Abstract
A dialectical structures is a special type of bipolarargumentation framework as developed by Cayrol and Lagasquie-Schiex (2005). Cayrol and Lagasquie-Schiex extend the abstract approach of Dung (1995) by adding support relations to Dung’s framework which originally considered attack relations between arguments only. A specific interpretation of Dung’s abstract framework that analyzes arguments as premiss–conclusion structures is carried out in Bondarenko et al. (1997). The theory of dialectical structures is more thoroughly developed in Betz (2008, 2009), and in particular in Betz (2010).
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- 1.
A dialectical structures is a special type of bipolarargumentation framework as developed by Cayrol and Lagasquie-Schiex (2005). Cayrol and Lagasquie-Schiex extend the abstract approach of Dung (1995) by adding support relations to Dung’s framework which originally considered attack relations between arguments only. A specific interpretation of Dung’s abstract framework that analyzes arguments as premiss–conclusion structures is carried out in Bondarenko et al. (1997). The theory of dialectical structures is more thoroughly developed in Betz (2008, 2009), and in particular in Betz (2010).
- 2.
As we shall see later, the evaluation procedures provided by the theory of dialectical structures appeal, however, to some minimal logical principles such as the principle of noncontradiction. Still, as far as I can see, one may even consistently claim that this principle should not be relied on when reconstructing individual arguments (the reconstruction logic should not imply the law of noncontradiction) while maintaining that, for the evaluation of proponent positions in complex controversies, this principle may very well be assumed. The theory of dialectical structures is compatible with all sorts of reconstruction logics.
- 3.
Cf. the online database on http://www.argunet.org
- 4.
With the pool of sentences being \(\{-14,\ldots ,-1, 1,\ldots , 14\}\), we tacitly assume that \(-14,-13,\)\(-12,-11,-10,-7,-5\) are assigned truth values complementary to those assigned to 14, 13, ….
- 5.
See Betz (2011b). Note also that degrees of partial entailment are defined with regard to partial positions and not with regard to sentences. The corresponding measure on the set of sentences—P(p) : = { Doj}([p])—does not necessarily satisfy the Kolmogoroff axioms. This is the problem: For every probability measure over a set of statements, it holds that \(\mathrm{P}(p \vee q) =\mathrm{ P}(p) +\mathrm{ P}(q)\) for contrary p, q. Now, assume that the three sentences p ∨ q, p, and q figure in some τ and that there is no dialectically coherent position according to which both p and q are true. Still, this does not guarantee that the (unconditional) degrees of partial entailment of the atomic positions according to which p and, respectively, q are true, add up to the (unconditional) degree of partial entailment of the atomic position which says that p ∨ q is true. This is because not every coherent complete position according to which p is true assigns p ∨ q the value true—unless an argument like (p;p ∨ q) is included in τ. A similar reasoning applies to conjuncts of single statements. Thus, degrees of partial entailment, when defined on sentences and not on partial positions, satisfy the probability axioms only if the respective dialectical structure is suitably augmented by simple arguments as indicated.
- 6.
Moreover, we effectively avoid counterexamples of the kind “The partial position \(\langle\)‘Our solar system contains 12 planets,’ ‘Helium is lighter than air’⟩ actually seems to be much closer to the truth than the partial position \(\langle\)‘Our solar system contains 12,000 planets,’ ‘Helium is lighter than air’⟩, yet both positions exhibit a verisimilitude of 0.5” by comparing positions with regard to their verisimilitude only if they range over one and the same set of sentences. Now, these advantages of our very simple notion of verisimilitude, however, go hand in hand with limitations and shortcomings. In particular, the straightforward and unambitious concept of verisimilitude, no matter how useful it might turn out to be in the following investigation, does not capture every aspect of our everyday concept of truthlikeness. So, the strong intuition that, for example, “Earth is 3 billion years old” is much closer to the truth than “Earth is 700,000 years old,” is not, at least not directly, accounted for.
- 7.
To see this in some more detail, note that the concept of a constituent, as used, for example, by Kuipers and Schurz (2011) and Niiniluoto (2011), corresponds to our notion of a complete position. As Kuipers and Schurz (2011) remark, the Hamming distance between constituents is fundamental for defining verisimilitude (see also Riegler and Douven 2009; de Lavalette and Zwart 2011). With respect to the so-called BF-approach developed by Cevolani, Crupi, and Festa (e.g., Cevolani et al. 2011), we may, more specifically, identify the verisimilitude of a complete position \(\mathcal{Q}\) with its “degree of true b-content”; the verisimilitude of a partial position \(\mathcal{P}\) (as defined here) is, moreover, proportional to the “degree of true b-content” of \(\mathcal{P}\) (understood as a “conjunctive theory” in line with Cevolani et al. 2011)—precisely, it equals n ∕ m times its “degree of true b-content.” Finally, a partial position \({\mathcal{P}}_{2}\) displays greater verisimilitude than a partial position \({\mathcal{P}}_{1}\) in terms of our framework, if \({\mathcal{P}}_{2}\) is “more verisimilar” than \({\mathcal{P}}_{1}\) according to the definition proposed by Cevolani et al. (2011). The reverse, it seems however, does not hold in general.
- 8.
Five of the arguments are not related to the main argumentation and thus omitted in the graph. Moreover, some of the arguments as reconstructed in Betz (2011a) are split up into several parts so as to make the dialectical role of preliminary conclusions explicit.
- 9.
See also Sect. 1.6.
- 10.
More specifically, the way initial proponent positions are assigned depends on whether the simulation serves to studyconsensus- ortruth-conduciveness, i.e. whether it is presented in this report’s first or second part. In simulations of consensus dynamics, half of the initial positions consist in randomly and independently determined truth values, which are assigned to the individual sentences in the pool. For each position \(\mathcal{P}\) specified in this way, an additional, corresponding proponent position is set up by (1) choosing a random number j with 0 ≤ j ≤ n and (2) altering j different truth value assignments of \(\mathcal{P}\). This procedure has the effect that the relative frequency of extreme distances between proponents positions—compared to a random determination of proponent positions without adjustment (i.e. every initial position is obtained by independent random assignments of truth values)—is relatively high, as the following figure illustrates.
- 11.
Cf. Hegselmann and Krause (2002, p. 9). “KISS” stands for “keep it simple, stupid”.
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Betz, G. (2013). An Introduction to the Theory of Dialectical Structures. In: Debate Dynamics: How Controversy Improves Our Beliefs. Synthese Library, vol 357. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4599-5_2
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