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Infinite Regress Arguments

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Abstract

Infinite regress arguments play an important role in many distinct philosophical debates. Yet, exactly how they are to be used to demonstrate anything is a matter of serious controversy. In this paper I take up this metaphilosophical debate, and demonstrate how infinite regress arguments can be used for two different purposes: either they can refute a universally quantified proposition (as the Paradox Theory says), or they can demonstrate that a solution never solves a given problem (as the Failure Theory says). In the meantime, I show that Black’s view on infinite regress arguments (1996, this journal) is incomplete, and how his criticism of Passmore can be countered.

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Notes

  1. Ignoring Russell (1903: §329).

  2. This does not exclude, of course, that such an argument schema is by and large a generalization of cases.

  3. As Black notes, one may say of course that, by the second theory, infinite regress arguments refute the proposition that an explanation is adequate. Yet, there remain substantive differences, as we shall see.

  4. A comparable view has been defended by Gratton (2010), who identifies some significant variations of the Paradox Schema. None of them, however, corresponds to the Failure Schema.

  5. The guardians case was introduced in Sect. 1 above, and the justification case derives at least from Aristotle’s Posterior Analytics (72b5-24) and Sextus Empiricus’ Outlines of Pyrrhonism (1.166-7). For many more filling instructions, cf. Black (1996) and my PhD dissertation ‘And so on. Two theories of regress arguments in philosophy’ (henceforth ‘And so on’).

  6. As the focus here is on the structure of infinite regress arguments, I will ignore some complications regarding the content of this instance, such as the difference between propositional and doxastic justification (cf. Klein 2007) and the difference between justification as a state and justification as an activity (cf. Rescorla 2012).

  7. Still, in case of regress arguments they are not always independently intuitive, cf. (2) of the guardians instance. For the term ‘paradox’ in this context, cf. Cling (2009).

  8. The latter would again be Juvenal’s and Sextus’ position, but in this case it requires further argumentation, i.e. a demonstration that any alternative solution fails as well.

  9. As I show in the Appendix, the Failure Schema does make use of a suppressed line (II). Yet, this line is rather different from the kind of premise required by the Paradox Schema.

  10. To be sure: FAIL is incompatible with ‘If you ψ any item of type i that you have to φ, then you φ at least one item of type i’ (cf. Johnstone 1996). Yet this line is not part of the Failure Schema.

  11. As there is no extra premise that can fail to be false, infinite regresses generated within the context of the Failure Schema are virtually always vicious. Yet, in some selected cases they may still be non-vicious if the inferences fail, cf. Peijnenburg (2010) and my dissertation ‘And so on’ for examples.

  12. This explanation is to be classified as non-causal: the colour of one’s eyes and one’s parents’ eyes are both effects of a common cause: the parents’ genotype.

  13. Geach seems to refer to a somewhat obscure remark by Wittgenstein (1967: §693).

  14. Cf. the same point in a different context: “Why should we believe that there is any explanation going on?” (Schnieder 2010: 296) Cf. also Klein (2003: 722).

  15. For interesting further criticism of this case, cf. Gettier (1965). Unsurprisingly perhaps, my position is that Gettier’s worries can be dealt with once the Failure Schema is at our disposal.

  16. Are there more kinds? Surely the Paradox and Failure Schema admit of variations, but I have found no regress argument that takes a substantially different form (see my dissertation ‘And so on’).

  17. ‘First’ and ‘before’ need not be read temporally. What matters is that solving the first problem depends upon solving the further problems, not vice versa. Cf. Black (1996: 118-9) and esp. Maurin (2007: 18-21).

  18. This schema has an important variant with a conclusion of the form ‘You never φ all items of type i’ rather than ‘You never φ any item of type i’. See my dissertation ‘And so on’.

References

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Acknowledgements

Thanks to: Anna-Sofia Maurin, Jonathan Sozek, Maarten Van Dyck, Erik Weber and the referees of the journal for advice. The author is PhD fellow of the Research Foundation Flanders.

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  1. Centre for Logic and Philosophy of Science, Department of Philosophy and Moral Science, Ghent University, Blandijnberg 2, 9000, Ghent, Belgium

    Jan Willem Wieland

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Appendix

Appendix

The Failure Schema has two premises, i.e. lines (1) and (3), one hypothesis, i.e. line (2), two main inferences, i.e. lines (4) and (C), and two suppressed lines (I) and (II). The latter are suppressed, as they are completely general and their truth does not usually depend on specific instances. In the following I briefly explain the two main inferences.

1.1 Inference of (4)

  1. (1)

    You have to φ at least one item of type i.

  2. (2)

    For any item x of type i, if you have to φ x, you ψ x.

  3. (3)

    For any item x of type i, if you ψ x, then there is a new item y of type i, and you φ x only if you φ y first.

  1. (I)

    For any item x of type i, if you have to φ x, and you φ x only if you φ y first, then you have to φ y first.

(I) says only that: if you have to do something, then you have to fulfil all necessary conditions to do that thing. For example, if you have to write a paper and if you write a paper only if you sit before your screen, write all of its subsections, drink enough coffee, do not check your e-mail all the time, etc., then you have to do all these things as well. By these four lines plus Existential Instantiation, Universal Instantiation, Modus Ponens and Conjunction, we can generate a regress:

$$ \matrix{ {({\text{a}})} \hfill &{{\text{You}}\,{\text{have}}\,{\text{to}}\,φ\,{\text{a}}{.}} \hfill &{1;{\text{EI}}} \hfill \\ {({\text{b}})} \hfill &{{\text{You}}\,ψ\,{\text{a}}{.}} \hfill &{{\text{a,}}\,{\text{2; UI, MP}}} \hfill \\ {{(}{{\text{b}}^{{*}}}{)}} \hfill &{{\text{You}}\,φ\,{\text{a}}\,{\text{only \ if \ you \ φ}}\,{\text{y}}\,{\text{first}}{.}\,} \hfill &{{\text{b, 3; UI, MP}}} \hfill \\ {{(}{{\text{b}}^{{{**}}}})} \hfill &{{\text{You}}\,{\text{have}}\,{\text{to \ φ}}\,{\text{y}}\,{\text{first}}{.}} \hfill &{{\text{a,}}\,{{\text{b}}^{{*}}},{\text{I;}}\,{\text{CONJ, UI, MP}}} \hfill \\ {{\text{(c)}}} \hfill &{{\text{You}}\,{\text{have}}\,{\text{to}}\,\,φ\,{\text{b \ first}}{.}} \hfill &{{{\text{b}}^{{{**}}}};{\text{EI}}} \hfill \\ }<!end array> $$

etc.

From this and Induction (i.e. what holds for the first lines of the regress, holds for the rest as well), it follows that you always have to φ a further, new item of type i. By Transitivity, all these problems are to be solved in order to φ the initial item a. By Universal Generalization, what holds for item a, which was an arbitrarily selected item of type i, holds for any other item of type i as well. And so we have (4): For any item x of type i, you always have to φ a further item of type i first (i.e. before φ-ing x).Footnote 17

1.2 Inference of (C)

  1. (II)

    For any item x of type i, if you always have to φ a further item of type i first (i.e. before φ-ing x), then you never φ x.

Again, this line is suppressed in the schema, for it seems completely general. An instance of (II) is for example: If you always have to make a further, new decision before making any decision, then you never make any decision. From (II) and (4) we obtain by Modus Ponens: You never φ any item of type i.Footnote 18 Mark this line as (5). Last step: (C) follows from (1)-(5) by Conditional Proof. It says: If you consider line (2) for hypothesis (and take (1) and (3) as premises), then you obtain line (5). That is: If you ψ any item of type i that you have to φ, then you never φ any item of type i.

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Wieland, J.W. Infinite Regress Arguments. Acta Anal 28, 95–109 (2013). https://doi.org/10.1007/s12136-012-0165-1

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