Default-assumption consequence relations in a preferential

Default-assumption consequence relations in a
preferential setting: Reasoning about normality.
Giovanni Casini1 and Hykel Hosni2
?
1
2
University of Pisa, giovanni.casini[at]gmail.com
Scuola Normale Superiore, Pisa, hykel.hosni[at]sns.it
Abstract. We attempt at a logical characterization of reasoning about
normality based on the intuition that a normal situation is represented
by a stable set of default assumptions. Within the context of preferential
consequence relations we shall define a normality operator B and study
its logical properties. Finally we shall focus on the AGM-like postulates
for the normality expansion, contraction and revision.
1
Introduction
The study of non monotonic (or defeasible) inference is dotted with considerations about normality. Non monotonic conditionals (or defaults) are often
attached an intuitive semantics to the effect that “in normal situations if θ then
φ”. This intuitive reading can be represented semantically in terms of an ordering over the set of (propositional) valuations leading to the so-called preferential
semantics for non monotonic reasoning, see eg. [9, 6, 7]. The idea here is that
an agent performs non monotonic inferences under a two-fold assumption on its
knowledge base: On the one hand the agent behaves as if the information at its
disposal is “complete”, that is to say, all the relevant information available to
the agent at that particular time is explicitly represented in the knowledge base.
On the other hand the agent reasons under the assumption that the current
situation is in fact a normal situation. In a realistic scenario, however, those
assumptions could be violated: new or more refined information might become
available to the agent to the effect that either some previously held belief turns
out to be contradicted by new evidence, or the newly acquired evidence leads
the agent to believe that the situation at hand is in fact not normal. In both
cases some of the previously inferred conclusions might need to be abandoned
making the agent’s reasoning essentially defeasible.
Thus, under the assumption that an agent is identified with a consequence
relation, there is a very tight connection between the non-monotonicity of an
intelligent agent’s reasoning and its reasoning about normality. This paper intends to shed further light on this connection by characterizing normality in
terms of preferential semantics. Given the amplitude of the field we shall restrict
ourselves to a particularly relevant class of non monotonic logics, namely those
?
We would like to thank David Makinson for useful conversations on this topic.
based on Default-assumption consequence relations (Dacr) [8]. The key idea underlying the Dacr approach consists in defining a consequence relation in such
a way that the conclusions of a given set of premisses are established modulo
a maxiconsistent subset of a given set of background assumptions. Indeed the
role of this latter set is to represent the situation or context K under which a
set of sentences Γ can be said to normally entail a sentence θ. Since a Dacr
is defined relative to a set of background assumptions K, each such set determines a distinct consequence relation. The purpose of this paper is to identify a
principled set of epistemic operations on a set of background assumptions under
which a given consequence relation stays fixed. We shall identify such a stable
set of default assumptions with a normal situation. Thus our approach consists
in characterizing normality in terms of the stability of a given preferential consequence relation. This puts us in a position to define a normality operator based
on the amount of variation that can be “absorbed” or “tolerated” by a given
preferential order.
The paper is organized as follows. We shall begin by reviewing the key notions leading to the correspondence between finite default assumption consequence and preferential reasoning. This will provide us with an adequate setting
to move on to the central topic of the paper, namely a preferential characterization of normality. The key step in our formalization consists in characterizing
normality in terms of stable ordering relations. Intuitively this amounts to say
that a logical agent holds a situation as normal just in case the available information does not force him to revise his preferential structure and therefore move
on to a new consequence relation. Building on this intuitions we shall define a
normality operator and investigate its logical properties. The final part of the
paper is devoted to a discussion of our normality operator in the light of the
epistemic operations of Expansion, Contraction and Revision, as defined within
the standard AGM approach [1].
2
Preferential Framework
The purpose of this section is to briefly recall some important features of Defaultassumption consequence relations and to point out the correspondence, in the
finite case, between those and the family of preferential consequence relations.
While our work was under way we realized that this latter correspondence had
already been fully investigated by Freund [3]. As a consequence we shall presently
confine ourselves only to the key properties, directing the interested reader to
Freund’s paper for detailed proofs.
Let us begin by fixing our notation. Let P = {p1 , . . . , pn } be a finite set
of propositional letters, and let ` be the propositional language generated from
P in the usual, recursive way. The sentences of ` will be denoted by low-case
Greek letters and subsets of ` will be denoted by capital Roman letters. Let
W = {w1 , . . . , wm } be the set of all the classical (two-valued) valuations on `,
without repetitions. Clearly, since P is finite, W is finite too with m = 2n . As
usual will denote the classical (Tarskian) consequence/satisfaction relation.
For A ⊆ `, we shall write [A]W for set valuations in W which satisfy the all
sentences in A:
[A]W = {w ∈ W | w φ for every φ ∈ A}.
Since W is finite, there is a bijection between the sentences in ` and the
subsets of W : given any sentence we can find a set of valuations (all the valuations
which satisfy that sentence), and, vice versa, given a set of valuations, we can
find a sentence that is satisfied precisely by those valuations. Finally, given a
set of sentences A, we denote by Aw the subset of A which is satisfied by the
valuation w, that is to say, Aw = {φ ∈ A | w φ}.
It is customary to assume in logic-based AI that the key features of an intelligent agent’s reasoning can be modelled using a consequence relation. In this
perspective it is natural to interpret a given set of premisses A as the relevant
information which is actually available to the agent at a specific time. For example, in the case of a robotic agent performing a navigation task, the set A might
encode its sensory data, the map of its current working space and so on. This
is entirely standard in the knowledge representation and reasoning based on the
classical (tarskian) consequence relation. On top of this, the Default-assumption
consequence relation approach adds the ability to make use also of a set of background information, that is to say a set K which encodes the agent’s information
about what normally holds - its default information. Thus, for example, the set
K for the above mentioned robot might include information such as “slippery
surfaces normally impede correct motion” and the like. Of course task-specific
and default information can interact within any reasoning task performed by the
agent and such an interaction is subject to the constraint of maxi-consistency:
Definition 1 (Maximally A-consistent sets). Given two sets of sentences,
K and A, we say that a set K 0 is a maximally A-consistent subset of K iff K 0
is consistent with A and for no K 00 s.t. K 0 ⊂ K 00 ⊆ K, K 00 is consistent with A.
Given a premise set A and a default set K, the maximally A-consistent
subsets of K represent all the default-information which is compatible with the
agent’s knowledge. Thus it is natural to think of those sets as representing what
the agent might expect or presume to be true in a situation in which A holds.
This intuitive consideration leads to the central definition:
Definition 2 (Default-assumption consequence relation). φ is a defaultassumption consequence of the set of premises A, given a set of default-assumptions
K, (written A |∼K φ) if and only if φ is a classical consequence of the union of
A and every maximally A-consistent subset of K:
A |∼K φ ⇔ A ∪ K 0 φ, for all maximally A-consistent K 0 ⊆ K.
Let us now move to preferential semantics for consequence relations. The
central idea here consists in ordering all the valuations on the language according to the normality of the situations which they describe. Hence given a set
of premises A, we can define its logical consequences by taking the set of the
classical consequences of the most normal situations (according to the above
ordering) classically satisfying A.
To put this more precisely, let δ ⊆ W × W be an irreflexive and transitive
ordering relation. As usual we shall write w <δ v for (w, v) ∈ δ). Given U ⊆ W ,
let minδ (U ) denote the set of minimal valuations in U with respect to δ, that
is:
min(U ) = {w ∈ U | 6 ∃v ∈ U, s.t.v <δ w}.
δ
Definition 3 (Preferential consequence relation). φ is a preferential consequence of the set of premises A, with respect to a strict order δ, (written
A |∼δ φ) if and only if φ is classically satisfied by every δ-minimal valuation
in |A|W :
A |∼δ φ ⇔ A φ, ∀w ∈ minδ ([A]W ).
In what follows we shall assume that each valuation in W appears only once
in the preferential ordering δ. This amounts to say that our results are restricted
to the class of preferential injective models (see e.g [2] for the technical details
and Sec. 3.3. of [8] for an extensive discussion on the issue).
As already pointed out above, the correspondence between preferential injective models and (what amounts to) default-assumption consequence relations is
known in the literature at least since [3], thus we shall only recall the key result.
Let us begin by defining the notion of generated strict order.
Definition 4 (Generated strict order). Given a set K of sentences, a relation δ is generated by K (written δK ) if and only if δ = {(w, v) ∈ W ×W | Kw ⊃
Kv }.
It is immediate to note that δK inherits irreflexivity and transitivity from ⊃.
Theorem 1. (Freund, 1998)
1. Given a default-assumption consequence relation |∼K , we can define an injective preferential consequence relation |∼δK such that A |∼δK φ holds just
if A |∼K φ and conversely,
2. given an arbitrary injective preferential consequence relation |∼δ , we can define a default-assumption consequence relation |∼K such that A |∼K φ holds
just if A |∼δ φ.
Proof. Observation 3.4.11 of [7], provides a key lemma to prove part 1 of the
theorem, while its converse can be proved using Theorems 13 and 14 of [3].
Note that Theorem 1 can be immediately extended to preorders (i.e. reflexive,
transitive relations) ε ⊆ W 2 which give rise to the notion of generated preorders:
εK = {(w, v) ∈ W × W | Kw ⊇ Kv }
(or equally ε = {(w, v) ∈ W × W | v ψ ⇒ w ψ for every ψ ∈ K}).
Then it is immediate to see that εK is an extension of δK as
εK = δK ∪ {(w, v) | Kw = Kv }.
3
Reasoning about normality
Let us now turn to the central topic of this paper, namely a characterization
of the normality based on the epistemic variations that a default-assumption
consequence relation is capable of tolerating.
3.1
Generated order stability
We begin by making the (presumably known) observation that each preferential
order δ determines uniquely the preferential consequence relation |∼δ .
Proposition 1. Given δ, δ 0 ⊆ W 2 and their associated consequence relations
|∼δ and |∼δ0 , then δ = δ 0 if and only if |∼δ =|∼δ0 .
Proof. The direction from left to right follows directly from Definition 3. As to
the converse, assume that |∼δ =|∼δ0 but δ 6= δ 0 . Then there is at least a pair
(w, v) such that either (w, v) ∈ δ and (w, v) ∈
/ δ 0 , or (w, v) ∈ δ 0 and (w, v) ∈
/ δ.
Assume, without loss of generality, that the first case holds. Let γ be the sentence
= αw ∨ αv which is satisfied only by w and v. Since (w, v) ∈ δ and (w, v) ∈
/ δ0
we have that
minδ ({w, v}) = {w},
while
(
{w, v},
minδ0 ({w, v}) =
{v},
if (v, w) ∈
/ δ0
if(v, w) ∈ δ 0 .
Either way, by the injectivity of the preferential model,
{φ | γ |∼δ φ} =
6 {φ | γ |∼δ0 φ}
contradicting the hypothesis that |∼δ =|∼δ0 .
Note that by Theorem 1 this extends immediately to default-assumption
consequence relations.
We now turn to the problem of how a default-assumption set can be expanded
while keeping its generated ordering fixed. Given a default assumption set K and
a sentence φ, we shall say that the generated strict ordering δK is stable with
respect to φ just if δK = δK∪{φ} . We begin by showing that a strict ordering δK
is stable w.r.t. φ ∈ ` just if the corresponding strict ordering εK is.
Proposition 2. δK = δK∪{φ} if and only if for every (w, v) ∈ δK and every w,
v s.t. Kw = Kv , v φ implies w φ.
Proof. The direction from left to right follows immediately from Definition 4.
As to the converse assume that v φ implies w φ for every (w, v) ∈ εK
and every w, v s.t. Kw = Kv . To avoid triviality, let φ ∈
/ K (otherwise clearly
δK = δK∪{φ} ).
Let δK 6= δK∪{φ} . To prove our result is enough to find w, v ∈ W with either
(w, v) ∈ δK or Kw = Kv , such that v φ but w 2 φ.
Note that the subset of K ∪ {φ} satisfied by w, (K ∪ {φ})w , can be defined
in terms of Kw and the value assigned to φ by w:
Kw ∪ {φ} if w φ
(K ∪ {φ})w =
Kw
otherwise
An immediate consequence of this observation is that (K ∪ {φ})w ⊇ Kw .
Returning to the proof proper suppose that (u, t) is such that (u, t) ∈ δK∪{φ}
but (u, t) ∈
/ δK . Then either ∃(w, v) ∈ εK or ∃w, v ∈ W with Kw = Kv such
that v φ and w 2 φ.
Notice that, since (u, t) ∈
/ δK and (u, t) ∈ δK∪{φ} , (K ∪ {φ})u ⊃ (K ∪ {φ})t ,
but Ku 6⊃ Kt . For (u, t) and φ as above there are four cases to check:
If u φ and t 2 φ we have (K ∪ {φ})u = Ku ∪ {φ} and (K ∪ {φ})t = Kt .
Since Ku ∪ {φ} ⊃ Kt , φ ∈
/ Ku and φ ∈
/ Kt hold it follows that Ku ⊇ Kt . In this
case Ku 6⊃ Kt just if Ku = Kt .
If u φ and t φ we have (K ∪{φ})u = Ku ∪{φ} and (K ∪{φ})t = Kt ∪{φ}.
Since Ku ∪{φ} ⊃ Kt ∪{φ}, φ ∈
/ Ku and φ ∈
/ Kt hold, if follows that (K)u ⊃ (K)t ,
and (u, t) must belong to δK too, contradiction.
Similar arguments show that the two remaining cases (u 2 φ and t φ; u 2 φ
and t 2 φ) similarly lead to contradiction hence we conclude that if u φ and
t 2 φ with Ku = Kt is not satisfied, then it is not the case that (u, t) ∈
/ δK and
(u, t) ∈ δK∪φ . So, if there is a pair (u, t) s.t. (u, t) ∈
/ δK and (u, t) ∈ δK∪φ , then
we can find w, v ∈ W such that Kw = Kv and v φ but w 2 φ.
It remains to check the failure of the converse inclusion. If δK 6⊂ δK∪{φ} ,
then ∃(u, t) s.t. (u, t) ∈ δK and (u, t) ∈
/ δK∪{φ} . Then either ∃(w, v) ∈ δK or
∃w, v ∈ W with Kw = Kv such that v φ but w 2 φ. In analogy with the
previous argument we can see that the only case consistent with the premisses
is u 2 φ and t φ. So if it is not the case that u 2 φ and t φ, then either
(u, t) 6∈ δK or (u, t) ∈ δK∪φ . We concluded that if ∃(u, t) such that (u, t) ∈ δK
and (u, t) ∈
/ δK∪φ , then ∃(w, v) ∈ δK , namely the pair (u, t) itself, such that
v φ and w 2 φ.
Summing up, we have shown that if δK 6= δK∪φ , then either ∃(w, v) ∈ δK
such that v φ does not imply w φ or ∃w, v ∈ W with Kw = Kv such that
v φ but w 2 φ.
Take the reflexive and the strict ordering generated by K εK and δK , respectively. As pointed out at the end of Section 2, εK = δK ∪ {(w, v) | Kw = Kv }.
Putting this observation together with Proposition 2 we get
Lemma 1. δK = δK∪{φ} if and only if v φ implies w φ for every (w, v) ∈
εK
As for preorders we have the following
Lemma 2. εK = εK∪{φ} if and only if v φ implies
(w, v) ∈ εK
w φ for every
Proof. The implication from left to right follows directly from the definition of
εK∪φ .
As to the other direction note that
εK = {(w, v) | v ψ ⇒ w ψ
εK∪φ = {(w, v) | v ψ ⇒ w ψ
for every ψ ∈ K}.
for every ψ ∈ K ∪ φ}.
(1)
(2)
Now, since v φ implies w φ for every (w, v) ∈ εK , then equations (1) and
(2) define exactly the same pairs. Thus εK = εK∪φ .
The upshot of Lemma 1 and Lemma 2 is the following:
Theorem 2. δK = δK∪{φ} if and only if εK = εK∪{φ} , that is, if and only if
v φ implies w φ, for every (w, v) ∈ εK .
As a consequence of Theorerm 2 we shall be freely swapping between δK and
εK in what follows. This will provide us with considerable technical simplification.
3.2
Reasoning About Defaults
Recall from Proposition 1 that every distinct default-assumption consequence
relation is semantically represented by a distinct strict preferential order. We
now define a preferential model and a corresponding notion of satisfiability, based
on the intuition that only those sentences which, if added to K keep |∼K fixed,
should be satisfied. This satisfiability relation gives us the building block to
construct our normality operator.
Let C be the class of models of the form M = (W, ε), with W and ε as above.
We say that the model M satisfies φ, written M φ, just if φ is compatible with
ε, that is
M φ iff v φ ⇒ w φ, ∀(w, v) ∈ ε.
(3)
On the basis of this notion of satisfaction, we can now define our normality
operator B by putting K B φ just if φ is satisfied by every model that satisfies
K:
K B φ iff ∀M ∈ C, if M ψ, ∀ψ ∈ K, then M φ.
The next Proposition justifies the intuitive reading of B as a normality operator.
Proposition 3.
K B φ iff εK = εK∪{φ} .
Proof. (⇒): suppose that K B φ. This means that in every preorder ε such that
for every ψ ∈ K and every (w, v) ∈ ε, v ψ ⇒ w ψ, it also holds that
v φ ⇒ w φ. Since εK is one of those preorders, we have v φ ⇒ w φ for
every (w, v) ∈ εK . So εK = εK∪{φ} .
(⇐): Suppose that εK = {(w, v) ∈ W × W | v ψ ⇒ w ψ for every
ψ ∈ K} = εK∪{φ} . Let M = (W, ε) be an arbitrary model in C. If M ψ for
every ψ ∈ K, then all the pairs (w, v) ∈ ε satisfy v ψ ⇒ w ψ for every
ψ ∈ K. But since all those pairs of valuations are in εK , it follows that ε ⊆ εK .
Since εK = εK∪{φ} , then v φ ⇒ w φ for every (w, v) ∈ εK . So it also holds
that v φ ⇒ w φ for every (w, v) ∈ ε, that is M φ.
Since M was arbitrary, we conclude that K B φ, as required.
Summing up, given a default-assumption consequence relation |∼K characterized by the generated preorder εK , B defines those sentences whose addition
to the default-assumption set K can be “absorbed” or “tolerated” by the consequence relation itself.
It is natural to ask, at this point, which kind of logical object is our normality
operator B.
We shall begin by showing that B is a Tarskian operator.
Proposition 4. B satisfies Reflexivity, Monotony and Cut.
Proof. We check the three properties in turn.
◦ Reflexivity (REFL):
K B φ for every φ ∈ K
This property is obviously satisfied: if M ψ for every ψ ∈ K, then M ψ
for every ψ ∈ K.
◦ Monotony (MON):
K Bφ
K ∪ {ψ} B φ
If M γ for every γ ∈ K 0 , then, since K ⊆ K 0 , it holds that M ψ for
every ψ ∈ K. Therefore, since K B φ, we obtain M φ.
◦ Cut (CT):
K ∪ {ψ} B φ K B ψ
K Bφ
Suppose that K ∪ {ψ} B φ and K B ψ. Then, Proposition 3, the equations
εK∪{ψ} = εK∪{ψ}∪{φ} εK = εK∪{ψ} hold. Thus, by transitivity we obtain
εK = εK∪{φ} . And from this we get, again by Proposition 3, that K B φ, as
required.
We now investigate the the behavior of B with respect to the classical consequence relation .
◦ Tautology (>):
K B>
To see that property holds note that every valuation in W satisfies >, so
for every pair (w, v) in every ordering it holds that v > ⇒ w >. This
means that we can add whichever tautology to our assumption set without
changing the consequence relation.
◦ Singleton Left Logical Equivalence (sLLE):
φ ≡ ψ K ∪ {φ} B γ
K ∪ {ψ} B γ
To see that sLLE is satisfied, it is enough to show that the set of models
M = (W, ε) satisfying K ∪ {φ} corresponds exactly to the set of models
satisfying K ∪ {ψ}. If M ρ for every ρ ∈ K ∪ {φ}, then for every pair
(w, v) ∈ ε, v ρ implies w ρ for every ρ ∈ K ∪ {φ}. Also, if φ ≡ ψ, then
w φ if and only if w ψ for every w ∈ W . So if v φ implies w φ, then
v ψ implies w ψ and conversely. Thus M ρ for every ρ ∈ K ∪ {φ}
holds if and only if M ρ for every ρ ∈ K ∪ {ψ}. So the set of consequences
of K ∪ {φ} is exactly the set of consequences of K ∪ {ψ}.
This property tells us that, even if we cannot substitute the entire set K
with a logically equivalent one (as illustrated by the failure of RW and I∨
proved below), B is closed under substitution of single sentences.
◦ Right Logical Equivalence (RLE):
φ≡ψ K Bφ
K Bψ
Suppose that φ ≡ ψ. Then w φ if and only if w ψ for every w ∈ W .
Assume K B φ. Then for every model M s.t. M γ for every γ ∈ K, then
M φ. Given that w φ if and only if w ψ, then each pair (w, v) s.t.
v φ ⇒ w φ satisfies also v ψ ⇒ w ψ. So M φ implies M ψ,
and we get K B ψ, as required
The properties that we have shown B to satisfy so far are all pretty standard for logical operators. The next property, however, will highlight a rather
odd feature of B, namely that it cannot distinguish between a tautology and a
contradiction.
◦ Contradiction (⊥):
K B⊥
To see that this property holds, note that no world satisfies ⊥, so for every
pair (w, v) in every order it holds that v ⊥ ⇒ w ⊥. This means
that we can add whichever contradiction to our assumption set without
changing the consequence relation. To understand why this is the case, let
us look at the following example. Let K = {p, q} and A = {¬q}. There
is only one maximally A-consistent subset of K, which is K1 = {p}. Let’s
now add a contradiction to K, so K 0 = {p, q, α ∧ ¬α}. Again,there is only
one maximally A-consistent subset of K 0 , which is still K1 = {p}. So, by
adding a contradiction we simply add sentences which play no role in the
construction of maximally consistent sets, and therefore in the construction
of our consequence relation.
If B is relatively well-behaved insofar as its structural properties are concerned, things appear to get worse once we introduce the standard propositional
connectives. Let us begin with the properties which rhd does satisfy.
◦ Disjunction in the premises (OR):
φBγ ψBγ
φ∨ψBγ
Let M = (W, ε) be a model. Assume φ B γ, ψ B γ and M φ ∨ ψ, which
means that for every (w, v) ∈ ε, if v φ ∨ ψ, then w φ ∨ ψ. Take one of
those pairs (w, v) ∈ ε. There can be three situations:
1) v φ ∨ ψ and w φ ∨ ψ.
Since w φ ∨ ψ, then either w φ or w ψ. Hence at least one of v φ
⇒ w φ and v ψ ⇒ w ψ is satisfied. Either way, v γ implies w γ
2) v 2 φ ∨ ψ and w φ ∨ ψ.
The same argument as (1) applies.
3) v 2 φ ∨ ψ and w 2 φ ∨ ψ.
We have v 2 φ, v 2 ψ, w 2 φ and w 2 ψ. Then v φ implies w φ and
v ψ implies w ψ. Hence v γ implies w γ.
Summing up, if we assume that φ B γ, ψ B γ, and M φ ∨ ψ, then, for every
pair (w, v) ∈ ε, v γ ⇒ w γ holds, that is M γ. So φ∨ψ Bγ, as required.
◦ Introduction of conjunction (I∧):
{φ} ∪ {ψ} B φ ∧ ψ
If an ordering ε satisfies φ and ψ, then for every (w, v) ∈ ε, if v φ, then
w φ, and if v ψ, then w ψ. Therefore, for such (w, v), if v φ ∧ ψ, we
have that v φ and v ψ, so also w φ and w ψ, i.e. w φ ∧ ψ.
This property together with, MON and CT give us the AND rule:
K Bφ K Bψ
K Bφ∧ψ
◦ Cautious Introduction of disjunction (CI∨):
{φ} ∪ {ψ} B φ ∨ ψ
If an ordering ε satisfies φ and ψ, that means that for every (w, v) ∈ ε, if
v φ, then w φ, and if v ψ, then w ψ. Then, for such (w, v), if
v φ ∨ ψ, we have that v φ or v ψ, so also w φ or w ψ, i.e. w φ ∨ ψ.
Note that we need both premises to derive the disjunction. In particular the
classical Introduction of disjunction (φ B φ ∨ ψ) is not valid.
To see this, take (w, v) ∈ ε s.t. v 2 φ, v ψ, w 2 φ and w 2 ψ, so v φ ∨ ψ
and w 2 φ ∨ ψ. For this pair v φ ⇒ w φ, but v φ ∨ ψ ; w φ ∨ ψ.
Finally, let us show that two important properties characterizing classical
negation and implication do not hold for B.
◦ Contraposition (CONTR):
ψBφ
¬φ B ¬ψ
Given ψ B φ, we need to find a model M s.t. M ¬φ and M 1 ¬ψ,
that is to say a pair (w, v) s.t. (w, v) satisfies v ¬φ ⇒ w ¬φ and
v ¬ψ ; w ¬ψ. Let (w, v) be v φ, v 2 ψ, w φ, and w ψ. This pair
satisfies v φ ⇒ w φ, v ψ ⇒ w ψ, and v ¬φ ⇒ w ¬φ, but it
does not satisfy v ¬ψ ⇒ w ¬ψ, as required.
◦ Right Weakening (RW):
φ→ψ K Bφ
K Bψ
Let (w, v) be v 2 φ, v ψ, w 2 φ, and w 2 ψ. This pair satisfies v φ ⇒
w φ, both v and w satisfy φ → ψ, but v ψ ; w ψ.
◦ Modus Ponens (MP):
K Bφ→ψ K Bφ
K Bψ
To see that MP does not hold for B it is enough to take the counterexample
used in the case of RW.
Note that failure of RW implies the failure of Supraclassicality. Supraclassicality (that is K φ, then K B φ) is a structural property which guarantees that
any given operator extends the tarskian consequence operator. One particularly
important consequence of the failure of Supraclassicality is that the normality
operator is not closed under classical consequence.
4
Default-revision
Recall that the main intuition underlying our characterization of normality is
based on the kind of “epistemic variation” that can be tolerated by a given preferential ordering (and therefore a given default-assumption consequence relation).
It is therefore natural to compare our normality operator with the epistemic
change operations constituting the AGM approach to theory change, namely
expansion, contraction and revision [1, ?].
The AGM model, which aims at characterizing the epistemic behaviour of
ideally rational agents, is centered around two key constraints: Logical closure
and consistency. The former imposes that, given a set K, an agent should behave
as if it accepted not only the information contained in K but also all its logical
consequence. The latter amounts to the requirement that no logical inconsistency should arise after the correct instantiation of any of the three epistemic
operations.
It is well known, however, (see, e.g. [8], Theor. 2.7.), that if the set of defaultassumptions K is closed under classical consequence, any default-assumption
consequence relation built up from K collapses to classical consequence. In order
to avoid this, we shall weaken the requirement of logical closure to closure under
the normality operator CB which takes a set of sentences as argument and returns
its closure under B as value, that is:
CB (A) = {α|A B α}.
In what follows, it will be useful to make the following terminological distinction. We shall refer to the finite set of default-assumptions K which determines
a default-assumption consequence relation as the default base, while we shall call
default sets those default bases D which are closed under the normality operator,
that is to say such that D = CB (D).
Besides the logical constraints of closure and consistency, the standard approach to theory revision adopts, as a heuristic constraint, the principle of socalled informational economy. Roughly speaking, this latter amounts to require
that any epistemic operation performed by an agent should result in the smallest
possible loss of information. This heuristic principle guides both the formalization of the normative postulates for expansion, contraction and revision, as well
as the explicit constructions, such as the Epistemic Entrenchment approach [5].
In the reminder we shall focus on the normality operators of Expansion, Contraction and Revision, leaving other constructions to future work.
4.1
Expansion
Expansion formalizes the epistemic operation of simply adding a sentence to a
default set D. That is, if an agent acquires the information that normally α
holds, then it will simply α to D and close this set under CB :
Dα+ := CB (D ∪ {α})
The normality expansion operator thus defined, satisfies the relevant AGM
postulates.
(+1) Dα+
is a default set.
(+ 1) forces the result of expansion to be again a default set, i.e. closed under
CB .
(+2) α ∈ Dα+ .
(+ 2) corresponds to the so-called success postulate and follows by REF.
(+3) D ⊆ Dα+ .
(+ 3) guarantees that no previous information is affected by the expansion of D
with α. Again, this follows by REF.
(+4) α ∈ D =⇒ Dα+ = D.
(+ 4) captures an aspect of informational economy: if the information to be
added is already in the agent’s default set, nothing changes. It clearly follows.
(+5) D ⊆ H =⇒ Dα+ ⊆ Hα+ .
(+ 5) is known as the monotonicity postulate. If D ⊆ H, then the adding of α
to D does not add anything that is not included also in Hα+ . It follows from the
monotonicity of B: D ⊆ H implies D∪α ⊆ H∪α, so CB (D∪{α}) ⊆ CB (H∪{α}).
(+6) For all belief sets D and all sentences α, Dα+ is the smallest belief set that
satisfies (+1) − (+5).
(+ 6) is known as the minimality postulate. It imposes that the new belief
set does not contain any extra information with respect to the addition of α to
D. To see that (+ 6) holds, let H be such that H ⊆ D. Assume H satisfies (+ 2)
and (+ 3), that is D ∪{α} ⊆ H. If H satisfies also (+ 1), then H = CB (H); given
Dα+ = CB (D ∪ {α}), we have Dα+ ⊆ H, by the monotonicity of B. Contradiction.
We conclude that the normality expansion operator qualifies as an AGMexpansion operator.
4.2
Contraction
Let’s now turn to the problem of removing a sentence from a given default set,
that is to say to the operation of normality contraction. As in the AGM case,
the problem of contraction is two-fold: on the one hand a specific sentence needs
removing from a default set; on the other hand, we need to to make sure that
its deduction is blocked in the new set. Of course there can be many ways to
achieve this latter result and the key heuristic principle to do so, is again the
principle of informational economy.
Given a default set D and a sentence α we define a function contr as follows:
∅
if α ∈
/ D or α = > or α = ⊥,
contrD (α) =
{α | α ∈
/ CB (D − contrD (α))} otherwise
Note that the clause relative to α = > and α = ⊥ accounts for the fact that
both classical tautologies and contradictions are B-valid sentences, and thus
cannot be removed from a default set. We can now define normality contraction
by letting
Dα− := CB (D − contrD (α)).
As for normality expansion, normality contraction is an AGM contraction
operation, as we can ascertain by checking the relevant postulates in turn.
(- 1) Dα− is a default set.
This immediately follows from the closure under CB .
(- 2) Dα− ⊆ K
Contraction simply eliminates information from D, so every sentence in Dα−
must already be in D. It follows from D − contrD (α) ⊆ D and the monotonicity
of B.
(- 3) If α ∈
/ D, then Dα− = D
If the sentence to be eliminated is not in D, then nothing needs changing. It
is satisfied by the clause that if α ∈
/ D, then contrD (α) = ∅.
(- 4) If 6 Bα, then α ∈
/ Dα−
This is the success postulate for contraction. The set resulting from a contraction must not contain the contracted sentence, unless it is a B-valid sentence (a
classical tautology or contradiction), which cannot be eliminated. This is guaranteed by the definition of contr(α).
(- 5) If α ∈ D, then D ⊆ (Dα− )+
α
Putting back a previously contracted sentence should not result in any loss of
information. This property is not always satisfied by the normality contraction
operator, but this need not worry us too much as this is probably the most
controversial principle of the AGM model. Furthermore, as we shall see in the
next section, its failure does not affect the desired properties of the normality
revision operator.
(- 6) If ` α ↔ β, then Dα− = Dβ−
Contraction should behave well with respect to classical equivalence. It follows from RLE that β ∈
/ Dα− and α ∈
/ Dβ− , but this alone does not guarantee
−
−
that Dα = Dβ . However this can be ensured in various ways depending on the
particular construction at hand. For space limitations, we shall postpone the
details to a follow-up paper.
4.3
Revision
The problem of revising a default set D consists in adding to D a sentence which
is potentially inconsistent with it, without affecting the consistency of D. We
formalize normality revision by means of a function *, which, takes a default set
D and a sentence α as inputs and returns a new consistent default set containing
α. Here, as in the AGM approach, consistency means classical consistency.
First of all, then, we need to make sure that our closure operator is consistency preserving, that is, for any given classically consistent default base K, the
closure operator should return a consistent default set. This is not immediate, in
our setting, since classical contradictions are B-valid sentences and so CB is not
consistency preserving. However we can easily constrain the closure operator B
to ensure consistency preservation. This is done by letting
KB0 φ ⇔ ∀M ∈ C if M ψ∀ψ ∈ K and ∃w ∈ W s.t. Kw = K, then M φ and w φ.
We can prove the following proposition.
Proposition 5.
CB0 (K) =
CB (K) − ⊥ if K is consistent
`
otherwise
Proof. If K is inconsistent, then there is no valuation satisfying it and vacuously
allows us to everything to follow Thus B0 is an explosive operator.
Otherwise, if K is consistent, there is a valuation w such that w ψ for
any ψ ∈ K. Since Kw = K, then w ≤εK v holds for every v ∈ W . We have to
show that CB0 (K) = CB (K) − ⊥. CB0 (K) ⊆ CB (K) − ⊥ is immediate, since
CB0 (K) ⊆ CB (K) and ⊥ ∈
/ CB0 (K) hold. To show that CB (K) − ⊥ ⊆ CB0 (K),
assume that there is a ψ 6= ⊥ s.t. ψ ∈ CB (K) and ψ ∈ CB0 (K). Since ψ 6= ⊥,
there must be a valuation v ∈ W such that v ψ. Moreover, the fact that
ψ ∈ CB (K) forces t ψ for every t ≤εK v. Now it follows from w ≤εK v, that
w ψ. Hence we get ψ ∈ CB0 (K), contradiction.
The above Proposition shows that, in the case of a consistent default base,
the constrained closure operator does preserve consistency, but it does so at the
price of eliminating only contradictions, which are irrelevant to the construction
of maximally consistent sets.
Consistency preservation and explosion give make B0 an intuitively appealing
operator in the characterization of normality: it would surely be counterintuitive
if an ideally rational agent could hold a sentence as a normal contradiction.
Returning to the formal properties, it is immediate to note that B0 satisfies
exactly the same structural properties as B, apart, obviously, from Contradiction. It is likewise easy to see that B0 behaves exactly as rdh insofar as the
properties of expansion and contraction are concerned. As a consequence we can
use CB0 characterize normality revision via the so-called Levi Identity (see e.g.
[4] p.69), which defines revision by means of a combination of expansion and
contraction:
− +
Kα∗ := (K¬α
)α .
The Levi Identity formalizes the two-step operation of revision, where the initial
contraction guarantees the consistency of the result while the final expansion
guarantees the success of the revision.
It often happens that revision is studied as a primitive operation ∗ which is
required to satisfy the following postulates:
(* 1) Dα∗ is a default set.
Closure for revision.
(* 2) α ∈ Dα∗ .
The success postulate of revision.
(* 3) Dα∗ ⊆ Dα+ . (* 4) If ¬α ∈
/ D, then Dα+ ⊆ Dα∗ .
(* 3) and (* 4) define the relation between revision and expansion. If the sentence
α is consistent with D, revision amounts to expansion.
(* 5) Dα∗ = D⊥ if and only if B0 ¬α.
The preservation of consistency is always satisfied, except in the case α is itself
a contradiction.
(* 6) If ` α ↔ β, then Dα∗ = Dβ∗ .
Revision is syntax-independent.
A major result in the field points out that the revision operator defined via
the Levi Identity satisfies the AGM postulates. Of course, in the AGM model
this result is obtained taking classical consequence as closure operator.
Theorem 3. If the normality contraction function - satisfies (- 1)-(- 4) and
(- 6), and the normality expansion function + satisfies (+ 1)-(+ 6), then the
normality revision function ∗, defined via the Levi identity, satisfies (* 1)-(* 6).
The proof is a straightforward adaptation from the original (see, e.g. [4]
theor.3.2). (Note that the satisfaction of (- 5) is not required for the representation theorem.)
Whether Theorem 3 can be extended to the postulates for Iterated revision
is a matter for future work.
5
Conclusions and further work
We have argued in favour of a characterization of normality based on defaultassumption consequence relations and we have shown how a normal situation can
be formalized by means of a default set with a stable preferential ordering. The
key notion intervening in our characterization is the normality operator B. Although we have shown how B arises very naturally in the preferential framework,
its interpretation is as yet not entirely satisfactory. Our initial investigations on
its logical properties, summed up in Section 3.2, have already put forward its
rather standard behaviour with respect to some key structural properties, while
highlighting some oddities with respect to the propositional connectives of negation and implication. We postpone fuller investigations on this to a follow-up
paper.
As to our general approach, it should be noted that one well-known drawback of the default-assumption framework is that logically equivalent assumption
sets might give rise to distinct consequence relations. For example, if we take
K = {p, p → q} and K 0 = {p, p → q, q}, or K = {p} and K 0 = {p, p ∨ q}, we generate distinct preferential orders, and therefore distinct consequence relations.
Indeed, our normality operator is not closed under classical logical consequence.
Whether this latter is a reasonable requirement for the logical characterization
of normality is open to debate.
We believe that one of the most interesting aspects of our characterization
of normality is its applicability to the general area of theory revision which we
have only begun to sketch in Section 4. One particularly interesting development
would be to extend Theorem 3 to the case of default bases as opposed to default
sets. This would allow us to characterize the case in which any kind of closure
fails. Also, our characterization of Default-revision seems to give rise to a revision
procedure which looks particularly promising from the computational point of
view. We shall pursue these matters in further work.
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