Download beliefrevision , epistemicconditionals andtheramseytest

Sten Lindström and Wlodzimierz Rabinowicz
BEL IEF REVIS ION, EP IS T EMIC CONDIT IONAL S AND
T H E RAMS EY T ES T *
1. Introduction
One often attributes to Frank Ramsey the following intuitive test for the acceptability of conditionals: The conditional proposition “If A, then B” is accepted (or acceptable) in a given state
of belief G just in case B should be accepted if G were revised with the new information A. To
give an example, anyone who accepts the notorious conditional “If Oswald did not kill
Kennedy, then someone else did” should be prepared to accept its consequent upon learning its
antecedent. Conversely, one might think that a person should accept the conditional if she is so
disposed as to accept that someone else killed Kennedy upon learning that Oswald did not.
It is clear, however, that the Ramsey test is not applicable to all conditionals. While it might
seem plausible for the indicative conditionals, it does not work so well for some subjunctive
ones. A person who accepts “If Oswald had not killed Kennedy, then no one else would have”
would most likely reject the consequent of this conditional upon learning that Oswald was innocent. This apparent difference in acceptability conditions suggests that there might be a semantical — and not only grammatical — distinction between two kinds of conditionals. On the one
hand, we have the epistemic (or doxastic) ones that express our dispositions to change our beliefs in the light of new information. These are the ones for which the Ramsey test appears
plausible. On the other hand, there are the “ontic” ones that are used to make factual claims
about the world. This distinction between two kinds of conditionals does not exclude the possibility that one and the same conditional statement may be used both to make an objective claim
and to express the speaker’s dispositions to change her beliefs.1
2
But even if we restrict our attention to epistemic conditionals, the Ramsey test turns out to be
problematic. Gärdenfors (1988) shows that this test, despite its initial attractiveness, is incompatible with certain plausible conditions on belief revision — at least as long as belief states are
viewed as sets of propositions (or sentences) and epistemic conditionals are treated as possible
members of such sets.
In view of Gärdenfors’ impossibility result, one can do as Levi (1988) suggests: deny that
epistemic conditionals have truth-values and for this reason deny that they can be members of
belief sets. Alternatively, one can continue to treat epistemic conditionals as possible members
of belief sets and instead investigate what happens if we either (a) keep the Ramsey Test but
weaken Gärdenfors’ axioms on belief revision, or (b) keep Gärdenfors’ axioms but replace the
full Ramsey test with its weaker variants — variants that still make the epistemic conditionals
essentially dependent on belief revision. In this paper, we are going to study the alternatives (a)
and (b). Unlike Gärdenfors, however, we are not assuming that for every belief state G and
every proposition A, there is a unique revision of G with A. Instead of looking upon belief revision as a function, we treat it as a relation: there might be several reasonable ways of revising
G with A.
We shall develop a semantic modelling in which one and the same relation is used both to
model belief revision and to give truth-conditions for epistemic conditionals. Thereby, we get a
bridging principle between the two notions, which we call the principle of normality (with respect to conditionals). Normality yields a weak version of the Ramsey test (WRR) which ties
the acceptance of a conditional “If A, then B” in a belief state to revisions of all the maximally
consistent extensions of that state with A.
As far as the alternative (a) is concerned, we present a modelling which validates the full
Ramsey test while preserving almost all of the Gärdenfors axioms for belief revision. This gives
us a consistency result. However, the modelling in question is quite unintuitive. We believe that
it is the alternative (b) that is philosophically more promising. In this connection, we show that
all of the Gärdenfors axioms are compatible with the Weak Ramsey test together with the full
Ramsey test from left to right: the conditional “if A, then B” is accepted in a belief state G only
3
if B is accepted in all the possible A-revisions of G. In the last section of the paper, we argue
that it is this direction of the Ramsey test, not the opposite one, that is intuitively correct.2
We also consider the following “Ramsey test” for epistemic “might”-conditionals:
The “might”-conditional “If A, then (for all that we know) B might be the case” is accepted in a belief state G just in case B is compatible with some possible revision of G
with A.
It is shown that this principle gives rise to an impossibility result that is similar to the one proved
by Gärdenfors. However, if “might”-conditionals are interpreted as the duals of “if—then”conditionals, the principle of normality gives us a weak version of the Ramsey test for
“mights”. At the same time, our modelling for alternative (b) validates the full Ramsey test for
“mights” only in the left-to-right direction: from conditionals to revisions.
2. Logics
We consider the sentential language L with the following symbols: (i) atomic sentences: p1,
p2,... ; (ii) classical connectives: ⊥ (falsity), → (the material conditional); (iii) the connective >,
thought of as representing the epistemic conditional; (iv) parentheses. The set Φ of sentences is
the smallest set such that: (i) all atomic sentences are in Φ; (ii) ⊥ ∈ Φ; (iii) if A, B ∈ Φ, then
(A → B), (A > B) are in Φ. The connectives ¬ (negation), ∧ (conjunction), ∨ (disjunction) and
↔ (the material biconditional) are introduced as abbreviations in the metalanguage in the usual
way. For instance, ¬A = A → ⊥.
A logic L is a set of sentences such that:
(i)
all truth-functional tautologies are in L;
(ii)
L is closed under modus ponens, that is, if A ∈ L and (A → B) ∈ L, then
B ∈ L.
For every logic L and set Γ of sentences, we define CnL(Γ) to be the smallest set of sentences
that includes Γ ∪ L and is closed under modus ponens. We say that A is an L-consequence of
4
Γ (also written as Γ ªL A) if A ∈ CnL(Γ). Of course, CnL: P(Φ) → P(Φ) (where P(Φ) is the
power set of Φ) is a consequence operation over Φ: that is, for all Γ, ∆ ⊆ Φ,
(i)
Γ ⊆ CnL(Γ);
(ii)
CnL(Γ) = CnL(CnL(Γ));
(iii)
if Γ ⊆ ∆, then CnL(Γ) ⊆ CnL(∆).
Moreover, every logic satisfies compactness and the deduction theorem:
(iv)
if Γ ªL A, then for some finite ∆ ⊆ Γ, ∆ ªL A;
(v)
if Γ ∪ {A} ªL B, then Γ ªL A → B.
Let L be a logic. L is (absolutely) consistent if and only if L ≠ Φ. A set Γ ⊆ Φ is said to be Lconsistent if CnL(Γ) ≠ Φ. A is said to be L-consistent if {A} is L-consistent. Γ is an L-theory
if and only if L ⊆ Γ and Γ is closed under modus ponens. In other words, Γ is an L-theory just
in case Γ = CnL(Γ). Note that every L-theory is a logic in its own right. A set Γ is L-maximal if
and only if Γ is L-consistent and for every ∆, if Γ ⊆ ∆ and ∆ is L-consistent, then Γ = ∆. Notice
that every L-maximal set is an L-theory. The following holds for every logic L:
Lindenbaum’s Lemma: Every L-consistent set is included in an L-maximal theory.
We write ML, TL for the set of all L-maximal theories and the set of all L-theories, respectively.
m, m’, m”,... are variables ranging over L-maximal theories and G, H, K, T, T’,... range over Ltheories. We also introduce the following notation:
for any Γ ⊆ Φ, |Γ|L = {m ∈ ML: Γ ⊆ m};
for A ∈ Φ, |A|L = |{A}|L = {m ∈ ML: A ∈ m}.
In what follows, we shall often suppress the subscript L in contexts where the logic is assumed
to be fixed.
The following statements are immediate consequences of Lindenbaum’s lemma:
(i)
L = ∩(ML), that is, A is a theorem of L (i.e., A belongs to L) iff A belongs to
every L-maximal theory;
5
(ii)
Cn(Γ) = ∩(|Γ|). That is, A is an L-consequence of Γ iff A belongs to every Lmaximal extension of Γ.
In particular, if G is an L-theory, then G = ∩ |G|. As a consequence, the function |...|L yields a
one-to-one mapping from TL into P(ML). It should also be noted that:
|⊥| = ∅
|A → B| = (ML - |A|) ∪ |B|.
The other truth-functional connectives behave analogously.
3. Belief Revision Systems and the Impossibility Result
Gärdenfors (1988) treats belief revision as a function on belief sets (i.e., theories). Here we
shall consider a generalization of this approach in which belief revision is seen as a relation
between theories. This generalization was already suggested in Lindström and Rabinowicz
(1989).
DEFINITION 3.1.
(a) A belief revision system (b. r. s) is an ordered pair <L, R> such that L is a consistent
logic, R ⊆ TL × Φ × T L and the following requirements are satisfied for all theories G, H and
sentences A, B:
(R0)
(∃H ∈ TL)(GRAH).
(Seriality)
(R1)
If GRAH, then A ∈ H.
(Success)
(R2)
If A as well as G are L-consistent and GRAH, then H is L-consistent.
(Consistency Preservation)
(R3)
If A ↔ B ∈ L and GRAH, then GRBH. (Substitutivity of Logical Equivalents)
(b) A b. r. s. is said to be functional if, in addition to (R0) - (R3), it satisfies:
(Funct) If GRAH and GRAK, then H = K.
It is said to be preservative, if it satisfies:
6
(Pres)
If ¬A ∉ G and GRAH, then G ⊆ H.
The system is Ramsey, if it satisfies the relational version of the Ramsey test:
(RR)
If G is L-consistent, then A > B ∈ G iff for every H such that GRAH, B ∈ H.3
(c) If a belief revision system <L, R> is functional, then, following Gärdenfors, we can write
it in the form <L, ∗>, where ∗ is the operation from TL × Φ into TL such that for every G and A,
G ∗ A is the unique theory H such that GRAH.
Of course, conversely, every such
“operational” b. r. s. can be regarded as a “relational” system which satisfies functionality.4
If a belief revision system is both functional and Ramsey, then it satisfies Gärdenfors’
Ramsey test:
(RT)
if G is L-consistent, A > B ∈ G iff B ∈ G ∗ A.
This principle implies in turn that the belief revision system is monotonic, that is, it satisfies:
(Mon)
if G ⊆ H and H is L-consistent, then G ∗ A ⊆ H ∗ A.
Two sentences A and B are completely L-independent if neither A nor ¬A is an L-consequence of B or ¬B. We say that a logic L is non-trivial, and a belief revision system <L, R> is
non-trivial, if there exist two completely L-independent sentences.
Gärdenfors (1988) has proved a theorem that in our framework may roughly be expressed as
follows: no functional and non-trivial b. r. s. can be preservative and monotonic at the same
time. Or to put it differently, no such b. r. s. can be both preservative and Ramsey. The natural
question that one might ask is whether this impossibility theorem essentially depends on the
Functionality assumption. The answer is that it does not.
THEOREM 3.2. There is no non-trivial b. r. s. <L, R> that is both preservative and Ramsey.
Proof: In the same way as (RT) implies (Mon) in the functional case, (RR) implies the relational version of Monotonicity:
(RMon) If G ⊆ H, H is L-consistent, and B ∈ K for all K such that GRAK, then B ∈ K
for all K such that HRAK.
7
It is easy to see that, in the presence of Functionality, (RMon) and (Mon) are equivalent.
We are going to show that (Pres) and (RMon) are mutually incompatible conditions on nontrivial belief revision systems. Our proof is essentially an adaptation to the relational case of
Segerberg’s proof (1989) of Gärdenfors’ theorem (see also Gärdenfors (1988), pp. 59 f).
We assume that <L, R> is a non-trivial b. r. s. satisfying (Pres) and (RMon). In particular,
suppose that A and B are completely L-independent. Consider the theories:
G = Cn({A}), H = Cn({B}) and K = Cn({A, B}).
Let C = ¬A ∨ ¬B. Clearly, given the complete L-independence of A and B, we have:
(1)
¬C ∉ G and ¬C ∉ H.
(1) together with (Pres) and Success (R1) yields, for all theories T:
(2)
if GRCT, then A, C ∈ T;
(3)
if HRCT, then B, C ∈ T.
Since G, H ⊆ K and K is L-consistent (given that A and B are completely L-independent), (2)
and (3) imply, by (RMon), that for all T:
(4)
if KRCT, then A, B, C ∈ T.
But the set {A, B, C} is L-inconsistent, so T must be L-inconsistent as well. By Consistency
Preservation (R2), this means that either K or C is inconsistent. But this is impossible, given the
complete L-independence of A and B.
M
The Ramsey test is an attractive principle for epistemic conditionals, insofar as it expresses a
close and, at first sight, very natural relationship between such conditionals and belief revision.
On the other hand, Preservation is an appealing condition on belief revision. It seems to follow
from the idea that one should keep one’s beliefs if one does not have to change them. Or to
quote Gärdenfors’ version of Harman’s conservativity principle: “When changing your beliefs
in response to new evidence, you should continue to believe as many of the old beliefs as possible”.5 This principle is especially appealing when we think of the theory as being applied to
rather primitive artificial agents.
8
A possible objection to Preservation might be formulated by inductivists: sometimes when
you receive new evidence consistent with your old beliefs, some of the old beliefs have to go —
not because they are logically incompatible with the new information but because this information disconfirms them. They are no longer sufficiently probable given the new evidence. If
the assumption here is that belief consists in high degree of confidence, then this objection rests
on an essentially different interpretation of the notion of belief than the one that we consider in
this paper. Here, belief should be understood as certainty, and not just as a high probability.
Belief sets are assumed to be theories, that is, they are closed under classical consequence. This
closure assumption is difficult to combine with the high degree of confidence-interpretation of
beliefs: the conjunction of premises, each of which is highly probable, may be highly
improbable (the Lottery Paradox).6
The objection may, however, be interpreted in another way. Even if belief is identified with
certainty, new evidence may undermine one’s original certainties without being logically incompatible with them. For instance, if you are certain that most F’s are G’s, your certainty may
well be undermined by repeated observations of F’s that are non-G’s. Clearly, such a nonBayesian behavior would make you vulnerable to diachronic Dutch-books, but it may be
questioned whether such diachronic vulnerability is a sign of irrationality. If this objection is
well-taken, then Preservation should be rejected as a general rationality requirement on belief
revision.
The above described counterexample to Preservation may be seen as a special case of a more
general objection to Preservation which can be formulated within Harman’s (1986) coherence
approach to belief revision. Harman writes (1986, p. 32):
“The coherence theory is conservative in a way the foundations theory is not. The coherence theory supposes one’s present beliefs are justified as they are in the absence of special reasons to change them,
where changes are allowed only to the extent that they yield sufficient increases in coherence.
...According to the coherence theory, if one’s beliefs are incoherent in some way, because of outright
inconsistency or simple ad hocness, then one should try to make minimal changes in those beliefs in
order to eliminate the incoherence. More generally, small changes are justified to the extent these
changes add to the coherence of one’s beliefs.
For present purposes, I do not need to be specific as to exactly what coherence involves, except to say
that it includes not only consistency but also a network of relations among one’s beliefs, especially relations of implication and explanation.
It is important that coherence competes with conservatism. It is as if there were two aims or
tendencies of reasoned revision, to maximize coherence and to minimize change.”
9
Consider now the case of revising a theory H with a sentence A which is consistent with H.
According to Harman’s coherence view, it may very well be the case that some sentences in H
are given up in the process. One method of revising H with A may be, first, to expand H with A
(i.e., add A and close the result under logical consequence), and then, to make small adjustments
in the resulting theory, in order to increase coherence. Obviously, a process of this kind would
not guarantee that Preservation is fulfilled. As a matter of fact, even Success may be violated,
since, after it has been added, A may be given up for the sake of coherence.7
However, the force of the above objection to Preservation (and Success) is diminished
somewhat by the following observation: belief changes that seem to violate these postulates can
be accommodated within a theory of Gärdenfors type by describing them not as revisions but as
sequences of revisions and contractions.8 In other words, we may still want to keep the postulates of Preservation and Success for the formal operation of belief revision while explaining
apparent counterexamples to these principles by invoking the operation of contraction. One
reason for contracting a theory with a sentence may be that such a step would lead to a theory
with greater coherence.
Given the incompatibility of the relational Ramsey test (RR) and Preservation, there are two
ways to proceed: either we can weaken Preservation — as little as possible, or we can try to find
weaker versions of the Ramsey test — versions that are compatible with Preservation but still
express the idea that epistemic conditionals depend on belief revision. In view of the discussion
above, both alternatives may be worth pursuing.
However, before exploring these two
possibilities, we are going to prepare the ground by developing a general semantics that ties
epistemic conditionals to belief revision.
4. Semantics for Conditionals and Belief Revision
DEFINITION 4.1. A structure (or a belief-relation structure ) is a four-tuple S = <W, Π, r, V>
such that W is a non-empty set; Π is is an algebra of sets on W, i.e., Π is a subset of P(W) such
that W ∈ Π and Π is closed under complements (relative to W) and finite unions; r ⊆ P(W) ×
10
Π × P(W); V is a function that assigns a value V(p) in Π to every atomic sentence p. Structures
are assumed to satisfy the following conditions:
(Comp) if F ⊆ Π is such that for every finite G ⊆ F, ∩G ≠ ∅, then ∩F ≠ ∅;
(compactness)
(r0)
for all X ⊆ W and all P ∈ Π, there exists a Y ⊆ W such that r(X, P, Y);
(seriality)
(r1)
if r(X, P, Y), then Y ⊆ P;
(r2)
if r(X, P, Y), X ≠ ∅ and P ≠ ∅, then Y ≠ ∅; (consistency preservation)
(r3)
if X ≈ Y and Z ≈ U and r(X, P, Z), then r(Y, P, U).
(success)
(congruence)
In (r3), ≈ is the relation between subsets of W which is defined by:
X ≈ Y iff (∀P ∈ Π)(X ⊆ P iff Y ⊆ P).
We say that X and Y are equivalent if X ≈ Y holds. ≈ is, of course, an equivalence relation on
P(W). According to (r3) it is also a congruence relation with respect to r.
The elements of W may be thought of as the doxastically possible worlds. The elements of
Π are the propositions of the model. (Some sets of worlds may not correspond to propositions). V, the valuation function of S, is an assignment of propositions to the atomic sentences
of the language. Subsets of W may be thought of as representing states of belief, where the
propositions accepted in a given state of belief X are all those P in Π that include X. W corresponds to total ignorance and ∅ to the absurd, or inconsistent belief state. Note that, given this
interpretation, elements w of W may be viewed as “fully opinionated”, maximally consistent
belief states: for every proposition P either P or its negation is accepted in a world w (or,
equivalently, in the belief state {w}). Intuitively, the relation ≈ holds between two subsets X and
Y of W if and only if they represent the same belief state, that is, just in case the same
propositions are accepted in X and Y. The belief states may be thought of as the equivalence
classes of subsets of W with respect to ≈. Alternatively, we may identify belief states with
closed subsets of W: For every X ⊆ W, let the closure of X, Cl(X), be ∩{P ∈ Π: X ⊆ P}. We
say that X is closed if X = Cl(X). Notice that, for any X, Cl(X) = ∪{Y: Y ≈ X}. Also, for any X
11
and Y, X ≈ Y if and only if Cl(X) = Cl(Y). In what follows, we shall, however, somewhat loosely
refer to any subset of W as a belief state.
If r(X, P, Y), then Y may be viewed as a state of belief that is a possible result of revising the
belief state X with the new information P. Condition (r1) guarantees that the new information is
always accepted in the revised belief state Y. Condition (r0) says that it is always possible to
revise any belief state with any proposition. (r2) guarantees that the revised belief state Y is
consistent provided that the original belief state as well as the new information is consistent.
According to (r3), subsets X and Y of W that represent the same belief state behave in the same
way with respect to the revision relation r. Finally, the compactness condition (Comp) says that
a family of propositions is consistent — has a non-empty intersection — if each of its finite
subsets is consistent. This condition is the semantic equivalent of the assumption that the underlying logic of a belief revision system is compact.9
We now proceed to the specification of the truth-conditions of sentences in our language
relative to a structure S = <W, Π, r, V>. We define what it means for a sentence A to be true at a
world w in S (in symbols, S ‚w A). Note that the following should hold: if a sentence A corresponds to a proposition P, then A is true in a world w if and only if P is accepted in the belief
state {w}.
If A is a atomic sentence, then S ‚w A holds if and only if w ∈ V(A). The clauses for ⊥ and
→ are the standard ones. We adopt the following truth condition for the epistemic conditional:
(>r)
S ‚w A > B iff for every Y ⊆ W, if r(w, ≠A≠S, Y), then Y ⊆ ≠B≠S.
Here, ≠A≠S = {w ∈ W: S ‚w A}. We also write r(w, P, Y) instead of r({w}, P, Y).
Intuitively, A > B is true in w or, what amounts to the same, is accepted in the belief state {w}
if and only if B is true in every A-revision of w (iff B is accepted in every belief state that we can
reach when we revise w with A). It should be noted that (>r) is nothing but the semantic
correlate of the relational Ramsey test restricted to “fully opinionated” belief states. We regard
this condition as a sine qua non for an epistemic interpretation of the conditional connective.
Note that r(w, ≠A≠S, Y) can hold only when ≠A≠S ∈ Π. That this is the case is, however, not
guaranteed by our definition of truth at a world in a structure. When ≠A≠ S ∉ Π, then S ‚ w A
12
> B will hold vacuously. In order to avoid such vacuous cases, we must restrict our attention to
structures S in which, for any sentence A, ≠A≠S is a proposition. Thus, we define the notion of
a model in the following way:
Consider a structure S = <W, Π, r, V>. For all P, Q ∈ Π, define the set P ⇒ Q as follows:
P ⇒ Q = {w ∈ W: for all Y ⊆ W, if r(w, P, Y), then Y ⊆ Q}.
In particular, if ≠A≠S = P and ≠B≠S = Q, then ≠A > B≠S = P ⇒ Q. We say that S is a model,
or a belief-relation model, just in case its set of propositions Π is closed under ⇒, that is, for all
P, Q ∈ Π, P ⇒ Q ∈ Π. We use letters M, M’,... for models. It is easy to show that for any
model M, ≠A≠M is a proposition for any sentence A.
We say that the model M = <W, Π, r, V> determines the b. r. s. <L, R> if the following
conditions are satisfied:
(i)
for every Γ and A, Γ ªL A iff ≠Γ≠M ⊆ ≠A≠M;
(ii)
for all L-theories G, and H and all sentences A,
GRAH iff r(≠G≠M, ≠A≠M, ≠H≠M).
Here, ≠Γ≠M stands for {w: for every B ∈ Γ, M ‚w B}.
In view of the compactness of M, (i) is equivalent to the simpler condition:
(i’)
L = {A: for all w ∈ W, M ‚w A}.
That (i’) follows from (i) is trivial. The other direction is proved as follows: Assume that
Γ ªL A. Then, there are sentences B1,..., Bn ∈ Γ such that B1 ∧... ∧ Bn → A ∈ L. Let w ∈
≠Γ≠M. Then M ‚w Bi for i = 1,..., n. Hence, M ‚w A. Thus, ≠Γ≠ M ⊆ ≠A≠M . For the converse, assume that Γ ªL A does not hold. Then, Γ ∪ {¬A} is L-consistent. Consider the
family of propositions:
F = {≠C≠M: C ∈ Γ ∪ {¬A}}.
Let G be a finite subfamily of F, say G = {≠B1≠M,..., ≠Bn≠M, ≠¬A≠M}, where B1,..., Bn ∈ Γ.
Then, ∩G ≠ ∅, since otherwise B1 ∧ ... ∧ Bn → A ∈ L contrary to the assumption. Hence, by
the compactness of M, ∩F ≠ ∅. That is, ≠Γ≠M ⊆ ≠A≠M does not hold.
M
13
A model M = <W, Π, r, V>is said to be meager (with respect to the language L) if every
proposition P in M is expressible by some set Γ of sentences, that is, if for every P in Π, there is
some set Γ such that P = ≠Γ≠M. Instead of saying that M is meager with respect to L, we could
just as well say that L is sufficiently rich with respect to M.
If X, Y ⊆ W, we shall say that X and Y are L-equivalent (in symbols, X ≈L Y) if for all
sentences A in L, X ⊆ ≠A≠M iff Y ⊆ ≠A≠M. It can be shown that M is meager if and only if Lequivalence coincides with the equivalence relation ≈ on M.
LEMMA 4.2. Every meager model M = <W, Π, r, V> determines a unique b. r. s. <L, R>.
This lemma, like several of the theorems and lemmas below, is proved in the Appendix.
A belief revision system <L, R> shall be said to be >-normal if it satisfies the condition:
(>-norm)
for every L-maximal theory m, A > B ∈ m iff for every L-theory H, if
mRAH, then B ∈ H.
(>-norm) is a syntactic correlate of the semantic principle (>r) which, as we remember,
establishes a minimal Ramsey-type connection between epistemic conditionals and belief revision. Via Lindenbaum’s lemma it is easy to show that >-normality is equivalent to what might
be called the Weak Relational Ramsey Test:
(WRR)
A > B ∈ G iff for every L-maximal theory m such that G ⊆ m and for
every H such that mRAH, B ∈ H.
If a belief revision system <L, R> is >-normal, then the logic L can be shown to satisfy the
following principles for the conditional connective >:
(Classicality)
If A ↔ B, C ↔ D ∈ L, then (A > C) ↔ (B > D) ∈ L;
(K)
[A > (B ∧ C)] ↔ [(A > B) ∧ (A > C)] ∈ L;
(N)
A > Τ ∈ L (where Τ is ¬⊥);
(1)
A > A ∈ L;
(2)
(A > ⊥) → (B > ¬A) ∈ L.
14
The first three principles make L a normal conditional logic in the sense of Chellas (1975).
Classicality follows from >-normality together with (R3). (K) and (N) follow directly from >normality. (1) is due to Success (R1). (2), finally, follows, in the presence of >-normality, from
Consistency Preservation (R2) and Seriality (R0).
LEMMA 4.3. If a belief revision system is determined by a meager model, then it is >-normal.
THEOREM 4.4. Let B = <L, R> be a >-normal b. r. s.. Then, there exists a meager model
which determines B. More specifically, this is the case for the canonical model M =
<W, Π, r, V> for B, which is defined by the conditions:
(i)
W = ML, the set of all L-maximal theories;
(ii)
Π = {|A|L: A is a sentence};
(iii)
for all X, Y ⊆ W and all P ∈ Π, r(X, P, Y) iff for some sentence A and some Ltheories G, H,
P = |A|L, X ≈ |G|L, Y ≈ |H|L and GRAH.
(iv)
for every atomic sentence p, V(p) = |p|L.
Thus, the >-normal belief revision systems are exactly those that are determined by meager
models. This result may also be described in another way. Let us say that a belief revision
system <L, R> corresponds to a model M = <W, Π, r, V> if
(1)
M determines <L, R>;
(2)
for every closed set X ⊆ W, there is an L-theory G such that X = ≠G≠M.
As we remember, closed sets may be seen as “proper” representatives of belief states. Thus,
(2) amounts to the claim that every belief state in M is expressed by some theory in <L, R>. It
can be shown that (2) holds if and only if M is meager. Hence, what we have proved above can
also be formulated as follows: The >-normal belief systems are exactly those that correspond to
models.
What about those belief revision systems that are determined by models but do not correspond to any model? It can be shown that any such b. r. s. can be expanded to one that is >-
15
normal and therefore corresponds to a model by an appropriate expansion of the language. The
expansion in question may demand the addition of uncountably many new atomic sentences.
THEOREM 4.5. A belief revision system <L, R> in the original language L is determined by a
model M if and only if there exists an extension L’ of the language L obtained by adding new
atomic sentences to L and a belief revision system <L’, R’> in L’ such that:
(i)
<L, R> is the restriction of <L’, R’> to the language L; and
(ii)
<L’, R’> is >-normal.
Sketch of the Proof: The if-part of the theorem is trivial. To prove the only-if-part, suppose
that <L, R> is determined by M = <W, Π, r, V>. We expand the original language L to a new
language L’ by adding to L a new atomic sentence P
for each proposition P ∈ Π. In the
model, the valuation function is correspondingly expanded by letting V( P ) = P. Call the resulting model M’. Let
L’ = {A in L’: for all w ∈ W, M’ ‚w A}.
For all L’-theories G, H and all sentences A in L’, let
GR’AH iff r(≠G≠M’, ≠A≠M’, ≠H≠M’).
Then it can be verified that M’ determines <L’, R’>. Since M’ is meager, <L’, R’> is >-normal.
M
Also, it is clear that <L, R> is the restriction of <L’, R’> to the language L.
For future use, we introduce a number of possible conditions on belief revision systems and
corresponding conditions on models:
(Funct)
If GRAH and GRAK, then H = K.
(Functionality)
(Pres)
If ¬A ∉ G and GRAH, then G ⊆ H.
(WPres)
If ¬A ∉ G, then for some H, GRAH and G ⊆ H.
(Preservation)
(Weak
Preservation)
(TPres)
If A ∈ G, ⊥ ∉ G and GRAH, then G ⊆ H. (Trivial Preservation)
16
If GRAH, then H ⊆ G+A,
(Min)
(Minimality)
where G+A is the expansion of G with A, i.e., G+A = Cn(G ∪ {A}).
(StepRev)
If GRAH and HRBK and H ⊆ K, then GRA∧BK.
(SCons) If ¬A ∉ L, and GRAH, then ⊥ ∉ H.
G ⊕ A ⊆ ∩{m ⊕ A: G ⊆ m}
(CU)
(Stepwise Revision)
(Strong Consistency)
(Connection Upwards)
where for any L-theory G, G ⊕ A = ∩{H: GRAH}. That is, G ⊕ A is the common part of all
the A-revisions of G.
(CD)
If ⊥ ∉ G, ∩{m ⊕ A: G ⊆ m} ⊆ G ⊕ A.
(RCU)
If ¬A ∈ G, then G ⊕ A ⊆ ∩{m ⊕ A: G ⊆ m}.
(Connection Downwards)
(Restricted Connection Upwards)
Conditions on models:
(funct)
If r(X, P, Y) and r(X, P, Z), then Y ≈ Z.
(functionality)
(pres)
If X ∩ P ≠ ∅ and r(X, P, Y), then Y ⊆ Cl(X).
(preservation)
(wpres)
If X ∩ P ≠ ∅, then for some Y ⊆ W, r(X, P, Y) and Y ⊆ Cl(X).
(weak preservation)
(tpres)
If ∅ ≠ X ⊆ P and r(X, P, Y), then Y ⊆ Cl(X).
(trivial
preservation)
(min)
If r(X, P, Y), then X ∩ P ⊆ Cl(Y).
(minimality)
(steprev) If r(X, P, Y), r(Y, Q, Z) and Z ⊆ Y, then r(X, P ∩ Q, Z).
(stepwise revision)
(scons)
If P ≠ ∅ and r(X, P, Y), then Y ≠ ∅.
(cu)
◊{w ⊕ P: w ∈ X} ⊆ X ⊕ P,
(strong consistency)
(connection upwards)
where for any family F of subsets of W, ◊F = Cl(∪F) and where X ⊕ P = ◊ {Y: r(X, P, Y)}.
(We write w ⊕ P instead of {w} ⊕ P.) Thus, ◊ F is the belief state that represents what is
17
common to all the belief states in F. Correspondingly, X ⊕ P is the belief state that represents
what is common to all the P-revisions of X.
(cd)
If X ≠ ∅ and X = Cl(X), then X ⊕ P ⊆ ◊{w ⊕ P: w ∈ X}.
(connection downwards)
(rcu)
If X ∩ P = ∅, then ◊{w ⊕ P: w ∈ X} ⊆ X ⊕ P.
(restricted connection upwards)
THEOREM 4.6. If <L, R> is a belief revision system that corresponds to the model M and M
satisfies a condition (cond) from the above list, then <L, R> satisfies the corresponding condition (Cond).
The proof of the theorem is straightforward but tedious. As an example, we show how to
derive (CD) from (cd). Suppose that M satisfies (cd) and that <L, R> corresponds to M. We
assume that ⊥ ∉ G and that B ∈ ∩ {m ⊕ A: G ⊆ m}. This means that ≠G≠M ≠ ∅ and that for
every m and H, if G ⊆ m and r(≠m≠M, ≠A≠M, ≠H≠M), then ≠H≠ M ⊆ ≠B≠M . We want to
show that:
(∗)
◊{w ⊕ P: w ∈ X} ⊆ ≠B≠M.
By the properties of closure, (∗) will follow if we can prove:
(∗∗)
∪{w ⊕ P: w ∈ X} ⊆ ≠B≠M.
Suppose that w ∈ ≠G≠M and r(w, P, Y). Clearly, w ∈ ≠m≠M for some m ⊇ G. By the meagerness of M, w ≈ ≠m≠M and Y ≈ ≠H≠M, for some L-theory H. By (r3), therefore, r(w, P, Y)
implies that r(≠m≠M, ≠A≠M, ≠H≠M). But ≠H≠M ⊆ ≠B≠M by the assumption. Therefore, Y ⊆
≠B≠M. This proves (∗∗).
Since ≠G≠ M = Cl(≠G≠M ) and ≠G≠ M ≠ ∅, (∗) and (cd) imply that ≠G≠ M ⊕ P ⊆ ≠B≠ M .
That is, if r(≠G≠M, P, Y), then Y ⊆ ≠B≠M. But then, for every H, if GRAH, B ∈ H.
M
THEOREM 4.7. If the b. r. s. <L, R> satisfies one of the conditions (Cond), then the canonical
model for <L, R> satisfies the corresponding condition (cond).
18
Again, we only carry out the proof for one of the cases. Thus, assume that <L, R> satisfies
(CD), that is:
If ⊥ ∉ G and for every m such that G ⊆ m, B ∈ ∩{H: mRAH}, then B ∈ ∩{H:
GRAH}.
This condition is easily seen to imply in the canonical model:
If X = Cl(X) and X ≠ ∅, then ∀Q ∈ Π[∀m∀Y(m ∈ X ∧ r(m, P, Y) → Y ⊆ Q) →
∀Y(r(X, P, Y) → Y ⊆ Q)].
Hence,
if X = Cl(X) and X ≠ ∅, then for all Q in Π,
if ∪w∈X{Y: r(w, P, Y)} ⊆ Q, then ∪{Y: r(X, P,Y)} ⊆ Q.
However, this means that:
If X = Cl(X) and X ≠ ∅, then ∪w∈X{Y: r(w, P, Y)} ⊆ Cl[∪{Y: r(X, P,Y)}],
which, by the properties of closure, is equivalent to the condition (cd).
M
Together, the two theorems above state that the relation of correspondence between models
and belief revision systems is a function mapping the class of all meager models satisfying
(cond) onto the class of all >-normal belief revision systems satisfying (Cond). This means that
each “syntactical” condition (Cond) has (cond) as its precise semantical counterpart. (cond)
validates (Cond) and it does not validate any syntactical condition that is logically stronger than
(Cond).
DEFINITION 4.8.
(a) A b. r. s. <L, R> is Gärdenfors-type if it satisfies Preservation, Minimality, Stepwise
Revision and Strong Consistency. If, in addition, <L, R> is functional, then R satisfies all the
axioms on belief revision imposed by Gärdenfors (1988), section 3.3 (axioms (K*1) (K*8)).10 See also Alchourrón, Gärdenfors and Makinson (1985) and the references therein.
Correspondingly, a model is Gärdenfors-type if it satisfies preservation, minimality, stepwise
19
revision and strong consistency. Clearly, if <L, R> corresponds to M, then <L, R> is
Gärdenfors-type iff M is Gärdenfors-type.
(b) A b. r. s. <L, R> is almost Gärdenfors-type if it satisfies Weak Preservation, Trivial
Preservation, Minimality, Stepwise Revision and Strong Consistency.
Trivial Preservation
immediately follows from Preservation. Given Seriality, the latter condition also implies Weak
Preservation. Thus, one might say that an almost Gärdenfors-type b. r. s. differs from a
Gärdenfors-type one in having weaker forms of preservation. However, this distinction can only
be maintained for non-functional belief revision systems.
Given Functionality, Weak
Preservation implies full Preservation.
A model M is almost Gärdenfors-type if it satisfies weak preservation, trivial preservation,
minimality, stepwise revision and strong consistency.
In the next section, we are going to consider almost Gärdenfors-type belief revision systems
and models when trying to combine the Ramsey test for conditionals with weaker forms of
preservation.
5. Relational Ramsey and Weakened Preservation
In order to investigate Relational Ramsey (RR), we need first to “deconstruct” it into a number
of simpler components.
LEMMA 5.1. The condition
(RR)
If ⊥ ∉ G, then A > B ∈ G iff for every H such that GRAH, B ∈ H.
is equivalent to to the conjunction of three conditions: >-normality, Connection Upwards and
Connection Downwards.
To be more specific, given >-normality, Connection Upwards is equivalent to:
(RR⇐) A > B ∈ G if for every H such that GRAH, B ∈ H,
while Connection Downwards is equivalent to:
20
(RR⇒) If ⊥ ∉ G and A > B ∈ G, then for every H such that GRAH, B ∈ H.
As we know, (RR) and (Pres) are incompatible on pain of triviality. Therefore, a non-trivial
Gärdenfors-type b. r. s. cannot satisfy (RR). Could we, perhaps, accommodate (RR) by
weakening Preservation? In particular, are there any non-trivial almost Gärdenfors-type systems
that satisfy (RR)? Or to put it differently, are there any non-trivial meager almost Gärdenforstype models that satisfy connection upwards and downwards? As we already know, the b. r. s.
that corresponds to such a model is almost Gärdenfors-type and satisfies (RR).
Our next
theorem says that such models exist.
THEOREM 5.2. Let M = <W, Π, r, V> be a model in which r is defined by the following
equivalence:
r(X, P, Y) iff (at least) one of the following conditions holds:11
(1) either X ⊆
/ P or X = ∅, and Y = P;
(2) X ∩ P ≠ ∅ and Y ≈ X ∩ P.
Then M is almost Gärdenfors-type and satisfies conditions (cu) and (cd). In addition, nontrivial and meager models of this kind exist.
(M is non-trivial if for some sentences A and B, neither ≠A≠ M nor ≠¬A≠ M is included in
≠B≠ M or ≠¬B≠ M ).
COROLLARY 5.3. There exist non-trivial almost Gärdenfors-type belief revision systems that
satisfy the Relational Ramsey test (RR).
It might be of some interest to note that in the models that we have described in the theorem
the conditional > satisfies the following simple truth clauses:
If M ‚/ w A, then M ‚w A > B iff ≠A≠M ⊆ ≠B≠M
If M ‚w A, then M ‚w A > B iff M ‚w A → B.
Thus, in these artificial models, the conditional > behaves like strict implication when the antecedent is false and like material implication when the antecedent is true.
21
6. Weakening Ramsey and Preserving Preservation
We have seen that the Ramsey test, while incompatible with preservation, can be accommodated
by almost Gärdenfors-type models — the models that are “almost preservative”. If we believe
that revision should be Gärdenfors-type, and therefore fully preservative, it is natural to
investigate weakenings of the Ramsey test that can be satisfied by Gärdenfors-type models.
In particular, we are going to show that there are non-trivial Gärdenfors-type models that
validate (RR⇒) — the Relational Ramsey Test in one direction: from conditionals to revisions.
As we have noted before, (RR⇒) is equivalent to Connection Downwards, in the presence of >normality.12 In order to prove that such models exist, we describe a class K of Gärdenfors-type
models that satisfy (cd), the semantic equivalent of Connection Downwards. We then show that
K contains at least one non-trivial member.
In order to construct our class K, we first introduce the notion of a world-revision choice
model.
DEFINITION 6.1.
(a) A world-revision choice structure is a 5-tuple S = <W, Π, r, C, V> where W, Π, and V
are as before (cf. Definition 4.1); r is a relation of belief revision that takes nothing but worlds
in its first argument place; and C is a choice function from closed subsets of W to their subsets.
More precisely, r is a subset of W × Π × P(W) and it satisfies the following conditions:
(r0)
for some Y, r(w, P, Y);
(seriality)
(r1)
if r(w, P, Y), then Y ⊆ P;
(r2)
if P ≠ ∅ and r(w, P, Y), then Y ≠ ∅;
(r3)
if w ≈ v, Y ≈ Z and r(w, P, Y), then r(v, P, Z); (congruence)
(r4)
if r(w, P, Y) and w ∈ P, then Y ≈ {w};
(r5)
if r(w, P, Y) and Y ∩ Q ≠ ∅, then r(w, P ∩ Q, Y ∩ Q).
(success)
(consistency preservation)
(centering)
(revision by conjunction)
C is a function which takes closed sets as its arguments and which, for every such closed X,
satisfies:
22
(C1)
C(X) ⊆ X;
(C2)
If X ≠ ∅, then C(X) ≠ ∅.
One way of obtaining such a C might be to derive it from some underlying “plausibility ordering” on worlds: One may look at C(X) as the set of the most plausible worlds in X. That is
C(X) = {w ∈ X: w ≥ v, for all v ∈ X}, where ≥ means “is at least as plausible as”. The plausibility ordering ≥ may, but need not, depend on the belief state X in question. In other words,
C(X) may be seen either as the set of X-worlds that are most plausible period, or as the set of Xworlds that are most plausible according to the plausibility ordering ≥X induced by the belief
state X. Given either interpretation, it might be natural to impose an additional requirement on
C: for any w ∈ C(X), if w ≈ v, then v ∈ C(X) (Closure of C(X) under ≈).
(b) A world-revision choice structure M = <W, Π, r, C, V> is said to be a model if Π is closed
under the operation ⇒ defined by:
P ⇒ Q = {w ∈ W: for every Y ⊆ W, if r(w, P, Y), then Y ⊆ Q}.
THEOREM 6.2. Let M = <W, Π, r, C, V> be a world-revision choice model. Let r be defined
as follows: for all X ⊆ W,
(1)
if X ∩ P ≠ ∅, r(X, P, Y) iff Y ≈ X ∩ P;
(2)
if X ∩ P = ∅ and X ≠ ∅, r(X, P, Y) iff for some w ∈ C(Cl(X)) r(w, P, Y);
(3)
if X = ∅, r(X, P, Y) iff Y = P.
Then, for all w, P, and Y, r(w, P, Y) iff r(w, P, Y) and M’ = <W, Π, r, V> is a Gärdenfors-type
model satisfying connection downwards (cd). (We shall call M’ the Gärdenfors-type model
corresponding to M.)13
The class K that we are going to consider is the class of all Gärdenfors-type models that can
be constructed in this way from world-revision choice models.
What happens to the Ramsey Test if the choice function C in the world-revision choice
model M picks out the whole of X for each X (C(X) = X)? Then it can be shown that the corresponding Gärdenfors-type model M’ not only satisfies (cd) but also (rcu). We have noted
23
before that (rcu) is a semantic counterpart of Restricted Connection Upwards (RCU). The latter
condition, in the presence of >-normality, is easily seen to be equivalent to the following restriction of (RR⇐):
(RestrRR⇐)
If ¬A ∈ G and B ∈ H for every H such that GRAH, then A > B ∈ G.
Thus, if we wish, we can accommodate almost all of Relational Ramsey in Gärdenfors-type
models. But can such models be non-trivial? To see that they can, we now construct a nontrivial world-revision choice model M in which C(X) = X, for all X. Hence, C may be omitted
from the model. (Clearly, if M is non-trivial, the corresponding Gärdenfors-type model M’ will
also be non-trivial.) We let W = {1,..., 10}, Π = P(W), and we define V as follows: V(pi) = {i},
if i = 1, ..., 10; and V(pi) = {10}, otherwise. r is defined by the following equivalence:
r(w, P, Y) iff P = Y = ∅ or Y is the set of all numbers (worlds) in W that are
closest to w among those in P.
It is easy to verify that r satisfies the required conditions (r0) - (r5) and that M = <W, Π, r, V>
is indeed a world-revision (choice) model. Finally, it is clear that M is non-trivial (see the corresponding argument in the proof of theorem 5.2.)
We have seen that the belief relation model M’ = <W, Π, r, V> that is generated from a
world-revision choice model M = <W, Π, r, C, V> must satisfy (cd) and must be Gärdenforstype. Under what conditions is M’ going to be functional as well, thereby validating all of the
Gärdenfors axioms? It is easy to see that the functionality of M’ would follow from two assumptions on M:
(i)
r is functional;
(ii)
for every closed X ⊆ W, if w, v ∈ C(X), then w ≈ v.
Given our earlier suggestion that C(X) be closed under ≈, (ii) amounts to the assumption that
C(X) = Cl({w}), for some w ∈ X.
It is easy to show that non-trivial models that satisfy (i) and (ii) exist. This means that there
exist non-trivial Gärdenfors-type models that satisfy both (cd) and functionality. Or to put it
24
differently, there exist non-trivial belief revision systems that satisfy (RR⇒), >-normality, and
all of the Gärdenfors axioms, including functionality.14
However, the assumption (ii) is incompatible with letting C(X) be X itself, for all X. Thus,
our construction does not answer the question whether (RestrRR⇐) can be satisfied by nontrivial functional Gärdenfors-type systems. But perhaps this question is philosophically not so
important. As will be seen in the last section, the restricted version of (RR⇐) is no more plausible than the unrestricted one.
7. Might-Conditionals
Consider conditional constructions of the form: “If A, then (for all that we know) it might be the
case that B”. In symbols, A N→ B. It is natural to think of the conditional connective N→ as
the dual of >:
A N→ B =df ¬(A > ¬B).
Given the truth-clause (>r) for >, the above definition gives us the following truth-clause for
N→:
(N→r) M ‚w A N→ B iff for some Y ⊆ W, r(w, ≠A≠M, Y) and Y ∩ ≠B≠M ≠ ∅.15
In the same way as (>r) gives us >-normality for belief revision systems, (N→r) corresponds
to N→-normality:
(N→norm)
for every L-maximal theory m, A N→ B ∈ m iff for some L-theory H,
mRAH and ¬B ∉ H.
Clearly, >-normality and N→-normality are equivalent conditions given the definition of N→ in
terms of >. Just as >-normality implies the Weak Relational Ramsey test (WRR), N→-normality yields the weak Ramsey test for might-conditionals:
(WRRN→)
A N→ B ∈ G iff for every L-maximal theory m such that
is some H such that mRAH and ¬B ∉ H.
G ⊆ m, there
25
But what about the full Ramsey test for might-conditionals?
(RRN→)
If G is L-consistent, then A N→ B iff for some H such that G RAH,
¬B ∉ H.
Like (RR), (RRN→) is incompatible with Preservation on pain of triviality. In fact, we do not
need full Preservation to obtain this result, Trivial Preservation is enough. In order to see this,
we first note that (RRN→) implies the following principle of inverse monotonicity:
(InvMon)
If G ⊆ H and H is L-consistent, then H ⊕ A ⊆ G ⊕ A.
As we remember, G ⊕ A = ∩{K: GRAK}.
Proof: Suppose that G ⊆ H and H is L-consistent. If B ∉ G ⊕ A, then for some K, GRAK
and B ∉ K. Therefore, by (RRN→), A N→ ¬B ∈ G. Since G ⊆ H, A N→ ¬B ∈ H. Given
that H is L-consistent, (RRN→) implies that for some K, HRAK and B ∉ K. But then B ∉ H
⊕ A.
M
A logic L is said to be minimally non-trivial if for some sentence A, neither A nor ¬A is in
L. Consider the following condition of Weak Consistency Preservation::
(WCons)
If ⊥ ∉ G + A and GRAH, then ⊥ ∉ H.
LEMMA 7.1. If L is minimally non-trivial and R satisfies Seriality and Weak Consistency
Preservation, then R cannot satisfy both (RRN→) and Trivial Preservation.16
Proof: By the minimal non-triviality of R, there is a sentence A such that both G = Cn({A})
and H = Cn({¬A}) are L-consistent. By Trivial Preservation and Seriality, A ∈ G ⊕ t and ¬A
∈ H ⊕ t, where t is any tautology. Since L is included both in G and in H, Inverse
Monotonicity (which follows from (RRN→)) yields: A, ¬A ∈ L ⊕ t. Therefore, since L + t =
L, Weak Consistency Preservation implies that L is inconsistent, which contradicts the assumption of minimal non-triviality.
M
In view of this result, (RRN→) must be considered clearly unacceptable. Trivial Preservation
and Weak Consistency Preservation are completely self-evident conditions on belief revision.
26
But can’t we at least save one direction of (RRN→), the one from left to right, in analogy
with our suggestion for (RR)?
(RRN→ ⇒)
If G is L-consistent and A N→ B ∈ G, then for some H such that
GRAH, ¬B ∉ H.17
In the presence of N→-normality, this principle is equivalent to the following condition on R,
which we might call Weak Connection Upwards:
(WCU) If G is L-consistent, then G ⊕ A ⊆ ∪{m ⊕ A: G ⊆ m}.
Proof: That (WCU) implies (RRN→ ⇒) is easy to show. Thus, assume (WCU) and suppose that G is L-consistent and A N→ B ∈ G. If the consequent of (RRN→ ⇒) does not hold,
then ¬B ∈ G ⊕ A. Therefore, by (WCU), there is some m such that G ⊆ m and ¬B ∈ m ⊕ A.
By N→-normality, this implies that A N→ B ∉ m. However, this is impossible in view of the
fact that A N→ B ∈ G and G ⊆ m.
For the other direction, assume (RRN→ ⇒) and suppose that G is L-consistent. We want to
show that if B ∉ ∪{m ⊕ A: G ⊆ m}, then B ∉ G ⊕ A. B ∉ ∪{m ⊕ A: G ⊆ m} amounts to
the claim that B ∉ m ⊕ A, for all m G. That is, by N→-normality, A N→ ¬B ∈ m, for all m
G. But then A N→ ¬B ∈ G. Applying (RRN→ ⇒), we get: B ∉ G ⊕ A.
M
Is (WCU) a condition that we can impose on Gärdenfors-type belief revision systems without trivialization? Yes, and in fact we can impose it together with Connection Downwards. In
order to see this, we observe that (WCU) has the following semantic correlate:
(wcu)
for all X and P, if X ≠ ∅ and X = Cl(X), then, for some w ∈ X,
w ⊕ P ⊆ X ⊕ P.
As we remember, X ⊕ P = ◊{Y: r(X, P, Y)} = Cl(∪{Y: r(X, P, Y)}).
In the previous section, we have seen that the belief-relation models that are generated from
world-revision choice models are all Gärdenfors-type and satisfy connection downwards. Now,
we can show that they also satisfy (wcu) — weak connection upwards. In order to see this,
consider any world-revision choice model M = <W, Π, r, C, V> and let M’ = <W, Π, r, V> be
27
the belief-relation model generated by M. We want to show that r satisfies (wcu). Suppose that
X ≠ ∅, X = Cl(X) and that X ⊕ P ⊆ Q. There are two cases to consider:
Case 1. X ∩ P ≠ ∅. Then, by the definition of r, (1) r(X, P, Y) iff Y ≈ X ∩ P. (1) implies
that (2) X ⊕ P ≈ Cl(X ∩ P). Now, consider any w in X ∩ P, which we have assumed to be
non-empty. As we have seen in the previous section (Theorem 6.2),
(∗)
for all Y, r(w, P, Y) iff r(w, P, Y).
Therefore, w ⊕ P = ◊{Y: r(w, P ,Y)} = ◊{Y: r(w, P ,Y)}. Since w ∈ P, the centering condition
(r4) on r implies that r(w, P, Y) iff Y ≈ {w}. It follows that, (3) w ⊕ P = Cl({w}). Since, w ∈
X ∩ P, Cl({w}) ⊆ Cl(X ∩ P). Therefore, by (2) and (3), w ⊕ P ⊆ X ⊕ P.
Case 2: X ∩ P = ∅. Since X ≠ ∅ and X is closed, the definition of r implies
(i)
r(X, P, Y) iff ∃w ∈ C(X) such that r(w, P, Y).
Given (∗) above, (i) is equivalent to
(ii)
r(X, P, Y) iff ∃w ∈ C(X) such that r(w, P, Y).
Pick any w ∈ C(X). Then (ii) implies that for all Y, if r(w, P, Y), then r(X, P, Y). Therefore, w ⊕
P ⊆ X ⊕ P.
M
To conclude: All >-normal belief revision systems satisfy the Ramsey test for maximal theories, both for > and for N→. In addition, we can impose on such systems all the Gärdenfors
axioms and still have the full Ramsey test in one direction — from conditionals to revisions, for
both kinds of conditionals. In other words, we may combine the Gärdenfors axioms (including
functionality) with the following modified Ramsey tests:
(MR)
A > B ∈ G iff for every L-consistent extension H of G, B ∈ H ∗ A.
(MRN→)
A N→ B ∈ G iff for every L-consistent extension H of G, ¬B ∉
H ∗ A.18
28
8.
The Intuitive Plausibility (Implausibility) of the Different Weakenings of the
Ramsey Test
Gärdenfors (1988) presents two intuitive arguments against the Ramsey test: one for each direction of the test. We shall consider his two objections and argue that only one of them is
convincing — the one directed against (RR⇐). The other one, directed against (RR⇒), is in
our view based on a conflation between two kinds of conditionals: the epistemic and the “ontic”
ones.
The intuitive argument against (RR⇐) in Gärdenfors (1988) goes as follows:
“Consider Victoria and her alleged father Johan. Let us assume that Victoria, in her present state of belief
G, believes that her own blood group is O and that Johan is her father, but she does not know anything
about Johan’s blood group. Let A be the proposition that Johan’s blood group is AB, and C the proposition that Johan is Victoria’s father. If she were to revise her beliefs by adding the proposition A, she
would still believe that C, that is C ∈ G ∗ A. But, in fact, she now learns that a person with blood
group AB can never have a child with blood group O. This information, which entails C → ¬A, is consistent with her present state of belief G, and thus her new state of belief, call it H, is an expansion of G.
If she then revises H by adding the information that Johan’s blood group is AB, she will no longer believe that Johan is her father, that is, C ∉ H ∗ A. Thus (Mon) is violated.
Note that this example does not depend on the presence of any conditionals in G or in H. In fact, if
we assume (RT) and not just (Mon), then Victoria would have believed A > B in G. But then the information that a person with blood group AB can never have a child with blood group O would contradict
her beliefs in G, which violates our intuitions that this information is indeed consistent with her beliefs
in G.
This violation can also be taken as an argument against (RT). The most problematic implication of
(RT) is the one saying that, if B ∈ G ∗ A, then A > B ∈ G. In a sense, this implication requires that too
many conditionals be elements of a belief set G because it must contain conditionals related to all possible revisions that G must undergo.”19
Gärdenfors seems to view the argument as an objection to the right-to-left direction of the
Ramsey test. However, as such, his argument does not seem quite conclusive. The monotonicity rule (Mon) that is used in the argument in order to derive an inconsistency is a logical
consequence of both the directions of the Ramsey test taken together, not just of the right-to-left
direction. In order to show that it is indeed the latter direction that leads to trouble, we shall now
reformulate the argument in our relational framework and discuss it in some detail.
We assume that G, Victoria’s state of belief, is consistent and contains the propositions:
O:
Victoria’s blood group is O;
C:
Johan is Victoria’s father.
and
In addition, the following propositions do not belong to G:
29
A:
Johan’s blood group is AB
LAW:
A person with blood group AB cannot have a child with blood group O.
The same applies to the negations of these propositions, they do not belong to G either.
Letting H = Cn(G ∪ {LAW}), the following assumptions seem to be intuitively correct in
Victoria’s case:
(a)
if GRAK, then C ∈ K;
(b)
for at least some K such that HRAK, C ∉ K.
Now, given the Ramsey test, we can derive a contradiction. By (RR⇐), (a) implies
(1)
A > C ∈ G.
Since G ⊆ H, (1) immediately implies:
(2)
A > C ∈ H.
By (RR⇒), (2) yields:
(3)
if HRAK, then C ∈ H.
But, given the seriality of R, (3) conflicts with (b).
Where did we go wrong in this argument? We would suggest that the incorrect step is the
one from (a) to (1). The principle (RR⇐) used in this step is unacceptable: although (a) is true,
(1) seems to be false. In order to see the falsity of (1) in the example, we note that (2), which
immediately follows from (1), implies:
(i)
A > C ∈ m, for every L-maximal theory m such that H ⊆ m.
This means, by >-normality, that:
(ii)
if H ⊆ m and mRAK, then C ∈ K.
Since H is consistent, such maximal extensions m of H must exist. Clearly, any such fully
opinionated belief set m will contain LAW, O and C. And in some such m, LAW and O may
well be more “epistemically entrenched” among Victoria’s beliefs than C. Therefore, for some
30
such m, it may very well be the case that Victoria would keep LAW and O, but give up C, when
she learns A. To put this assumption in formal terms:
(c)
for some m such that H ⊆ m and some K, mRAK and C ∉ K.
But (c) is incompatible with (ii).
Thus, if we accept >-normality, it is the step from (a) to (1) that is incorrect. That is, (RR⇐)
must be rejected.20
Note that a slightly modified Victoria-Johan case could also be used as a counterexample to
the restricted right-to-left direction of the Ramsey test:
(RestrRR⇐)
A > B ∈ G if ¬A ∈ G and B ∈ K for every K such that GRAK.
The required modification would consist in supposing that Victoria’s original belief set G contains ¬A (Johan’s blood group is not AB). This change would not make the assumptions (a) (c) less intuitive, but it would allow using (RestrRR⇐) at the crucial step from (a) to (1).21
As far as the other direction of the Ramsey test is concerned, Gärdenfors (1988) presents the
following objection:
“...this half of (RT) is not without problems. The following troublesome example is borrowed from
Stalnaker (1984, p. 105):
‘Suppose I accept that if Hitler had decided to invade England in 1940, Germany would have won the
war. Then suppose I discover, to my surprise, that Hitler did in fact decide to invade England in 1940
(although he never carried out his plan). Am I now disposed to accept that Germany won the war? No,
instead I will give up my belief in the conditional. In this case, my rejection of the antecedent was an
essential presupposition of my acceptance of the counterfactual, and so gives me reason to give up the
counterfactual rather than to accept its consequent, when I learn that the antecedent is true.’”
The example is supposed to show that one may very well accept a conditional and at the same
time be disposed to reject the consequent upon learning its antecedent — a clear violation of
(RR⇒).
However, according to Stalnaker (1984), this example only shows that counterfactual conditionals do not satisfy the left-to-right direction of the Ramsey test. Such conditionals do not
represent “conditional beliefs”, that is, dispositions to change one’s beliefs upon receiving new
information. After presenting the example, Stalnaker therefore continues with the following
claim:
“but for every counterfactual conditional which does not represent a rational disposition to change one’s
beliefs, there will be a contrasting conditional, which I will call an open conditional, with the same antecedent and consequent which does, or would if accepted. If I accept that if Hitler did decide to invade
31
England in 1940, then Germany did win the war, then I will be rationally disposed to accept that
Germany won the war upon learning that Hitler decided to invade England in 1940. This example fits a
pattern of examples, the originals invented by Ernest Adams, which shows that there must be a semantic,
and not merely a pragmatic difference between so-called subjunctive and indicative conditionals. To use
Adams’s example, we all accept that if Oswald didn’t shoot Kennedy then someone else did, since we are
disposed to conclude that someone other than Oswald shot Kennedy on learning that Oswald did not. But
it would be quite a different matter to accept that if Oswald hadn’t shot Kennedy, someone else would
have.”
In this paper, we are concerned with what Stalnaker calls “open conditionals”. For these, as
Stalnaker points out, the left-to-right direction of the Ramsey test seems valid. Thus, (RR⇒)
survives our scrutiny.22
Instead of talking about “counterfactual” vs “open” conditionals, we prefer to speak of, respectively, the ontic and the epistemic ones. (Instead of “epistemic” one could also use the
term “doxastic”.) “Counterfactual” does not seem to be the correct label, since some of the
conditionals in this group do not presuppose the falsity of their antecedents. Thus in the wellknown Newcomb’s paradox, the agent who accepts the epistemic conditional: “If I am to take
both boxes, then the black one will turn out to be empty”, may well doubt the truth of the corresponding ontic conditional: “If I were to take both boxes, the black one would turn out to be
empty”. Clearly the latter conditional, while non-epistemic, does not presuppose the falsity of
its antecedent.
What is the reason, then, behind our terminological proposal? We would like to suggest the
following somewhat vague answer: ontic conditionals concern hypothetical modifications of the
world, but epistemic conditionals have to do with hypothetical modifications of our beliefs about
the world.
APPENDIX: PROOFS OF LEMMAS AND THEOREMS
Proof of Lemma 4.2:
Clearly, every model determines at most one b. r. s..
Let M =
<W, Π, r, V> be a model and let L be defined by:
L = {A: for all w ∈ W, M ‚w A}.
L is clearly a logic and satisfies (i’). Therefore, in view of the equivalence noted above, L
satisfies (i).
32
Let R be defined by:
(ii)
GRAH iff r(≠G≠M, ≠A≠M, ≠H≠M).
It is easy to show that R satisfies success (R1), consistency preservation (R2) and substitutivity
of logical equivalents (R3), using conditions (r1), (r2) and (r3) on M. In the proof we make use
of the compactness of M.
If M is assumed to be meager, we can also prove that R is serial (R0): Let G be any L-theory
and A any sentence. We have to show that for some H, GRAH. By the definition of R, GRAH
iff r(≠G≠M, ≠A≠M, ≠H≠M). By the seriality of r (r0), for some Y ⊆ W, r(≠G≠ M , ≠A≠ M , Y).
Consider any such Y. Let H = {B: Y ⊆ ≠B≠M }. Clearly, H is an L-theory. If M is meager,
then Y ≈ ≠H≠M. But then by (r3), r(≠G≠M, ≠A≠M, ≠H≠M).
Proof of Lemma 4.3:
M
Let <L, R> be a belief revision system and M = <W, Π, r, V> a
meager model that determines <L, R>. We prove that for every L-maximal theory m, A > B ∈
m iff for every H, if mRAH, then B ∈ H.
Observation: If M is meager, a set Γ of sentences is L-maximal iff ≠Γ≠M ≠ ∅ and for every
w ∈ ≠Γ≠M, ≠Γ≠M ≈ {w}.
Assume first that A > B ∈ m and that mRAH. Let w ∈ ≠m≠M . From A > B ∈ m, we conclude that M ‚w A > B, which in turn yields:
(∗)
for all Y, if r(w, ≠A≠M, Y), then Y ⊆ ≠B≠M.
mRAH yields r(≠m≠M, ≠A≠M, ≠H≠ M ).
Meagerness together with (r3) then yields
r(w, ≠A≠ M , ≠H≠ M ) (see the observation above). By (∗) we get ≠H≠ M ⊆ ≠B≠M, that is,
B ∈ H.
For the opposite direction, assume that for every H, if mRAH, then B ∈ H. Suppose further
that A > B ∉ m. Then, for some w ∈ ≠m≠M, it is not the case that M ‚w A > B. This implies
that for some Y, r(w, ≠A≠M, Y) and Y ⊆
/ ≠B≠M. Let H = {C: Y ⊆ ≠C≠M }. Clearly, ≠H≠ M ≈L
Y. But then it follows by meagerness that ≠H≠M ≈ Y. Hence, for some H, r(w, ≠A≠M, ≠H≠ M )
and ≠H≠ M ⊆/ ≠B≠ M . That is, given the observation above and (r3), for some H, r(≠m≠M,
≠A≠M, ≠H≠M) and B ∉ H, contrary to the assumption.
M
33
Proof of Theorem 4.4: Let B = <L, R> be any >-normal b. r. s. and let M = <W, Π, r, V>
be its canonical model. Since L is consistent, W, the set of all L-maximal theories, is nonempty. Clearly, Π = {|A|L: A is a sentence} contains W and is closed under complements and
finite unions. In addition, by >-normality and (R3), Π is closed under the conditional-forming
operation: P ⇒ Q = {m: ∀Y(r(m, P, Y) → Y ⊆ Q)}. The compactness of the canonical model
immediately follows from the compactness of the logic L.
In the same way, each of the conditions (r0) - (r2) follows from the corresponding condition
among (R0) - (R2). As an example, let us consider the proof of (r0). Let X ⊆ W and P ∈ Π.
We want to show that for some Y, r(X, P, Y). We know that for some L-theory G, |G|L ≈ X.
Also, for some A, |A|L = P. By (R0), we have, for some H, that GRAH. Therefore, by (iii)
above, r(X, P, |H|L). Notice also that (iii) immediately validates (r3).
Thus, we have shown that the canonical model M is indeed a model. That M is meager immediately follows from the definition of Π. It remains to be shown that M determines the b. r.
s. <L, R>. We have to show that in the canonical model M,
(a) for every Γ and A, Γ ªL A iff ≠Γ≠M ⊆ ≠A≠M;
(b) for all L-theories G, and H and all sentences A,
GRAH iff r(≠G≠M, ≠A≠M, ≠H≠M).
Both conditions will follow given that we can show that for every set Γ of sentences,
(∗∗)
≠Γ≠M = |Γ|L.
(∗∗) would immediately follow from the following equivalence:
for all L-maximal theories m, M ‚m A iff A ∈ m.
The proof of this equivalence proceeds by induction on the complexity of A. The only nontrivial step in the proof concerns sentences of the form A > B. By (>r), we have:
(1)
M ‚m A > B iff for all Y ⊆ W, if r(m, ≠A≠M, Y), then Y ⊆ ≠B≠M.
By the induction hypothesis, we have that ≠A≠M = |A|L, for all A. Hence, the RHS of (1) holds
iff
34
(2)
for all Y ⊆ W, if r(m, |A|L, Y), then Y ⊆ |B|L.
That is, iff
(3)
for all H, if mRAH, then B ∈ H.
But, by >-normality, (3) is equivalent to
(4)
A > B ∈ m.
M
Proof of Lemma 5.1: First, we note that >-normality follows from (RR) when we substitute
m for G in (RR).
Next, we prove that (RR⇐) together with >-normality implies (CU). If B ∈ G ⊕ A, then, by
(RR⇐), A > B ∈ G. But then, if G ⊆ m, A > B ∈ m. Therefore, by >-normality, if mRAH, B
∈ H. Thus, B ∈ m ⊕ A.
Further, we have to show that (RR⇒) together with >-normality implies (CD). If for every m
such that G ⊆ m, B ∈ m ⊕ A, then, by >-normality, A > B ∈ m, for every such m. But then A >
B ∈ G. By (RR⇒), this means that B ∈ H, whenever GRAH, provided that ⊥ ∉ G. In other
words, if ⊥ ∉ G, then B ∈ G ⊕ A.
It remains to be shown that:
(i)>-normality together with Connection Upwards imply (RR⇐). In order to see this, note
that if B ∈ H, for every H such that GRAH, then — by Connection Upwards — B ∈ H, if G ⊆
m and mRAH. But then, by >-normality, A > B ∈ m, for all m such that G ⊆ m. Then, if ⊥ ∉
G, A > B ∈ G. On the other hand, if ⊥ ∈ G, then it is trivial that A > B ∈ G.
(ii) >-normality and Connection Downwards imply (RR⇒). If A > B∈ G, then A > B ∈ m,
for every m such that G ⊆ m. Therefore, by >-normality, B ∈ H, for every H and m such that G
⊆ m and mRAH. But, then, if ⊥ ∉ G, Connection Downwards implies that B ∈ H whenever
GRAH.
M
Proof of Theorem 5.2: Suppose that M = <W, Π, r, V> is a model satisfying the assumptions of the theorem. First, we have to prove that M is almost Gärdenfors-type. The proof is
35
straightforward, except for the rather cumbersome verification of the fact that r satisfies stepwise
revision: If r(X, P, Y), r(Y, Q, Z) and Z ⊆ Y, then r(X, P ∩ Q, Z). Given that r satisfies (r3), full
stepwise revision would follow from stepwise revision restricted to closed sets X, Y, Z.
Therefore, it is sufficient to prove the restricted condition. If its antecedent holds, then, by the
definition of r, we have four possible cases:
(1) Y = P and Z = Q;
(2) Y = P, P ∩ Q ≠ ∅ and Z = P ∩ Q;
(3) X ∩ P ≠ ∅, Y = X ∩ P and Z = Q;
(4) X ∩ P ≠ ∅, Y = X ∩ P, Y ∩ Q ≠ ∅ and Z = Y ∩ Q.
We have to show that, in each case, r(X, P ∩ Q, Z). First, we note that, by the definition of r, r(X,
P ∩ Q, P ∩ Q) always holds. Therefore, in case (2) where Z = P ∩ Q, the wanted conclusion,
r(X, P ∩ Q, Z), immediately follows. In case (1), the conclusion follows since Z = Q and Z ⊆ Y
(= P), which means that Z = P ∩ Q. In case (3), since Z = Q ⊆ Y and Y = X ∩ P, Z = Y ∩ Q =
X ∩ P ∩ Q. This would imply the wanted conclusion provided that Q ≠ ∅. If Q ≠ ∅, then X
∩ P ∩ Q = Y ∩ Q = Q ≠ ∅. On the other hand, if Q = ∅, then Z = Q = P ∩ Q = ∅ and either
X⊆
/ P ∩ Q or X = ∅. In both cases, the wanted conclusion once again follows. In case (4), we
get: Z = Y ∩ Q = X ∩ P ∩ Q. The conclusion follows because X ∩ P ∩ Q = Y ∩ Q ≠ ∅.
Now, we want to show that r in our model satisfies (cu). (cu) will follow if we can prove the
following stronger statement:
(∗)
∪w∈X{Y: r(w, P, Y)} ⊆ ∪{Y: r(X, P,Y)}.
Suppose that w ∈ X, r(w, P, Y) and v ∈ Y. We want to show that for some Z, r(X, P, Z) and v ∈
Z. We consider two cases:
(1) X ⊆
/ P or X = ∅. Then, by the definition of r, r(X, P, P). Since r(w, P, Y) and r satisfies
success (r2), Y ⊆ P. Thus, v ∈ P.
(2) X ∩ P ≠ ∅ and X ⊆ P. Then, w ∈ P and r(X, P, X ∩ P). Since w ∈ P, Y = {w}, by the
definition of r. But then v = w and v ∈ X ∩ P.
Just as (cu) follows from (∗), (cd) follows from
36
(∗∗) If X ≠ ∅ and X = Cl(X), then ∪{Y: r(X, P,Y)} ⊆ ∪w∈X{Y: r(w, P, Y)}.
In order to prove (∗∗), suppose that X ≠ ∅, X = Cl(X), r(X, P, Y) and v ∈ Y. We want to show
that for some Z and some w ∈ X, r(w, P, Z) and v ∈ Z. We have two possible cases: (1) X ⊆
/ P;
and (2) X ∩ P ≠ ∅ and X ⊆ P. In case (1), there is some w ∈ X such that w ∉ P. By the
definition of r, r(w, P, P). By success (r2), Y ⊆ P. Therefore v ∈ P. In case (2), Y ≈ X ∩ P.
Hence, Y ⊆ Cl(X ∩ P). But, since both X and P are closed, Cl(X ∩ P) = X ∩ P. Hence, Y ⊆ X
∩ P. This in turn implies that Y ⊆ Cl(X). Therefore, v ∈ Cl(X). Hence, v ∈ X and v ∈ P. But
if v ∈ P, then by the definition of r, r(v, P {v}).
It remains to be shown that there exists a non-trivial and meager model of this kind. This,
however, is straightforward. Let, for instance, W = {1,..., 10}, Π = P(W), V(pi) = {i}, for i =
1,...,10, and V(pi) = {10}, if i > 10. r is defined as in the theorem. It is easy to check that the
resulting structure M = <W, Π, r, V> is a meager model. In order to see that M is non-trivial,
note that neither ≠p1 ∨ p 2 ≠ M nor its complement is included in either ≠p 2 ∨ p 3 ≠ M or its
M
complement.
Proof of Theorem 6.2: Suppose that M and M’ are as in the theorem. First, we show that
for all w, P and Y,
(∗)
r(w, P, Y) iff r(w, P, Y).
Case 1: w ∈ P. Then, by (1) in the definition of r, r(w, P, Y) iff Y ≈ {w}. But, by the rconditions of centering, seriality, and congruence, if w ∈ P, Y ≈ {w} iff r(w, P, Y).
Case 2: w ∉ P. Then, r(w, P, Y) iff for some v ∈ C(Cl({w})) r(v, P, Y). But, by congruence, the
latter condition holds iff r(w, P, Y).
(∗) implies that the operation ⇒ is the same regardless of whether we define it in terms of r
or r. Therefore, in order to prove that M’ is a Gärdenfors-type model, it suffices to show that r
satisfies (r0) - (r3) together with preservation, minimality, strong consistency, and stepwise
revision. Since the proofs are mostly routine, we only take as an example the rather cumbersome case of stepwise revision. In fact, since ≈ can be shown to be a congruence relation with
respect to r (r3), it is sufficient to show that stepwise revision holds for closed sets X, Y and Z.
37
Suppose then that r(X, P, Y), r(Y, Q, Z) and Z ⊆ Y. Assume that X, Y, Z are closed. We want
to show that r(X, P ∩ Q, Z).
Case 1: Y ∩ Q = ∅. Since Z ⊆ Y, by the assumption, and Z ⊆ Q, by the fact that r satisfies
success, Z ⊆ Y ∩ Q = ∅. That is, Z = ∅. But then, since r(Y, Q, Z) and r satisfies strong
consistency, Q = ∅. Therefore, P ∩ Q = ∅ = Z. It follows that r(X, P ∩ Q, Z), by clause (3) in
the definition of r.
Case 2: Y ∩ Q ≠ ∅. Then, by the definition of r and the fact that Z and Y are closed, Z =
Y ∩ Q.
Case 2.1: X ∩ P ≠ ∅. Then, by the same reasoning as above, Y = X ∩ P. But then Z = Y ∩
Q = X ∩ P ∩ Q and X ∩ P ∩ Q = Y ∩ Q ≠ ∅. By clause (1) in the definition of r, it follows
that r(X, P ∩ Q, Z).
Case 2.2: X = ∅. Then Y = P, due to the fact that Y is closed. Therefore, Z = Y ∩ Q =
P ∩ Q. Hence, r(X, P ∩ Q, Z), by clause (3) in the definition of r.
Case 2.3: X ≠ ∅, but X ∩ P = ∅. Then, by clause (2) in the definition of r, r(w, P, Y), for
some w ∈ C(X). Since Z = Y ∩ Q ≠ ∅, (r5) implies that r(w, P ∩ Q, Z). But then, since X ∩ P
∩ Q = ∅ (because X ∩ P = ∅), clause (2) in the definition of r implies that r(X, P ∩ Q, Z).
This completes the proof that M’ is a Gärdenfors-type model. It remains to show that M ’
satisfies (cd). By the properties of closure, (cd) is equivalent to
(cd’) If X ≠ ∅ and X = Cl(X), then ∪{Y: r(X, P,Y)} ⊆ Cl[∪w∈X{Y: r(w, P, Y)}]
Assume the antecedent of (cd’) and suppose that r(X, P, Y) and v ∈ Y. We want to show that
for some w ∈ X and some Z such that v ∈ Z, r(w, P, Z).
Case 1: X ∩ P = ∅. Since X ≠ ∅, clause (2) in the definition of r implies that r(w, P, Y), for
some w ∈ C(X). But then, by (∗), r(w, P, Y). From this the wanted conclusion immediately
follows.
Case 2: X ∩ P ≠ ∅. Then, Y ≈ X ∩ P. Since X is closed, this implies that Y ⊆ X ∩ P.
Consider any v ∈ Y. Then v ∈ X ∩ P. By the r-conditions of congruence, centering and seriality, r(v, P, {v}). But then by (∗), r(v, P, {v}). This concludes the proof.
M
38
NOTES
*
An earlier version of this paper was presented at the conference on the dynamics of knowledge and belief at
Lund University, August 24-26, 1989. We wish to thank Sven Danielsson, Peter Gärdenfors, Sören Halldén,
David Makinson, Hugh Mellor, Michael Morreau, Nils-Eric Sahlin and Brian Skyrms for their very helpful
suggestions and remarks. We are also grateful for thought-provoking criticism and comments from two
anonymous referees.
1
Actually, the claim that English contains a distinction between the indicative and the subjunctive mood is quite
controversial. V. H. Dudman has vigorously argued against the mood-distinction in a series of influential
papers. (See, for example, Dudman (1984), (1988). Cf. also Smiley (1983-84).) According to Dudman, the
right distinction between conditionals in English should be drawn in another way, so that some “indicative” will
fall in the same group as some as the so-called “subjunctive” ones. For example, the sentence “If Oswald does
not kill Kennedy, then no one else will” uttered “before the event” would make the same claim as the sentence
“If Oswald had not killed Kennedy, then no one else would have” when uttered now.
According to Dudman,
conditionals of this kind — he refers to them simply as “conditionals” — are arrived at “...by envisaging a
developing sequence of events”, “by imaginatively projecting steadily futureward from v [the time referred to in
the utterance] in a fantasy in which the ‘if’-condition is satisfied” (Dudman (1984), pp. 153, 152). This is more
or less how we think of the ontic conditionals. The conditional sentences belonging to the other group, which
Dudman calls “hypotheticals”, are “arrived at by arguing from proposition to proposition...”, “Someone shot
Kennedy, Oswald didn’t, therefore someone else did” (1984, p. 153).
It seems quite clear that Dudman’s
semantical interpretation of his grammatical distinction between “hypotheticals” and “conditionals” resembles
our semantical distinction between epistemic and ontic conditionals.
2
A similar position is taken by Sahlin (1990), Chap 4, Section “Conditionals and the Ramsey test”.
3
Actually, the restriction to consistent theories G is normally omitted in formulations of the Ramsey test.
However, this restriction will considerably simplify our presentation and it does not diminish the
informativeness of the test, since it is quite clear under what circumstances a conditional belongs to an
inconsistent theory: it always does!
Note that, in the absence of the restriction, the Ramsey test would imply that all revisions of the inconsistent
theory must themselves be inconsistent.
39
4
Consider the following Intersection Condition on R:
(Inters)
for all G, A, GRA(∩{H: GRAH}).
According to this condition, upon receiving a new information A, it is always possible to revise the old theory
in such a way that one accepts nothing but the common part of all the possible A-revisions of the theory in
question.
Let us say that an A-revision of G is cautious if it does not properly include another A-revision of G. Clearly,
if the Intersection Condition is satisfied, there is a unique cautious A-revision of G, for every A and G.
In
addition, this cautious revision is included in every A-revision of G.
It should be noted that, even if R is not functional, it is possible to “extract” a belief revision operation ∗
from R, provided only that R satisfies (Inters). We can then simply let G ∗ A be ∩{H: GRAH}. Philosophers
who speak of a belief revision operation perhaps do not assume that the relation of belief revision is functional.
Instead it may be the case that they only assume (Inters) and that the operation they are talking about is the
“cautious” revision operation ∗ defined above.
However, note that (Inters) would imply Functionality, if we were to assume that all revisions must be
cautious. That is, Functionality follows from (Inters) together with the following condition on belief revision:
(Caution)
5
If GRAH, GRAK and H ⊆ K, then H = K.
Gärdenfors (1988), p. 67. Harman’s (1986, p. 46) own formulation of the principle is more cautions: “One is
justified in continuing fully to accept something in the absence of a special reason not to”.
6
For a further discussion of this issue, see Lindström and Rabinowicz (1989), pp. 76-77. The following
interpretation of belief and (functional) belief revision within non-standard (in the sense of A. Robinson)
probability theory was not considered in Lindström and Rabinowicz (1989). It is inspired by an analogous
interpretation of nonmonotonic consequence within non-standard probability theory, due to Lehman and Magidor
(to appear).
The agent’s belief state is represented by a a probability function — satisfying the usual
Kolmogorov axioms —that takes values in the non-standard interval [0, 1]. The agent’s belief set H consists of
just those sentences A whose probability is either 1 or infinitesimally close to 1. That is, A ∈ H iff 1 - P(A) is
either 0 or infinitesimal. Only those sentences that are logically true are assigned probability 1. We may say
that the agent is absolutely certain that A if P(A) is 1 and that she is virtually certain that A if P(A) is
40
*
infinitesimally close to 1. Revision proceeds by conditionalization: In case, P(A) = 0, we let PA be the absurd
*
probability function P⊥ that assigns probability 1 to all sentences B. Otherwise, P A(B) = P(B/A). The latter
equation holds even for the case when P(A) is infinitesimal, i.e., when ¬A is virtually certain. Thus, the virtual
certainties, although not the absolute ones, can be given up when the agent revises her beliefs in the light of
*
*
new information. The new belief set KA is defined as {B: PA(B) is either 1 or infinitesimally close to 1}. It
can be verified that this interpretation of belief revision satisfies Gärdenfors’ axioms (K*1) - (K*8) (Gärdenfors
(1988), pp. 54-56). The interpretation has the following advantages: (i) Belief revision can be modeled as
conditionalization, since the process of conditionalization may lead to old beliefs being given up. This is not
possible if belief is interpreted as absolute certainty. (ii) Belief sets are logically closed and belief revision
satisfies Preservation. Neither of these conditions is satisfied if the agent’s belief set is taken to be {A: P(A) > 1
- ε}, for some fixed standard number ε > 0.
7
The condition of Success was questioned by one of the referees.
8
Contraction is discussed in Gärdenfors (1988), Ch. 3.
9
Admittedly, our semantic modelling of belief revision in terms of a relation r is not very informative. There is
an alternative, more interesting, modelling of belief revision in terms of neighborhood systems — a modelling
that we have some predilection for.
A belief-neighborhood structure (or a neighborhood structure, for short) is a four-tuple S = <W, Π, N, V>
where W, Π and V are as before (see definition 4.1), and N is a function which to every X ⊆ W assigns a set NX
of subsets of W satisfying the following conditions:
(N0)
every N ∈ NX is closed, i.e., N = ∩{P ∈ Π: N ⊆ P};
(N1)
W ∈ NX;
(N2)
if X ≈ Y, then NX = NY
(N3)
for every X ⊆ W, P ∈ Π and every N ∈ NX, if N ∩ P ≠ ∅, then there exists an N’ in NX such
that N’ ⊆ N, N’ ∩ P ≠ ∅ and for every N” in N X , if N” ∩ P ≠ ∅, then N” ⊄ N’.
If N belongs to NX, then we say that N is a neighborhood of X. Thus NX is the set of all neighborhoods of X.
According to (N0), all neighborhoods of X are closed sets. (N1) says that W is the greatest neighborhood of X.
41
Condition (N2), which is a counterpart of condition (r3) for belief-relation structures, says that equivalent belief
states have identical systems of neighborhoods.
If N ∈ N X and N ∩ P ≠ ∅, we say that N is a P-neighborhood of X. (N3) amounts then to the assumption
that every P-neighborhood of X includes a minimal P-neighborhood of X. Clearly, a given P-neighborhood of
X may include several such minimal P-neighborhoods.
Note that if were to impose the following additional postulates on neighborhood systems:
(N4)
Cl(X) ∈ NX;
(preservation)
(N5)
for every N ∈ NX, Cl(X) ⊆ N;
(centering)
(N6)
For any N, N’ in NX, N ⊆ N’ or N’ ⊆ N,
(nestedness)
then the system of neighborhoods around X would become very similar to a system of spheres centered on X, in
the sense of Grove (1988). (Grove however, does not require the spheres to be closed. Therefore, instead of
assuming that Cl(X) is a sphere centered on X, he lets X itself be such a sphere. Furthermore, there is nothing
corresponding to our (N2) in his semantics.) In such a nested neighborhood system, there is always a unique
minimal P-neighborhood of X, for every non-empty P.
For any X ⊆ W, the elements of NX may be viewed as “fallback” states of belief: states of belief that one may
arrive at from X, when one withdraws some of the beliefs that are held in X.
We assume that when one
withdraws a proposition P from X, one wants to keep as many of the beliefs in X as possible.
Within the
neighborhood framework, this means that a proper fallback position should be a minimal non-P-neighborhood of
X.
Intuitively, when we revise a state of belief X with a proposition P, we do it in two steps: first we withdraw
non-P from X and then we add the information P to the result. This idea is due to Levi (1977). This amounts
to the following construction of the belief-revision relation r:
r(X, P, Y) iff either P = Y = ∅ or there exists a minimal P-neighborhood N of X such that Y ≈ N ∩ P.
It is easy to see that this definition implies that r satisfies the conditions (r0) - (r3) that we impose on beliefrelation structures. However, the neighborhood modelling, while being less trivial than the one we use in the
text, is also less general: it validates some extra postulates on belief revision that some belief-relation structures
need not satisfy. In particular, it validates the postulate of strong consistency which we introduce later in this
42
section and the condition (α) which is discussed in note 10 below. (α) is a closely related to the postulate of
stepwise revision which is going to be considered later in this section. (These ideas for a neighborhood modelling of relational belief revision are developed further in Lindström and Rabinowicz (to appear)).
10
In particular, it can be shown that Stepwise Revision, in the presence of Minimality, Weak Preservation, and
Strong Consistency (together with the general axioms on belief revision systems) is equivalent to the following
condition:
(α) If GRAH and ¬B ∉ H, then GRA∧B(H + B) (where H + B = CnL(H ∪ {B}).
As we remember, Stepwise Revision is the following principle:
(StepRev) If GRAH and HRBK and H ⊆ K, then GRA∧BK.
In order to derive (α) from Stepwise Revision, assume the latter condition and suppose also that GRAH and
¬B ∉ H. Weak Preservation and Minimality then yields HRB(H + B). Since, H ⊆ H + B, (StepRev) yields
GRA∧B(H + B).
For the derivation of Stepwise Revision from (α), assume (α) and let GRAH, HRBK and H ⊆ K.
We
consider two cases: Case (i): Either A or B is L-inconsistent. Then, A, B ∈ K, by Success, so K must be Linconsistent.
Furthermore, Seriality and Success yields GRA∧BK’, for some K’ such that A ∧ B ∈ K’.
However, then K’ must be L-inconsistent, so K’ = K. That is, GRA∧BK, in this case. Case (ii). Both A and
B are L-consistent. Then, H and K are L-consistent, by Strong Consistency.
Success yields that B ∈ K.
Hence, ¬B ∉ H (because otherwise ¬B ∈ K, which would contradict the L-consistency of K). But then, since
GRAH, (α) implies that GRA∧B(H + B). Since H ⊆ K, Success yields H + B ⊆ K.
On the other hand,
HRBK yields, via Minimality, that K ⊆ (H + B). Hence, K = H + B. We conclude that GRA∧BK.
The condition (α) is of interest for two reasons:
(1) Given functionality, (α) immediately entails the condition:
If ¬B ∉ G ∗ A, then G ∗ (A ∧ B) = (G ∗ A) + B.
This condition, in its turn, is equivalent to the conjunction of Gärdenfors’ conditions (K*7) and (K*8) — his
“non-basic axioms” on belief revision.
(2) Unlike Stepwise Revision, (α) is a “local” condition on belief revision in the sense that it only concerns
different revisions of one and the same theory: it does not relate revisions of different theories to each other.
43
This “local character” is something that (α) shares with the rest of Gärdenfors’ axioms (the distinction between
local and global conditions on belief revision is discussed in greater detail in Lindström and Rabinowicz (1989)).
Notice that, unlike (StepRev), (α) implies Weak Preservation, provided only that we assume Minimality and
Trivial Preservation. To see this, let A in (α) be any tautology. Seriality, Minimality and Trivial Preservation
then imply that GRAG. Therefore, (α) implies that if ¬ B ∉ G, GRB(H+B). But then Weak Preservation
follows.
There is another condition that is closely related to Stepwise Revision, namely:
(β) If G RAH, HRBK and ¬B ∉ H, then GRA∧BK.
This is the condition that was used in place of Stepwise Revision in Lindström and Rabinowicz (1989). (β)
immediately follows from Stepwise Revision together with Preservation. On the other hand, Stepwise Revision
can be derived from (β) in the presence of Strong Consistency, Minimality and Weak Preservation.
Condition (β) differs from Stepwise Revision and (α) in that it implies full Preservation (given Trivial
Preservation and Minimality). Therefore this condition is an inappropriate alternative to Stepwise Revision
when we consider belief revision systems that are not fully preservative. Similarly, (α) is inappropriate in
contexts where even Weak Preservation is questioned.
11
This definition is equivalent to the following definition by cases:
(a) if X ∩ P ≠ ∅ and X ⊆/ P, r(X, P, Y) iff Y = P or Y ≈ X ∩ P;
(b) if X ∩ P ≠ ∅ and X ⊆ P, r(X, P, Y) iff Y ≈ X ∩ P;
(c) if X ∩ P = ∅, r(X, P, Y) iff Y = P.
12
It might be noted that, in the presence of >-normality, Connection Downwards is not only equivalent to
(RR⇒) but also to the following Modified Relational Ramsey test (MRR):
A > B ∈ G iff, for every L-
consistent extension H of G, and for every K, if HRAK, then B ∈ K.
13
This idea of defining r in terms of an underlying revision-relation for worlds r is somewhat similar to a
construction in Segerberg (to appear). However, there are two important differences between our approach and
Segerberg’s. Firstly, Segerberg’s construction forces the revision relation to be functional, and, secondly, he
does not have anything corresponding to our choice function. As a result of these differences, Segerberg’s
modelling is not fully Gärdenfors-type: it fails to validate Stepwise Revision. However, it should be added that
44
Segerberg’s ambition is different from ours: he wants to define the operation of belief revision in terms of beliefs
in conditionals. Our goal is more modest: to give a semantics that ties belief revision and conditionals together.
14
Notice, that in the presence of functionality, >-normality and (RR⇒) are equivalent to the Modified Ramsey
test (MR): A > B ∈ G iff for every L-consistent extension H of G, B ∈ H ∗ A. Here, ∗ is the belief revision
operation.
15
As a matter of fact, in our semantical framework, there are other ways of interpreting a might-conditional.
The following strengthenings of the right-hand side in (N→r) are all more or less plausible:
(i)
for all Y ⊆ W, if r(w, ≠ A≠ M , Y) then Y ∩ ≠ B≠ M.≠ ∅;
(ii)
for some Y ⊆ W, r(w, ≠ A≠ M, Y) and Y ⊆ ≠ B≠ M;
(iii)
the conjunction of (i) and (ii).
Each of these alternative truth-clauses would make A > B entail A N→ B, but not vice versa. However, neither
of them would be compatible with the definition of N→ as the dual of >. Possibly, one could introduce mightconnectives corresponding to clauses (i) and (ii) as new primitives of the language.
16
For more triviality results in connection with the negative Ramsey test, see Gärdenfors, Lindström, Morreau
and Rabinowicz (to appear).
17
Just as (RR⇒) together with >-normality are equivalent to the Modified Relational Ramsey Test (MRR) for
>, N→-normality together with (RRN→ ⇒) may be stated in the form of a modified Relational Ramsey Test
for might-conditionals (MRRN→): A N→ B ∈ G iff for every L-consistent extension H of G, there is some K
such that HRAK and ¬B ∉ K.
18
This principle is the functional version of (MRRN→), which as we have pointed out above, is equivalent to
(RRN→ ⇒) in the presence of N→-normality. Analogously, (MR) is the functional version of (MRR), and we
have seen that (MRR) is equivalent to (RR⇒) in the presence of >-normality.
19
In this quotation, we have slightly adjusted Gärdenfors’ notation and terminology in order for it to conform to
that of the rest of our paper.
20
In fact, if we assume a slightly stronger, but still intuitive, version of (c),
(c’)
for some m such that H ⊆ m and for every K, if mRAK, then ¬C ∈ K,
45
then we would get a counterexample not just to (RR⇐) but also to (RRN→ ⇐). The argument goes as
follows: Given (RRN→ ⇐), the assumption (a) together with the seriality of R implies
(1’)
A N→ C ∈ G,
from which it immediately follows that
(2’)
A N→ C ∈ H.
By N→-normality, (2’) implies:
(ii’)
for every m such that H ⊆ m, there is some K such that mRAK and ¬C ∉ K.
But, (ii’) contradicts (c’).
21
Another counterexample to the restricted Ramsey test could be constructed out of the “Tweety”-example in
Lindström and Rabinowicz (1989), pp. 80 f.
22
Note that the situation is analogous for might-conditionals. Counterfactual might-conditionals, such as “If
Hitler had decided to invade England, Germany might have won the war”, do not satisfy (RRN→ ⇒), but the
open ones do.
REFERENCES
Alchourrón, C. E., Gärdenfors, P., and Makinson, D. (1985) ‘On the logic of theory change:
Partial meet contraction and revision functions’, Journal of Symbolic Logic 50, 510-530.
Chellas, B. (1975) ‘Basic conditional logic’, Journal of Philosophical Logic 4, 133-153.
Dudman, V. H. (1984) ‘Parsing if-sentences’, Analysis 44, 145-186.
Dudman, V. H. (1988) ‘Indicative and subjunctive’, Analysis 48, 114-122.
Gärdenfors, P. (1988)
Knowledge in Flux: Modeling the Dynamics of Epistemic States,
Bradford Books, MIT Press.
Gärdenfors, P., Lindström, S., Morreau, M., and Rabinowicz, W. (to appear) ‘The negative
Ramsey test: another triviality result’. Forthcoming in Fuhrmann/ Morreau (eds.) The Logic
of Theory Change, Lecture Notes in Computer Science. Berlin: Springer Verlag.
46
Grove, A. (1988) ‘Two modellings for theory change’, Journal of Philosophical Logic 17, 157170.
Harman, G (1986) Change in View: Principles of Reasoning. Bradford Books, MIT Press.
Lehman, D. and Magidor, M. (to appear) ‘What does a conditional knowledge base entail?’.
Levi, I. (1977) ‘Subjunctives, dispositions and chances’, Synthese 34, 423-455.
Levi, I. (1988) ‘Iteration of conditionals and the Ramsey test’, Synthese 76, 49-81.
Lindström, S., and Rabinowicz, W. (1989) ‘On probabilistic representation of non-probabilistic
belief revision’, Journal of Philosophical Logic 18, 69-101.
Lindström, S., and Rabinowicz, W. (to appear) ‘Epistemic entrenchment with incomparabilities
and relational belief revision’. Forthcoming in Fuhrmann/ Morreau (eds.) The Logic of
Theory Change, Lecture Notes in Computer Science. Berlin: Springer Verlag.
Sahlin, N.-E., (1990) The Philosophy of F. P. Ramsey, Cambridge University Press.
Segerberg, K. (1989) ‘A Note on an Impossibility Theorem of Gärdenfors’, Noûs 23, 351-354.
Segerberg, K. (to appear) ‘On the logic of small changes of theories I’, Acta Philosophica
Fennica.
Smiley, T. (1983-84) ‘Hunter on Conditionals’ Proceedings of the Aristotelian Society 84,
242-249.
Stalnaker, R. (1984) Inquiry, MIT Press.
Department of Philosophy
Uppsala University
Villavägen 5
S-752 36 Uppsala
Sweden