Abstract
The qualitative theory of nomic truth approximation, presented in Kuipers (From instrumentalism to constructive realism. On some relations between confirmation, empirical progress, and truth approximation. Synthese library, vol. 287. Kluwer Academic Publishers, Dordrecht, 2000), in which ‘the truth’ concerns the distinction between nomic, e.g. physical, possibilities and impossibilities, rests on a very restrictive assumption viz. that theories always claim to characterize the boundary between nomic possibilities and impossibilities. In Chap. 2 we have already shown that truth approximation is possible by ascribing, in Popperian spirit, theories only an excluding function. By fully recognizing two different functions of theories, viz. excluding and representing (including), this chapter drops the restrictive assumption by conceiving theories in development as tuples of postulates and models, where the postulates claim to exclude nomic impossibilities and the (not-excluded) models claim to represent nomic possibilities.
Revising such ‘two-sided’ theories becomes then a matter of adding or revising models and/or postulates in the light of increasing evidence, now captured by a special kind of theories, viz. ‘data-theories’. Under the assumption that the data-theory is true, achieving empirical progress on its basis provides good reasons for the abductive conclusion that truth approximation has been achieved as well. As in Chap. 2, the notions of truth approximation and empirical progress are formally direct generalizations of the earlier ones. However, truth approximation is now explicitly defined in two-sided terms of increasing truth-content and decreasing falsity-content of theories, whereas empirical progress is defined in terms of lasting increased accepted-content and decreased rejected-content in the light of increasing evidence. As already noted in Chap. 2, these definitions are strongly inspired by a paper of Gustavo Cevolani, Vincenzo Crupi and Roberto Festa, viz., “Verisimilitude and belief change for conjunctive theories” (Erkenntnis, 2011).
Revised version of: “Models , postulates , and generalized nomic truth approximation ”, Synthese, 193 (10), 2016, 3057–3077.
Acknowledgement: I like to thank David Atkinson , Erik Krabbe , and Jan-Willem Romeijn for their helpful remarks at a first presentation in Groningen (PCCP), Fred Muller as commentator and the rest of the audience in Rotterdam (EIPE), the audience in Cambridge (CamPoS) and Roberto Festa and, above all, Gustavo Cevolani for their constructive comments.
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Notes
- 1.
Note that as far as postulates represent, they represent unrealizable possibilities.
- 2.
See also Cevolani and Festa (to appear) as the most recent presentation, including a significant extension to the case of arbitrary theories, notably disjunctive theories.
- 3.
Note that such a set of specified (intended and interpreted) models is comparable to a set of exemplars in Kuhn’s sense or a set (of GSP-closures of tentative representations) of the intended applications in Sneed’s sense. Examples in the present context are falling objects, pendulae, ballistic and planetary curves.
- 4.
Subsets of Pcpm corresponding to systems with different kinds of forces may overlap, notably in the case of two kinds of forces, but they need not exhaust Pcpm, e.g. there may be room for as yet unknown (formally different) kinds of forces. Moreover, although it is (claimed to be) excluded that non-members of the subset of, say gravitational cpm-models , are realizable, it may, but need not be claimed that all members are realizable. It seems plausible that physicists before Einstein assumed that e.g. all gravitational cpm-models are (in principle) physically realizable by natural systems, but this had to be revised by revising (concepts and) postulates .
- 5.
This also provides a potential explanation of that behavior, where, in agreement with the DN-model of explanation, the bottom-up data function as initial conditions and the postulates as laws.
- 6.
Although we suggest that P is determined as the set of models of a given set of postulates Π, and M is just given as a set of members of U, it may well be that in some contexts both P and M are directly specified as subsets of U. Alternatively, whereas P may be based on a set of exclusion postulates, M may be based on a set of inclusion postulates. However, I prefer to speak of models because the term ‘model’ suggests to represent something and hence to be included in some target set, whereas the term ‘postulate’ suggests to exclude things by imposing a necessary condition to be satisfied in order to be included.
- 7.
A theory is inconsistent in the syntactic (logical) sense when it is possible to derive a contradiction or in the semantic (logical), but equivalent, sense when there is no model satisfying it.
- 8.
In the first view, the postulates (or axioms) are explicitly formulated in a specified language and the models are those structures of the language that satisfy all of them. In the second view, the language is not specified but the models are directly specified in such a way that the postulates are built into them. Note that in the basic format of a theory in the structuralist sense <U, M, I>, U corresponds with our U, M with our M and P, with M = P, and I with T. As suggested, both views on theories seem to conceive them as maximal. However, one may also argue that they are purely dealing with an exclusion claim , viz. (T=) I ⊆ M (=P), in the structuralist case.
- 9.
This leads to the original ‘symmetric difference’- definition (Miller 1978; Kuipers 1982), assuming the strong claim of maximal theories: M = P = T. By the way, Miller’s definition was focused on ‘the actual truth’, i.e., one particular possibility. However, both share the characteristic that they are intuitively similar to Popper’s original but faulty definition in terms of more true and less false consequences. The symmetric difference definition for maximal theories may be summarized as: (1) more correctly and less incorrectly included possibilities (as nomic) or, equivalently, (2) more correctly and less incorrectly excluded possibilities (from being nomic). In the generalized theory, applied to non-maximal theories, (1) characterizes ‘closer to’ on the M-side, and (2) on the P-side.
- 10.
A true consequence of <M, P> at the P-side can be represented as a superset of both P and T, i.e., a set having both P and T as subsets, for the following reason. Let P, T, and C (subset of U) represent the models of the (complex) sentences p, t, and c, respectively, and let P∪T ⊆ C. Then c is a consequence of p and of t, and hence a true consequence of p, for c is true in all models of P and T. The condition that P*∪T is included in P∪T now guarantees that all supersets of P∪T are supersets of P*∪T, and hence that all true consequences of <M, P> (at the P-side) are (true) consequences of <M*, P*>. For a detailed comparison between Popper’s failing consequence-based approach and the ‘model-based’ approach, both for maximal theories, the reader is referred to Kuipers (2000, Chap. 8.1), where the latter is also translated in terms of consequences, leading to the identification of Popper’s bad luck.
- 11.
Formally more neat, we should represent the set of inductive generalizations e.g. by Σ, such that S is the set of models of Σ.
- 12.
This and the following ‘(counter-) example terminology’ is a variation of similar terminology in Kuipers (2000, p. 158).
- 13.
True (false) positives are truly (falsely) claimed nomic possibilities , true (false) negatives are truly (falsely) claimed nomic impossibilities. This terminology assumes that we take the claim that an item in U is a nomic possibility as the claim to be ‘a positive (result)’ and that it is a nomic impossibility as the claim to be ‘a negative (result)’. Now, for example, regarding the members of M∩R, theory <M, P> truly claims that they are positives; hence they may be called ‘true positives’.
- 14.
The undecided M-content is that part of the content of M about which <R, S> is silent, that is, the M-content apart from its accepted and rejected parts: ‘M-content − (accepted M-content ∪ rejected M-content) = M − [(M∩R) ∪ (M∩cS)] = M−(R∪cS). Similarly for the undecided P-content.
- 15.
E.g. suppose U is finite and let <M*, P*> be closer to the truth than <M, P> (partly) due to the ‘strong’ TC-clause (see Table 4.2) that M ∩ T is a proper subset of M* ∩ T. If R results from random sampling in T, sooner or later one of the extra items in M* ∩ T (i.e., items in (M*∩T)−M) will show up, proving ‘more successful’. Similarly, for the possible strong TC-clause on the P-side and the two possible strong FC-clauses.
- 16.
The latter has to be excluded, for it is not only assumed to be true, it would provide a circular explanation of being the best theory. Moreover, opting for it would also amount to the conclusion that after all there was nothing of interest in the original theory.
- 17.
See Chap. 11 for an analysis of this kind of inference to the best explanation.
References
Cevolani, G., & Festa, R. (To appear). A partial consequence account of truthlikeness. To appear in a special issue of Synthese in honor of Gerhard Schurz.
Cevolani, G., Crupi, V., & Festa, R. (2011). Verisimilitude and belief change for conjunctive theories. Erkenntnis, 75(2), 183–202.
Kuipers, T. (1982). Approaching descriptive and theoretical truth. Erkenntnis, 18(3), 343–378.
Kuipers, T. (2000). From instrumentalism to constructive realism. On some relations between confirmation, empirical progress, and truth approximation. Synthese library (Vol. 287). Dordrecht: Kluwer Academic Publishers.
Miller, D. (1978). On distance from the truth as a true distance. In J. Hintikka et al. (Eds.), Essays on mathematical and philosophical logic (pp. 415–435). Dordrecht: Reidel.
Niiniluoto, I. (1987). Truthlikeness. Synthese library (Vol. 185). Dordrecht: Reidel.
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Kuipers, T.A.F. (2019). Models, Postulates, and Generalized Nomic Truth Approximation. In: Nomic Truth Approximation Revisited. Synthese Library, vol 399. Springer, Cham. https://doi.org/10.1007/978-3-319-98388-2_4
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