Abstract
In this paper, it is argued that the most fruitful approach to developing normative models of argument quality is one that combines the argumentation scheme approach with Bayesian argumentation. Three sample argumentation schemes from the literature are discussed: the argument from sign, the argument from expert opinion, and the appeal to popular opinion. Limitations of the scheme-based treatment of these argument forms are identified and it is shown how a Bayesian perspective may help to overcome these. At the same time, the contributions of the standard scheme-based approach are highlighted, and it is argued that only a combination of the insights of different traditions will yield a complete normative theory of argument quality.
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In first instance, probabilities—as degrees of belief—are subjective, and the probability calculus is about coherence, in the same way that classical logic is about the relationships between statements, not their truth or falsity per se. However, it is typically assumed that a rational agent should adopt as her subjective degree of belief objective probabilities (limit frequencies) where these are defined, see e.g., Lewis (1980).
Posterior odds convert into posterior degrees of belief via the simple relationship \(P(A) = Odds(A)/(1+Odds(A))\).
This is not to say, however, that alternative frameworks have not been put forward, see e.g., Schum (1994).
For continuous variables correlation is defined as \({\textit{rAB}}=\frac{E(A,B)-E(A)E(B)}{\sqrt{E(A^{2})-(E(A)})^{2}\sqrt{E(B^{2})-(E(B))^{2}}}\) where E(A) is the expected value of A, Specifically, independence implies zero correlation, but the converse is not necessarily true. Variables can be systematically related, and hence non-independent, in ways not captured by (linear) correlation (e.g., x and y in \({y=sin(x)}\)). This also suggests that independence, as the more general notion, is preferable to correlation as the basis for the argument from sign.
\(P(A{\vert }B)=P(A,B)/P(B)\)
To put this more formally, one might think of a generalization such as “If A, then generally B” as saying that \({P(B{\vert }A)}\) is high. On observing A, the probability we should now assign to B will be \({P(B{\vert }A)}\), exactly as (defeasible) modus ponens suggests. However, whether A provides a reason for believing B, depends on whether \({P(B{\vert }A)}\) is greater than P(B) in the first place, and that depends on the likelihood ratio being greater than 1, i.e., that \({P(B{\vert }A) > P(B{\vert } \lnot A)}\). On modus ponens and other conditional inferences from a probabilistic perspective see e.g., Oaksford and Chater (1994), Evans and Over (2004) and, specifically in an argumentation context Hahn and Oaksford (2012).
Bayesian Belief Networks simplify multi-variable computations by taking into account dependence and independence relations within a graphical representation (for an introduction see e.g., Pearl 1988 or Korb and Nicholson 2003). The nodes in a network such as that in Fig. 2 represent random variables. The directed arrows (links) between them signify (assumed) direct causal influences and the strengths of these influences are quantified by conditional probabilities. Each variable is assigned a link matrix that represents estimates of the conditional probabilities of the events associated with that variable given any value combination of the parent variables’ states. These matrices together provide a joint distribution function: a complete and consistent global model, on the basis of which all probabilistic queries can be answered.
Within the scheme-based tradition Hastings (1962, p. 143) also considers both schemes to be related to “causal relations which are used as generalizations to justify the conclusion on the basis of the premises”.
Further examples of the dissociation between logical validity and inductive strength to those given thus far are the so-called paradoxes of material implication, see Oaksford and Hahn (2007).
These seem largely based on consideration of characteristics of probability in the context of logical inference, rather than, as advocated here, Bayesian conditionalization. For example, Pollock’s arguments about how multiple, independent, premises lead rapidly to improbable conclusions assume that the relationship between premises and conclusions is conceived of as a logical inference from a conjunction, not as a conditional probability. In general, believing more things does not inherently imply greater risk of error, see e.g., Bovens and Olsson (2002).
Hahn and Oaksford (2007b) argue, among other things, that the notion of burden of proof is inherently tied to action, stemming in law from the need to make a decision. Where a decision is required, the utilities associated with various courses of action provide ‘burdens of proof’. Where a decision is not immediately required, the notion is forced, and there are no normatively compelling reasons for determining either levels of proof required, or who should carry them.
Carneades can handle such accrual of evidence for cumulative arguments if an argument for every member of the powerset of the pieces of evidence is included in the argument graph, see also Gordon and Walton (2009).
By contrast, Walton and Gordon (2014) explicitly highlight ‘relevance’ as a key issue that still needs to be formally modelled within Carneades.
This is not to deny that there may be contexts, such as the law, in which distinguishing between being ‘in a position to know’ and ‘being expert’ might be meaningful (see e.g., Godden and Walton 2006). However, in order to justify different argument schemes there must minimally be some consequential difference to either the basic inference or the critical questions.
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Acknowledgments
We would like to thank Frank Zenker for helpful comments on a draft of this manuscript, and Tom Gordon for helpful discussion.The first author was partially supported by the Swedish Research Council’s Hesselgren professorship, and the second author was partially supported by the Centre for Language Studies (Nijmegen).
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Hahn, U., Hornikx, J. A normative framework for argument quality: argumentation schemes with a Bayesian foundation. Synthese 193, 1833–1873 (2016). https://doi.org/10.1007/s11229-015-0815-0
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DOI: https://doi.org/10.1007/s11229-015-0815-0