in terms of
bodily movements and intentions [10, 12]. As an example let us consider an
agent who is able to call emergency services and provide first aid to another
person in need8. These two actions will be used as elements of the description
of the behaviour of the agent. The agent has to make the decision what to do
in a situation of an accident. Disregarding the sequence of actions we can
say that, if anybody is seriously injured, the agent should perform the action
of calling emergency services and providing first aid to the injured person.
Refraining from these actions or even performing only one of them would be
wrong in the considered situation. Thus, we can look at the possible actions
of the agent in a binary way, dividing them into two groups: those that are
right (they include calling emergency services and providing first aid) and
all others. Within these two groups actions are indiscernible from the moral
point of view. We propose to understand actions in concreto as such groups
of morally indiscernible actions for any particular situation.
Certainly, in a different situation, the correspondence between the de-
scription of actions and particular actions in concreto may be different.
Imagine that nobody is hurt in the accident. Then it is not possible to pro-
vide first aid to anybody. Moreover, calling emergency service would be an
abuse of it and therefore something wrong. Then, in terms of the elements
of the description of the behaviour of the agent it is right not to call emer-
gency services and not to provide first aid, and it is wrong to call emergency
services and not to provide first aid, and it is impossible to provide first aid.
When we look at actions abstracted from a particular situation we obtain
8Certainly the agent may able to do many other things, but they are not relevant to
our further considerations.
Jerzy Kalinowski’s logic of normative sentences revisited 9
a different perspective. We can trace their moral value in different circum-
stances. Thus, we have actions that (i) are always good, (ii) always bad,
(iii) sometimes good and sometimes bad, (iv) impossible in any situation.
Within our model actions in genere are just sets of actions in concreto. As
such they represent the four types of actions just mentioned.
1.4. Completeness of Kalinowski’s logic of normative sentences
In this section Kalinowski’s logic of normative sentences K1is introduced.
We shall first define its language and present axioms. Then an interpretation
function and satisfaction conditions will be defined in order to establish a
relation between K1and models based on DAF .
The language of the logic consists of the language of propositional cal-
culus (henceforward PC) extended by one deontic operator of the syntactic
category s/n, i.e., the category of operators that combine with names (n) to
yield sentences (s). The language can be defined in Backus-Naur notation
in the following way:
ϕ::= PK(α)| ¬ϕ|ϕ→ϕ(6)
α::= ai|α(7)
where aibelongs to a set of action names Act0={a1, a2, a3, . . . };PK(α)
stands for “αis permitted”; “¬” and “→” are operators of PC: negation
and implication respectively. Other commonly used connectives of PC such
as disjunction (“∨”), conjunction (“∧”) and equivalence (“≡”) are defined
in the standard way. For fixed Act0, the set of action names defined by (7)
will be denoted by ActK:
ActK=Act0∪ {a1, a2, a3, . . . , a1, a2, a3, . . . }
Logic K1consists of axioms of PC, rule of Modus Ponens and two specific
axioms:
ϕ(α)→ϕ(α//α) (8)
¬PK(α)→PK(α) (9)
The first axiom allows us to substitute some or all occurrences of doubly
negated action name “α” by “α” in any formula in which the doubly negated
action occurs. The second one is a formula characterising permission. It says
that either αor its negations is permitted. K1contains also definitions of
obligation (OK) and prohibition (FK). The concepts are characterised as
follows:
OK(α) =df ¬PK(α) (10)
10 Robert Trypuz, Piotr Kulicki
FK(α) =df ¬PK(α) (11)
Some characteristic theses of K1are listed below.
PK(α)∨PK(α) (12)
OK(α)→PK(α) (13)
¬(OK(α)∧FK(α)) (14)
OK(α)≡FK(α) (15)
The relations between the logic and the deontic action frame DAF de-
scribed in previous section are established by the interpretation function
and satisfaction conditions, in the way described below. The interpretation
function
IK:Act −→ Gen
assigns an action in genere to each action name. It is defined as follows9:
IK(ai)∈Gen \ {Con, ∅},for ai∈Act0(16)
IK(ai) = Con \ IK(ai) (17)
M |=PK(α)⇐⇒ IK(α)6⊆ Bad
M |=¬ϕ⇐⇒ M 6|=ϕ
M |=ϕ∧ψ⇐⇒ M |=ϕand M |=ψ
Operators of obligation and prohibition, defined by definitions (10) and (11),
have (by definition) the following satisfaction conditions:
M |=FK(α)⇐⇒ IK(α)⊆Bad
M |=OK(α)⇐⇒ IK(α)⊆Bad
Theorem 1.3.K1is sound and complete with respect to class Cof models
M=hCon, Bad, IKisatisfying properties (1), (4).
9By this definition of the interpretation function we do not attach any of the border-
case actions: neither ∅nor C on to any action name (basic nor compound). Thus, any
action name is connected with a neutral action and somehow we lose neutral actions from
the logic. We will regain them in the following section when we extend the language.
Jerzy Kalinowski’s logic of normative sentences revisited 11
Proof. To prove soundness it is enough to show that formulas (8) and (9)
are true in the models and the rules lead from true formulas to true formulas.
The proof is straightforward (it is enough to refer to condition (4)).
The completeness part of Theorem 1.3 is proved in standard way by
showing that each consistent set of formulas has a model. The canonical
model and the truth lemma crucial for this kind of proof are introduced
below.
Definition 1.4.(First canonical model) Let Φbe a maximally consistent
set of formulas of the language of K1. For α, β ∈ActKwe say that α'β
iff “ϕ(α)≡ϕ(β)” is provable, where by “ϕ(α)” we understand any formula
in which “α” occurs. We can observe that ‘'’ is an equivalence relation.
Let for α∈ActK,[α]'be an equivalence class, then hConΦ, B adΦ,IΦ
Kiis
a canonical model for this language. ConΦ,BadΦand IΦ
Kare defined as
follows:
•ConΦ={[α]':α∈ActK}
•BadΦ={[α]':¬PK(α)∈Φ}
• IΦ
K(α) = {[α]': [α]'∈ConΦ}
Lemma 1.5.hConΦ, BadΦ,IΦ
Kisatisfies properties (1), (4).
Proof. The proof of (1) is obvious. To prove (4) we assume that ConΦ⊆
BadΦ. That means that for every α∈ActK,¬PK(α)∈Φ. If it is so, we
find β, β ∈ActKs.t. ¬PK(β) and ¬PK(β) which is in contradiction with
axiom (9). It is also worth noting that IΦ
Khas the same formal properties
as its counterpart IK.
Lemma 1.6.(Truth lemma)
MΦ|=ϕ⇐⇒ ϕ∈Φ
Proof. The most interesting part of the proof concerns permission. It is
to be shown that
• MΦ|=PK(α)⇐⇒ PK(α)∈Φ
(=⇒) Assume that MΦ|=P(α). Then IΦ(α)6⊆ BadΦand by definition
of BadΦ,IΦ(α)6⊆ {[α]':¬PK(α)∈Φ}. If we notice that IΦ(α) is a
singleton for any α∈ActK, then it is obvious that ¬PK(α)6∈ Φ. Because
Φ is maximally consistent we get that PK(α)∈Φ.
12 Robert Trypuz, Piotr Kulicki
(⇐=) Let us take now PK(α)∈Φ. Because Φ is maximally consistent
¬PK(α)6∈ Φ and then by definition [α]'6∈ BadΦ. Finally IΦ(α)6⊆ BadΦ
and MΦ|=PK(α).
Theorem 1.7.K1is sound and complete with respect to class of models
M=hDAF,IKisatisfying properties (1), (2), (3), (4) and (5).
Proof. For soundness proof, Theorem 1.3 can be applied, since here we
have a narrower class of models. To prove completeness we need to enrich
the first canonical model by definition of a set of good actions and restrict
the definition of the set of actions in concreto ConΦ.
Definition 1.8.(Second canonical model) Let Φbe a maximally consistent
set of formulas of the language of K1. For α, β ∈ActKwe say that α'β
iff
1. both ¬PK(α)and ¬PK(β)are in Φor
2. both PK(α)and PK(β)are in Φ,
It is clear that the quotient set ActK/'of ActKby relation 'has one
or two elements.
•ConΦ={[α]':PK(α)∨ ¬PK(α)∈Φ}=ActK/'
•GoodΦ={[α]':PK(α)∈Φ}
•BadΦ={[α]':¬PK(α)∈Φ}
• IΦ
K(α) = {[α]': [α]'∈ConΦ}
(1) is obvious.
To prove (2) it is enough to assume that there exists an action in con-
creto [α]'∈ConΦwhich is at the same time good and bad, i.e., [α]'∈
GoodΦ∩BadΦ. So [α]'∈GoodΦand [α]'∈BadΦ. By the canonical
model’s definition we obtain that PK(α)∈Φ and ¬PK(α)∈Φ, which is a
contradiction.
To prove (3) let us assume that there exists an action in concreto [α]'∈
ConΦwhich escapes GoodΦ∪BadΦ, i.e., [α]'6∈ GoodΦ∪BadΦ. Then we
obtain that PK(α)6∈ Φ and ¬PK(α)6∈ Φ which means that [α]'6∈ ConΦ.
That is of course inconsistent with our assumption.
To prove (4) we assume that ConΦ⊆BadΦ. That means that for every
[α]'∈ConΦ,¬PK(α)∈Φ. (i) If C onΦconsists of elements that only
Jerzy Kalinowski’s logic of normative sentences revisited 13
satisfy ¬PK(α)∈Φ, we find β, β ∈ActKs.t. ¬PK(β) and ¬PK(β) which is
in contradiction with axiom (9). (ii) If C onΦconsists of elements that only
satisfy PK(α)∈Φ, then we obtain a contradiction with ¬PK(α)∈Φ.
To prove (5) let us take any set Xbeing a proper subset of ConΦdifferent
from ∅. Because ConΦhas one or two elements we have two cases. If ConΦ
is a singleton, then we cannot find a set Xsatisfying the aforementioned
conditions. If C onΦhas exactly two elements, then Xis a singleton. Let us
prove the implication from left to right for this case (the opposite direction
is to be proved analogically). So let us assume that X={[α]'} ⊆ GoodΦ.
It means that [α]'∈GoodΦand PK(α)∈Φ (from definition of GoodΦ).
Now let us take the second element of [β]'∈ConΦ, which is not an element
of GoodΦ. Thus, ¬PK(β)∈Φ and then [β]'∈BadΦ. Because {[β]'}=
−{[α]'}we obtain that −X⊆BadΦ.
The truth lemma can be proved similarly as in lemma 2.510.
2. Extending Kalinowski’s logic of normative sentences
The previous section has clearly shown that Kalinowski’s logic K1is really
weak. Many important facts about DAF and models based on it cannot
be captured by its language. Thus, there are philosophical assumptions ex-
pressed in the models which do not directly correspond to any formula/thesis
of K1. It is enough to compare theorems 1.3 and 1.7 to see that conditions
(2), (3), and (5) do not have their counterparts in the logic. For instance (3)
cannot be expressed in K1because it has no representation of (i.e., direct
reference to) actions in concreto in it. It is also true that we can freely
intersect and sum the elements of Gen in the models. The language of K1
does not have counterparts of these operations. K1also offers no reference
to Con and ∅. To summarise we can state that a deontic action frame and
the models based on it offer much more than can be expressed in K1. In this
section we introduce a language and a logic containing action operators and
constants missing in K1in order to get as close as possible to the models.
2.1. Krister Segerberg’s deontic action logic
The starting point is Krister Segerberg’s system of deontic action logic
B.O.D.[11]. The language of B.O.D.is defined in Backus-Naur notation
10It is easy to see that completeness part of Theorem 1.3 follows from the completeness
part of Theorem 1.7. We have presented both proofs to show respective canonical models
which we find informative.
14 Robert Trypuz, Piotr Kulicki
1
call-em ⊔ aid call-em ⊔ aid call-em ⊔ aid call-em ⊔ aid
call-em aid call-em
aid
call-em ⊕ aid call-em ⊕ aid
call-em ⊓ aid call-em ⊓ aid call-em ⊓ aid call-em ⊓ aid
0
Figure 1. The structure of action algebra with two basic actions: call-em and aid. “⊕” is
defined as follows: call-em ⊕aid =df (call-em uaid)t(call-em uaid).
in the following way:
ϕ::= α=α|P(α)|F(α)| ¬ϕ|ϕ→ϕ(18)
α::= ai|0|1|α|αtα|αuα(19)
where aibelongs to a set of basic actions Act0, “0” is the impossible action
and “1” is the universal action; “α=β” means that αis identical with
β; “P(α)” – αis strongly permitted (i.e., its performance is permitted in
combination with any action); “F(α)” – αis forbidden, “αtβ” – αor β(a
free choice between αand β); “αuβ” – αand β(parallel execution of αand
β); “α” – not α(negation of α). Further, for fixed Act0, by ActSwe shall
understand the set of formulae defined by (19). It is obvious that for fixed
Act0
ActK⊆ActS
The set of action names corresponding to Example 1 from section 1.3
is a set Act0={call-em, aid}—where call-em and aid stand for “calling
emergency services” and “providing first aid treatment” respectively—is il-
lustrated in Figure 1.
The actions from ActS(i.e., all actions constructed from Act0with the
use of the action connectives) can be ordered by a relation defined as follows:
αvβ=df αuβ=α(20)
Jerzy Kalinowski’s logic of normative sentences revisited 15
Structure hAct0, , t,u,0,1iis a Boolean algebra, i.e.. ,t,u,0,1are char-
acterised by (or in other words, satisfy) the standard axioms of this algebra.
The fact that it is possible to carry out actions call-em and aid simulta-
neously is easily expressible here by stating that action call-em uaid isn’t
impossible, formally:
call-em uaid 6=0(21)
Upon the Boolean algebra of actions Segerberg builds a deontic logic
B.O.D.by introducing the following three axioms:
P(αtβ)≡P(α)∧P(β) (22)
F(αtβ)≡F(α)∧F(β) (23)
α=0≡F(α)∧P(α) (24)
They express the fact that Pand Fare “strong” i.e., that any permitted
(prohibited) action is permitted (prohibited) in combination with any action,
formally:
P(α)→P(αuβ) (25)
F(α)→F(αuβ) (26)
or
P(β)∧αvβ→P(α) (27)
F(β)∧αvβ→F(α) (28)
It is worth noting here that formula
¬P(α)→P(α) (29)
is not a thesis of B.O.D.Thus “P” is not Kalinowski’s permission.
Segerberg has shown in [11] that B.O.D.is sound and complete with
respect to DAF =hC on, Good, Badisatisfying conditions (1) and (2)11.
Model Mfor B.O.D.is a structure hDAF ,Ii, where I:Act −→ Gen is an
interpretation function defined as follows:
I(ai)⊆Con, for ai∈Act0(30)
I(0) = ∅(31)
11Instead hC on, Good, Badi[11] uses different notation hU, Leg, Illi, where Uis a set of
possible outcomes, Leg and Ill are sets of legal and illegal outcomes, respectively.
16 Robert Trypuz, Piotr Kulicki
I(1) = Con (32)
I(αtβ) = I(α)∪ I(β) (33)
I(αuβ) = I(α)∩ I(β) (34)
I(α) = Con \ I(α) (35)
The definition of an interpretation function makes it clear that every
action from Act0is interpreted as a set of (its) actions in concreto, the
impossible action is interpreted as the empty set, the universal action covers
all actions in concreto, operations “t”, “u” between actions and “ ” on a
single action are interpreted as set-theoretical operations on interpretations
of actions.
A class of models defined as above will be represented by C0. Satisfaction
conditions for the primitive formulas of B.O.D.in any model M ∈ C0are
defined as follows:
M |=P(α)⇐⇒ I(α)⊆Good
M |=F(α)⇐⇒ I(α)⊆Bad
M |=α=β⇐⇒ I(α) = I(β)
M |=¬ϕ⇐⇒ M 6|=ϕ
M |=ϕ∧ψ⇐⇒ M |=ϕand M |=ψ
2.2. Embedding Kalinowski’s logic of normative sentences into
an extension of Segerberg-style deontic action logic
We have stated above that B.O.D.logic is adequate with respect to the
models satisfying conditions (1) and (2). Now we turn towards expressing
the other Kalinowski’s conditions: (3), (4) and (5) in B.O.D.The language of
B.O.D.enables us to do so. As a result we intend to obtain a deontic action
logic—an extension of B.O.D.—which (a) will correspond to the models
C, (b) which will embed Kalinowski’s logic of normative sentences and,
most importantly, (c) which will contain theses being counterparts of the
aforementioned model conditions.
2.2.1. Extending B.O.D.by three new axioms
First we assume that Act0={a1, a2, . . . , an}is a finite set and we shall call
its elements action generators. This assumption guarantees the existence
of a finite set of atomic actions. An atom is an action different from the
impossible action 0such that there is no action between it and 0(see Figure
Jerzy Kalinowski’s logic of normative sentences revisited 17
1, where there are four atoms). Each atom is a combination of all action
generators and has the form:
δ1u. . . uδn
where δkis a generator ak∈Act0or its complement. We also assume that
each atom corresponds with a singleton subset of Gen; in fact we can regard
the element of each singleton as an action in concreto. This assumption is
expressed by the following condition:
card(I(δ1u. . . uδn)) = 1 (36)
Having atoms we are able to express assumption (3) in the language of the
logic. The way of doing this is to exclude the actions in concreto which are
neither good nor bad from the models. To obtain the intended result we add
a new axiom to B.O.D.stating that each atom is either good or bad:
P(δ1u. . . uδn)∨F(δ1u. . . uδn) (37)
Secondly we shall express assumption (4) in the logic. Having a universal
action “1” being a counterpart of Con it is quite easy. We just assume that
not all actions are bad:
¬F(1) (38)
This formula is the next new axiom12.
Finally we need to express (5) in the logic, i.e., the assumption that the
complement of a good action is bad13. The last new axiom extending B.O.D.
is then:
(α6=1∧α6=0)→P(α)≡F(α) (39)
The antecedent of the formula above is very important because it prevents
inconsistency with other just added axioms (it is enough to notice that 1
and 0according to their interpretations should be neutral).
We shall call the logic obtained from B.O.D.by adding axioms (37), (38)
and (39) SK.
12One can find systematic presentation of five extensions of B.O.D.and formal relations
between them in [14, 15].
13Let us notice that formula (39), “deontic switch”, isnt normally accepted in deontic
literature as valid for complement of action (see [1, 3, 4]).
18 Robert Trypuz, Piotr Kulicki
2.2.2. Completeness of SK
Theorem 2.1.SK is complete with respect to the class of models C.
Proof. The proof of this theorem is similar to the one of Theorem 1.3.
The canonical model and the truth lemma are introduced below (see also
[14] and [2]).
Definition 2.2.Let Φbe a maximally consistent set of formulas of the
language of SK and [α]=be an equivalence class of relation =, for α∈ActS.
Then a canonical model for this language has the form:
•ConΦ={[α]=:αis an atom of Act}
• IΦ(α) = {[β]=∈ConΦ:βvα∈Φ}
•GoodΦ=S{IΦ(α) : P(α)∈Φ}
•BadΦ=S{IΦ(α) : F(α)∈Φ}
Lemma 2.3.MΦ=hDAFΦ,IΦi, where DAF =hConΦ, GoodΦ, BadΦi,
belongs to C.
Proof. We prove lemma 2.3 by showing that the canonical model satisfies
condition (1)–(4), (5) and (30)–(36). Conditions (30)–(35) are easily prov-
able also for IΦ. Condition (36) follows immediately from the definitions of
IΦand ConΦ.
(1) follows directly from definitions and the fact that each action in finite
Boolean algebra is a sum of atoms. To prove (2) it is enough to assume that
there exists an action token [α]=∈ConΦbeing at the same time good and
bad, i.e., [α]=∈GoodΦ∩BadΦ. So [α]=∈GoodΦand [α]=∈BadΦ. By
the definitions of the canonical model and theses (27) and (28) we obtain
that P(α)∈Φ and F(α)∈Φ, which then implies by axiom (24) that α=
0∈Φ. The last formula gives a contradiction because α, according to our
assumption, should be an atom (or an action identical with an atom). To
prove (3) let us assume that there exists an action token [α]=∈ConΦwhich
escapes GoodΦ∪BadΦ, i.e., [α]=6∈ GoodΦ∪BadΦ. Similarly to what we
have done above we obtain that P(α)6∈ Φ and F(α)6∈ Φ. But because
αwas assumed to be an atom, it is true for it (see 37) that P(α)∈Φ or
F(α)∈Φ. To prove (4) we assume that ConΦ⊆BadΦ. Taking into account
that IΦ(1) = ConΦ, we get that F(1)∈Φ which is in contradiction with
(38). To prove (5) let us take whatever α,IΦ(α)⊆GoodΦ. Then P(α)∈Φ.
From (39) it follows that F(α)∈Φ. Then we get IΦ(α)⊆BadΦand finally
Jerzy Kalinowski’s logic of normative sentences revisited 19
that −IΦ(α)⊆BadΦ. Similarly we prove that −IΦ(α)⊆GoodΦunder
assumption that IΦ(α)⊆BadΦ.
Lemma 2.4.∀α∈ActS,∀[β]=∈ IΦ(α) (βvα∈Φ)
Proof. The proof of that lemma is inductive, assuming that αcan have
the forms: α=ai,α=0,α=βtγ,α=βuγ,α=β(cf. the proof of
lemma 1 in [2])
Lemma 2.5.(Truth lemma) MΦ|=ϕ⇐⇒ ϕ∈Φ
Proof. The proof is inductive. For PC operators the proof is standard.
For the other ones we prove as follows:
• MΦ|=α=β⇐⇒ α=β∈Φ
(=⇒) Assume that MΦ|=α=β. Then IΦ(α) = IΦ(β). For IΦ(α) =
IΦ(β) = ∅we get α=0∈Φ and β=0∈Φ and finally that α=β∈Φ.
For IΦ(α) and IΦ(β) being nonempty sets we shall notice that they have
the same elements, which are all atoms “included” in αand also in β.
Let χbe a sum of all atoms γkfor which it is true that [γk]=∈ IΦ(α)
and [γk]=∈ IΦ(β). Then obviously: χ=α∈Φ and χ=β∈Φ and
finally α=β∈Φ.
(⇐=) Assume that α=β∈Φ. Then αvβ∈Φ and βvα∈Φ. If so,
all atoms “included” in αare “included” in βand vice versa. The last
implies that IΦ(α) = IΦ(β). Finally MΦ|=α=β.
• MΦ|=P(α)⇐⇒ P(α)∈Φ
MΦ|=P(α)⇐⇒ IΦ(α)⊆GoodΦ⇐⇒ P(α)∈Φ
• MΦ|=F(α)⇐⇒ F(α)∈Φ
MΦ|=F(α)⇐⇒ IΦ(α)⊆BadΦ⇐⇒ F(α)∈Φ
2.3. Further remarks on SK system
To complete the picture we can define neutrality in SK as follows:
N(α) =df ¬(P(α)∨F(α)) ∨α=0(40)
Action is (syntactically) neutral (N) iff it is neither good nor bad or it is
impossible. Neutrality of the impossible action is assumed for two reasons.
It follows from proposition 2 that if 1is not permitted, then the new operator
20 Robert Trypuz, Piotr Kulicki
“N” satisfies the desired property that αis neutral iff its complement is
neutral too (see proposition 2):
¬P(1)→(N(α)≡N(α)) (41)
Additionally it can be proved that:
P(α)∧F(β)→N(αuβ) (42)
(α6=0∧β6=0)→(P(α)∧F(β)→N(αtβ)) (43)
Property (41) can be equivalently expressed as the following disjunction:
P(1)∨(N(α)≡N(α)) (44)
However, none of the disjuncts of formula (44):
P(1) (45)
N(α)≡N(α) (46)
is itself valid. Thus we can formulate the following proposition concerning
Halld´en completeness (cf. [9]) of SK.
Proposition 3.System SK is Halld´en incomplete.
Halld´en incompleteness of a system is usually treated as a syndrome that
the system is badly formalised or at least suspicious. We believe that it is
not so in the case of SK. Halld´en incompleteness reflects the fact that the
system has two different (alternative) models and has the expressive power
to distinguish them in the language (formulas (45) and (46) are valid in
models (i) and (ii) from Theorem 1.2 respectively).
Kalinowski’s permitted action – as the one that is either good or neutral
– can be defined in the following way:
PK(α) =df P(α)∨N(α) (47)
One can easily check that axiom (9) of K1is a thesis of the system.
It is important that for “PK” it is valid that (compare with definitions
(10) and (11))
(α6=1∧α6=0)→F(α)≡ ¬PK(α) (48)
(α6=1∧α6=0)→P(α)≡ ¬PK(α) (49)
Jerzy Kalinowski’s logic of normative sentences revisited 21
Formula (48) makes it clear that “F” refers to Kalinowski’s forbiddance. For-
mula (49) shows that “P” in SK plays the role of Kalinowski’s obligation14.
They also satisfy semantical conditions restricting obligatory actions only to
the good ones and the forbidden actions only to the bad ones.
It is also worth noting that for any α6=1∧α6=0formula (38) is
equivalent with the counterpart of axiom (9) of K1which shows that they
both represent principle (4) of the model.
2.3.1. Action space filtration by deontic values
We have already explained that although the models of SK have two ele-
ment set universe at most, the space ActSof compound action names (of
SK language) is really unlimited. For instance in section 1.3 we presented
Example 1 describing an agent being able to call emergency services and
provide first aid treatment, i.e., Act0of the agent equals {call-em, aid}and
his ActSability space consisted of sixteen action names – see Figure 1. Now
we shall see that the identity sign “=” in SK behaves like a moral indis-
cernibility relation and partitions each space of action names from ActSinto
four or two (ordered) pieces.
We shall now show how Kalinowski’s actions in concreto, i.e., the groups
of morally indiscernible actions, are created. Our starting point is Figure 1.
Having two generators we have four formulas:
1. call-em uaid;
2. call-em uaid;
3. call-em uaid;
4. call-em uaid.
They are atoms if no extra restrictions (axioms) are added. In this
case, since card(Con) = 2 and an interpretation function satisfies Property
(36), they are grouped into two nonempty morally indiscernible sets; in the
one of them we find good atoms and in the other – the bad one(s). For
instance if call-em uaid and call-em uaid are morally indiscernible (i.e.,
call-em uaid =call-em uaid) and call-em uaid and call-em uaid are
indiscernible too, then the partition of ActSis such as depicted in Figure 4.
This situation shows that providing fist aid is irrelevant from a moral point
of view, while calling emergency services is either good or bad.
14In [8] it was shown that obligation can be understood as the weakest strong permission,
i.e., the permission which covers all legal/good outcomes.
22 Robert Trypuz, Piotr Kulicki
1
call-em ⊔ aid call-em ⊔ aid call-em ⊔ aid
call-em aid call-em ⊕ aid call-em ⊕ aid
call-em ⊓ aid call-em ⊓ aid
0
aid call-em
call-em ⊓ aid call-em ⊓ aid
call-em ⊔ aid
Figure 2. Exemplary partition with four cells: 1. {call-em uaid, call-em uaid, call-em},
2. {call-em uaid, call-em uaid, call -em}, 3. {0}, 4.
{1, call-emtaid, call-emtaid, call-emtaid, call-emtaid, aid, aid, call-em⊕aid, call-em⊕aid}.
The second possibility is to establish a correspondence between two ele-
ments of Con and some (and not all) chosen atoms. Then the atoms whose
interpretations are not a proper subset of Con, are impossible actions, i.e.,
each of them equals 0or some of them equals 0and the other equal 1. An
exemplary partition of this kind is shown in Figure 3. In this situation an
agent can either simultaneously carry out calling emergency services and
providing first aid or does nothing. Each of these actions is either good or
bad.
The special case of the second possibility is that the atoms whose inter-
pretations are not the proper subset of Con, are universal actions, i.e., each
of them equals 1. The space of action names here is reduced to two sets. In
the situation in Figure 4 an agent cannot do anything (e.g. we can imagine
that he/she is a casualty of a car accident and as a result of injuries cannot
move.)
Conclusions
We have considered Kalinowski’s logic of normative sentences with the use
of more recent formal tools. We have constructed a set-theoretical model
representing Kalinowski’s intuitions and proved that his K1logic is complete
with respect to it. Moreover, we have noticed that the language of K1is too
weak to capture some features of the model. We constructed a new logic,
which we called SK, based on Boolean algebra of actions, with the language
that is powerful enough to express the specificity of the model.
Jerzy Kalinowski’s logic of normative sentences revisited 23
1
call-em ⊔ aid
call-em aid call-em ⊕ aid
call-em ⊓ aid
0
aid
call-em ⊓ aid
call-em
call-em ⊓ aid call-em ⊓ aid
call-em ⊕ aid
call-em ⊔ aid call-em ⊔ aid
call-em ⊔ aid
Figure 3. Another exemplary partition with four cells: 1.
{call-em uaid, call-em, aid, call-em taid}, 2. {call-em uaid, call-em, aid, call-em taid},
3. {call-em taid, call-em taid, call -em ⊕aid, 1}, 4.
{0, call-em uaid, call-em uaid, call -em ⊕aid}
The crucial notion opening a gate to intuitive meaning of the logic is
moral indiscernibility. The relation groups all individual actions into two
groups of equally good ones (good or bad) on the basis of the agent’s ac-
tual situation. We traced how the whole space of actions from an agent’s
repertoire can be split into those categories.
We have also considered formal properties of SK and its place in the uni-
verse of deontic logics. We have proven that SK is sound and complete with
respect to the intended semantics and noticed that it is Halld´en incomplete.
We have related the system to other systems of deontic action logic based
on Boolean algebra.
Finally, it is worth noting that in [7] Kalinowski considered the exten-
sion of his theory by other action operators such as parallel execution or
indeterministic choice, but at the same time he was very sceptical about the
applicability of theories more expressive than his own deontic theory. In fact
we can see that the operators of parallel execution and indeterministic choice
are indeed very limited also in SK since they do not produce new actions
different from the actions that are good, bad, impossible or necessary in a
given situation.
Acknowledgements
This research was supported by the National Science Center of Poland (DEC-
2011/01/D/HS1/04445). The authors would like to thank Bart lomiej Krzos
24 Robert Trypuz, Piotr Kulicki
1
call-em ⊔ aid
call-em aid call-em ⊕ aid
call-em ⊓ aid
0
aid
call-em ⊓ aid
call-em
call-em ⊓ aid call-em ⊓ aid
call-em ⊕ aid
call-em ⊔ aid call-em ⊔ aidcall-em ⊔ aid
Figure 4. Another exemplary partition with four cells: 1.
{call-em uaid, call-em, aid, call-em taid}, 2. {call-em uaid, call-em, aid, call-em taid},
3. {call-em taid, call-em taid, call -em ⊕aid, 1}, 4.
{0, call-em uaid, call-em uaid, call -em ⊕aid}
for discussions on Kalinowski’s theory of norms.
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Robert Trypuz
Faculty of Philosophy
The John Paul II Catholic University of Lublin
Al. Raclawickie 14
Lublin, Poland,
trypuz@kul.pl
Piotr Kulicki
Faculty of Philosophy
The John Paul II Catholic University of Lublin
Al. Raclawickie 14
Lublin, Poland,
kulicki@kul.pl
