Abstract
The thesis of this paper is that we can justify induction deductively relative to one end, and deduction inductively relative to a different end. I will begin by presenting a contemporary variant of Hume (1739/1896, 1748/1993)’s argument for the thesis that we cannot justify the principle of induction. Then I will criticize the responses the resulting problem of induction has received by Carnap (1963b, 1968) and Goodman (1954), as well as praise Reichenbach (1938, Journal of Philosophy, 37, 97–103, 1940)’s approach. Some of these authors compare induction to deduction. Haack (Mind, 85, 112–119, 1976) compares deduction to induction, and I will critically discuss her argument for the thesis that we cannot justify the principles of deduction next. In concluding I will defend the thesis that we can justify induction deductively relative to one end, and deduction inductively relative to a different end, and that we can do so in a non-circular way. Along the way I will show how we can understand deductive and inductive logic as normative theories, and I will briefly sketch an argument to the effect that there are only hypothetical, but no categorical imperatives.
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Notes
To be sure, we do not need to have information about how induction works, or if it can be justified, in order to apply induction and be good at doing so. However, we do need to have such information if we want to delegate the task of making inductive inferences to computers. Besides our philosophical curiosity I take this engineering task to be the major reason for the interest in the justification of induction. To the extent that philosophy is a normative discipline it is a branch of engineering (cf. Quine 1986).
I do not assume that theorem proving is an a priori matter, and nothing in this paper depends on the assumption that means-end relationships can be established a priori.
There is another reason why Carnapian explication requires intuitive judgment to be a powerful source of a priori knowledge. According to Carnap (1962: 3) an explication is “the transformation of an inexact, prescientific concept, the explicandum, into a new exact concept, the explicatum.” An explicatum is to satisfy four requirements: similarity (in meaning), exactness, fruitfulness, and simplicity. “The explicatum is to be similar to the explicandum in such a way that, in most cases in which the explicandum has so far been used, the explicatum can be used; however, close similarity is not required, and considerable differences are permitted” (Carnap 1962: 7). In this respect explication differs from analysis, where the analysans is to be identical (in meaning) to the analysandum in such a way that, in all cases in which the analysandum has been used so far, the analysans can be used; strict identity (in meaning) is required, and no differences are permitted.
Explication requires intuitive judgment to be a powerful source of a priori knowledge, because the method of counterexamples does not work for explication. We can grant that Gettier (1963) shows that justified true belief is not identical in meaning to knowledge. What Gettier (1963) does not show is that justified true belief is not similar in meaning to knowledge, because a few counterexamples to the identity of meaning do not establish gross dissimilarity. Intuitive judgment already has to be a powerful source of knowledge if it can yield a priori knowledge that analysans and analysandum are not identical (in meaning). It has to be an even more powerful source of knowledge if it can yield a priori knowledge of gross dissimilarity (in meaning) between explicatum and explicandum.
I am grateful to André W. Carus for pointing me to, and providing me with, these three papers.
In the unpublished manuscript Carnap also talks about “inductive inertia”, which plays a role for the second problem of riches, and which seems to be similar to what contemporary decision theorists call “risk” (Buchak 2014).
Carnap (1962: 172f) distinguishes between explication and clarification. The latter is the process of clarifying the explicandum before it is explicated by an explicatum in explication. While the definition in an explication must not be circular, clarification indispensably is. I fail to see how clarification can be separated from explication, and Carnap (1968) does not seem to put much emphasis on this distinction anymore either.
Goodman’s positive theory of induction in terms of “entrenchment” is developed in Goodman (1983: sct. IV). If (inductive) logic is a normative theory it says that we ought to project or infer inductively what we do in fact project or infer inductively, and thus infers an Ought from an Is.
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Acknowledgements
I am grateful to, among others, Jake Chandler, Richard Dawid, Rory Harder, Ole Hjortland, David Hyder, Norbert Gratzl, Terence Irwin, Hannes Leitgeb, Paul Rusnock, Gregory Wheeler, Timothy Williamson, and, especially, Wolfgang Spohn and Marta Sznajder for many helpful comments and suggestions on earlier versions of this paper.
My research was supported by the Canadian SSHRC through its Insight program and by the Connaught Foundation through its New Researcher program.
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Appendix:
Appendix:
The following sentence schema S gives rise to a possible objection to the inductive completeness argument:
No instance of S is true in the model-theoretic sense in any finite model. However, the instance of S that replaces ‘P’ with ‘the natural number ... is a precursor of the natural number ...’ is true. That is, S is true in the model whose domain is the set of natural numbers and whose interpretation function interprets ‘P’ as the precursor relation on the natural numbers. Therefore every instance of the negation of S, ¬S, is true in every finite model, but some instance of ¬S is false in some infinite model. This means that the general rule of logic \(\therefore \neg S\) is not valid.
Here is the objection. Presumably the proponent of the inductive completeness argument is an empiricist who holds that every particular inference of which she, i.e. the empiricist, has the information that it conforms to this general rule, and whether its ingredient sentences are true or false, has (at least one false premise or) a true conclusion. After all, how could an empiricist claim something that is only true in infinite models. Yet if this is so, then this general rule comes out as valid on the inductive completeness argument, even though it is not!
My response to this objection is that it applies a double standard. Either the objector has the information that the instance of S described above (or some other instance) is true, or else she does not. If, as her objection suggests, she does, then she, i.e. the objector, does not have the information that every particular inference of which she, i.e. the objector, has the information that it conforms to this general rule, and whether its ingredient sentences are true or false, has (at least one false premise or) a true conclusion. In this case the above rule does not come out as valid on the inductive completeness argument. If, on the other hand, the objector does not have the information that the instance of S described above (or some other instance) is true, then she does not have the information that the rule is not valid. In this case she cannot raise her objection.
Thus this objection to the inductive completeness argument rests on applying a double standard. The objector assumes to have information about arithmetical claims that she does not allow herself to possess when carrying out the inductive completeness argument. Put differently, the objection denies the empiricist proponent of the inductive completeness argument information about arithmetical claims that she must suppose her to possess if the objection is to get off the ground.
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Huber, F. On the justification of deduction and induction. Euro Jnl Phil Sci 7, 507–534 (2017). https://doi.org/10.1007/s13194-017-0177-1
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DOI: https://doi.org/10.1007/s13194-017-0177-1