Abstract
Some arguments are good; others are not. How can we tell the difference? This article advances three proposals as a partial answer to this question. The proposals are keyed to arguments conditioned by different degrees of uncertainty: mild, where the argument’s premises are hedged with point-valued probabilities; moderate, where the premises are hedged with interval probabilities; and severe, where the premises are hedged with non-numeric plausibilities such as ‘very likely’ or ‘unconfirmed’. For mild uncertainty, the article proposes to apply a principle referred to as ‘Jeffrey’s rule’, for the principle is a generalization of Jeffrey conditionalization. For moderate uncertainty, the proposal is to extend Jeffrey’s rule for use with probability intervals. For severe uncertainty, the article proposes that even when lack of probabilistic information prevents the application of Jeffrey’s rule, the rule can be adapted to these conditions with the aid of a suitable plausibility measure. Together, the three proposals introduce an approach to argument evaluation that complements established frameworks for evaluating arguments: deductive soundness, informal logic, argumentation schemes, pragma-dialectics, and Bayesian inference. Nevertheless, this approach can be looked at as a generalization of the truth and validity conditions of the classical criterion for sound argumentation
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Notes
Skyrms writes of statements rather than propositions, perhaps to avoid the Platonic realism often associated with propositions. I will use the term ‘statement’ throughout this paper to mean that which is asserted by a sentence. Some of the ontological questions raised by propositions and statements can be sampled in a classic exchange between Cartwright (1962, 1968) and Stroll (1967).
This and the following paragraph are responses to highly constructive comments by an anonymous reviewer.
Inductive cogency is actually a generalization of deductive soundness. Deductive validity is a limiting case of inductive strength, and the truth requirement is the same in both criteria (Welch 2014, pp. 26–28).
According to this rule, a conclusion of the form ‘Possibly A’ is less probative than one of the form ‘Actually A’, which is in turn less probative than one of the form ‘Necessarily A’.
I am indebted to Geoff Goddu for spotting a difficulty with an earlier version of this argument.
Probability intervals have been objects of study at least as far back as Keynes: “Many probabilities, which are incapable of numerical measurement, can be placed nevertheless between numerical limits” (1921, p. 160, Keynes’ emphasis). On the extent to which Keynes did or should have rejected Ramsey’s influential critique of his theory, see Runde (1994).
Like Pfeifer and Kleiter, Douven and Verbrugge (2013) endorse the thesis of conditional probability and offer empirical results in support.
Bradley’s (2009, pp. 240–241) complementary analysis identifies four sources of these failures: unawareness of existing prospects, consciously imposed limits on deliberation (suspension of judgment, for example), lack of relevant information, and unresolved conflict. Unresolved conflict coincides with Levi’s sense of indeterminacy.
“[A]ll [elicitation] subjects report, or otherwise reveal, that they do not know their own preferences; they experience wavering and indecision that cannot be identified with mere indifference” (Savage 1971, p. 795).
This rule can be seen as a generalization of a maximum probability rule for conjunctions proposed by Hahn et al. (2013, p. 23).
The descriptions of non-probabilistic approaches are based on Halpern (2003, ch. 2), but they adapt Halpern’s set-theoretic formulations to the idiom of statements employed throughout this paper.
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Acknowledgements
This paper grew out of a presentation at the Eleventh Conference of the Ontario Society for the Study of Argumentation, University of Windsor, Canada, on May 19, 2016. I am grateful to the organizers for the invitation, and to the participants, particularly Frank Zenker, David Godden, and Geoff Goddu, for stimulating exchanges. I am also indebted to two anonymous reviewers for this journal, whose comments were invaluable in the maturation of this paper.
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Appendix
Appendix
This appendix includes the details of three probabilistic calculations referred to in the body of the paper. The first and third calculations rely on Jeffrey’s rule; the second uses Bayes’ theorem. For easy reference, the relevant theorem is restated for each calculation.
- 1.
point-valued \( p(A \to B) \) in Sect. 2
Consider the argument A, therefore \( A \to B \) . Truth-table analyses show that there are logical grounds for the following probabilities:
To determine the epistemic probability of the conclusion \( A \to B \), we can apply Jeffrey’s rule:
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2.
point-valued \( p(A \to B|A) \) in Sect. 2
Once again, the argument is A, therefore \( A \to B \). Truth-table analyses can verify that there are logical grounds for these probabilities:
Then Bayes’ theorem can be used to calculate the inductive probability of the conclusion \( A \to B \) given the premise A:
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3.
interval-valued p(A) in Sect. 3
Here the argument is \( A \to B \), B, therefore A. Suppose that there are empirical grounds for the following probabilistic intervals:
We want to determine the epistemic probability of the conclusion A. Hence we consider the possible combinations of minimum (min) and maximum (MAX) values of these probabilities in the light of Jeffrey’s rule: \( p\left(H \right) = p\left(E \right)p\left({H|E} \right) + p(\overline{E} )p(H|\overline{E} ) \):
Since the lowest of these values is .46 and the highest is .57, the epistemic probability of A is the interval .46–.57.
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Welch, J.R. Credence for conclusions: a brief for Jeffrey’s rule. Synthese 197, 2051–2072 (2020). https://doi.org/10.1007/s11229-018-1782-z
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DOI: https://doi.org/10.1007/s11229-018-1782-z