Normatively Determined Propositions

Abstract

In the present work we provide a logical analysis of normatively determined and non-determined propositions. The normative status of these propositions depends on their relation with another proposition, here named reference proposition. Using a formal language that includes a monadic operator of obligation, we define eight dyadic operators that represent various notions of “being normatively (non-)determined”; then, we group them into two families, each forming an Aristotelian square of opposition. Finally, we show how the two resulting squares can be combined to form an Aristotelian cube of opposition.KeywordsNormatively determined propositionsAristotelian squaresAristotelian cubesModal logicDeontic logic

Full text

Normatively determined propositions⋆
Matteo Pascucci1and Claudio E.A. Pizzi2
1Department of Analytic Philosophy, Institute of Philosophy, Slovak Academy of
Sciences, v.v.i., Bratislava, Slovakia
matteopascucci.academia@gmail.com
2Emeritus, University of Siena, Italy
pizzi4@gmail.com
Abstract. In the present work we provide a logical analysis of norma-
tively determined and non-determined propositions. The normative sta-
tus of these propositions depends on their relation with another proposi-
tion, here named reference proposition. Using a formal language that in-
cludes a monadic operator of obligation, we define eight dyadic operators
that represent various notions of “being normatively (non-)determined”;
then, we group them into two families, each forming an Aristotelian
square of opposition. Finally, we show how the two resulting squares can
be combined to form an Aristotelian cube of opposition.
Keywords: Normatively determined propositions ·Aristotelian squares
·Aristotelian cubes ·modal logic ·deontic logic.
1 Introduction
Formal logic has been used for decades in the analysis of normative concepts,
shedding light on their properties and relations. In the area of normative rea-
soning, logic has been employed to deal with several families of concepts. Just
to mention a few of these, much has been written about the notions of obliga-
tion, permission, prohibition (see, e.g., the surveys by ˚
Aqvist [1] or Hilpinen and
McNamara [6]), right and duty (see, e.g., Lindahl [7] or Makinson [8]). Other
concepts, such as the ones of power, liability and responsibility, are receiving
increasing attention (see, e.g., Glavaniˇcov´a and Pascucci [5], Markovich [9] or
Pascucci and Sileno [10]). In the present work we propose an inquiry on a topic
that has not received attention in the logical literature, namely the formal char-
acterization of normatively determined propositions.
Saying that a proposition Bis determined by a proposition A, in general,
means that either (i) Bholds in all circumstances in which Aholds or (ii) B
⋆This is a preprint version of the work appeared in Diagrammatic Representation and
Inference. DIAGRAMS 2022. The published version is available here:
https://doi.org/10.1007/978-3-031-15146- 0_6
Matteo Pascucci was supported by the ˇ
Stefan Schwarz Fund for the project “A fine-
grained analysis of Hohfeldian concepts” (2020-2023) and by the VEGA project no.
2/0125/22 “Responsibility and modal logic”. The article results from a joint work
of the two authors.

2 M. Pascucci and C.E.A. Pizzi
holds in no circumstances in which Aholds. In this informal definition Acan be
said to be the reference proposition (i.e., the proposition with reference to which
the status of Bis assessed). It can be easily checked that “being determined” is
abilateral notion, since, according to the above definition, Bis determined by
Aiff ¬Bis determined by A(where ¬is the classical operator of negation).3
Here we propose to focus on a deontic variant of the notion at issue. Let
Nbe a set of normatively relevant scenarios: a proposition Bis normatively
determined by a proposition Ain Niff either (i) the truth of Aimplies the truth
of Bin every scenario of Nor (ii) the truth of Aimplies the falsity of Bin every
scenario of N. Consequently, a proposition Bis normatively non-determined by
a proposition Ain Niff the truth of Ais conjoined with the truth of Bin some
scenarios of Nand with the falsity of Bin other scenarios of N. The framework
introduced here will not be committed to any particular choice of normatively
relevant scenarios; in the simplest interpretation, one can take them to be the
normatively ideal scenarios (along the lines of ˚
Aqvist [1]).
Determination is a crucial issue in the normative domain, since sets of norms
are associated with layers: norms belonging to one layer may depend on norms
belonging to an upper-level layer. Logical accounts of normative conditionals and
contrary-to-duty reasoning (see Hilpinen and McNamara [6] for an extended dis-
cussion) capture some aspects of this; yet, the notion of normative determination
offers a broader perspective, since it covers other forms of dependency among
norms. In fact, one can distinguish various kinds of determination for a propo-
sition Bon the basis of the normative status of a proposition A(the reference
proposition). Here we will focus on the following two issues:
–whether the normative status assigned to Ais that of an obligation or of a
permission;
–whether the normative status of Ais claimed to be of a certain kind or
simply supposed to be of a certain kind.
Combining these options with the two possible ways in which the normative
status of Bdepends on the normative status of A, namely “being determined by
A” or “being non-determined by A”, one gets as a result a set of eight normative
relations between Aand B.
The following are a few examples of claims taken from everyday normative
discourse illustrating the meaning of some of the notions at issue:
1. it is obligatory to pay for the goods at the time of delivery and it is permitted
(although not obligatory) to use a credit card.
2. it is obligatory to pay for the goods in advance but it is forbidden to pay via
a bank transfer.
3. if an online payment for the goods is permitted, then one can pay via a bank
transfer or with a different method.
3More precisely, “being determined” is a dyadic notion of non-contingency. An ax-
iomatic characterization of dyadic non-contingency has been recently proposed by
Pizzi [13]. For more on the logic of (non-)contingency, see Cresswell [2].

Normatively determined propositions 3
In example (1) the proposition that one pays with a credit card is not normatively
determined by the proposition that one pays at the time of delivery. Indeed, we
know that a customer has to pay at the time of delivery but also that she can
choose whether to pay with a credit card or not (neither of the two options
is forced). In example (2) the proposition that one pays via a bank transfer is
normatively determined by the proposition that one pays in advance. Indeed, we
know that customers have to pay in advance and this forces them to avoid using
a bank transfer; hence, this option is ruled out.4In example (3) the proposition
that one uses a bank transfer is not normatively determined by the proposition
that one pays online. Indeed, if we suppose that a customer is allowed to pay
online, then she can choose between using a bank transfer or a different method.
These relations will be analysed via a language of propositional modal logic
in terms of dyadic deontic operators. Two Aristotelian squares of opposition will
be drawn which, in turn, will clarify logical connections between pairs of the
normative relations at issue. Finally, we will show how the two squares can be
combined in order to form an Aristotelian cube of opposition. Aristotelian dia-
grams are known for their didactic efficacy. A long-term objective of the present
work is contributing to the development of graphical interfaces based on these
diagrams for human-machine interaction. For instance, imagine that a user spec-
ifies a set of normative statements in a simplified language (as initial hypotheses)
and gives it as input to a program that builds an Aristotelian diagram out of this
set; then, the user can explore the displayed diagram and make inferences from
the nodes associated with the initial hypotheses to other nodes, by following the
available paths of edges. For the user, this may be an effective help in reasoning
on normative problems (e.g., on the content of a contract).
2 Formal setting
The formal language which will be used here consists of (i) a set of propositional
variables, denoted by a,b,c, etc., (ii) the monadic operators ¬(negation) and □
(obligation), (iii) the dyadic operator →(material implication). We take ♢(per-
mission) to be a shorthand for ¬□¬and other propositional connectives, such as
∧(conjunction), ∨(disjunction) and ↔(material equivalence) to be defined in
terms of the primitive ones, as usual. Arbitrary formulas will be denoted by A,
B,C, etc. If Nis the chosen set of normatively relevant scenarios, we read □A
as “Ais true in all scenarios of N”, which essentially means that Ais obligatory.
4Since “being normatively determined” is a bilateral notion, in examples (1)-(3), if a
proposition Bis normatively (non-)determined by a proposition A, then so is ¬B.
For instance, in example (2) both the proposition that one pays via a bank transfer
and the proposition that one does not pay via a bank transfer are normatively
determined by the reference proposition that one pays in advance, since the latter
excludes one of the two alternatives and forces the other. Furthermore, we highlight
that our analysis covers also cases of (non-)determination with respect to forbidden
propositions, as long as one defines “Ais forbidden” as “¬Ais obligatory” (□¬A).

4 M. Pascucci and C.E.A. Pizzi
Furthermore, we read ♢Aas “Ais true in some scenarios of N”, which essentially
means that Ais permitted.
The following is a list of definitions for the auxiliary modal operators repre-
senting the eight dyadic modalities that will be the object of our inquiry:
–△(A, B) := ♢A∧(□(A→B)∨□(A→ ¬B)), meaning that Ais permitted
and Bis normatively determined by A.
–△∗(A, B) := □A∧(□(A→B)∨□(A→ ¬B)), meaning that Ais obligatory
and Bis normatively determined by A.
–▲(A, B) := ♢A→(□(A→B)∨□(A→ ¬B)), meaning that if Ais permit-
ted, then Bis normatively determined by A.
–▲∗(A, B) := □A→(□(A→B)∨□(A→ ¬B)), meaning that if Ais
obligatory, then Bis normatively determined by A.
–▽(A, B) := ♢A∧(♢(A∧B)∧♢(A∧ ¬B)), meaning that Ais permitted and
Bis not normatively determined by A.5
–▽∗(A, B) := □A∧(♢(A∧B)∧♢(A∧ ¬B)), meaning that Ais obligatory
and Bis not normatively determined by A.
–▼(A, B) := ♢A→(♢(A∧B)∧♢(A∧ ¬B)), meaning that if Ais permitted,
then Bis not normatively determined by A.
–▼∗(A, B) := □A→(♢(A∧B)∧♢(A∧ ¬B)), meaning that if Ais obligatory,
then Bis not normatively determined by A.
Notational conventions for dyadic operators are as follows: (i) operators express-
ing the claim that the reference proposition Ahas a certain status are white
triangles (∧is the main operator in the definiens), whereas those expressing
the supposition that Ahas a certain status are black triangles (→is the main
operator in the definiens); (ii) operators saying that Bis determined by Aare
up-pointing triangles, whereas those saying that Bis non-determined by Aare
down-pointing triangles (cf. the use of the symbols ‘delta’ and ‘nabla’ by authors
working on contingency logic, such as Cresswell [2]); (iii) operators treating the
reference proposition Aas an obligation (rather than a permission) are distin-
guished by ∗.
The three examples discussed in Section 1 can be rendered as follows (d=
one pays for the goods at the time of delivery, c= one pays with a credit card, b
= one pays via a bank transfer, a= one pays in advance, o= one pays online):
1. □d∧(♢(d∧c)∧♢(d∧ ¬c));
2. □a∧□¬b;
3. ♢o→(♢(o∧b)∧♢(o∧ ¬b)).
Notice that the formula encoding (1) corresponds to ▽∗(d, c) and the formula
encoding (3) corresponds to ▼(o, b); moreover, in any normal modal system, the
formula encoding (2) entails △∗(a, b).
There are several ways of grouping the eight dyadic modalities. Here we take
△(A, B) and ▽(A, B ) to be the basic notions: they express the conjunction of a
5In normal modal systems ▽(A, B) boils down to ♢(A∧B)∧♢(A∧ ¬B).

Normatively determined propositions 5
statement describing a normative relation between Aand Bwith the statement
that Arepresents a permission. We will use each of these two operators to build
an Aristotelian square of opposition.
3 Geometrical representations
In Pizzi [12] an Aristotelian square of opposition over a logical system S, or
simply an Aristotelian S-square, is a 4-tuple of formulas Q= (W, X, Y, Z ), where
each formula in the 4-tuple is said to be a vertex of Qand the pairs of formulas
(W, X), (Y , Z), (W, Y ) and (X, Z ) are the edges of Q. The first formula Win Q
is said to be its origin. The logical relations in an Aristotelian S-square Qare
as follows: Wand Xrepresent contrary propositions in S;Yand Zsubcontrary
propositions in S;Wand Y, as well as Xand Zconnected propositions (more
precisely, Yis a subalternant of Wand Zis a subalternant of X) in S;Wand Z,
as well as Xand Y,contradictory propositions in S.6We will be here working
within modal system KD, i.e., the smallest normal system closed under the
schema □A→♢Aand whose models are serial7
△(A, B)▽∗(A, B )
▲∗(A, B)▼(A, B )
W X
Y Z
▽(A, B)△∗(A, B )
▼∗(A, B)▲(A, B )
W X
Y Z
Fig. 1. △-rooted (left) and ▽-rooted (right) Aristotelian KD-squares
Fig. 1 graphically represents two Aristotelian KD-squares of opposition, one
having △(A, B) at its origin, the other having ▽(A, B) at its origin. The former
will be said to be a △-rooted square, the latter a ▽-rooted square. In each square
an arrow from one vertex to another stands for subalternation, a full line between
two vertices for contradiction, a dashed line between two vertices for contrariety
and a dotted line between two vertices for subcontrariety.
The construction of the △-rooted square can be justified as follows. In KD,
from the assumption △(A, B), namely ♢A∧(□(A→B)∨□(A→ ¬B)), one can
6We assume familiarity with the meaning of the Aristotelian relations at issue. For
details, see Pizzi [11, 12].
7For details, see ˚
Aqvist [1]. For Aristotelian squares built on non-normal modal sys-
tems, see Demey [3].

6 M. Pascucci and C.E.A. Pizzi
infer □A→(♢A∧(□(A→B)∨□(A→ ¬B))) via the Propositional Calculus
(PC), whence □A→(□(A→B)∨□(A→ ¬B)), namely ▲∗(A, B), again via
PC. By contrast, the inference from ▲∗(A, B) to △(A, B) is not supported by
KD, since any KD-model including a world wthat has access to no worlds
is such that ▲∗(A, B) is true at wand △(A, B) is false at w. This means that
▲∗(A, B) is a subalternant of △(A, B) in KD. Moreover, ♢A→(♢(A∧B)∧♢(A∧
¬B)) is equivalent to ¬(♢A∧(□(A→B)∨□(A→ ¬B))) thanks to the definition
of ♢and PC. This means that △(A, B) and ▼(A, B ) are contradictories in KD.
Finally, ♢A∧(□(A→B)∨□(A→ ¬B)) entails ¬(□A∧(♢(A∧B)∧♢(A∧¬B))),
since □(A→B)∨□(A→ ¬B) is equivalent to ¬(♢(A∧B)∧♢(A∧¬B)); however,
¬(♢A∧(□(A→B)∨□(A→ ¬B))) is equivalent to ♢A→(♢(A∧B)∧♢(A∧¬B))
and the latter does not entail □A∧(♢(A∧B)∧♢(A∧ ¬B)); for instance, any
KD-model including a world whaving access to a unique world vwhere Ais
false is such that ♢A→(♢(A∧B)∧♢(A∧¬B)) is true at wand □A∧(♢(A∧B)∧
♢(A∧ ¬B)) is false at w. This means that △(A, B) and ▽∗(A, B ) are contraries
in KD. The fact that ▲∗(A, B) and ▼(A, B ) are sub-contraries in KD follows
from the rest.8
In the case of the ▽-rooted square, the construction can be justified as follows.
In KD, starting with the assumption ▽(A, B), namely ♢A∧(♢(A∧B)∧♢(A∧
¬B)), one can infer □A→(♢A∧(♢(A∧B)∧♢(A∧ ¬B))) via PC, whence
(again, via PC)□A→(♢(A∧B)∧♢(A∧ ¬B)), namely ▼∗(A, B). By contrast,
the inference from ▼∗(A, B) to ▽(A, B ) is not supported by KD, since any KD-
model including a world wthat has access to a single world vwhere Ais false
is such that ▼∗(A, B) is true at wand ▽(A, B) is false at w. This means that
▼∗(A, B) is a subalternant of ▽(A, B) in KD. Moreover, ♢A∧(♢(A∧B)∧
♢(A∧ ¬B)) is equivalent to ¬(♢A→(□(A→B)∨□(A→ ¬B))) thanks to the
definition of ♢and PC. This means that ▽(A, B) and ▲(A, B ) are contradictories
in KD. Finally, ♢A∧(♢(A∧B)∧♢(A∧¬B)) entails ¬(□A∧(□(A→B)∨□(A→
¬B))), since □(A→B)∨□(A→ ¬B) is equivalent to ¬(♢(A∧B)∧♢(A∧ ¬B));
however, ¬(♢A∧(♢(A∧B)∧♢(A∧ ¬B))) is equivalent to ♢A→(□(A→
B)∨□(A→ ¬B)) and the latter does not entail □A∧(□(A→B)∨□(A→ ¬B));
for instance, any KD-model including a world wthat has access to a unique
world vwhere Ais false is such that ♢A→(□(A→B)∨□(A→ ¬B)) is true at
wand □A∧(□(A→B)∨□(A→ ¬B)) is false at w. This means that ▽(A, B)
and △∗(A, B) are contraries in KD. The fact that ▼∗(A, B) and ▲(A, B ) are
sub-contraries in KD follows from the rest.
The two squares can be combined in system KD to form an Aristotelian
cube. The notion of an Aristotelian cube has been defined in various ways (see,
for instance, Dubois, Prade and Rico [4]). Here we follow Pizzi [11] and first
introduce the notion of a semiaristotelian square. A semiaristotelian S-square is
a 4-tuple Q= (W, X, Y, Z ), where each edge represents one of the Aristotelian
8As observed by Pizzi [13], the notion of absolute (non-)determination may be defined
in terms of dyadic (non-)determination by replacing the reference proposition with
a tautology ⊤. For instance: △(⊤, B ).

Normatively determined propositions 7
relations of connectedness, contrariety, subcontrariety and contradiction.9An
Aristotelian S-cube is a set K={Q1, ..., Q6}where:
–every Qi, for 1 ≤i≤6, is a semiaristotelian square;
–for some j, k s.t. 1 ≤j=k≤6, Qj,Qk are Aristotelian S-squares;
–each edge of each square in Kis also an edge of some other square in K.
Looking at the graphical representation of the cube in Fig. 2, the two formulas
occupying corners Wand W′, as well as the two formulas occupying corners
Xand X′are contraries in KD. For instance, in such system △(A, B) logically
entails ¬▽(A, B), whereas ¬△(A, B) does not entail ▽(A, B). Furthermore, the
two formulas occupying corners Yand Y′, as well as the two formulas occu-
pying corners Zand Z′are sub-contraries in KD. For instance, in such system
¬▲∗(A, B) entails ▼∗(A, B), whereas ▲∗(A, B ) does not entail ¬▼∗(A, B). Thus,
the squares (W, W ′, X, X ′), (W, W ′, Y , Y ′), (X, X ′, Z, Z′) and (Y, Y ′, Z, Z′) are
all semiaristotelian and one can conclude that the cube at issue is an Aristotelian
KD-cube, according to the definition in Pizzi [11].
Yet, the 16 relations graphically represented in Fig. 2 are not the only rela-
tions between pairs of vertices of the two squares. Indeed, the total number of re-
lations on 8 formulas is (8×(8−1))/2 = 28 and the following 12 hold too: ▽(A, B)
is a subalternant of ▽∗(A, B); △(A, B) is a subalternant of △∗(A, B); ▼∗(A, B) is
a subalternant of ▼(A, B); ▲∗(A, B) is a subalternant of ▲(A, B ); △(A, B) and
▼∗(A, B) are subcontraries; ▽(A, B) and ▲∗(A, B) are subcontraries; ▲(A, B)
and ▽∗(A, B) are contraries; ▼(A, B) and △∗(A, B ) are contraries; ▲(A, B) is
a subalternant of △(A, B); ▼(A, B) is a subalternant of ▽(A, B ); ▲∗(A, B) is a
subalternant of △∗(A, B); ▼∗(A, B) is a subalternant of ▽∗(A, B ).
△(A, B)▽∗(A, B )
▲∗(A, B)▼(A, B )
W X
Y Z
▽(A, B)△∗(A, B )
▼∗(A, B)▲(A, B )
W’ X’
Y’ Z’
Fig. 2. Aristotelian KD-cube for the eight dyadic operators
9Thus, a semiaristotelian S-square is a square whose edges are associated with some
of the relations holding between the edges of an Aristotelian S-square.

8 M. Pascucci and C.E.A. Pizzi
4 Final Remarks
Our logical analysis of normatively determined and non-determined propositions
can be extended in many respects. From the point of view of Aristotelian dia-
grams, alternative combinations of the operators introduced here can be taken
into account. For instance, consider the formulas △(A, B) and ▽(A, B) at the
origins of the two Aristotelian squares in Fig. 1: it might be possible to build
other Aristotelian squares having the same origins, by finding new formulas C
and Dthat are respectively contrary to △(A, B) and to ▽(A, B) in system KD
or in stronger systems. Dyadic operators can be also used to define the monadic
operators □and ♢, as shown by Pizzi [13]. For instance, in normal modal sys-
tems ♢Ais definable as △(A, A). Moreover, △(A, B) entails △(A, A) (i.e., ♢A)
whereas there is no entailment in the opposite direction: thus, ♢Ais a subalter-
nant of △(A, B). In the light of the latter observation, one can check whether it
is possible to build Aristotelian squares and cubes involving both monadic and
dyadic operators.
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