Abstract
Inductive reasoning, initially identified with enumerative induction (inferring a universal claim from an incomplete list of particular cases) is nowadays commonly understood more widely as any reasoning based on only partial support that the premises give to the conclusion. This is a tad too sweeping, for this includes any inconclusive reasoning. A more moderate and perhaps more adequate characterization requires that inductive reasoning not only includes generalizations, but also any (ideally, rational) predictions or explanations obtained in absence of suitable deductive premises. Inductive logic is meant to provide guidance in choosing the most supported from a given assembly of conjectures. (Some authors think that this has to be done by capturing the notion of partial support, but this conviction is by no means universally accepted.)
The authors would like to express gratitude to Mathieu Beirlaen and Frederik Van De Putte for reading and commenting on an earlier draft of this paper. Work on this paper was supported by the Special Research Fund of Ghent University through project [BOF07/GOA/019] and by Polish National Science Centre grant 2016/22/E/HS1/00304.
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Notes
- 1.
- 2.
In case no evidence is available, hypothesis H is evaluated against any logical theorem ⊤, so that c(H, ∅) = c(H, ⊤) = m(H).
- 3.
This holds as long as the evidence does not contain any constant occurring in the hypothesis.
- 4.
Most notably, regularity (every state description has non-zero probability) and symmetry (complete permutations of individual constants and predicates of the same type do not change the value of the function).
- 5.
See [79] for historical remarks.
- 6.
- 7.
- 8.
- 9.
The paradox is slightly better known in the version from 1953, where Goodman speaks of ‘blue’ and ‘grue’ [27].
- 10.
Reichenbach developed a slightly unorthodox probability calculus, see [17] for details.
- 11.
- 12.
As almost always in philosophy, the devil is in the details, and various worries arise when one really wants to measure degrees of belief in terms of bets, but those issues lie beyond the scope of our survey.
- 13.
- 14.
For a discussion, see [1].
- 15.
That is, the assumption that they know all logical consequences of what they know.
- 16.
A twist to this problem is that once classical logic becomes the underlying logic, Bayesianism is unable to account for the possibility of the underlying logic being revised and to explain how evidence might motivate a change of underlying logic [72].
- 17.
It is also worth mentioning that one of the strength of Bayesianism lies in various applications of the framework to classical philosophical problems. For instance, the framework is used to describe and assess more precisely various arguments in the philosophy of religion (see e.g., [29]).
- 18.
Compare this to the fact that if the number of constants is infinite, then every measure function m from Carnap’s λ-continuum gives m(h) = 0 whenever h is a universally quantified formula, and gives c(h, e) = 0 whenever h is a universally quantified formula and e is the conjunction of finitely many singular formulas.
- 19.
In the appendix of (1979) Popper moreover rejects the common sense ‘bucket theory’ of knowledge.
- 20.
Compare this to Quine’s arguments in “Two dogmas of empiricism” [63], which led Quine to a holistic position.
- 21.
Many of Popper’s ideas stem from (what since Kuhn is called) revolutionary science and this requires conceptual change. Yet Popper’s formal criteria (like all approaches discussed in the previous sections) presuppose a given language.
- 22.
But Popper hastens to relativize this ‘demarcation criterion’. ‘Metaphysical’ ideas play a central role in generating scientific theories.
- 23.
Compare also “all heavenly bodies move in circles” to “all planets of the sun move in ellipses”, remembering that all circles are ellipses.
- 24.
Note this entails they’re contingent.
- 25.
Popper introduced also two quantitative notions of verisimilitude, which employed the notion of probability. Tichý [74] argues that both attempts have highly counterintuitive consequences. This is not to say that the project of defining truthlikeness is doomed. There are various interesting attempts to define the concept after Popper’s initial failure (see e.g., [54, 58]). Even though no particular account is currently agreed on by everyone, certain progress has been made, and the issue is a lively topic (for a survey, see [55, 56]).
- 26.
To complete the picture, we would need adaptive logics that enable one to derive hypotheses of the form Pr(A∣B) = r, in which Pr is an objective probability, and we would need adaptive logics that enable one to derive predictions that do not follow from derived general hypotheses.
- 27.
There is a mechanical procedure which for any particular inconsistent premise set will show that it is inconsistent. But there is no mechanical procedure which for any particular premise set will decide whether it is inconsistent. So, technically speaking, the set of inconsistent sets of formulas is semi-recursive (or semi-decidable) but not recursive (not decidable).
- 28.
We do not say “on the condition that the generalization is not falsified by the premises”. Soon, the reason will become clear.
- 29.
This is the so-called Reliability strategy. The Minimal Abnormality strategy is slightly different from Reliability and offers a few more consequences than Reliability (and never less consequences). We shall not introduce it here.
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Urbaniak, R., Batens, D. (2018). Induction. In: Hansson, S., Hendricks, V. (eds) Introduction to Formal Philosophy. Springer Undergraduate Texts in Philosophy. Springer, Cham. https://doi.org/10.1007/978-3-319-77434-3_5
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