Abstract
A dynamic descriptor carries information on how an agent’s beliefs are disposed to be changed in response to potential input(s). A particularly important class of dynamic descriptors are the Ramsey descriptors that have the form \(\Psi \Rightarrow \Xi \) where \(\Psi \) and \(\Xi \) are (static) descriptors of the types introduced in Chapter 4. For example, \(\mathfrak {B}(p \vee q) \Rightarrow \lnot \mathfrak {B}r\) denotes that if the agent changes her beliefs to believe that \(p \vee q\), then she will not believe in r. Ramsey descriptors are axiomatically characterized with a set of plausible postulates that are generalizations of postulates commonly used in the logic of conditional sentences. It is also shown that Ramsey descriptors can unproblematically be inserted into belief sets. Revision by Ramsey descriptors does not give rise to the problems that arise when Ramsey test conditionals are inserted into the AGM framework or related models of belief change. The special case represented by the Ramsey descriptor \(\mathfrak {B}p \Rightarrow \mathfrak {B}q\) corresponds to standard Ramsey test conditionals, and we can define the epistemic conditional \(p \rightarrowtail q\) (“if p then q”) to hold if and only if \(\mathfrak {B}p \Rightarrow \mathfrak {B}q\). However, this is not the only way to derive a sentential conditional from a Ramsey descriptor. Two alternatives to the standard approach are introduced. Furthermore, a formula for deriving non-monotonic inference from a Ramsey descriptor is presented. This proposal is offered as an improvement over the common view that the logic of non-monotonic inference is a fragment of the logic of conditional sentences. The chapter also explores various methods to introduce modalities and autoepistemic beliefs into the belief change framework. The introduction of modalities serves to connect descriptor revision with Dynamic Doxastic Logic (DDL) and related systems.
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Notes
- 1.
- 2.
On what it means to know one’s own beliefs, see [231]. On logics employing an autoepistemic belief operation, see [158, 190, 241, 259].
- 3.
Isaac Levi has expressed a similar view with respect to the dynamic information contained in conditional sentences. See [161] and [163, pp. 49–50]. See also [61, 65, 102].
- 4.
The effects of justificatory relationships on patterns of belief change has been investigated with models employing belief bases, see for instance [91, 104].
- 5.
This term was used with essentially the same meaning by Hans Rott [212].
- 6.
The same type of cognitive unrealism is inherent in standard probability theory. If an agent with a probability function \(\mathfrak {p}\) learns that q, then (provided that \(\mathfrak {p}(q)\ne 0\)) her new probability function \(\mathfrak {p}'\) is derivable from \(\mathfrak {p}\) through the simple formula \( \mathfrak {p}'(x) = \mathfrak {p}(x\mid q) = \mathfrak {p}(x \& q)/\mathfrak {p}(q)\).
- 7.
This example is an improvement by Hans Rott [216] of an example first published in [81].
- 8.
On the Ramsey test, see also Section 3.6 and [6, 85].
- 9.
It is an interesting issue whether a rational agent can have the autoepistemic belief \(\mathfrak {B}p\vee \mathfrak {B}\lnot p\Rightarrow \mathfrak {B}p\) without also having the (static) belief p. This relates to the discussion in Section 7.1 on the connection between static and dynamics beliefs.
- 10.
In the AGM framework such a connection was introduced in [184]. See also [13, 176, 214, 217, 228].
- 11.
This is the solution commonly chosen for contraction by a tautology [1], for shielded contraction in which some non-tautologous sentences are not contractible [51], and for non-prioritized revision in which some sentences cannot be incorporated into the belief set [137, 179]. In our presentation of descriptor revision, we have followed this tradition. (See for instance Definitions 5.2 and 5.9.)
- 12.
The ex falso quodlibet principle is seldom mentioned in presentations of conditional logic, but it follows from the common principle that if p logically implies q, then \(p\rightarrowtail q\) holds in all belief states. See e.g. [26].
- 13.
Technically, in the logic of descriptors a belief set X is interchangeable with a descriptor \(\Pi _X\) that is satisfied by X but not by any other belief set. (See Definition 4.14.) Therefore unitarity can equivalently be expressed by a requirement that the descriptor \(\bigcup \{\Xi \mid \Psi \Rightarrow \Xi \}\) is satisfied by exactly one belief set; this is also why the name “unitarity” was chosen for this postulate.
- 14.
See [177] or [217, pp. 111–119] for useful overviews of properties of sentential conditionals.
- 15.
This postulate is called “cautious monotonicity” in [153, p. 178].
- 16.
See [177, p. 45] and [57, pp. 164–165].
- 17.
Let \(p\rightarrowtail q_1\) and \(p\rightarrowtail q_2\). Cumulative monotony yields \( p \& q_1\rightarrowtail q_2\). Reflexivity yields \( p \& q_1 \& q_2\rightarrowtail p \& q_1 \& q_2\), and with right weakening we obtain \( p \& q_1 \& q_2\rightarrowtail q_1 \& q_2\). Applying cut to \( p \& q_1\rightarrowtail q_2\) and \( p \& q_1 \& q_2\rightarrowtail q_1 \& q_2\) we obtain \( p \& q_1\rightarrowtail q_1 \& q_2\). Finally, we apply cut to \(p\rightarrowtail q_1\) and \( p \& q_1\rightarrowtail q_1 \& q_2\), and obtain \( p\rightarrowtail q_1 \& q_2\) [153, p. 179].
- 18.
This was pointed out to me by John Cantwell.
- 19.
The Gärdenfors theorem is based on the combination of two properties of a belief revision framework: (1) If a sentence p is logically compatible with a belief set K, i.e. \(\lnot p\notin K\), then the revision \(K*p\) does not remove anything from K, i.e. \(K\subseteq K*p\). (2) All Ramsey test conditionals are included in the belief sets at which they hold, i.e. \(p\rightarrowtail q \in K\) if and only if \(q\in K*p\). The combination of (1) and (2) implies that if \(q\in K*r\) and \(\lnot p\notin K\), then \(r\rightarrowtail q \in K\subseteq K*p\), thus \(q\in K*p*r\). Counterexamples to this pattern are easily found; see for instance the taxi driver example in Section 3.5. Gärdenfors showed that the combination of (1) and (2) is incompatible with a set of plausible formal properties of a belief revision framework [68]. Descriptor revision avoids these problems since it does not satisfy (1). For arguments against (1), see Section 3.5 and [85, 212].
- 20.
The context dependence of conditionals has been referred to as the shiftability problem [79]. Other early discussions can be found in [166, p. 465] and [202, pp.134–135]. Several other examples have been given in the literature: “If frogs were mammals, they would have mammae.” − “If frogs were mammals, they would be the only ones not to have mammae.” [256]. “If I had been John Keats, I should not have been able to write the Ode to a Nightingale.” − “If I had been John Keats, then I should have been the man who wrote the Ode to a Nightingale.” [79, pp. 5–6].
- 21.
This is also an illustration of the difficulties involved in representing an actual or hypothetical input (element of \(\mathbb {I}\)) by a single sentence. (Cf. Section 4.1.) Serious considerations of what would happen if kangaroos had no tails do not come out of the blue, but would typically take place in some context that makes it clear whether physical or biological principles are under scrutiny.
- 22.
On inputs that cannot be processed due to vagueness, see also [117, pp. 1021–1025].
- 23.
In their respective contexts, \( p \& s\) and \( p \& e\) are more adequate representations than p of the hypothetical input whose effect on the belief state, specifically with respect to q, is under consideration.
- 24.
This criterion does not preclude the existence of belief change outcomes in which \( p \& \lnot q\) holds. There can be some sentence r, less plausible than p, such that \( p \& \lnot q\) holds in some or all of the r-satisfying belief set outcomes. For an example, let p denote that Bitsy is a female mammal, q that Bitsy can give birth to live young, and r that Bitsy is a platypus.
- 25.
From a formal point of view, this proposal is related to the proposals by Nute [201] and Schlossberger [230, p. 80] that in possible world semantics, the assessment of a conditional sentence should refer not only to the antecedent-satisfying possible worlds that are most similar to the actual world but to all those that are sufficiently similar.
- 26.
This construction has the property that if \(K*p_1=K*p_2\) then \(p_1\) and \(p_2\) are evaluated with the same set of belief sets. Another plausible property of \(\ell \) is: If \(X\leqq Y\) then \(\ell (X)\leqq \ell (Y)\). It will not be needed here.
- 27.
A well-ordering is a linear ordering such that every non-empty subset of its domain has at least one minimal element. That the strict part < of \(\leqq \) has an order type that is either finite or \(\omega \) means that < is either isomorphic with a finite string \(\langle 0,1,\dots n\rangle \) of natural numbers or with the full infinite series \(\langle 0,1, 2\dots \rangle \) of natural numbers. This is a stronger requirement than wellfoundedness. For instance, let \(\mathbb {X}\) consist of all sets \(X_k\) where k is a natural number, and let \(X_k<X_m\) hold if and only if either (a) \(X_k\) is even and \(X_m\) is odd, or (b) \(X_k\) and \(X_m\) are either both even or both odd, and \(k<m\). (This is the sequence \(X_0,X_2,X_4\dots X_1, X_3,X_5\dots \).) This relation is well-founded since every subset of \(\mathbb {X}\) has a <-minimal element. However, it does not satisfy the criterion of Theorem 7.7.
- 28.
Logical necessity and physical necessity may both have the same (S5) logic, but that is no reason to conflate them. ([74, pp. 104–105], cf. [27, 59].) In social choice theory, we usually assume that the preferences of different persons satisfy the same logical rules, but in all non-trivial cases they differ in substance.
- 29.
\(\rightarrowtail \) satisfies CS in any model such that \(*\) satisfies confirmation. CS holds in many systems of conditional logic, see for instance [167, pp. 26–31], [207, p. 249], and [203]. However, it has also been criticized, for instance by Bennett [11, pp. 386–388] and Nozick [200, p. 176].
- 30.
Actually, baboons can even hear tones that are an octave above the upper limit of what a human can hear [243].
- 31.
The introduction of modal notions with similar definitions into the AGM framework is less promising. Due to the success property (\(p\in K*p\)), \(\diamond \)
would hold in AGM for all sentences p. - 32.
Due to the general nature of this definition, the term “theory” for a logically closed set of sentences is used rather than “belief set” that is limited to epistemological interpretations.
- 33.
See [91] for a study of corresponding modal notions in a belief base framework.
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Hansson, S.O. (2017). Dynamic Descriptors. In: Descriptor Revision. Trends in Logic, vol 46. Springer, Cham. https://doi.org/10.1007/978-3-319-53061-1_7
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