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 diference 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.
est known (preference-based) dyadic deontic logic. E
corresponds to the most general case, involving no
commitThe present paper ([
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 [
Reference is made to Åqvist [
The syntax is generated by adding the following primitive operators to the syntax of propositional logic: □ (for historical necessity); ○ (− /− ) (for conditional oblipreference relation ⪰ ⊆ empty set of worlds. Intuitively, ⪰ comparative goodness relation; “ ⪰
× “world is at least as good as world ”. and are equally good (indiferent), if ⪰ and ⪰ . is strictly better is a betterness or ” can be read as gation). The main ingredient of a preference model is a
These conditions are studied in relation with four
sys , where is a non- tems of increasing strength. The base system is Åqvist’s
) if ⪰
and ̸⪰
. In that
plementing F with the principle of cautious monotony
framework, ○
than (notation: ≻
(/) is true if the best -worlds are all
(CM): (○
(/) ∧ ○
(/)) → ○
(/ ∧ ). Finally,
-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
terms of maximality or− following Bradley [
(/)). We have E ⊂ F+(DR).1 (D⋆) rules out the possibility of conflicting
F ⊂
F+(CM) ⊂
worse than any other worlds. And is strongly maximal
obligations for a “consistent” context . (CM) tells us
if no world equally good as 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 afairs triggers an
obliBradley calls it) “Indiference based choice” (IBC): two
gation, then at least one disjunct triggers this obligation.
system E, shown in Fig. 2 (labels are from [
Suitable axioms for propositional logic S5 schemata for □ and ♢ ○ ( → /) → (○ (/) → ○ (/)) ○ (/) → □ ○ (/) □ → ○ (/) □ ( ↔ ) → (○ (/) ↔ ○ (/)) ○ (/ ∧ ) → ○ ( → /) ○ (/) If ⊢ and ⊢ → then ⊢
If ⊢ then ⊢ □ can be violated, if ⪰ is no longer assumed to be transitive.
Consider three worlds , and with ⪰ , ⪰ and . and are equally good; is maximal (and hence chosen), but not . Maximality and strong maximality coincide when ⪰
is transitive. may be defined thus:
The weakened forms of transitivity mentioned above ; transitive closure of ≻ ); • ⪰ is Suzumura consistent, if ⪰ • ⪰ is quasi-transitive, if ≻ is transitive; • ⪰ is acyclic, if ≻ ⋆ implies ̸≻ (≻ ⋆ is the
⋆ implies ̸≻ • ⪰ is an interval order, if ⪰ is reflexive and Ferrers ( ⪰ and ⪰ imply ⪰ or ⪰ ).
Intuitively, quasi-transitivity demands that the strict part of the betterness relation be transitive. A-cyclicity rules out the presence of strict betterness cycles. Suzumura consistency rules out the presence of cycles with at least one instance of strict betterness. The interval order condition makes room for the idea of non-transitive equal goodness relation due to discrimination thresholds.
The relationships between these conditions may be described thus (an arrow represents implication):
Quasi-transitivity
Suzumura consistency Interval order
Transitivity
Acyclicity then given the interval order condition ⪰ is max-smooth and hence max-limited. Hence (D⋆)–the distinctive axiom of F–is validated, and so is (CM).
As a spin-of, one gets that the theoremhood problem in F+(DR) is decidable.
The completeness result below is shown to hold under a rule of interpretation in terms of maximality and of strong maximality.2 Such a result tells us that transitivity, quasi-transitivity, acyclicity and Suzumura consistency make less diference to the logic of ○ (− /− ) than one would have thought. The axiomatic system remains the same whether or not these conditions are introduced.
Th.1 tells us that plain transitivity, quasi-transitivity, acyclicity and Suzumura consistency make less diference 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 efect 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 is non-empty, 26, 22]). then there is a world that is (strongly) maximal in this set. Acknowledgments • F+(CM) with respect to models in which ⪰ meets the so-called (strong-)max-smoothness condition.
It says: if satisfies , then either is (strongly) maximal in the set of worlds that satisfy , or it is worse than some that is (strongly) maximal in the set of worlds that satisfy .
The paper also points out that Th.1 carries over to models with a reflexive betterness relation.
Xavier Parent was funded in whole, or in part, by the Austrian Science Fund (FWF) [M3240 N]. For the purpose of open access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission. I acknowledge the following individuals for valuable comments: R. Booth, W. Bossert, J. Carmo and P. McNamara. I also thank three anonymous reviewers for their comments.
A model is said to be finite, if its universe has finitely many worlds. The following result is established for a rule of interpretation in terms of maximality.3 This result may fruitfully be compared to the representation result reported by [24] for models with a strict preference relation.
Theorem 2 (Weak completeness, finite preference models). Under the max rule F+(DR) is weakly sound and complete with respect to the class of finite preference models = (, ⪰ , ) in which ⪰ is an interval order.
The assumption of finiteness is used in the arguments
for both soundness and completeness. For soundness,
ifniteness comes into play as follows. If the model is finite,
2The proof draws on the work of [