## Abstract

The aim of this paper is to show that the logical approach to philosophy of science could be further improved with tools like the ones put forward by Schurz in his proposal to model scientific theory development. Section 2 is a presentation of the basics in AGM epistemology of logical abduction and of their connection. In Sect. 3 several operations for theory change proposed by Schurz (2011; 2018) are presented, followed by my own proposal of a further case of hypothesis refinement, consisting of the instantiation of an existential abduction. Section 4 offers a discussion and conclusions.

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## Notes

Note that Schurz (2011, 85) draws a distinction between

*settled*and*unsettled*hypotheses (taken in turn from Levi 1980); the former become part of background beliefs—only after sufficiently many confirmations—the latter are those hypotheses still under investigation. Let us recall that for Peirce every hypothesis must be subject to further testing.Some examples: ∀xFx, ∀x(Fx ⊃ Gx), ∀yGx are elementary, while \(\forall {\text{x}}\forall {\text{y}}({\text{Fx}} \wedge {\text{Gy}})\) is not; note the latter formula is logically equivalent with \(\forall {\text{xFx}} \wedge \forall {\text{yGy}}\).

As noted by Schurz, the contraction operation is “intended to be some preferred T-subset which does no longer entail s” (Schurz 2018, 470) and several ways of dealing with this operation are possible.

A belief set (i) is a set of sentences from a logical language L closed under logical consequence. The second approach (ii) emerged in reaction to the first. It represents the theory T as a

*base for a belief set*B_{T}, where B_{T}is a finite subset of T satisfying Cn(B_{T}) = T. (That is, the set of logical consequences of B_{T}is the classical belief state). The intuition behind this is that some of the agent's beliefs have no independent status, but arise only as inferences from more basic beliefs. Finally, the more semantic approach (iii) moves away from syntactic structure, and represents theories as sets W_{T}of possible worlds (i.e., their models). For more details and various equivalences between these approaches see (cf. Gärdenfors 1988; Hansson 1999; Rott 1995).A constructive approach is more appropriate for algorithmic models of belief revision, the one based on postulates serves as a logical description of the properties that any such operations should satisfy. The two can also be combined. For example, an algorithmic contraction procedure may be checked for correctness according to given postulates.

These choices may be interdependent. Say, a constructive approach might favor a representation by belief bases, and hence define belief revision operations on some finite base, rather than the whole background theory. Moreover, the epistemological stance determines what constitutes

*rational epistemic change*.Except in those models of non-prioritized belief revision (Niiniluoto 2018, 146).

A theory T is trivial if it is the case that for every formula s, T ⊨ s. In classical logic consistency and triviality are indeed equivalent but this is not so in other logics, such as in paraconsistent logics. For an account of abduction in one of these logical systems, see Carnielli (2006).

The minimality criterion can change depending on the abductive problem that is being handled. In certain contexts, it can refer to the best explanation (add to the theory as few formulas as possible) and in others to the preferred explanation (which requires a pre-defined preference order).

The syntactic restriction serves as a way to establish if the explanation should be an atomic formula or maybe a conjunction of atomic formulas. This condition is necessary in computational frameworks, those in which a restriction on the possibly large number of explanations is needed.

Accepted beliefs are not considered, since these do not call for explanation. Moreover, note that abductive and epistemic notions are equivalent when the theory is closed under logical consequence.

In many cases, several formulas and not just one must be removed from the theory. As explained, the reason is that sets of formulae which entail (explain) e

_{i}should be removed. That is, T′ = T – {s_{1}, …, s_{l}}_{(1 ≤ i ≤ l)}. For example, given T = {s_{1}→ s_{2}, s_{1}, s_{2}} and \({\text{e}}_{{\text{i}}} = \neg {\text{s}}_{{2}}\), in order to make T and \(\neg {\text{s}}_{{2}}\) consistent, one needs to remove either {s_{2}, s_{1}} or {s_{2}, s_{1}→ s_{2}}.Even so, one might speculate whether facts which are merely probable on the basis of T might still need explanation of some sort to further cement their status.

An equivalent formulation of abductive revision is

*ab-initio*revision, its core idea being to construct h_{r}from scratch, given that the original hypothesis h would be forgotten after theory contraction. (Schurz 2011, 95).The

*Rule of Success*(RS) states that a theory is replaced by another one when the latter has been “sufficiently confirmed”.Kuipers’ (2005, 56–58) further argues for his non-naive falsificationism, which together with his evaluation methodology, is compatible with truth-approximation.

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## Acknowledgements

I am grateful to two anonymous reviewers for their detailed analysis and suggestions on a previous version of this paper. I also thank the support of the research project (PAPIIT IN 403219) headed by Ambrosio Velasco-Gómez at the Institute for Philosophical Research, Universidad Nacional Autónoma de México, (UNAM).

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Aliseda, A. On Schurz’s *Construction Paradigm* of Scientific Theory Development.
*J Gen Philos Sci* **54**, 473–490 (2023). https://doi.org/10.1007/s10838-021-09593-z

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DOI: https://doi.org/10.1007/s10838-021-09593-z