Abstract
Theory change is a central concern in contemporary epistemology and philosophy of science. In this paper, we investigate the relationships between two ongoing research programs providing formal treatments of theory change: the (post-Popperian) approach to verisimilitude and the AGM theory of belief change. We show that appropriately construed accounts emerging from those two lines of epistemological research do yield convergences relative to a specified kind of theories, here labeled “conjunctive”. In this domain, a set of plausible conditions are identified which demonstrably capture the verisimilitudinarian effectiveness of AGM belief change, i.e., its effectiveness in tracking truth approximation. We conclude by indicating some further developments and open issues arising from our results.
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Notes
The first full-fledged account of verisimilitude was provided by Karl Popper (1963). Later, Miller (1974) and Tichý (1974) showed that Popper’s account was untenable, thus opening the way to post-Popperian theories of verisimilitude, emerging ever since 1975. For an excellent survey of the modern history of verisimilitude, see Niiniluoto (1998).
There are exceptions, however: Hans Rott, for instance, recently remarked that AGM theorists “should worry more about truth” meant as one of the basic aims of scientific inquiry; see Rott (2000, p. 513, 518 and ff., and in particular note 38).
See Festa (2007) for early motivation and Cevolani et al. (2011b) for a complete exposition; for discussion of some applications see Cevolani and Festa (2009), Cevolani et al. (2010), and Cevolani and Calandra (2010). Some of the main ideas of the BF-approach were anticipated by Kuipers’ notion of “descriptive verisimilitude” (Kuipers 1982).
See for instance Hintikka (1973, p. 152).
This terminology is inspired by Carnap (1950, p. 67), who calls “basic sentences” the literals of \({\fancyscript{L}}_{n}.\)
The notation “T ?” is motivated by the fact that T ? can be seen as the “question mark area” of T.
Cf. Schurz and Weingartner (2010, section 2) and, in particular, the discussion of “Popper’s intuitions” on pp. 417–418. Indeed, one may note that Definition 1 is structurally identical to Popper’s comparative definition of verisimilitude (Popper 1963, p. 233). The crucial difference is that, instead of the “true b-content” and “false b-content” of c-theories, Popper’s definition concerns the “truth-content” and the “falsity-content” of logically closed theories, defined, respectively, as the classes of their true and false classical logical consequences.
The expression “contrast measures” refers to the fact that these measures can be seen as an application of the “contrast model” of similarity introduced by Tversky (1977) in his study of the similarity of psychological stimuli. Contrast measures have been introduced (without using this name) by Cevolani and Festa (2009) and Cevolani et al. (2010) and are fully discussed in Cevolani et al. (2011b).
In other words, not only all these measures are c-monotonic, but, when applied to the evaluation of the verisimilitude of c-theories, they also turn out to be ordinally equivalent to (and sometimes identical with) Vs ϕ. A measure Vs is ordinally equivalent to another measure Vs′ just in case, for any pair of theories T 1 and \(T_2, Vs(T_1)\gtreqqless Vs(T_2)\) iff \(Vs'(T_1)\gtreqqless Vs'(T_2). \)
For a comparison between contrast measures of verisimilitude and the other measures mentioned above, see Cevolani et al. (2011b).
See Gärdenfors (1988) for a standard introduction to the BS-version.
The largest part of the BB-version has been developed by Hansson (1999).
Package change, along with other kinds of multiple change, has been studied, in particular, by Fuhrmann and Hansson (1994); see also Hansson (1999, pp. 134–139 and 258–261). Multiple change—i.e., change with respect to a input consisting of more than one sentence—should not be conflated with iterated change, where a belief base is repeatedly modified with respect to a number of single inputs.
For a detailed critical examination of this principle see Rott (2000).
As a remark, one should note that the definition in (9) includes, for the sake of generality, also the case where A logically contradicts B, and hence B + A is inconsistent. In the following, we shall always assume, in agreement with (C) and with the informal notion of expansion, that expansion is performed only with respect to compatible inputs. When A is incompatible with B, then a revision, not an expansion, of B by A should be performed.
Hansson (1999, p. 37).
Suppose for instance that B = {p, p → q, q → r} and that \({\fancyscript{X}}\) receives the eliminative input A = {q, r}. In order to remove A from his beliefs, \({\fancyscript{X}}\) has to withdraw some elements of B, since both \(q\in\hbox{Cn}(B)\) and \(r\in\hbox{Cn}(B). \) However, one can check that there are two remainders of B by A, i.e., {p → q, q → r} and {p, q → r}. Thus, \({\fancyscript{X}}\) may remove either p or p → q in order to perform the required contraction. Philosophers of science are familiar with discussions of this issue under the heading of “the Duhem problem.”
In the AGM literature, relative importance is usually represented by the degree of “entrenchment” of \({\fancyscript{X}}\)’s beliefs: when withdrawing some sentences from \({B,\fancyscript{X}}\) will choose the less entrenched in agreement with appropriate selection rules. See Hansson (1999, pp. 11–15) for a discussion of the basic idea.
Hansson (1999, p. 259) calls this the “sentential negation” of A.
See Niiniluoto (1999, sections 4 and 5, in particular Eqs. 10, 17 and 20 and the corresponding discussion).
See Cevolani et al (2010a).
See Niiniluoto (1999, Eqs. 11 and 20).
By the way, one may note that the AGM analysis of theory change has so far been limited only to propositional theories; on this, cf. also Niiniluoto (2010).
Moreover, the BF-approach can be applied to any kind of non-propositional and non-conjunctive theory T, for instance a logically closed set of sentences in a first-order monadic language. Even in cases like this, in fact, one is often interested only in what T entails about the basic features of the world, e.g., about which prototypical properties characterize a certain class of entities. For other relevant examples of this kind, see Kuipers (2011b).
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Acknowledgments
The authors wish to thank Theo Kuipers, Ilkka Niiniluoto, Gerhard Schurz, and all the participants in the workshop on Probability, Confirmation, and Verisimilitude (Trieste, September 2009) and in the EPSA2009-symposium on Belief Revision Aiming at Truth Approximation (Amsterdam, October 2009) for their comments. Two anonymous referees suggested a number of improvements which we gratefully acknowledge. Vincenzo Crupi also acknowledges support by a grant from the Spanish Department of Science and Innovation (FFI2008-01169/FISO).
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Appendix: Proofs
Appendix: Proofs
Proof of Theorem 1
The following remark will be useful in proof:
Remark 1
If X is set of literals and x is a literal of \({\fancyscript{L}_{n},}\) then \(x\in \hbox{Cn}(X)\) iff \(x\in X. \)
-
(i)
Recall first that we are assuming that A is logically compatible with T. By the definition in (9), (T + A)b = T b + A b = T b∪ A b. According to Definition 4, T b may be written as T b oA ∪ T b cA ∪ T b xA and, likewise, A b = A b oT ∪ A b cT ∪ A b xT . Since by hypothesis A is compatible with T, \(T_{cA}^{b}=A_{cT}^{b}=\varnothing; \) moreover T b oA = A b oT by Definition 4. Hence, T b = T b oA ∪ T b xA and A b = T b oA ∪ A b xT . It follows that T b∪ A b = T b∪ A b xT , i.e., T + A = T∧ A xT .
-
(ii)
According to the definition in (10), \(T^{b} \ast A^{b}=((T^{b}-\neg(A^{b}))+A^{b}),\) where \(\neg(A^{b})\) is the negation of A b. Note that, since A is a c-input, any \(a\in A^{b}\) is a literal. We start by proving that T b oA ∪ T b xA is the unique remainder of T b by \(\neg(A^{b})\) (see Definition 3), and hence the unique result of the contraction of T b by \(\neg(A^{b}).\) By Definition 4, \(T_{oA}^{b}\cup T_{xA}^{b}\subseteq T^{b}. \) Moreover, T b oA ∪ T b xA does not imply \(\neg(A^{b}).\) In fact, by Definition 4, each element of T b oA ∪ T b xA is either the negation of a disjunct of \(\neg(A^{b})\) or a logically independent literal. Finally, any set Y such that \(T_{oA}^{b}\cup T_{xA}^{b}\subset Y\subseteq T^{b}\) will contain some element of T oA , again by Definition 4. Any such element is a negation of an element of A b and then implies \(\neg(A^{b}). \) Thus, T b oA ∪ T b xA is a remainder of T b by A b. To see that T b oA ∪ T b xA is the unique remainder, note that any other subset X of T b is such that either \(X\subseteq T_{oA}^{b}\cup T_{xA}^{b}\) or X overlaps T cA and then implies \(\neg(A^{b}). \) In other words, either X is not maximal or implies \(\neg(A^{b}). \) It follows that \((T^{b}-\neg(A^{b}))=T_{oA}^{b}\cup T_{xA}^{b}. \) According to definition 9, \((T^{b}-\neg(A^{b}))+A^{b} = T_{oA}^{b}\cup T_{xA}^{b}\cup A^{b}. \) Since, by Definition 4, \(T_{oA}^{b}=A_{oT}^{b}\subseteq A_{oT}^{b}, \) it follows that T b oA ∪ T b xA ∪ A b = T b xA ∪ A b, i.e., T * A = T xT ∧ A.
-
iii)
To prove the theorem it will suffice to prove that T b cA ∪ T b xA is the unique remainder of T b by A b (cf. Definition 3), and hence the unique result of the contraction of T b by A b. First, let us prove that \(T_{cA}^{b}\cup T_{xA}^{b}\in T^{b}\bot A^{b}. \) By Definition 4, \(T_{cA}^{b}\cup T_{xA}^{b}\subseteq T^{b}. \) Moreover, by Remark 1 and Definition 4, T b cA ∪ T b xA implies no element of A b. Finally, any set Y such that \(T_{cA}^{b}\cup T_{xA}^{b}\subset Y\subseteq T^{b}\) will contain some element of T cA = A oT and then will imply some element of A b. Thus, T b cA ∪ T b xA is a remainder of T b by A b. To see that T b cA ∪ T b xA is the unique remainder, note that any other subset X of T b is such that either \(X\subseteq T_{cA}^{b}\cup T_{xA}^{b}\) or X overlaps T oA = A oT and then implies some element of A b. In other words, X is not maximal or implies some element of A b. Thus, T b − A b = T b cA ∪ T b xA and then T − A = T cA ∧ T xA .
Proof of Theorem 2
First, note that if A is true then A oT , A cT and A xT are also true, and that if A is completely false then A oT , A cT and A xT are also completely false. Moreover, recall that we leave cases of vacuous change aside.
-
(i)
By hypothesis, A is compatible with T, i.e., \(A_{cT}^{b}=T_{cA}^{b}=\varnothing. \) By Theorem, T + A = T∧ A xT = T oA ∧ T xA ∧ A xT . Thus, \(T^{b}\subset(T+A)^{b}=T^{b}\cup A_{xT}^{b}; \) moreover, since A xT is true by hypothesis, \(t(T, C_{\star})\subset t(T+A, C_{\star}), \) whereas \(f(T, C_{\star})=f(T+A, C_{\star}). \) It follows from condition (M t ) of Definition 1 that Vs(T + A) > Vs(T).
-
(ii)
By Theorem 1, T − A = T cA ∧ T xA . Thus, \( (T-A)^{b}\subset T^{b}; \) moreover, since A oT is true by hypothesis, \(t(T-A, C_{\star})\subset t(T, C_{\star}), \) whereas \(f(T, C_{\star})=f(T-A, C_{\star}). \) It follows from condition (M t ) of Definition 1 that Vs(T) > Vs(T − A).
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(iii)
By Theorem, T * A = A ∧ T xA . Note that T can be written as \(A_{oT}\wedge\widetilde{A_{cT}}\wedge T_{xA}\) and T * A as A oT ∧ A cT ∧ A xT ∧ T xA . In the limiting case \( A_{cT}^{b}=\varnothing\) then T * A = A oT ∧ A xT ∧ T xA = T∧ A xT = T + A; thus Vs(T* A) > Vs(T) by the first clause of the present theorem (proved above). Otherwise, note first that since A cT is true by hypothesis, \(\widetilde{A_{cT}}\) is completely false by Definition 4. It follows that \(t(T, C_{\star})\subset t(T+A, C_{\star})=t(T, C_{\star})\cup (A_{oT}\wedge A_{xT})^{b}, \) and that \(f(T \ast A, C_{\star})\subset f(T, C_{\star})=f(T \ast A, C_{\star})\cup (\widetilde{T_{oT}})^{b}. \) Then, from condition (M f ) of Definition 1 that Vs(T* A) > Vs(T).
The proofs of the remaining three clauses of the theorem can be obtained from the ones above in a straightforward manner.
Proof of Theorem 4
Let us first state without proof the following useful lemma:
Lemma 1
Given a c-theory \(T, Vs_{\phi}(T)=\sum_{x\in T^{b}} Vs_{\phi}(x). \) It follows that, given a c-input A, Vs ϕ(T) = Vs ϕ(T oA ) + Vs ϕ(T cA ) + Vs ϕ(T xA ) and Vs ϕ(A) = Vs ϕ(A oT ) + Vs ϕ(A cT ) + Vs ϕ(A xT ).
Then:
-
1.
Vs ϕ(T + A) > Vs ϕ(T) iff (by Theorem 1) Vs ϕ(T∧ A xT ) > Vs ϕ(T) iff (by Definition 4 and Lemma 1) Vs ϕ(T) + Vs ϕ(A xT ) > Vs ϕ(T) iff Vs ϕ(A xT ) > 0.
-
2.
Vs ϕ(T − A) > Vs ϕ(T) iff (by Theorem 1) Vs ϕ(T cA ∧ T xA ) > Vs ϕ(T) iff (by Definition 4 and Lemma 1) Vs ϕ(T cA ) + Vs ϕ(T xA ) > Vs ϕ(T oA ) + Vs ϕ(T cA ) + Vs ϕ(T xA ) iff Vs ϕ(A oT ) = Vs ϕ(T oA ) < 0.
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3.
Vs ϕ(T * A) > Vs ϕ(T) iff (by Theorem 1) Vs ϕ(T xA ∧ A) > Vs ϕ(T) iff (by Definition 4 and Lemma 1) Vs ϕ(T xA ) + Vs ϕ(A oT ) + Vs ϕ(A cT ) + Vs ϕ(A xT ) > Vs ϕ(T oA ) + Vs ϕ(T cA ) + Vs ϕ(T xA ) iff (recalling the definition of specular) \(Vs_{\phi}(A_{xT})>Vs_{\phi}(\widetilde{A_{cT}})-Vs_{\phi}(A_{cT}). \)
Proof of Theorem 5
The following results will be useful in proof:
Lemma 2
For any c-theory T:
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1.
T is verisimilar iff (by Definition 5) Vs ϕ(T) > 0 iff (by Definition 6) \(cont_{t}(T,C_{\star})>\phi cont_{f}(T,C_{\star}). \)
-
2.
\(Vs_{\phi}(T)>Vs_{\phi}(\widetilde{T})\) iff (by Definition 6 and the definition of specular) \(cont_{t}(T,C_{\star})-\phi cont_{f}(T,C_{\star})> cont_{f}(T,C_{\star})-\phi cont_{t}(T,C_{\star})\) iff \( cont_{t}(T,C_{\star})> cont_{f}(T,C_{\star}). \)
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3.
By the two previous results, it follows that: For ϕ = 1, T is verisimilar iff \(Vs_{\phi}(T)>Vs_{\phi}(\widetilde{T})\) iff \(\widetilde{T}\) is t-distant. For ϕ < 1, if T is t-distant then \(Vs_{\phi}(T)<Vs_{\phi}(\widetilde{T})\) and \(\widetilde{T}\) is verisimilar. For ϕ > 1, if T is verisimilar then \(Vs_{\phi}(T)>Vs_{\phi}(\widetilde{T})\) and \(\widetilde{T}\) is t-distant.
Accordingly:
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1.
If A cT and A xT are verisimilar, then Vs ϕ(A cT ) > 0 and Vs ϕ(A xT ) > 0 by Definition 5. Thus, if ϕ ≥ 1, then (by Lemma 2) \(Vs_{\phi}(A_{cT})>Vs_{\phi}(\widetilde{A_{cT}})\) and \(Vs_{\phi}(\widetilde{A_{cT}})-Vs_{\phi}(A_{cT})<0<Vs_{\phi}(A_{xT}); \) it follows that Vs ϕ(T* A) > Vs ϕ(T) by Theorem 4.
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2.
If A cT and A xT are t-distant, then Vs ϕ(A cT ) < 0 and Vs ϕ(A xT ) < 0 by Definition 5. Thus, if ϕ ≤ 1, then (by Lemma 2) \(Vs_{\phi}(A_{cT})<Vs_{\phi}(\widetilde{A_{cT}})\) and \(Vs_{\phi}(\widetilde{A_{cT}})-Vs_{\phi}(A_{cT})>0>Vs_{\phi}(A_{xT}); \) it follows that Vs ϕ(T* A) < Vs ϕ(T) by Theorem 4.
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Cevolani, G., Crupi, V. & Festa, R. Verisimilitude and Belief Change for Conjunctive Theories. Erkenn 75, 183–202 (2011). https://doi.org/10.1007/s10670-011-9290-2
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DOI: https://doi.org/10.1007/s10670-011-9290-2