Abstract
I erect a framework within the semantic view of theories for explaining the empirical success of internally inconsistent models and theories, with scientific realism in mind. The framework is an instance of the ‘content-driven’ approach to inconsistency, advocated by both Norton (Philos Sci 54:327–350, 1987) and Smith (Stud Hist Philos Sci 19:429–445, 1988a, In: Fine A, Leplin J (eds) PSA1988, 1988b), whose ideas my analysis aims to clarify and substantiate.
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Notes
It would be foolishly optimistic, however, to think that it successfully deals with each and every instance of inconsistency in science.
We need to look at the explicit content of the theoretical assumptions, of course, since these assumptions imply everything by virtue of being inconsistent.
Making ‘more or less the same assumptions’ about the world is a contextual notion: the content of the respective assumptions must be compared in the appropriate theoretical context, taking into account the empirical success in question, for example. Here’s an artificial toy example. From assumptions regarding the speed of light and the length of a stick one can calculate the time (\(T\)) it takes for light to travel back and forth the stick’s length. Let’s assume that both the speed of light (\(c=1\)) and the stick’s length (\(l=1\)) are constant. It follows that \(T=2\). It would be inconsistent to assume that (i) the stick’s length is constant; (ii) the stick’s length is 0.99999 units when light travels one way; and (iii) 1.00001 units when light travels the other way. From these three assumptions one can calculate the time for light to travel back and forth. Although everything follows from an inconsistent set of assumptions, clearly most natural calculations give as the answer either 1.99998, or 2, or 2.00002 units, depending on which assumptions are actually employed in the calculation. All of these are very close to the actual value of \(T\). This can be explained by pointing to the fact that each assumption in the inconsistent set {(i), (ii), (iii)} is (in a natural, intuitive sense) approximately true, given the aim of the calculation.
Unfortunately Smith doesn’t say much about this kind of weakening in general, apart from the suggestive idea that such weakening is appropriate if “the evidence bears more directly on a consequence of the original hypothesis (and certain auxiliary claims)” (Smith 1988b, p. 247, my italics).
These restrictions may not be articulated and made explicit.
In a different context—Newtonian cosmology—Norton (2002) frames the issue in slightly more general terms, but now the content-driven approach seems to have a slightly different twist: the new idea is that “meta-level arguments” (concerning symmetry, say) can pick out some features that an unknown consistent replacement theory would have, and those features are then used to isolate some consequences of the inconsistent theory as worthy of scientific interest. I personally can’t see how to appropriate this recipe in the case of the old quantum theory.
Examples of more specific and less specific properties are afforded by determinate–determinable pairs of properties: being red is one way of being coloured; weighing 10 kg is one way of weighing between 9 and 11 kg; being an equilateral triangle is one way of being a triangle; being neuro-toxic is one way of being poisonous; etc. See, Funkhouser (2006), for example, on the determination relation and how it differs from other kinds of the specification relations, such as multiple realization. My use of ‘more specific’ and ‘less specific’ is meant to cover all of these determination relations.
Or, at least they have given a description of the model system that is unnecessarily specific, in case they are not committed to taking those aspects of the model as representing the target.
Inconsistent property attributions imply everything if taken jointly, of course.
The realist requires a further positive argument to take any given consistent sub-model to give an actual realist explanation of the empirical success of some related inconsistent model.
This is of course manifested in the fact that mathematically both transverse and longitudinal waves can be described by the same mathematical equations.
This example provides a useful illustration of the distinction between more specific and less specific properties. I don’t take it to be a good example of an internally inconsistent model, however, given that the mutually inconsistent hypotheses did not feature in any particular derivation of, or argument for, a particular theoretical result. The example that follows is more pertinent in this regard, but it gives a less intuitive handle on the kinds of properties at play.
The three properties given here are enough to recover the Stefan–Boltzmann law, for example.
There’s no logical inconsistency yet, of course, given that the first piece of information strictly speaking concerns only the past.
What Smith says is suggestive, but begs for a general account: “the evidence bears more directly on a consequence of the original hypothesis (and certain auxiliary claims)” (1988a,b, p. 247).
Cf. also Smith (1988a) on the old quantum theory case.
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Saatsi, J. Inconsistency and scientific realism. Synthese 191, 2941–2955 (2014). https://doi.org/10.1007/s11229-014-0466-6
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DOI: https://doi.org/10.1007/s11229-014-0466-6