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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