Abstract
In earlier chapters we have defined the general notion of correspondence relation and its special cases, the most important of which is counterfactual correspondence. This relation, in turn, was seen to be a global form of explanatory translation. In what follows, I investigate some cases of scientific change that are well known from the history of science, and point out that our notions are applicable to them — at least if appropriate pragmatic conditions are assumed to obtain. As we have seen earlier, whether an application of the correspondence relation, particularly of the counterfactual one, is of explanatory import is very much dependent on relevant pragmatic and hermeneutic conditions. I also examine the intricate — and perhaps not equally well known — problem whether between the phlogiston theory and modern chemistry one can define a reduction or correspondence sketch, and we shall see that the answer is not quite evident, since the alleged informal correspondence relation seems to split into local relations. On the other hand, I study a structuralist logical reconstruction of the former theory, which might yield a possibility to define an exact correspondence relation. The reconstruction may not be intuitively evident, however.
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Notes
This case was worked out in detail in Pearce and Rantala (1984a); see also Pearce (1987).
See Chapters 8–9. The definitions of certain notions in Sections 8.6 and 9.1 require L kω) (where k ≤ beth-ω), a many-sorted infinitary logic admitting of disjunctions and conjunctions of length less than k but only finite strings of quantifiers. As pointed out by J. van Bentham, L ω1ω is not sufficient, contrary to what was claimed in Pearce and Rantala (1984a). For simplicity, we shall try to avoid extensions that are unnecessarily strong in expressive power, such as second order logic.
See Section 9.1.
By ‘velocity’, ‘acceleration’, ‘particle’, etc., is here meant the informal interpretations of respective formal terms occurring in axioms. They have their natural meanings in intended models, but in some models they mean velocity, etc., only in a pathological sense. Similarly for the scientific terms occurring in the other case studies below.
Recall Section 5.1.
For the notion, see Section 9.1.
For the principle, recall Section 5.7; see also Pearce and Rantala (1983b).
For more exact effects of that assumption, see Section 9.1.
Example 1 in Section 6.5.
That is, interpreted as a classical mass of a particle in an intended or standard model, which means that m = m0; and similarly for the other symbols here.
Cf. Sections 6.5 and 9.1.
More precisely, the term denoting the derivative of the position.
For a detailed treatment, see that article.
See Note 2, above.
For the meanings of the terms ‘mass’ and ‘planet’ in this context, see Note 4, above.
Recall Chapter 6.
Consult, e.g., Matthews (1968).
We assume that only one state function is present.
E.g., the momentum and energy operators and Planck’s constant.
As can be seen from Section 9.1.
Gasiorowicz (1996), pp. 125–128. Notation is slightly different here.
See the relevant formal considerations and proofs in Chapters 8–9.
Assuming similar conditions on the second derivative and on the other functions involved.
Matthews (1968), pp. 157–158.
Recall Kuhn’s (1983) strict notion of translation; Chapter 1.
Recall Chapters 2–3 and 6.
Originated by Suppes (1957) and advanced, e.g., by Adams (1959); Sneed (1971); Stegmüller (1973), (1976); Moulines (1975); Balzer (1982); see especially Balzer, Moulines and Sneed (1987).
This is, of course, a counterfactual assumption, but this time on the side of the reduced theory and not of the correspondence relation itself.
There are more complicated and involved forms of the structuralist reduction to which Pearce’s result does not necessarily apply.
See Rantala and Vadén (1997).
Their semantics is sometimes given from the outside, as in the case of digital machines. Then a system is not what Fetzer (1990) calls a semiotic system.
See also Vadén (1995) and (1996) for a clarification of Smolensky’s position. It is not always very clear what Smolensky means by his various notions (to be briefly discussed below).
Cf. Smolensky (1988), pp. 10–11. (7.5.1.1) and (7.5.1.2) are not direct quotations.
This interpretation is presented in Rantala and Vadén (1994).
E.g. in Smolensky (1988), p. 12 (end of Section 5). There seems to be ambiguity here of the kind I referred to above.
A detailed and formal investigation is presented in Section 9.3, below.
Cf. (9.3.1) and (9.3.8), Section 9.3, for more precise formulations.
Cf. (9.3.2) and (9.3.12), where the input argument is omitted, however, to simplify the exposition.
Self-organization means that given an input, N will adjust its connection weights so that an appropriate output follows.
In actual reasearch on neural networks researchers are often aiming at smaller nets, not bigger. But this practical work has nothing to do with the problem concerning the conceptual relationship between the symbolic and subsymbolic.
The conditions were presented in Section 6.4.
P. 112.
P. 282.
p. 59.
See also Section 1.2.
See Rantala (1997).
See Churchland (1992); Bechtel and Abrahamsen (1991).
In Smolensky’s terminology, this is a higher level than the neural level, but lower than the conceptual one.
However, Smolensky says that subsymbolic models that are more likely to be reducible to the neural level should be favored in the coming research.
Recall our discussions in Section 1.2 and Chapter 6.
The nature of the correlation F makes this theoretically obvious in any case.
I shall use the term ‘knowledge’ here (in agreement with computer scientists and many cognitive scientists) even though ‘information’ migth often be more appropriate, since philosophically more neutral.
See, e.g., Bechtel and Abrahamsen (1991), p. 290.
If this assumption is given up, the following considerations are relevant to artificial networks only.
For the latter distinction, see below.
Smolensky (1988), pp. 12 and 20.
What relation he means is not quite clear.
This would mean that S licenses θ’ in something like the sense discussed in Section 8.3, below.
Bechtel and Abrahamsen (1991), p. 163.
See Bechtel and Abrahamsen (1991) for interpreting the kinds of knowledge in this way.
Smolensky (1988), p. 5.
It is obvious that here cp must be an appropriate translation (e.g., of the form ‘the output is close to 1’) of the sentence (e.g., of the form ‘such-and-such a sequence of symbols is an expression of such-and-such a language’) describing the propositional knowledge in question into the (mathematical) language to which S and NI belong.
Smolensky (1988), pp. 4–5.
For a more exact treatment of Nœ, see Section 9.3.
See Bechtel and Abrahamsen (1991).
Quine’s arguments are evidently the best known.
The above studies give rise to further distinctions. For instance, they can be used to explore a subsymbolic distinction between competence and performance in Smolensky’s sense. He invites us to analyze a subsymbolic system in two ways, first at the subconceptual level and then at the conceptual level with and without suitable idealizations. If its processing is characterized at the former level (e.g., by means of S), a description of its performance is obtained, whereas desriptions of the latter kinds (like θ’ and θ’+m is infinite) are about competence. Thus the distinction seems to be roughly the same as the distinction between nonpropositional and proposiotional knowledge of a connectionist system.
This simplification should not influence the results below, but, again, make the possible relation of M to a Turing machine more obvious.
Cf. θ’ in Section 7.5.1. E would now be replaced by a set of true sentences.
Cf. the theory S mentioned in Section 7.5.3, above.
If more than one output unit were needed here, it would slightly modify (7.5.1.5).
See Hintikka (1975).
It is an idealization to assume that S2 is nonempty, but for logical purposes such idealizations are unavoidable. There are others here, of course.
The present Kripke model can be considered a special case of the model defined there.
Recall Section 5.10.
Section 5.10.
Here, of course, implication is defined in terms of negation and conjunction as usual.
See Hughess and Cresswell (1984). ‘Correspondence’ has there a meaning that is different from its meaning in this book.
The latter information I owe to Ken Manders.
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Rantala, V. (2002). Case Studies. In: Explanatory Translation. Synthese Library, vol 312. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1521-8_7
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