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| id | Vol-3197/short3 |
| wikidataid | →Q117341786 |
| title | On Some Weakened Forms of Transitivity in the Logic of Norms |
| pdfUrl | https://ceur-ws.org/Vol-3197/short3.pdf |
| dblpUrl | https://dblp.org/rec/conf/nmr/Parent22 |
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On Some Weakened Forms of Transitivity in the Logic of Norms
On Some Weakened Forms of Transitivity in the Logic of
Norms
(Extended Abstract)
Xavier Parent
Technological University of Vienna, Institute of Logic & Computation, Theory and Logic Group, Favoritenstrasse 9, A-1040 Wien, Austria
Abstract
The paper investigates the impact of weakened forms of transitivity of the betterness relation on the logic of conditional
obligation, originating from the work of Hansson, Lewis, and others. These weakened forms of transitivity come from the
rational choice literature, and include: quasi-transitivity, Suzumura consistency, a-cyclicity, and the interval order condition.
The first observation is that plain transitivity, quasi-transitivity, acyclicity and Suzumura consistency make less difference to
the logic of ○(−/−) than one would have thought. The axiomatic system remains the same whether or not these conditions
are introduced. The second observation is that unlike the others the interval order condition corresponds to a new axiom,
known as the principle of disjunctive rationality. These two observations are substantiated further through the establishment
of completeness (or representation) theorems.
Keywords
Deontic conditional, betterness, transitivity, quasi-transitivity, Suzumura consistency, acyclicity, interval order
1. Introduction est known (preference-based) dyadic deontic logic. E cor-
responds to the most general case, involving no commit-
The present paper ([1], under review) continues a project ment to any structural property of the betterness relation
started in [2] and pursued further in [3, 4]. It deals with in the models. E offers a simple solution to the contrary-
the problem of axiomatizing the logic of conditional obli- to-duty paradoxes and allows to represent norms with
gation (aka dyadic deontic logic) with respect to prefer- exceptions. As is well-known (e.g. [12]), deontic logicians
ence models. Two types of consideration are thoroughly have struggled with the problem of giving a formal treat-
investigated: the choice of properties of the betterness ment to contrary-to-duty (CTD) obligations. These are
(or preference) relation in the models, and the choice of obligations that come into force when some other obliga-
the evaluation rule for the conditional obligation opera- tion is violated. According to Hansson [13], Lewis [14]
tor. Here my focus is on weakened forms of transitivity and others, the problems raised by CTDs call for an order-
discussed in the related area of rational choice theory: ing on possible worlds in terms of preference (or relative
quasi-transitivity, Suzumura consistency, a-cyclicity and goodness, or betterness), and Kripke-style models fail
the interval order condition [5, 6]. in as much as they do not allow for grades of ideality.
An important task in Knowledge Representation and The use of a preference relation has also been advocated
Reasoning (KRR) is to understand what new axiom cor- for the analysis of defeasible conditional obligations. In
responds to a given semantic property in the models (as particular, Alchourrón [15] argues that preference mod-
identified by the expert of the domain). This is relevant els provide a better treatment of this notion than the
for the design of the reasoner itself: the conclusions this usual Kripke-style models do. Indeed, a defeasible con-
one will be able to draw from a KB vary depending on ditional obligation leaves room for exceptions. Under a
the logical system being used. This paper focuses on the preference-based approach, we no longer have the deon-
property of transitivity of betterness and its weakenings tic analogue of two laws, the failure of which constitutes
thereof. Transitivity seems entrenched in our conceptual the main formal feature expected of defeasible condition-
scheme, if not analytically true. However, the question als: “deontic” modus-ponens; and Strengthening of the
of whether it holds, in what form, and in what context, Antecedent. ○(𝐵/𝐴) may be read as “𝐵 is obligatory,
has been much debated over the years [5, 6, 7, 8, 9, 10]. given 𝐴”. The first is the law: ○(𝐵/𝐴) and 𝐴 imply ○𝐵.
Reference is made to Åqvist [11]’s system E, the weak- The second is the law: ○(𝐵/𝐴) entails ○(𝐵/𝐴 ∧ 𝐶).
NMR 2022: 20th International Workshop on Non-Monotonic Reasoning,
August 07–09, 2022, Haifa, Israel 2. Framework
" x.parent.xavier@gmail.com (X. Parent)
~ https://xavierparent.co.uk/ (X. Parent) The syntax is generated by adding the following prim-
� 0000-0002-6623-9853 (X. Parent) itive operators to the syntax of propositional logic: □
© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0). (for historical necessity); ○(−/−) (for conditional obli-
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073
CEUR Workshop Proceedings (CEUR-WS.org)
147
�gation). The main ingredient of a preference model is a These conditions are studied in relation with four sys-
preference relation ⪰ ⊆ 𝑊 × 𝑊 , where 𝑊 is a non- tems of increasing strength. The base system is Åqvist’s
empty set of worlds. Intuitively, ⪰ is a betterness or system E, shown in Fig. 2 (labels are from [2]). Next
comparative goodness relation; “𝑎 ⪰ 𝑏” can be read as we have Åqvist’s system F; it is obtained by supple-
“world 𝑎 is at least as good as world 𝑏”. 𝑎 and 𝑏 are equally menting E with the law (D⋆ ): ♢𝐴 → (○(𝐵/𝐴) →
good (indifferent), if 𝑎 ⪰ 𝑏 and 𝑏 ⪰ 𝑎. 𝑎 is strictly better 𝑃 (𝐵/𝐴)). Then comes F+(CM); it is obtained by sup-
than 𝑏 (notation: 𝑎 ≻ 𝑏) if 𝑎 ⪰ 𝑏 and 𝑏 ̸⪰ 𝑎. In that plementing F with the principle of cautious monotony
framework, ○(𝐵/𝐴) is true if the best 𝐴-worlds are all (CM): (○(𝐵/𝐴) ∧ ○(𝐶/𝐴)) → ○(𝐶/𝐴 ∧ 𝐵). Finally,
𝐵-worlds. There is variation among authors regarding we have F+(DR); it is obtained by supplementing F with
the definition of “best”. Here I assume “best” is cast in the principle of disjunctive rationality: ○(𝐶/𝐴 ∨ 𝐵) →
terms of maximality or−following Bradley [16]−strong (○(𝐶/𝐴)∨○(𝐶/𝐵)). We have E ⊂ F ⊂ F+(CM) ⊂
maximality. A world 𝑎 is maximal if it is not (strictly) F+(DR).1 (D⋆ ) rules out the possibility of conflicting
worse than any other worlds. And 𝑎 is strongly maximal obligations for a “consistent” context 𝐴. (CM) tells us
if no world equally good as 𝑎 is worse than any other that complying with an obligation does not modify our
worlds. The role of strong maximality is to ensure that other obligations arising in the same context. (DR) tells
the agent’s choice meets the natural requirement of (as us that if a disjunction of state of affairs triggers an obli-
Bradley calls it) “Indifference based choice” (IBC): two gation, then at least one disjunct triggers this obligation.
alternatives that are equally good should always either It is noteworthy that (CM) is a theorem of F+(DR).
both be chosen or both not chosen. Such a requirement
can be violated, if ⪰ is no longer assumed to be transitive.
Consider three worlds 𝑎, 𝑏 and 𝑐 with 𝑎 ⪰ 𝑏, 𝑏 ⪰ 𝑐 and Suitable axioms for propositional logic (PL)
𝑐 ⪰ 𝑏. 𝑏 and 𝑐 are equally good; 𝑐 is maximal (and hence S5 schemata for □ and ♢ (S5)
chosen), but not 𝑏. Maximality and strong maximality ○ (𝐵 → 𝐶/𝐴) → (○(𝐵/𝐴) → ○(𝐶/𝐴)) (COK)
coincide when ⪰ is transitive.
○ (𝐵/𝐴) → □ ○ (𝐵/𝐴) (Abs)
The weakened forms of transitivity mentioned above
may be defined thus: □𝐴 → ○(𝐴/𝐵) (O-nec)
• ⪰ is quasi-transitive, if ≻ is transitive; □(𝐴 ↔ 𝐵) → (○(𝐶/𝐴) ↔ ○(𝐶/𝐵)) (Ext)
• ⪰ is acyclic, if 𝑎 ≻⋆ 𝑏 implies 𝑏 ̸≻ 𝑎 (≻⋆ is the ○ (𝐶/𝐴 ∧ 𝐵) → ○(𝐵 → 𝐶/𝐴) (Sh)
transitive closure of ≻); ○ (𝐴/𝐴) (Id)
• ⪰ is Suzumura consistent, if 𝑎 ⪰⋆ 𝑏 implies 𝑏 ̸≻
If ⊢ 𝐴 and ⊢ 𝐴 → 𝐵 then ⊢ 𝐵 (MP)
𝑎;
• ⪰ is an interval order, if ⪰ is reflexive and Ferrers If ⊢ 𝐴 then ⊢ □𝐴 (Nec)
(𝑎 ⪰ 𝑏 and 𝑐 ⪰ 𝑑 imply 𝑎 ⪰ 𝑑 or 𝑐 ⪰ 𝑏).
Intuitively, quasi-transitivity demands that the strict part Figure 2: Åqvist’s system E
of the betterness relation be transitive. A-cyclicity rules
out the presence of strict betterness cycles. Suzumura
A few comments on the axioms of E are in order. (COK)
consistency rules out the presence of cycles with at least
is the conditional analogue of the familiar distribution
one instance of strict betterness. The interval order con-
axiom K. (Abs) is the absoluteness axiom of [14], and
dition makes room for the idea of non-transitive equal
reflects the fact that the ranking is not world-relative.
goodness relation due to discrimination thresholds.
(O-nec) is the deontic counterpart of the familiar necessi-
The relationships between these conditions may be
tation rule. (Ext) permits the replacement of necessarily
described thus (an arrow represents implication):
equivalent sentences in the antecedent of deontic con-
Interval order Transitivity ditionals. (Sh) is named after Shoham [17, p. 77], who
seems to have been the first to discuss it. (Id) is the de-
ontic analogue of the identity principle. The question of
@ @
@ @
R
@ R
@ whether (Id) is a reasonable law for deontic conditionals
Quasi-transitivity Suzumura consistency has been much debated. A defence of (Id) can be found
in [13, 18] (see also [19]).
@ For an automation of reasoning tasks in E in Is-
abelle/HOL, see [20, 21].
@
R
@
Acyclicity
Figure 1: Implication relations
1
F+(CM) corresponds to the KLM system P supplemented with the
principle of consistency preservation (if 𝐴 ̸⊢ ⊥, then 𝐴 ̸|∼ ⊥).
148
�3. Quasi-transitivity, Suzumura then given the interval order condition ⪰ is max-smooth
and hence max-limited. Hence (D⋆ )–the distinctive ax-
consistency and a-cyclicity iom of F–is validated, and so is (CM).
The completeness result below is shown to hold under As a spin-off, one gets that the theoremhood problem
a rule of interpretation in terms of maximality and of in F+(DR) is decidable.
strong maximality.2 Such a result tells us that transitivity,
quasi-transitivity, acyclicity and Suzumura consistency
make less difference to the logic of ○(−/−) than one
5. Wrap-up
would have thought. The axiomatic system remains the Th.1 tells us that plain transitivity, quasi-transitivity,
same whether or not these conditions are introduced. acyclicity and Suzumura consistency make less difference
Theorem 1. E is sound and complete with respect to the to the logic of ○(−/−) than one would have thought.
following classes of preference models: The determined logic is E whether or not these condi-
(i) The class of all preference models; tions are introduced. Th. 2 tells us that (in the finite case)
(ii) The class of those in which ⪰ is transitive; the interval order condition boosts the logic to F+(DR),
(iii) The class of those in which ⪰ is quasi-transitive; obtained by supplementing F with the principle of dis-
(iv) The class of those in which ⪰ is Suzumura consistent; junctive rationality (DR).
(v) The class of those in which ⪰ is quasi-transitive and Topics for future research include the following: to
Suzumura consistent; study the interval order condition in conjunction with the
(vi) The class of those in which ⪰ is acyclic. other candidate weakenings of transitivity; to study the
effect of using variant evaluation rules for the conditional,
An analogous result is shown to hold for like maximality-in-the-limit or variations thereof, where
• Åqvist’s system F with respect to models in which there are no best worlds, but (non-empty) sets of ever-
⪰ meets the condition of max-limitedness. It says: better ones, which approximate the ideal (see, e.g., [25,
if the set of worlds that satisfy 𝐴 is non-empty, 26, 22]).
then there is a world that is (strongly) maximal
in this set.
• F+(CM) with respect to models in which ⪰ meets Acknowledgments
the so-called (strong-)max-smoothness condition.
Xavier Parent was funded in whole, or in part, by the
It says: if 𝑎 satisfies 𝐴, then either 𝑎 is (strongly)
Austrian Science Fund (FWF) [M3240 N]. For the purpose
maximal in the set of worlds that satisfy 𝐴, or it
of open access, the author has applied a CC BY public
is worse than some 𝑏 that is (strongly) maximal
copyright licence to any Author Accepted Manuscript
in the set of worlds that satisfy 𝐴.
version arising from this submission. I acknowledge the
The paper also points out that Th.1 carries over to models
following individuals for valuable comments: R. Booth,
with a reflexive betterness relation.
W. Bossert, J. Carmo and P. McNamara. I also thank three
anonymous reviewers for their comments.
4. Interval order
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