Abstract
Criteria of empirical significance are supposed to state conditions under which (putative) reference to an unobservable object or property is “empirically meaningful”. The intended kind of empirical meaningfulness should be necessary for admissibility into the selective contexts of scientific inquiry. I defend Justus’s recent argument that the reasons generally given for rejecting the project of defining a significance criterion are unpersuasive. However, as I show, this project remains wedded to an overly narrow conception of its subject matter. Even the most cutting edge significance criteria identify empirical significance with predictive power, and thereby rule out vocabulary with legitimate scientific functions. In a nutshell, the problem is that there are terms that reduce the computational burden of extracting predictions from theory, and that may therefore be scientifically useful, but that do not produce any additional predictions, and so are ruled scientifically inadmissibility by existing significance criteria. I spell out this objection by specifying terms of this kind that are ruled inadmissible by Creath’s and Schurz’s criteria. Having objected in this way to extant criteria, and to the equation of empirical significance with predictive power in general, I discuss an approach to defining empirical significance that is capable of avoiding my objection and, more ambitiously, that may break the cycle of “punctures and patches” that has plagued the project from the beginning: I gloss Goldfarb and Ricketts’s idea of “case-by-case” delineations of empirically significant terms as the provision of special rather than general explications of the informal concept of empirical significance.
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Notes
Carnap provides a “rough explanation” of the notion of an observation term, which grounds the distinction between \(\hbox {V}_{\mathrm{O}}\) and \(\hbox {V}_{\mathrm{T}}\):
A predicate ‘P’ of a language L is called observable for an organism (e.g. a person) N, if, for suitable arguments, e.g. ‘b’, N is able under suitable circumstances to come to a decision with the help of few observations about a full sentence, say ‘P(b)’, i.e. to a confirmation of either ‘P(b)’ or ‘\(\sim \hbox {P(b)}\)’ of such a high degree that he will either accept or reject ‘P(b)’ (Carnap 1936–1937, pp. 454–455).
Degrees of observationality, so understood, lie on a continuum. Carnap proposes that the sharp line between \(\hbox {V}_{O}\) and \(\hbox {V}_{T}\) be drawn at the most convenient point on the continuum (Carnap 1966, p. 226).
The distinction between observational and theoretical language is contested, but I will presuppose it for the sake of argument. For an argument that Carnap only requires an innocuous version of this distinction, see Justus (2014, p. 420).
Creath specifies that the criterion is meant to apply to primitive terms and suggests that “[f]or defined terms we need only require in addition that every descriptive term in the definiens thereof must be antecedently shown to be significant” (1976, p. 395).
Carnap’s (1956a) criterion has an analogous requirement that the observation sentence be logically derivable within the theoretical context. He seems to treat this requirement as a simplifying assumption. A more complete formulation of his criterion would encompass also probabilistic relations between theory and observational prediction (Carnap 1956a, p. 49). This point can be applied mutatis mutandis to subsequent criteria.
It is worth noting that by beginning with term significance and working up to sentence significance, the definition avoids Church’s (1949) famous counter-example to Ayer’s (1946) criterion. Let \(\hbox {O}_{1}\), \(\hbox {O}_{2}\), and \(\hbox {O}_{3}\) be primitive observation predicates and A (for ‘Absolute’) a primitive “metaphysical” predicate, i.e., a predicate every attribution of which the logical empiricists are determined to reject as nonsense. Translating Church’s argument from sentential to predicate logic, we get the sentences
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1.
\( [(\sim (\hbox {x})\hbox {O}_{1}\hbox {x}) \& (\hbox {x})\hbox {O}_{2}\hbox {x}] \bigvee [(\hbox {x})\hbox {O}_{3}\hbox {x} \& (\sim (\hbox {x})\hbox {Ax})]\) and
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2.
\((\hbox {x})\hbox {O}_{1}(\hbox {x})\)
which together imply the observation sentence (x)\(\hbox {O}_{3}\hbox {x}\). According to Ayer’s criterion, a sentence is “directly verifiable” if, with other observation sentences, it entails an observation sentence that these other observation sentences alone do not entail (Ayer 1946, p. 16). So sentence (1) is directly verifiable. And a sentence is empirically significant if, together with other directly verifiable sentences, it entails an observation sentence that these other directly verifiable sentences alone do not entail. Therefore, since (x)Ax and (1) together entail (x)\(\hbox {O}_{2}\hbox {x}\), and since (x)Ax is essential to this entailment, (x)Ax is empirically significant. Of course, any sentence can be put in the truth-functional position of (x)Ax in (1) and thereby rendered empirically significant by Ayer’s (1946) criterion.
Creath’s (1976) criterion avoids this problem. The set of primitive descriptive predicates occurring in (1) has a proper subset—\(\{\hbox {O}_{1}, \hbox {O}_{2}, \hbox {O}_{3}\}\)—whose members occur in sentences that non-vacuously imply (x)\(\hbox {O}_{2}\hbox {x}\). So A’s role in the derivation of (x)\(\hbox {O}_{2}\hbox {x}\) does not make it empirically significant, according to this criterion. And since the criterion builds sentence significance out of term significance, (1) is not empirically significant according to Creath’s criterion. (The basic strategy here originates in Carnap’s (1956a) criterion).
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1.
Schurz (1991) gives a more formally rigorous inductive definition of ‘relevant consequence’ on pp. 410–412.
Schurz, writing with Lambert, indicates one way for theories to meet this requirement, viz., inclusion of “many \(T_{i}\)’s [i.e., laws of the form \(\forall x(F_{i} x \supset Tx\)) for various i] plus many \(T'_{k}\)’s [i.e., laws of the form \(\forall x( \textit{Tx} \supset G_{k}x)\) for various k]” (Schurz and Lambert 1994, p. 88), where each \(F_{i}\) and each \(G_{k}\) is an observational predicate. They comment that
[t]his is the typical situation where one conjectures in science that there is a certain ‘intrinsic’ and not directly observable property of objects, say T(x), common to all [observational] properties \(F_{1}\), . . . , \(F_{\mathrm{m}}\), which has as its empirical effects the [observational] properties \(G_{1}\), . . . , \(G_{\mathrm{n}}\). T(x) may e.g. mean that ‘x has metallic structure’; then the \(F_{i}\) are different kinds of metals, and the \(G_{k}\) are typical properties of metals (Schurz and Lambert 1994, p. 87).
Christian Feldbacher-Escamilla expressed to me the worry that, without a condition like this, something could count as a shortcut term on the basis of such an idiosyncracy.
\(\hbox {Q}_{1}\)(x,y) and R(x,y) might each be defined as the functions \(\hbox {q}_{1}(\hbox {x})=\hbox {y}\) and \(\hbox {r(x)}=\hbox {y}\), respectively. Thanks to an anonymous reviewer for pointing this out.
Goldbach’s conjecture says that every even integer greater than three is the sum of two primes.
To be concrete, we might suppose that the agent uses the algorithm developed by e Silva et al. (2013), which has computed Goldbach partitions for every even number between six and \(4\times 10^{18}\) (which entails the validity of the conjecture up to \(8.37\times 10^{26})\). This algorithm contains a sub-algorithm (a sieve of Eratosthenes) that generates the prime numbers. And such a sub-algorithm is of course less complex, in regards to computational time or space, than its containing super-algorithm. So since using \(\hbox {Q}_{1}\) would require a sub-algorithm and eschewing it the super-algorithm, using \(\hbox {Q}_{1}\) eases the computational burden of deriving the prediction in question.
I do so to address an argument, put to me by an anonymous reviewer, that contends that there are no meta-linguistic concepts that apply to all languages; rather, such concepts must be relativized to a logical type.
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Acknowledgements
I would like to thank Gerhard Schurz, Giovanni Valente, Raja Rosenhagen, and audiences at the 2016 Gesellschaft für Wissenschaftsphilosophie in Düsseldorf, Germany and at the 2016 Central Texas Philosophy workshop for useful feedback.
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Surovell, J. Empirical significance, predictive power, and explication. Synthese 196, 2519–2539 (2019). https://doi.org/10.1007/s11229-017-1554-1
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DOI: https://doi.org/10.1007/s11229-017-1554-1