Appendix A — The Tipping Line of Linear Model Variants
Linear model variants involve a linear function \(G(x) = 1-x\) in the calculation of account (\(A\)), faithfulness (\(F\)) and systematicity (\(S\)) instead of the quadratic function \(G(x) = 1-x^2\) used in Beisbart, Betz, and Brun (2021). For the linear models, we observed a tipping line in ternary plots that marks off configurations of weights that lead to drastically different behaviour with respect to the attainment of full RE states (see, e.g., Figure 5.7) consistency considerations (see, e.g., Figure 6.4 and Figure 6.5), or the maximization of measures such as account or faithfulness (see, e.g., Figure 7.9 and Figure 7.10).
This tipping line is characeterized by the following equation: \[ \alpha_{A} = \frac{1-\alpha_{S}}{2} \tag{A.1}\]
The boundary condition \(\alpha_{A} + \alpha_{S} + \alpha_{F} = 1\) allows us to rewrite Equation A.1 in an even simpler form:
\[ \alpha_{A} = \alpha_{F} \]
Consequently, the tipping line splits the space of weight configurations into the two regions \(\alpha_{A} < \alpha_{F}\) and \(\alpha_{A} > \alpha_{F}\).
There are interesting analytical results for both regions. The following propositions and their corollaries help to explain the salient change in the behaviour of linear model variants when crossing the tipping line.1
- Proposition 1: For the linear model variants all global optima are full RE states if \(\alpha_{A} > \alpha_{F}\).
- Proposition 2: For the linear model variants all global-optimum commitments maximize faithfulness if \(\alpha_{F} > \alpha_{A}\).
A.1 Proposition 1
Let \(\tau\) be a dialectical structure and \(\mathcal{C}_{0}\) some initial commitments. Moreover, assume \(\alpha_{A} > \alpha_{F}\) for a configuration of weights \((\alpha_{A}, \alpha_{S}, \alpha_{F})\) in a linear model variant. Then, all global optima (relative to \(\mathcal{C}_{0}\)) are full RE states.
Corollaries The linear model variants exhibit the following behaviour for \(\alpha_{A} > \alpha_{F}\).
- For global optima, there are no inconsistency-preserving cases.
- Consistency-eliminating cases do not occur for global optima.
Proof sketch
Intuitively, \(\alpha_{A} > \alpha_{F}\) means that account trumps faithfulness. Accordingly, the process can maximize account during the adjustment step of commitments without caring about faithfulness.
Assume that an epistemic state \((\mathcal{C}, \mathcal{T})\) is a global optimum according to the achievement function \(Z\) given some initial commitments \(\mathcal{C}_{0}\) and a configuration of weights \((\alpha_{A}, \alpha_{S}, \alpha_{F})\) such that \(\alpha_{A} > \alpha_{F}\). We need to show that \((\mathcal{C}, \mathcal{T})\) is a full RE state, i.e., that \(\mathcal{T}\) fully and exclusively accounts for \(\mathcal{C}\), or equivalently, \(A(\mathcal{C}, \mathcal{T}) = 1\).
For a proof by contradiction, assume that \[ A(\mathcal{C}, \mathcal{T})=G\left(\frac{D_{0,\,0.3,\,1,\,1}(\mathcal{C},\overline{\mathcal{T}})}{n}\right) < 1 \]
This holds only if \(D_{0,\,0.3,\,1,\,1}(\mathcal{C},\overline{\mathcal{T}}) > 0\). In other words, there is at least one sentence \(s\) (negated or unnegated) for which there is a positive contribution to the Hamming distance (penalty). In particular, we have the following cases:
- \(\overline{\mathcal{T}}\) extends \(\mathcal{C}\) with respect to \(s\): There is \(s \in \overline{\mathcal{T}}\), but \(s\) and \(\neg s\) are not in \(\mathcal{C}\).
- penalty: \(0.3\)
- \(\overline{\mathcal{T}}\) contracts \(\mathcal{C}\) with respect to \(s\): There is \(s \in \mathcal{C}\), but \(s\) and \(\neg s\) are not in \(\overline{\mathcal{T}}\).
- penalty: \(1\)
- \(\overline{\mathcal{T}}\) and \(\mathcal{C}\) contradict each other with respect to \(s\): Either \(s \in \overline{\mathcal{T}}\) and \(\neg s \in \mathcal{C}\) or \(\neg s \in \overline{\mathcal{T}}\) and \(s \in \mathcal{C}\)
- penalty: \(1\)
Each case of changing \(\mathcal{C}\) with respect to \(s\), yielding new commitments \(\mathcal{C}'\), impacts the contributions to the Hamming distances for account and faithfulness. Note that systematicity is not affected by changing the commitments.
The complete linearity of the achievement function allows us to distribute (“push in”) the weights \(\alpha_{A}\) and \(\alpha_{F}\) over the individual contributions of the hamming distances.
\[ \begin{aligned} & Z(C, T\vert C_{0})\\ & = \alpha_{A}\cdot A(C, T) + \alpha_{F}\cdot F(C\vert C_{0}) + \alpha_{S}\cdot S(T)\\ & = \alpha_{A}\cdot (1-\frac{D_{0,\, 0.3,\, 1,\, 1}(C, \overline{T})}{n}) + \alpha_{F}\cdot (1-\frac{D_{0,\, 0,\, 1,\, 1}(C_{0}, C)}{n}) + \alpha_{S}\cdot (1-\frac{\vert T\vert-1}{\vert\overline{T}\vert})\\ & = \alpha_{A} - \frac{\alpha_{A}\cdot D_{0,\, 0.3,\, 1,\, 1}(C, \overline{T})}{n} + \alpha_{F} - \frac{\alpha_{F}\cdot D_{0,\, 0,\, 1,\, 1}(C_{0}, C)}{n} + \alpha_{S} - \frac{\alpha_{S}\cdot(\vert T\vert-1)}{\vert\overline{T}\vert}\\ & = 1 - \frac{\alpha_{A}\cdot D_{0,\, 0.3,\, 1,\, 1}(C, \overline{T})+ \alpha_{F}\cdot D_{0,\, 0,\, 1,\, 1}(C_{0}, C)}{n} - \frac{\alpha_{S}\cdot(\vert T\vert-1)}{\vert\overline{T}\vert} \end{aligned} \]
Changing the commitments has no effect on \[ \frac{\alpha_{S}\cdot(\vert \mathcal{T}\vert-1)}{\vert\overline{\mathcal{T}}\vert}, \]
and \(n\) is fixed. Consequently, \(Z\) can be optimised by changing the commitments such that the following term is minimized:
\[ \begin{aligned} &\alpha_{A}\cdot D_{0,\, 0.3,\, 1,\, 1}(\mathcal{C}, \overline{\mathcal{T}})+ \alpha_{F}\cdot D_{0,\, 0,\, 1,\, 1}(\mathcal{C}_{0}, \mathcal{C})\\ &= \alpha_{A}\cdot\sum_{i=1}^{n} d_{0,\, 0.3,\, 1,\, 1}(\mathcal{C}, \overline{\mathcal{T}}, \lbrace s_{i}, \neg s_{i}\rbrace) + \alpha_{F}\cdot\sum_{i=1}^{n} d_{0,\, 0,\, 1,\, 1}(\mathcal{C}_{0}, \mathcal{C}, \lbrace s_{i}, \neg s_{i}\rbrace)\\ &= \sum_{i=1}^{n} \alpha_{A}\cdot d_{0,\, 0.3,\, 1,\, 1}(\mathcal{C}, \overline{\mathcal{T}}, \lbrace s_{i}, \neg s_{i}\rbrace) + \alpha_{F}\cdot d_{0,\, 0,\, 1,\, 1}(\mathcal{C}_{0}, \mathcal{C}, \lbrace s_{i}, \neg s_{i}\rbrace) \end{aligned} \]
Since the achievement function is optimized for minimal contributions and \(\alpha_{A} > \alpha_{F}\), it is always more attractive to change the commitments to increase account rather than faithfully respecting the initial commitments. To see this, consider the change in contributions multiplied by the corresponding weights in the table below. This argument can be repeated for every sentence for which \(\mathcal{C}\) and \(\overline{\mathcal{T}}\) differ.
| account penalty | faithfulness penalty (worst case) | |
|---|---|---|
| adjusted commitmetments | \(d_{0,\,0.3,\,1,\,1}(\mathcal{C}',\overline{\mathcal{T}}, \lbrace s, \neg s\rbrace)\) | \(d_{0,\,0,\,1,\,1}(\mathcal{C}_{0}, \mathcal{C}', \lbrace s, \neg s\rbrace)\) |
| -old commitments | \(- d_{0,\,0.3,\,1,\,1}(\mathcal{C},\overline{\mathcal{T}}, \lbrace s, \neg s\rbrace)\) | \(- d_{0,\,0,\,1,\,1}(\mathcal{C}_{0}, \mathcal{C}, \lbrace s, \neg s\rbrace)\) |
| change | difference | difference |
| remove contradicting element from \(\mathcal{C}\) | -1 | +1 |
| revise contradicting element in \(\mathcal{C}\) | -1 | +1 |
| add missing element to \(\mathcal{C}\) | -0.3 | 0 |
| remove additional element from \(\mathcal{C}\) | -1 | +1 |
In summary, if \((\mathcal{C}, \mathcal{T})\) is a global optimum but \(A(\mathcal{C}, \mathcal{T}) < 1\), then there is a position \((\mathcal{C}', \mathcal{T})\) such that \(A(\mathcal{C}, \mathcal{T}) < A(\mathcal{C}', \mathcal{T})\) contradicting \((\mathcal{C}, \mathcal{T})\) being a global optimum. Consequently, we must have \(A(\mathcal{C}, \mathcal{T}) = 1\), i.e., \(\mathcal{T}\) accounts fully and exclusively for \(S\). This shows that \((\mathcal{C}, \mathcal{T})\) is a full RE state.
Remark: Note that this argument does not work for quadratic model variants, and in particular, the default model of Beisbart, Betz, and Brun (2021). The Hamming distance \(D\) is a summation of penalties. Consequently, squaring the hamming distance yields a polynomial expression where every contributing penalty “interferes” due to multiplication with the others. The resulting multiplicative terms block the above strategy of comparing the contributions and distributing the weights \(\alpha_{A}\) or \(\alpha_{S}\) over these expressions. This is why the quadratic models’ share of full RE states among global optima changes gradually with a change in \(\alpha\)-weights (see Chapter 5).
A.2 Proposition 2
Assume that a dialectical structure \(\tau\) and some initial commitments \(\mathcal{C}_{0}\) are given. Moreover, assume \(\alpha_{A} < \alpha_{F}\) for a configuration of weights \((\alpha_{A}, \alpha_{S}, \alpha_{F})\) in a linear model variant. Then, for all global optima:
\[ F(\mathcal{C}\,\vert\,\mathcal{C}_{0}) = 1. \]
Corollaries
The linear model variants exhibit the following behaviour for \(\alpha_{A} < \alpha_{F}\):
- The relative share of inconsistency-eliminating cases among global optima is 0.0.
- Explanation: Removing or revising an initial inconsistency requires deviating from the initial commitments, which is incompatible with maximal faithfulness.
- Similarly, the relative share of inconsistency-preserving cases in this region of weight configurations corresponds to the relative share of inconsistent initial commitments.
- In turn, the relative share of global optima with maximal value for faithfulness is 1.0.
Proof sketch of Proposition 2
The proof of Proposition 2 is highly similar to that of Proposition 1.
For a proof by contradiction, assume that \((\mathcal{C}, \mathcal{T})\) is a global optimum according to \(Z\), but \(F(\mathcal{C}\,\vert\,\mathcal{C}_{0}) < 1\).
This holds only if \(G\left(\frac{D_{0,0,1,1}(\mathcal{C}_{0}, \mathcal{C})}{n}\right) < 1\), i.e. only if \(D_{0,0,1,1}(\mathcal{C}_{0}, \mathcal{C}) > 0\). In other words, there is at least one sentence for which there is a positive contribution to the Hamming distance. In particular, there are two cases:
- \(\mathcal{C}\) contracts \(\mathcal{C}_{0}\) with respect to \(s\): +1 (there is \(s \in \mathcal{C}_{0}\), but \(s\) and \(\neg s\) are not in \(\mathcal{C}\))
- \(\mathcal{C}\) and \(\mathcal{C}_{0}\) contradict each other with respect to \(s\): +1
Consider the impacts on individual contributions to the Hamming distances for account and faithfulness of changing \(\mathcal{C}\) with respect to \(s\), yielding new commitments \(\mathcal{C}'\), in particular the difference \(d(\mathcal{C}_{0}, \mathcal{C}', \lbrace s, \neg s\rbrace) - d(\mathcal{C}_{0}, \mathcal{C}, \lbrace s, \neg s\rbrace)\). In the following subcases, (*) will denote the worst cases.
Case 1
There is an \(s\) in \(\mathcal{C}_{0}\), but \(s\) and \(\neg s\) are not in \(\mathcal{C}\). We can now define a new \(\mathcal{C}'\) by \(\mathcal{C}':=\mathcal{C}\cup \{s\}\)
Faithfulness
- agreement (new) - contraction (old): -1
Account
- Case: \(s \in \overline{\mathcal{T}}\): agreement (new) - expansion (old): -0.7
- Case \(\neg s \in \overline{\mathcal{T}}\): contradiction (new) - expansion (old): + 0.7
- Case \(s\) and \(\neg s \notin \overline{\mathcal{T}}\) (*): contraction (new)- agreement (old): +1
That is, adding \(s\) to \(\mathcal{C}\) yields a +1 contribution to the account penalties in the worst case. This is counterbalanced by a -1 improvement in the faithfulness penalties.
Case 2
Without loss of generality, we can assume that \(s \in \mathcal{C}_{0}\) and \(\neg s \in \mathcal{C}\). We can now either remove \(\neg s\) from \(\mathcal{C}\) (Subcase A) or revise \(\mathcal{C}\) with \(s\) (Subcase B).
Subcase A: \(\mathcal{C}':=\mathcal{C}\setminus \{\neg s\}\)
Faithfulness
- contraction (new) - contradiction (old): +0
Account
- \(s \in \overline{\mathcal{T}}\): expansion (new) - contradiction (old): -0.7
- Case \(\neg s \in \overline{\mathcal{T}}\) (*): expansion (new) - agreement (old): +0.3
- Case \(s\) and \(\neg s \notin \overline{\mathcal{T}}\): agreement(new) - contraction(old): -1
Now, removing \(\neg s\) from \(\mathcal{C}\) leads to a worsening in the account penalties of +0.3 in the worst case. This is contrasted with no differences in the contributions to faithfulness.
Subcase B: \(\mathcal{C}':=(\mathcal{C}\setminus \{\neg s\}) \cup\{s\}\)
Faithfulness
- agreement (new) - contradiction (old): -1
Account
- Case: \(s \in \overline{\mathcal{T}}\): agreement (new) - contradiction (old): -1
- Case \(\neg s \in \overline{\mathcal{T}}\) (*): contradiction (new) - agreement (old): +1
- Case \(s\) and \(\neg s \notin \overline{\mathcal{T}}\): contraction (new) - contraction (old): +0
In this case, revising \(\mathcal{C}\) with \(s\) leads to a +1 contribution to the account penalties in the worst case. This is counterbalanced by an improvement of -1 in the faithfulness penalties.
The complete linearity of the achievement function allows us to distribute (push in) the weights \(\alpha_{A}\) and \(\alpha_{F}\) over the individual contributions of the hamming distances in \(Z\). Hence, the weights also apply to the individual contributions considered above. Moreover, changing the commitments does not affect the systematicity of the theory, i.e. \(S(\mathcal{T})\) is identical for \((\mathcal{C}, \mathcal{T})\) and \((\mathcal{C}', \mathcal{T})\). Hence, the achievement function is optimized for minimal contributions in the measures for account and faithfulness and \(\alpha_{F} > \alpha_{A}\).
Consequently, it is always (Case 1, Case 2 (A and B)) more attractive to stay faithful to the initial commitments rather than to change the commitments in order to increase account.
This argument can be repeated for every sentence, for which \(\mathcal{C}_{0}\) and \(\mathcal{C}\) differ.
In summary, if \((\mathcal{C}, \mathcal{T})\) is a global optimum but it is assumed that \(F(\mathcal{C}\,\vert\, \mathcal{C}_{0}) < 1\), then there is a position \((\mathcal{C}', \mathcal{T})\) such that \(Z(\mathcal{C}, \mathcal{T} \,\vert\, \mathcal{C}_{0}) < Z(\mathcal{C}', \mathcal{T} \,\vert\, \mathcal{C}_{0})\), contradicting \((\mathcal{C}, \mathcal{T})\) being a global optimum. Consequently, we must have \(F(\mathcal{C} \,\vert\, \mathcal{C}_{0}) = 1\).
A.3 Generalization to Fixed Points
The results we prooved for the linear model variants hold not only for global optima but also for fixed points, which requires but a slight modification of the above proofs. The following proof sketch shows how to generalize Proposition 1 to fixed points for the semi-globally optimizing model variant LinearGlobalRE. Proposition 2 can be generalized similarly.
Proof sketch
Let \(\tau\) be a dialectical structure and \(\mathcal{C}_{0}\) some initial commitments. Moreover, assume \(\alpha_{A} > \alpha_{F}\) for a configuration of weights \((\alpha_{A}, \alpha_{S}, \alpha_{F})\).
For a proof by contradiction, we assume that \((\mathcal{C}_{i}, \mathcal{T}_{i})\) is a fixed point with \(A(\mathcal{C}_{i}, \mathcal{T}_{i}) < 1\).
\((\mathcal{C}_{i}, \mathcal{T}_{i})\) being a fixed point implies that \((\mathcal{C}_{i-1}, \mathcal{T}_{i-1})=(\mathcal{C}_{i}, \mathcal{T}_{i})\) and hence that \(A(\mathcal{C}_{i-1}, \mathcal{T}_{i-1}) < 1\) as well. However, during the last adjustment step (from \(i-1\) to \(i\)), all minimally consistent positions were available as candidates. Since \(A(\mathcal{C}_{i-1}, \mathcal{T}_{i-1}) < 1\), the process could have found other commitments \(\mathcal{C}_{i}'\) which would have resulted from changing \(\mathcal{C}_{i-1}\) with respect to \(s\) following the same line of reasoning we used to prove Proposition 1. Again, there would have been at least one sentence \(s\) for which there is a positive contribution to the Hamming distance in the measure of account. Hence, there would have been \((\mathcal{C}_{i}', \mathcal{T}_{i})\) with \(\mathcal{C}_{i}'\neq \mathcal{C}_{i-1}\) that would have performed better than \((\mathcal{C}_{i}, \mathcal{T}_{i})\) according to the achievement functon. This shows that \((\mathcal{C}_ {i}, \mathcal{T}_{i})\) cannot be a fixed point (contradicting the assumption).
Local Model variants
Finally, we can also generalize Proposition 1 and Proposition 2 to fixed points of the LinearLocalRE model variant. The difference to the semi-globally optimizing RE process of LinearGlobalRE is that locally optimizing models (with a neighborhood depth of \(1\)) proceed by changing at most one sentence per adjustment step. But this is all we need to get the above proofs by contradiction off the ground, where we only considered hypothetical adjustments of the commitments with respect to a single sentence. Accordingly, the propositions will also hold if we enlarge the \(d\)-neighborhood to more than one sentence.