Abstract
In this paper, we investigate the problem of truth approximation via belief merging, i.e., we ask whether, and under what conditions, a group of inquirers merging together their beliefs makes progress toward the truth about the underlying domain. We answer this question by proving some formal results on how belief merging operators perform with respect to the task of truth approximation, construed as increasing verisimilitude or truthlikeness. Our results shed new light on the issue of how rational (dis)agreement affects the inquirers’ quest for truth. In particular, they vindicate the intuition that scientific inquiry, and rational discussion in general, benefits from some heterogeneity in opinion and interaction among different viewpoints. The links between our approach and related analyses of truth tracking, judgment aggregation, and opinion dynamics, are also highlighted.
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Notes
In a multiset, each element can appear more than once: thus, while the two sets \(\{T_1,T_2,T_2\}\) and \(\{T_1,T_2\}\) are the same, the two multisets \([T_1,T_2,T_2]\) and \([T_1,T_2]\) are different. In both cases the order of elements is irrelevant.
The discussion of arbitration merging, and of other kinds of operators, in connection with truth approximation deserves further research, but has to be left for another occasion.
After Miller (1974) and Tichý (1974) independently proved that, on the basis of Popper’s own definition of verisimilitude, a false theory can never be closer to the truth than another (true or false) theory, such authors as Niiniluoto (1987, 1999b), Kuipers (1987, 2000), Oddie (1986), Festa (1987, 2007) and Schurz and Weingartner (1987, 2010) developed a number of post-Popperian theories of verisimilitude that succeed in avoiding the problems encountered by Popper’s definition. A survey of the history of theories of verisimilitude is provided by Niiniluoto (1998); see also Oddie (2008, 2013) for a comparison and an assessment of different accounts, and Cevolani and Tambolo (2013) for an introduction to the verisimilitudinarian approach to scientific progress.
Two examples of such distance measures are the “average” measure proposed by Oddie (1986)—who defines \(\varDelta _{}(T,c_\star )\) as the average distance of the constituents in \(\fancyscript{R}_{T}\) from \(c_\star \)—and the “min-sum” measure proposed by Niiniluoto (1987)—defined as a weighted sum of the minimum distance of \(T\) and of the normalized sum of all distances of the constituents in \(\fancyscript{R}_{T}\) from \(c_\star \).
A survey of the main results obtained so far can be found in the introductory essay by Kuipers and Schurz (2011) to a special issue of Erkenntnis entirely devoted to the topic of Belief Revision Aiming at Truth Approximation; see in particular the contributions by Niiniluoto (2011), Cevolani et al. (2011), Kuipers (2011), and Schurz (2011). See also Cevolani et al. (2013) and Cevolani (2013) for further work in this direction.
Indeed, \( Vs _{\phi }\) satisfies the stronger condition that, among true theories, the one with the greater number of matches is more verisimilar than the other; i.e., if \(T_1\) and \(T_2\) are true and \( cont _{t}(T_1,c_\star )> cont _{t}(T_2,c_\star )\) then \( Vs _{\phi }(T_1)> Vs _{\phi }(T_2)\).
Note the difference between the merging and the AGM revision of c-theories \(T_1\) and \(T_2\), which amounts to \(T_1*T_2 = T_2 \wedge X_{T_1T_2}\) (Cevolani et al. 2011, p. 193).
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Acknowledgments
Earlier versions of this paper were presented at the workshop on “Realism, Antirealism, and the Aims of Science” in Trieste (30 June 2012) and at the 5th Copenhagen Lund Workshop on Social Epistemology in Lund (6–7 December 2012). I’d like to thank the participants in those conferences, and in particular Gregor Betz, Jesús Zamora Bonilla, Vincenzo Crupi, Vincent Hendricks, Erik Olsson, Carlo Proietti, and Frank Zenker, for their useful comments. I’m also grateful to Roberto Festa, Theo Kuipers, Gabriella Pigozzi, Luca Tambolo and to two anonymous referees for providing valuable feedback on the first draft of the paper. Support is acknowledged from the Deutsche Forschungsgemeinschaft (priority program “New Frameworks of Rationality”, SPP 1516, grant CR 409/1-1) and from the Italian Ministry of Scientific Research (FIRB project “Structures and Dynamics of Knowledge and Cognition”, Turin unit, D11J12000470001).
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Appendix: Proofs
Appendix: Proofs
We first introduce some notation and terminology. Given a c-theory \(T\), a completion of \(T\) is any constituent \(c_i\) of \(\fancyscript{L}_{n}\) which agrees with all the claims of \(T\). One can check that \(c_i\in \fancyscript{R}_{T}\) iff \(c_i\) is a completion of \(T\). The reversal of a constituent is another constituent, which is the conjunction of all the negations of the conjuncts of the former constituent. By extension, the reversal of a sentence \(X\) is the sentence which contains, in its range, the reversal of each constituent appearing in \(\fancyscript{R}_{X}\) (Oddie 1986, p. 50). One can check that, given a c-theory \(T\), the reversal of \(T\) is just the conjunction of the negations of the claims of \(T\); this was called the “specular” of \(T\) by Cevolani et al. (2011, p. 186).
Proof of result(8) Let be \(E=[T_1,\dots ,T_k]\) a profile of \(k\) c-theories. Recall from (3) that \(E^{\circ } = \bigvee \{c_i: \varDelta _{ min }({E},{c_i})\text { is minimal}\}\), where \(\varDelta _{}(E,c_i)= \sum _{T_j\in E} \varDelta _{ min }({T_j},{c_i})\). Note that, if \(T_j\) is a c-theory, the closest constituent in \(\fancyscript{R}_{T_j}\) to \(c_i\) is the completion of \(T_j\) which agree with \(c_i\) on all the basic sentences which are indeterminate in \(T_j\); thus, \(\varDelta _{ min }({T_j},{c_i})\) is just the number of mismatches between \(T_j\) and \(c_i\). In turn, \(\varDelta _{}(E,c_i)\) is the sum of the number of mismatches between \(c_i\) and each \(T_j\) in \(E\). Moreover, one can check that this sum can be equivalently expressed as the the sum, for each basic sentence \(b\) appearing in \(c_i\), of the number of theories in \(E\) rejecting \(b\). This means that \(c_i\) is the closer to \(E\), the smaller the number of theories in \(E\) rejecting each of its claims.
Let be \(c_i\) and \(c_h\) two constituents differing only because \(b\) is accepted in \(c_i\) and rejected in \(c_h\). Then \(c_i\) is closer than \(c_h\) to \(E\) iff: 1) either some \(T_j\) in \(E\) accepts \(b\) and no \(T_j\) in \(E\) rejects \(b\); 2) or there are in \(E\) both theories accepting \(b\) and theories rejecting \(b\), but the former are more than the latter. This means that all the constituents closest to \(E\) will agree on each basic sentence \(b\) such that the number of theories in \(E\) accepting \(b\) is greater than the number of those rejecting it. The conjunction of such \(b\) is exactly the c-theory \(E^{\circ }\).
Proof of result(9) Let be \(T_1\) and \(T_2\) two c-theories. As Fig. 2 shows, they can be expressed, respectively, as \(T_1\equiv O_{T_1T_2}\wedge C_{T_1T_2}\wedge X_{T_1T_2}\) and \(T_1\equiv O_{T_2T_1}\wedge C_{T_2T_1}\wedge X_{T_2T_1}\). Note also that, on the one hand, the claims of \(O_{T_1T_2}=O_{T_2T_1}\), of \(X_{T_1T_2}\), and of \(X_{T_2T_1}\) are rejected neither by \(T_1\) nor by \(T_2\). On the other hand, each claim of \(C_{T_1T_2}\) and of \(C_{T_2T_1}\) is rejected by one theory and accepted by the other. From result (8), it follows that \(T_1\circ T_2 = O_{T_1T_2}\wedge X_{T_1T_2} \wedge X_{T_2T_1}\).
Proof of result(10) Let be \(T_1\) and \(T_2\) two c-theories. Note first that, as one can easily check, measure \( Vs _{\phi }\) as defined in (7) is additive in the sense that, given a c-theory \(T\), \( Vs _{\phi }(T)=\sum _{b} Vs _{\phi }(b)\) where \(b\) is a claim of \(T\). Thus, e.g., \( Vs _{\phi }(T_1)= Vs _{\phi }(O_{T_1T_2})+ Vs _{\phi }(C_{T_1T_2})+ Vs _{\phi }(X_{T_1T_2})\). From result (9), it follows that \(T_1\circ T_2 = O_{T_1T_2}\wedge X_{T_1T_2} \wedge X_{T_2T_1}\). Then \( Vs _{\phi }(T_1\circ T_2)> Vs _{\phi }(T_1)\) iff \( Vs _{\phi }(O_{T_1T_2})+ Vs _{\phi }(X_{T_1T_2})+ Vs _{\phi }(X_{T_2T_1})> Vs _{\phi }(O_{T_1T_2})+ Vs _{\phi }(C_{T_1T_2})+ Vs _{\phi }(X_{T_1T_2})\) iff \( Vs _{\phi }(X_{T_2T_1})> Vs _{\phi }(C_{T_1T_2})\). Similarly, \( Vs _{\phi }(T_1\circ T_2)> Vs _{\phi }(T_2)\) iff \( Vs _{\phi }(X_{T_1T_2})> Vs _{\phi }(C_{T_2T_1})\).
Proof of result(11) Let be \(T_1\) and \(T_2\) two c-theories and suppose that there is no weak disagreement between \(T_1\) and \(T_2\), i.e., that their extra parts are “empty”. From result (9) it follows that \(T_1\circ T_2 = O_{T_1T_2}\). Moreover, from result (10) it follows that \( Vs _{\phi }(T_1\circ T_2) > Vs _{\phi }(T_1)\) iff \( Vs _{\phi }(C_{T_1T_2})<0\) and \( Vs _{\phi }(T_1\circ T_2)> Vs _{\phi }(T_2)\) iff \( Vs _{\phi }(C_{T_2T_1})<0\). Note that, by definition, the conflicting part of \(T_1\) is the reversal of the conflicting part of \(T_2\), and vice versa. One can check that, if \(\phi \ge 1\), then \( Vs _{\phi }(C_{T_1T_2})>0\) iff \( Vs _{\phi }(C_{T_2T_1})<0\). In fact, \( Vs _{\phi }(C_{T_1T_2})>0\) iff, by definition (7), \( cont _{t}(C_{T_1T_2},c_\star )>\phi cont _{f}(C_{T_1T_2},c_\star )\), i.e., if \(T\) makes more matches than mistakes. But since, by definition, \( cont _{t}(C_{T_1T_2},c_\star )= cont _{f}(C_{T_2T_1},c_\star )\) and \( cont _{f}(C_{T_1T_2},c_\star )= cont _{t}(C_{T_2T_1},c_\star )\), this means that \(C_{T_2T_1}\) makes more mistakes than matches and hence, given \(\phi \ge 1\), that \( Vs _{\phi }(C_{T_2T_1})<0\). It then follows that \( Vs _{\phi }(T_1\circ T_2) > Vs _{\phi }(T_1)\) iff \( Vs _{\phi }(T_1\circ T_2)< Vs _{\phi }(T_2)\), and vice versa.
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Cevolani, G. Truth approximation, belief merging, and peer disagreement. Synthese 191, 2383–2401 (2014). https://doi.org/10.1007/s11229-014-0486-2
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DOI: https://doi.org/10.1007/s11229-014-0486-2