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Abstraction without exceptions

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Abstract

Wright claims that “the epistemology of good abstraction principles should be assimilated to that of basic principles of logical inference”. In this paper I follow Wright’s recommendation, but I consider a different epistemology of logic, namely anti-exceptionalism. Anti-exceptionalism’s main contention is that logic is not a priori, and that the choice between rival logics should be based on abductive criteria such as simplicity, adequacy to the data, strength, fruitfulness, and consistency. This paper’s goal is to lay down the foundations for an application of the anti-exceptionalist methodology to abstraction principles. In Sect. 1 I outline my strategy. I show that anti-exceptionalism has a bearing on the so-called Bad Company problem (Sect. 2), and explore some versions of anti-exceptionalism about abstraction (Sect. 3). I then consider new criteria for choosing between rival abstraction principles, and, in particular, between consistent revisions of Frege’s Basic Law V (Sect. 4). I finally consider how the abstractionist can compare between competing criteria (Sect. 5), and argue that anti-exceptionalists should be pluralists about abstraction (Sect. 6).

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Notes

  1. ‘\(F \approx G\)’ abbreviates the statement that there is a relation R such that every object falling under F is R-related to a unique object falling under G, and every object falling under G is such that there is a unique object falling under F which is R-related to it.

  2. That is, biconditionals stating that the abstract of an item \(\alpha\) is identical to the abstract of an item of the same type \(\beta\) if and only if \(\alpha\) and \(\beta\) stand in a given equivalence relation over entities of that type.

  3. The second-order comprehension axiom is predicative if the comprehension formula contains no bound second-order variables, and impredicative otherwise.

  4. Hale and Wright (2001, 26).

  5. Logical pluralism faces a similar challenge; see Woods (2019).

  6. Priest (2016, 359).

  7. Of course, the Bad Company problem may be regarded as solved by the moves proposed by the neo-logicists provided one is prepared (1) to regard any conflicting abstraction principles as introducing different kinds of abstracts, and (2) to forego the ambition of neo-logicist foundations for set theory. In Sect. 5 I argue, however, that the neo-logicist solution is not motivated (I wish to thank an anonymous reviewer for this suggestion).

  8. Quine (1951, 58–60).

  9. Rayo (2003, 308); cf. also Boolos (1997). For a related argument, cp. Shapiro and Weir (2000). Note that Rayo’s views might have changed since then, cp. Rayo (2013, 180–187).

  10. Rayo (2003, 318).

  11. Note that some of the criteria mentioned below—in particular adequacy to the data, strength, and fruitfulness—can be understood as relating either to actual mathematical theories or to legitimate mathematical theories, with different results (I wish to thank an anonymous reviewer for this suggestion).

  12. More precisely, anti-exceptionalists uniformly deny logical Euclideanism, according to which “at the foundations of logic are certain immediately obvious, certain, a priori truths” (Wright, “Basic (propositional) Knowledge of (truths of) Logic”, quoted in Priest 2016, 359).

  13. For example, Burgess (2005, 1) claims that “the assessment of the ultimate significance of any neo-Fregean approach must await a determination of just how much mathematics [emphasis in the text] can be reconstructed, without resort to ad hoc hypotheses, on that approach”.

  14. For a similar defence of HP, cf. Mancosu (2016, 190).

  15. For a similar approach, cp. Panza (2016).

  16. For a similar defence of HP, cp. Hale (2018, 165).

  17. Priest (2016, 9–10).

  18. Cp. Gödel (1964)’s characterization of the “intrisic” justification of new axioms of set theory.

  19. As suggested, for example, by Priest (2006, 28–30).

  20. Cp. Gödel (1964)’s characterization of the “extrinsic” justification of new axioms of set theory.

  21. Cf. Williamson (2017, 337), Hjortland (2017, §5.2), and Russell (2019) for discussion.

  22. For an analogy with HP, see Wright (1999, 14–15).

  23. Wright (1998, 301).

  24. Cp. Quine (1970)’s “Maxim of Minimal Mutilation”.

  25. I wish to thank Maria Paola Sforza Fogliani for this suggestion.

  26. The term “good company”, in this sense, is due to Mancosu (2016, Ch. 4).

  27. Hale and Wright (2001, 436–437).

  28. Cf. e.g. Incurvati (2020).

  29. For a similar case, see Mancosu (2016, Ch. 4).

  30. Recall that the set-membership can be defined in terms of the extension operator in the usual way as follows: \(x \in y \leftrightarrow \exists F \ (F(x) \wedge y = \text {Ext}(F))\).

  31. Cp. Cook (2003, 66–67).

  32. Cp. section 6.1.2 of F. Boccuni and M. Panza, “Frege’s Theory of Real Numbers: A Consistent Rendering”—forthcoming in The Review of Symbolic Logic.

  33. I developed this view in joint work with Andrea Sereni and Maria Paola Sforza Fogliani; cp. Sereni, Sforza Fogliani, and Zanetti, “For Better and for Worse: Abstractionism, Good Company, and Pluralism”—under review.

  34. To the best of my knowledge, the only mention of many-sorted languages in this context is in Rayo (2013, 80–82).

  35. One might wonder how opposed Frege himself would have been to anti-exceptionalism about abstraction (or, at least, to pluralism about criteria). In Grundlagen (Frege 1884), Frege suggests that HP may provide an implicit definition of the concept of cardinal number (cp. also May and Wehmeier 2019). However, he oftentimes motivates BLV as an inference pattern that is generally accepted within the practise (cp. Heck 2011, 126). Finally, Frege’s account of the real numbers in Grundgesetze (Frege 1903), unlike his account of the natural numbers, appears to be motivated by considerations of mathematical fruitfulness (cp. Dummett 1991, 283–285). I wish to thank Francesca Boccuni and an anonymous referee at Philosophical Studies for this suggestion.

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Acknowledgements

I wish to thank Francesca Boccuni, Maria Paola Sforza Fogliani, Andrea Sereni, and two anonymous referees for helpful comments on previous versions of this paper.

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  1. Department of Humanities and Life Sciences, Scuola Universitaria Superiore IUSS Pavia, Piazza della Vittoria, 15, 27100, Pavia, Italy

    Luca Zanetti

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Zanetti, L. Abstraction without exceptions. Philos Stud 178, 3197–3216 (2021). https://doi.org/10.1007/s11098-020-01597-7

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