Abstract
In this paper I reconstruct and compare criteria of theoreticity that have been developed by Carnap, Sneed and proponents of the Munich school of structuralist philosophy of science. For this purpose I develop a unified framework in which one can transform model-theoretic theory representations into linguistic ones, and vice versa. This bridges the gap between statement and non-statement view and allows a precise comparison of linguistic and model-theoretic criteria of theoreticity. In the final part I suggest a system of improved definitions of theoreticity and pre-theoreticity.
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Notes
BR's for permanent dispositions entail the synthetical consequence ∃t(Txt∧Rxt) → ∀t(Txt → Rxt)); so they can be considered as "analytic" only relative to this assumption.
Single quotation marks are used for stylistic and double ones for literal quotation.
The implication R(T)→T is called the "Carnap-sentence" and expresses the analytic content of T in a global way, which is not "divisible" among the theory's axioms (s. Tuomela 1973, 59).
Otherwise one incurs the so-called Newman problem according to which R(T) does not imply any restrictions on T's empirical submodels, except from possible cardinality restrictions on the domain, which means that the content of R(T) does not go "essentially" beyond T's empirical content (for further discussion see Ladyman and Ross 2007, 126f; Schurz 2014, chap. 5.8.2).
More adequately one should also divide the domain(s) D into non-T-theoretical and T-theoretical individuals and restrict T's partial models to the subdomain Dn ⊆ D of non-T-theoretical individuals. This refinement is not made in structuralism. Balzer et al. (1987), 49, give an argument for why the notion of a particle is generally non-T-theoretical in classical particle mechanics. I don't think that their argument holds for microscopic entities—whether they behave particle-like or (e.g.) wave-like can only be determined using theories.
The axiomatic changes are these: (EXT-in-ZFU): ∀x, y ∉ U(∀z(z ∈ x ↔ z ∈ y) → x = y). (FOUND-in-ZFU): ∀x ∉ U∪{Ø}∃y ∈ x(x ∩ y = Ø)). (UREL-1): ∀x ∈ U(∀y:y ∉ x). (UREL-2):∃B(BIJEC(ω, U)), where BIJEC(ω, U) is the formula asserting that B is a bijection from ω to U. Cf. Löwe (2006).
Cf. van Dalen et al. (1978); Machover (1996). The formal translations can be lengthy. E.g., f:A→ B is translated as: ∀x∃y∃z(x ∈ f → x = (y, z)) ∧ ∀y∃x∃z(y ∈ A → x = (y, z)) ∧ ∀y∀z1, z2((y, z1) ∈ f ∧ (y, z2) ∈ f → z1 = z2), where subformulas of the form "x = (y, z)" have to be replaced by "∀u(u ∈ x ↔ u = y ∨ ∀w(w ∈ u ↔ w = y ∨ w = z))" (according to Kuratowski's definition of the pair (x, y) as {x, {x, y}}).
To handle the problem of non-unique referents, Andreas (2010, def. 11) suggests partial truth values.
The domain unification axiom should also hold for theory nets. Since the partial potential models of different theory lattices do not have all non-theoretical concepts in common, we must restrict this axiom here to those non-theoretical relations which are common to these models.
I have put "essentially" in brackets, because if the additional axioms do not contain the concept rs essentially, this means by definition that there exist a formulation Ax++ of Ax+ which doesn't contain rs and is equivalent with Ax+ given the remainder axioms of T.
A legitimate objection to def. 8 (suggested by a referee) could point out that the equivalent conditions (a) and (a*) are too strong, because several specializations of CPM are characterized by axioms which essentially involve the theoretical term whose values are to be determined. An example is the specialization of CPM by Hooke's law in regard to determinations of the force function. A similar problem is discussed in Balzer et al. (1987, 74f). It emerges from this discussion that if condition (a*) would be omitted, too many terms of a theory T would come out as T-theoretical; for example, the position function would come out as CPM-theoretical.
This disadvantage applies also to the improved Sneed-criterion.
Alternatively, one could allow that pre-theories contain new theoretical concepts and assume a well-founded ordering of "theoreticity-levels" of concepts. We cannot deal here with these complications. The idea of a pre-theory has been explicated in a slightly different way by Balzer and Mühlhölzer (1982). Similar ideas can also be found in Balzer et al. (1987), 58f, where "pre-theoreticity" is modeled by intertheoretical links.
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Acknowledgments
For valuable comments I am indebted to Ulises Moulines, Holger Andreas, Hannes Leitgeb, Ulrich Gädhe, Wolfgang Balzer, and two anonymous referees.
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Schurz, G. Criteria of Theoreticity: Bridging Statement and Non-Statement View. Erkenn 79 (Suppl 8), 1521–1545 (2014). https://doi.org/10.1007/s10670-013-9581-x
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DOI: https://doi.org/10.1007/s10670-013-9581-x