On Degrees of Justification

Abstract

This paper gives an explication of our intuitive notion of strength of justification in a controversial debate. It defines a thesis’ degree of justification within the theory of dialectical structures as the ratio of coherently adoptable positions according to which that thesis is true over all coherently adoptable positions. Broadening this definition, the notion of conditional degree of justification, i.e. degree of partial entailment, is introduced. Thus defined degrees of justification correspond to our pre-theoretic intuitions in the sense that supporting and defending a thesis t increases, whereas attacking it decreases, t’s degree of justification. Moreover, it is shown that (conditional) degrees of justification are (conditional) probabilities. Eventually, the paper explains that it is rational to believe theses with a high degree of justification inasmuch as this strengthens the robustness of one’s position.

Appendices

Appendix 1: Proofs of Propositions

1.1 Proof of Proposition 1

Assume $a \rightarrow t$. Let σ (σ′) denote the number of all dialectically coherent, complete positions on τ (τ′), and σ _p (σ _p ′) the number of dialectically coherent, complete positions on τ (τ′) corresponding to which p is true. Now, consider an arbitrary dialectically coherent, complete position ${\mathcal{P}}$ on τ corresponding to which p is true. Because argument a is assumed to be independent, and because its conclusion is true in ${\mathcal{P}}$, any truth value assignment to its premisses will extend ${\mathcal{P}}$ to a dialectically coherent, complete position on τ′. If a has n premisses, there will be 2ⁿ dialectically coherent, complete position on τ′ which extend ${\mathcal{P}}$. As a next step, consider an arbitrary dialectically coherent, complete position $\mathcal{Q}$ on τ corresponding to which p is false. Those and only those truth value assignments to premisses of a according to which not all premisses are true will extend $\mathcal{Q}$ to a dialectically coherent, complete position on τ′. So, there will be 2ⁿ − 1 dialectically coherent, complete position on τ′ which extend $\mathcal{Q}$. According to Lemma 1, every dialectically coherent, complete position on τ′ extends a dialectically coherent, complete position on τ. Hence, we can calculate the number of positions on τ′ as follows:

$$ \begin{aligned} \sigma_p' &= 2^n \cdot \sigma_p \\ (\sigma'-\sigma_p') &= (2^n-1) \cdot (\sigma-\sigma_p). \end{aligned} $$

Therefore:

$$ \begin{aligned} {\textsc{Doj}}_{\tau'}(p) &= \frac{\sigma_p'}{\sigma'}\\ &= \frac{\sigma_p'}{(\sigma'-\sigma_p')+\sigma_p'}\\ &= \frac{2^n \cdot \sigma_p}{(2^n-1) \cdot (\sigma-\sigma_p)+2^n \cdot \sigma_p}\\ &= \frac{2^n \cdot \sigma_p}{2^n \cdot \sigma -\underbrace{(\sigma- \sigma_p)}_{>0}}\\ &> \frac{2^n \cdot \sigma_p}{2^n \cdot \sigma}\\ &= {\textsc{Doj}}_{\tau}(p). \end{aligned} $$

For symmetrical reasons, ${\textsc{Doj}}_{\tau'}(p)<{\textsc{Doj}}_{\tau}(p)$ if $a \rightsquigarrow t$.

1.2 Proof of Proposition 2

We calculate to how many different dialectically coherent, complete positions on τ′ the respective positions on τ can be extended. (In this proof, all positions are understood to be dialectically coherent and complete.) We shall assume that a contains n premisses, and b contains m premisses. Consider the first case, i.e. $a \rightarrow t$ and $b\rightarrow a$, and let q be the conclusion of b (and therefore a premiss of a). Since a is independent in τ, the ratio of (1) positions on τ according to which p and q are true over (2) positions on τ according to which p is true but q is false equals 2ⁿ⁻¹:2ⁿ⁻¹. This is because all 2ⁿ truth value assignments to a’s premisses satisfy the coherence constraint if a’s conclusion, p, is true, and q is true in exactly half of these. Yet, if a’s conclusion is false, there is one truth value assignment to its premisses which will not figure in a position on τ, namely the one which considers all premisses true. So, in that case, there are only 2ⁿ − 1 corresponding truth value assignments, 2ⁿ⁻¹ of which regard q as false and 2ⁿ⁻¹ − 1 take q as true. The ratio of (1) positions on τ according to which p is false and q is true over (2) positions on τ according to which p and q are false equals therefore $2^{n-1}-1:2^{n-1}.$

Every position on τ with true q can be extended to 2^m different positions on τ′. In other words, the positions with true q are multiplied by 2^m when introducing b. Still, a position on τ with false q can only be extended to 2^m − 1 different positions on τ′.

Given (a) the ratio of positions on τ with p and q true over positions with p true and q false, and (b) the respective multipliers, the number of positions on τ with p true is multiplied by the following factor when introducing b:

$$ m_1 = \frac{2^{n-1}\cdot 2^m + 2^{n-1}\cdot (2^m-1) }{2^n}. $$

Likewise, the number of positions on τ with p false is multiplied by the following factor when introducing b:

$$ \begin{aligned} m_2 &= \frac{(2^{n-1}-1)\cdot 2^m + 2^{n-1}\cdot (2^m-1) }{2^{n}-1}\\ &=\frac{2^{n-1}\cdot 2^m + 2^{n-1}\cdot (2^m-1) -\frac{2^m+2^m}{2}}{2^{n}-1}\\ &<\frac{2^{n-1}\cdot 2^m + 2^{n-1}\cdot (2^m-1) -\frac{2^m+(2^m-1)}{2}}{2^{n}-1}\\ &=\frac{2^{n-1}\cdot 2^m + 2^{n-1}\cdot (2^m-1)}{2^{n}}\\ &= m_1. \end{aligned} $$

So the number of positions on τ with p true is multiplied by a greater factor than the number of positions with p false, and that is why p’s degree of justification increases when introducing b.

We will briefly consider the second case, that is $a \rightarrow t$ and $b\rightsquigarrow a$. (Cases (3) and (4) hold for analogous reasons.) Let $\lnot q$ be the conclusion of b. The ratios of positions on τ as calculated in the first case do apply. Yet, because $b\rightsquigarrow a$, a position on τ with q true can be extended to 2^m − 1 different positions on τ′. Every position on τ with q false yields 2^m positions on τ′ when introducing b. This implies for the corresponding factors m ₂ and m ₁:

$$ \begin{aligned} m_2 &=\frac{(2^{n-1}-1)\cdot (2^m-1) + 2^{n-1}\cdot 2^m }{2^{n}-1}\\ &=\frac{2^{n-1}\cdot (2^m-1) + 2^{n-1}\cdot 2^m -\frac{(2^m-1)+(2^m-1)}{2}}{2^{n}-1}\\ &> \frac{2^{n-1}\cdot (2^m-1) + 2^{n-1}\cdot 2^m -\frac{(2^m-1)+2^m}{2}}{2^{n}-1}\\ &= \frac{2^{n-1}\cdot (2^m-1) + 2^{n-1}\cdot (2^m)}{2^{n}}\\ &= m_1. \end{aligned} $$

Thus, a position in τ with p false can be extended to, on average, more positions in τ′ than a position in τ with p true. As a consequence, p’s degree of justification decreases when introducing b.

1.3 Proof of Proposition 3

We prove, in this appendix, statements (1) and (2) of Proposition 3 only, claims (3) and (4) follow symmetrically. We consider statement (1), first. So, let us assume that $\tau,\,\tau',\,a,\,\mathcal{B}$ and ${\mathcal{P}}$ are given as assumed in the theorem. We shall denote the number of complete and coherent position on τ and τ′ as follows,

σ	number of complete and coherent positions on τ;
σ′	number of complete and coherent positions on τ′;
σ c	number of complete and coherent positions on τ which assign c the value t;
$\sigma^{\prime}_c$	number of complete and coherent positions on τ′ which assign c the value t;
$\sigma_{\lnot c}$	number of complete and coherent positions on τ which assign c the value f;
$\sigma'_{\lnot c}$	number of complete and coherent positions on τ′ which assign c the value f;
$\sigma_{\mathcal{P}}$	number of complete and coherent positions on τ which extend ${\mathcal{P}}$;
$\sigma_{\lnot{\mathcal{P}}}$	number of complete and coherent positions on τ which don’t extend ${\mathcal{P}}$;
	$\ldots$
$\sigma_\mathcal{B}$	number of complete and coherent positions on τ which extend $\mathcal{B}$;
	$\ldots$
$\sigma_{{\mathcal{P}}c}$	number of complete and coherent positions on τ which extend $\mathcal{B}$ and assign c the value t;
	$\ldots$

As these distinctions are mutually exclusive and collectively exhaustive, we have,

$$ \begin{aligned} \sigma &= \sigma_{{\mathcal{P}}}+\sigma_{\lnot{{\mathcal{P}}}} \\ \sigma_{{\mathcal{P}}} &= \sigma_{{{\mathcal{P}}}c}+\sigma_{{{\mathcal{P}}}\lnot c}\\ \sigma_{\mathcal{PB}} &=\sigma_{{\mathcal{PB}}c}+\sigma_{{\mathcal{PB}}\lnot c}\\ &\ldots \\ \end{aligned} $$

(1)

The introduction of the new argument a with its n premisses that don’t figure in τ multiplies the number of coherent positions. For every complete and coherent position $\mathcal{Q}$ in τ which considers c true, there are 2ⁿ corresponding complete and coherent positions in τ′ (which extend $\mathcal{Q}$). A complete and coherent position $\mathcal{Q'}$ in τ which considers c false, however, is merely extended by 2ⁿ − 1 complete and coherent positions in τ′, because assigning all premisses the value t doesn’t yield a coherent position if the conclusion is false. In sum, we obtain the following equations,

$$ \begin{aligned} \sigma'_c &= 2^n \sigma_c \\ \sigma'_{{{\mathcal{P}}}c} &= 2^n \sigma_{{{\mathcal{P}}}c}\\ \sigma'_{\lnot{{\mathcal{P}}}c} &= 2^n \sigma_{\lnot{{\mathcal{P}}}c}\\ \sigma'_{\lnot c} &= (2^n-1) \sigma_{\lnot c}\\ \sigma'_{{{\mathcal{P}}}\lnot c} &= (2^n-1) \sigma_{{{\mathcal{P}}}\lnot c}\\ \sigma'_{\lnot{{\mathcal{P}}}\lnot c} &= (2^n-1) \sigma_{\lnot{{\mathcal{P}}}\lnot c}\\ \end{aligned} $$

(2)

The number of conditionally complete and coherent positions on τ′ relative to the background knowledge $\mathcal{B}$ can be related to the number of positions on τ, as well. If the truth-value of on of a’s premisses is fixed, truth-values can be assigned only to the remaining (n − 1) premisses. That’s why every complete and coherent position $\mathcal{Q}$ in τ which considers c true, yields 2ⁿ⁻¹ corresponding complete and coherent positions in τ′ given $\mathcal{B}$, and every complete and coherent position $\mathcal{Q'}$ in τ which considers c false is extended by 2ⁿ⁻¹ − 1 complete and coherent positions. Accordingly,

$$ \begin{aligned} \sigma'_{c{\mathcal{B}}} &= 2^{n-1} \sigma_{c} \\ \sigma'_{{{\mathcal{P}}}c{\mathcal{B}}} &= 2^{n-1} \sigma_{{{\mathcal{P}}}c} \\ \sigma'_{\lnot{{\mathcal{P}}}c{\mathcal{B}}} &= 2^{n-1} \sigma_{\lnot{{\mathcal{P}}}c}\\ \sigma'_{\lnot c{\mathcal{B}}} &= (2^{n-1}-1) \sigma_{\lnot c} \\ \sigma'_{{{\mathcal{P}}}\lnot c{\mathcal{B}}} &= (2^{n-1}-1) \sigma_{{{\mathcal{P}}}\lnot c} \\ \sigma'_{\lnot{{\mathcal{P}}}\lnot c{\mathcal{B}}} &= (2^{n-1}-1) \sigma_{\lnot{{\mathcal{P}}}\lnot c}\\ \end{aligned} $$

(3)

We can now derive the sought-after inequality, starting with the assumption that the introduction of a has in fact increased the degree of justification of ${\mathcal{P}},$

$$ \begin{aligned} {\textsc{Doj}}_{\tau}({{\mathcal{P}}}) &< {\textsc{Doj}}_{\tau'}({{\mathcal{P}}}) \\ \frac{\sigma_{{{\mathcal{P}}}}}{\sigma} &< \frac{\sigma'_{{{\mathcal{P}}}}}{\sigma'} \end{aligned} $$

with (1),

$$ \begin{aligned} \frac{\sigma_{{{\mathcal{P}}}}}{\sigma_{{{\mathcal{P}}}}+\sigma_{\lnot{{\mathcal{P}}}}} &<\frac{\sigma'_{{{\mathcal{P}}}}} {\sigma'_{{{\mathcal{P}}}}+\sigma'_{\lnot{{\mathcal{P}}}}} \end{aligned} $$

as $\sigma_{\mathcal{P}}>0$ and $\sigma_{\lnot {\mathcal{P}}}>0,$

$$ \begin{aligned} \frac{1}{1+\frac{\sigma_{\lnot{{\mathcal{P}}}}}{\sigma_{{{\mathcal{P}}}}}}&<\frac{1}{1+\frac{\sigma'_{\lnot{{\mathcal{P}}}}} {\sigma'_{{{\mathcal{P}}}}}}\\ \frac{\sigma_{{{\mathcal{P}}}}}{\sigma_{\lnot{{\mathcal{P}}}}}& <\frac{\sigma'_{{{\mathcal{P}}}}}{\sigma'_{\lnot{{\mathcal{P}}}}}\\ \frac{\sigma'_{\lnot{{\mathcal{P}}}}}{\sigma_{\lnot{{\mathcal{P}}}}}&<\frac{\sigma'_{{{\mathcal{P}}}}} {\sigma_{{{\mathcal{P}}}}} \end{aligned} $$

with (1) we obtain,

$$ \begin{aligned} \frac{\sigma'_{\lnot{{\mathcal{P}}}c}+\sigma'_{\lnot{{\mathcal{P}}}\lnot c}}{\sigma_{\lnot{{\mathcal{P}}}}}<\frac{\sigma'_{{{\mathcal{P}}}c}+\sigma'_{{{\mathcal{P}}}\lnot c}}{\sigma_{{{\mathcal{P}}}}} \end{aligned} $$

substituting (2) and applying (1), again, yields,

$$ \begin{aligned} \frac{2^n\sigma_{\lnot{{\mathcal{P}}}c}+(2^n-1)\sigma_{\lnot{{\mathcal{P}}}\lnot c}}{\sigma_{\lnot{{\mathcal{P}}}}} &< \frac{2^n\sigma_{{{\mathcal{P}}}c}+(2^n-1)\sigma_{{{\mathcal{P}}}\lnot c}} {\sigma_{{{\mathcal{P}}}}} \\ \frac{2^n(\sigma_{\lnot{{\mathcal{P}}}}-\sigma_{\lnot{{\mathcal{P}}}\lnot c})+(2^n-1)\sigma_{\lnot{{\mathcal{P}}}\lnot c}} {\sigma_{\lnot{{\mathcal{P}}}}}&<\frac{2^n(\sigma_{{{\mathcal{P}}}}-\sigma_{{{\mathcal{P}}}\lnot c})+(2^n-1)\sigma_{{{\mathcal{P}}}\lnot c}}{\sigma_{{{\mathcal{P}}}}} \\ \frac{2^n\sigma_{\lnot{{\mathcal{P}}}}-\sigma_{\lnot{{\mathcal{P}}}\lnot c}} {\sigma_{\lnot{{\mathcal{P}}}}} &< \frac{2^n\sigma_{{{\mathcal{P}}}}-\sigma_{{{\mathcal{P}}}\lnot c}} {\sigma_{{{\mathcal{P}}}}}\\ 2^n-\frac{\sigma_{\lnot{{\mathcal{P}}}\lnot c}}{\sigma_{\lnot{{\mathcal{P}}}}}&< 2^n-\frac{\sigma_{{{\mathcal{P}}}\lnot c}}{\sigma_{{{\mathcal{P}}}}}\\ \frac{\sigma_{\lnot{{\mathcal{P}}}\lnot c}}{\sigma_{\lnot{{\mathcal{P}}}}}&>\frac{\sigma_{{{\mathcal{P}}}\lnot c}} {\sigma_{{{\mathcal{P}}}}} \end{aligned} $$

as $\sigma_{\mathcal{P}}>0$ and $\sigma_{\lnot {\mathcal{P}}}>0,$

$$ \begin{aligned} {\sigma_{{{\mathcal{P}}}}}{\sigma_{\lnot{{\mathcal{P}}}\lnot c}} &> {\sigma_{\lnot{{\mathcal{P}}}}}{\sigma_{{{\mathcal{P}}}\lnot c}}\\ {\sigma_{{{\mathcal{P}}}}}{\sigma_{{{\mathcal{P}}}\lnot c}}+{\sigma_{{{\mathcal{P}}}}}{\sigma_{\lnot{{\mathcal{P}}}\lnot c}}&> {\sigma_{{{\mathcal{P}}}}}{\sigma_{{{\mathcal{P}}}\lnot c}}+{\sigma_{\lnot{{\mathcal{P}}}}}{\sigma_{{{\mathcal{P}}}\lnot c}}\\ {\sigma_{{{\mathcal{P}}}}}({\sigma_{{{\mathcal{P}}}\lnot c}}+{\sigma_{\lnot{{\mathcal{P}}}\lnot c}}) &> {\sigma_{{{\mathcal{P}}}\lnot c}}({\sigma_{{{\mathcal{P}}}}}+{\sigma_{\lnot{{\mathcal{P}}}}}) \end{aligned} $$

according to (1),

$$ \begin{aligned} {\sigma_{{{\mathcal{P}}}}}{\sigma_{\lnot c}} &> {\sigma_{{{\mathcal{P}}}\lnot c}}{\sigma}\\ (2^{n+1}-2^n){\sigma_{{{\mathcal{P}}}}}{\sigma_{\lnot c}} &>(2^{n+1}-2^n){\sigma_{{{\mathcal{P}}}\lnot c}}{\sigma} \\ 2^{n+1}{\sigma_{{{\mathcal{P}}}}}{\sigma_{\lnot c}}-2^n{\sigma_{{{\mathcal{P}}}}}{\sigma_{\lnot c}} &> 2^{n+1}{\sigma_{{{\mathcal{P}}}\lnot c}}{\sigma}-2^n{\sigma_{{{\mathcal{P}}}\lnot c}}{\sigma} \end{aligned} $$

we rearrange,

$$ -2^n{\sigma_{{{\mathcal{P}}}}}{\sigma_{\lnot c}}-2^{n+1}{\sigma_{{{\mathcal{P}}}\lnot c}}{\sigma} >-2^{n+1}{\sigma_{{{\mathcal{P}}}}}{\sigma_{\lnot c}}-2^n{\sigma_{{{\mathcal{P}}}\lnot c}}{\sigma} $$

add two additional terms,

$$ 2^n{\sigma_{{{\mathcal{P}}}}}2^n{\sigma}-2^n{\sigma_{{{\mathcal{P}}}}}{\sigma_{\lnot c}}-2^{n+1}{\sigma_{{{\mathcal{P}}}\lnot c}}{\sigma}+2{\sigma_{{{\mathcal{P}}}\lnot c}}{\sigma_{\lnot c}} >2^n{\sigma_{{{\mathcal{P}}}}}2^n{\sigma}-2^{n+1}{\sigma_{{{\mathcal{P}}}}}{\sigma_{\lnot c}}-2^n{\sigma_{{{\mathcal{P}}}\lnot c}}{\sigma}+2{\sigma_{{{\mathcal{P}}}\lnot c}}{\sigma_{\lnot c}} $$

and may then factorize,

$$ \begin{aligned} (2^n{\sigma_{{{\mathcal{P}}}}}-2\sigma_{{{\mathcal{P}}}\lnot c})(2^n{\sigma}-{\sigma_{\lnot c}}) &> (2^n{\sigma_{{{\mathcal{P}}}}}-\sigma_{{{\mathcal{P}}}\lnot c})(2^n{\sigma}-{2\sigma_{\lnot c}})\\ \frac{2^n{\sigma_{{{\mathcal{P}}}}}-2\sigma_{{{\mathcal{P}}}\lnot c}} {2^n{\sigma}-{2\sigma_{\lnot c}}}&>\frac{2^n{\sigma_{{{\mathcal{P}}}}}-\sigma_{{{\mathcal{P}}}\lnot c}} {2^n{\sigma}-{\sigma_{\lnot c}}} \end{aligned} $$

applying (1),

$$ \begin{aligned} \frac{2^{n-1}(\sigma_{{{\mathcal{P}}}c}+\sigma_{{{\mathcal{P}}}\lnot c})-\sigma_{{{\mathcal{P}}}\lnot c}}{2^{n-1}(\sigma_{c}+\sigma_{\lnot c})-{\sigma_{\lnot c}}} &> \frac{2^n(\sigma_{{{\mathcal{P}}}c}+\sigma_{{{\mathcal{P}}}\lnot c})-\sigma_{{{\mathcal{P}}}\lnot c}}{2^n(\sigma_{c}+\sigma_{\lnot c})-{\sigma_{\lnot c}}}\\ \frac{2^{n-1}\sigma_{{{\mathcal{P}}}c}+(2^{n-1}-1)\sigma_{{{\mathcal{P}}}\lnot c}}{2^{n-1}\sigma_{c}+(2^{n-1}-1)\sigma_{\lnot c}} &> \frac{2^n\sigma_{{{\mathcal{P}}}c}+(2^n-1)\sigma_{{{\mathcal{P}}}\lnot c}} {2^n\sigma_{c}+(2^n-1)\sigma_{\lnot c}} \end{aligned} $$

with (3) and, respectively, (2), this becomes,

$$ \frac{\sigma'_{{{\mathcal{P}}}c{\mathcal{B}}}+\sigma'_{{{\mathcal{P}}}\lnot c{\mathcal{B}}}} {\sigma'_{c{\mathcal{B}}}+\sigma'_{\lnot c{\mathcal{B}}}} > \frac{\sigma'_{{{\mathcal{P}}}c}+\sigma'_{{{\mathcal{P}}}\lnot c}} {\sigma'_{c}+\sigma'_{\lnot c}} $$

and, according to (1), we finally have,

$$ \begin{aligned} \frac{\sigma'_{{{\mathcal{P}}}{\mathcal{B}}}}{\sigma'_{{\mathcal{B}}}} &> \frac{\sigma'_{{{\mathcal{P}}}}}{\sigma'}\\ {\textsc{Doj}}_{\tau'}({{\mathcal{P}}}|{\mathcal{B}}) &> {\textsc{Doj}}_{\tau'}({{\mathcal{P}}}). \end{aligned} $$

This proves statement (1) of Proposition 3. It is comparatively easy to justify substatement (2). If a premiss of the argument a is fixed as false according to the background knowledge, any assignment of truth-values to the remaining premisses extends a complete and coherent position $\mathcal{Q}$ on τ to a complete and coherent position on τ′ given the background knowledge—no matter whether $\mathcal{Q}$ considers the conclusion c true or false. But then, ${\textsc{Doj}}_{\tau'}({\mathcal{P}}|\mathcal{B})$ equals ${\textsc{Doj}}_{\tau}({\mathcal{P}})$, which is, by assumption, smaller than ${\textsc{Doj}}_{\tau'}({\mathcal{P}})$.

1.4 Proof of Proposition 4

Since neither s nor $\lnot s$ figures in any arguments in τ, every complete and coherent position on τ can be extended to a complete and coherent position on τ′ in two ways, namely both by assigning s the value t or by assigning it the value f. We thus have, for an arbitrary partial position ${\mathcal{P}}$ on τ, 

$$ {\textsc{Doj}}_\tau({{\mathcal{P}}}) =\frac{\sigma_{{\mathcal{P}}}}{\sigma} =\frac{2\sigma_{{\mathcal{P}}}}{2\sigma} =\frac{\sigma'_{{\mathcal{P}}}} {\sigma'} ={\textsc{Doj}}_{\tau'}({{\mathcal{P}}}). $$

1.5 Proof of Proposition 5

Let us consider a dialectical structure τ′′ which is obtained from τ by adding the sentences $p_1,\ldots,p_n$ and c to the respective sentence pool (e.g. by way of introducing theses), without introducing the argument a. Every complete and coherent position $\mathcal{Q}$ on τ is, correspondingly, extended by 2ⁿ⁺¹ complete and coherent positions on τ′′. Because of the argument a, exactly one of these complete and coherent positions on τ′′ is not coherent on τ′. That is why a complete and coherent position $\mathcal{Q}$ on τ is extended by (2ⁿ⁺¹ − 1) complete and coherent positions on τ′. We thus obtain, for arbitrary partial positions ${\mathcal{P}}$ on τ, 

$$ {\textsc{Doj}}_\tau({{\mathcal{P}}}) =\frac{\sigma_{{\mathcal{P}}}}{\sigma} =\frac{(2^{n+1}-1)\sigma_{{\mathcal{P}}}}{(2^{n+1}-1)\sigma} =\frac{\sigma'_{{\mathcal{P}}}}{\sigma'} ={\textsc{Doj}}_{\tau'}({{\mathcal{P}}}). $$

Appendix 2: Dialectically Coherent Positions and Complete, Closed Subdebates in Equilibrium

Before we relate the notion of a dialectically coherent position to the concept of a complete, closed subdebate in equilibrium, I repeat, without comment, the relevant definitions from Betz (2009).

Definition 17

(Validity-function) Let $\tau = \langle T,A,U \rangle$ be a dialectical structure. A function v: T→ {0,1} is called a validity-function on τ iff for all $a \in T$: $(v(a)=0 \leftrightarrow \exists b\in T: b\rightsquigarrow a \land v(b)=1)$.

If the validity-function exists on τ and is unique, it is labelled “ϑ” and an argument $a \in T$ is called “τ-valid” iff ϑ(a) = 1, “τ-invalid” otherwise.

Definition 18

(Free premiss) Let $\tau = \langle T,A,U \rangle$ be given. A premiss p of an argument in τ is called “bound in τ” iff

$$ \exists a \in T: \left[\vartheta(a)=1 \land \left((p \Leftrightarrow C(a))\lor(p \Leftrightarrow \lnot C(a))\right)\right]. $$

If and only if a premiss is not bound in τ, it is “free in τ”. The set of all free premisses of τ is called $\Uppi_\tau$.

Definition 19

(Equilibrium) A dialectical structure $\tau = \langle T,A,U \rangle$ is said to be in equilibrium iff not

$$ (p\in \Uppi_\tau \lor \Uppi_\tau \vdash_\tau p ) \land (\lnot p\in \Uppi_\tau \lor \Uppi_\tau \vdash_\tau \lnot p ) $$

for some sentence p.

Definition 20

(Stance-attribution) Let $\tau = \langle T,A,U \rangle$, and $O=\{o_1, \ldots, o_k\}$ be a set of proponents. A function $S:O\rightarrow{\bf P}(T)$ which assigns each proponent a subset $T_i\subseteq T$ is called a stance-attribution on τ. $\tau_i = \langle S(o_i),A|_{S(o_i)},U|_{S(o_i)}\rangle$ is the subdebate accepted by o _i . A proponent o _i claims that

• All $p\in \Uppi_{\tau_i}$ are true.

• All C(a) (with $a \in S(o_i)$ is τ _i -valid) are true.

Definition 21

(Closed subdebates) Let $\tau = \langle T,A,U \rangle$ be a dialectical structure and $S:O\rightarrow{\bf P}(T)$ a stance-attribution on τ. A subdebate τ _i induced by S is called “closed” iff there is no $a \in (T\setminus T_i)$ such that $\Uppi_{\tau_i}=\Uppi_{\tau'},\,\tau'=\langle S(o_i)\cup \{a\},A|_{S(o_i)\cup \{a\}},U|_{S(o_i)\cup \{a\}} \rangle$.

Betz (2009) stipulated that a subdebate has to be complete in order to represent a position a proponent can rationally adopt in a debate, for otherwise the status assignment might not even exist on her subdebate. The following attempt to relate the concept of a coherent position (as truth value assignment) and the notion of a closed, complete subdebate in equilibrium will show that subdebates have to satisfy an additional condition in order to represent rational positions: for each sentence whose negation occurs in the debate as well, the proponent has to assert exactly one of both in a thesis. As the completeness condition already required that specific theses exist in a dialectical structure, I propose to modify and extend the definition of a complete stance-attribution as follows instead of introducing a further condition.

Definition 22

(Complete stance-attribution) Let $\tau = \langle T,A,U \rangle$ be a dialectical structure. The stance-attribution $S:\{o_1,\ldots,o_k\}\rightarrow {\bf P}(T)$ is called “complete” iff for every induced subdebate τ _i ($i=1\ldots k$) there is a τ _i -valid thesis $t\in T_i$ stating either p or $\lnot p$

• 1. for every pair of contradictory sentences $p,\lnot p$ which both occur in τ while neither p nor $\lnot p$ occurs in τ _i ,

• 2. for every conclusion p of a τ _i -invalid argument which neither attacks nor supports another argument in τ _i , and

• 3. for every red circle C in τ _i such that

  • (a) t attacks one of C’s arguments,

  • (b) t is neither part of a red circle itself nor connected to a red circle via a red directed path from that circle to a _C , and

  • (c) t is assigned the validity value 1 according to a partial evaluation of $\tau_i,\,\vartheta_{\rm partial}$, which excludes all arguments in red circles.

As a final preliminary concept, we introduce

Definition 23

(Generated v-function) Let $\tau = \langle T,A,U \rangle$ be a dialectical structure and $\mathcal{Q}$ be a complete position on τ. A function $v:T\rightarrow\{0,1\}$ is generated by $\mathcal{Q}$ iff for every argument $a\in T$ with premisses $p_1,\ldots,p_n$:

$$ v(a)=1 \iff {\mathcal{Q}}(p_1)= \ldots {\mathcal{Q}}(p_n)={\bf t}. $$

v is called a v-function.

With these definitions at hand, we can now proof

Proposition 9

(Construction of dialectically coherent position) Let $S: \{o_1, \ldots ,o_k\}\rightarrow {\bf P}(T)$ be a complete stance-attribution on the dialectical structure $\tau=\langle T,A,U \rangle$. If the induced subdebate τ _i is closed, in equilibrium, and the validity function ϑ exists on τ _i , then there is a dialectically coherent, complete position $\mathcal{Q}$ on τ such that for the v-function v which is generated by $\mathcal{Q}$:

$$ \forall a \in T: v(a)=1 \iff \vartheta(a)=1. $$

Proof

Let $\Uppi_{\tau_i}$ denote the set of free premisses of τ _i . We construct $\mathcal{Q}$ on τ _i first, show that $\mathcal{Q}|_{\tau_i}$ satisfies the coherence constraints on τ _i , and then proceed by extending $\mathcal{Q}$ to those sentences that do not occur in τ _i .

Step 1: We set for every sentence p that occurs in τ _i

$$ {\mathcal{Q}}(p) :=\left\{ \begin{array}{ll} {\bf t} & p\in\Uppi_{\tau_i} \lor \exists a \in T_i:[(p\leftrightarrow C(a))\land \vartheta(a)=1]\\ {\bf f} & {\rm otherwise} \end{array} \right. $$

To see that $\mathcal{Q}|_{\tau_i}$ assigns complementary truth values to contradictory sentences (constraint 2 in Definition 5), consider p, q with $q\leftrightarrow \lnot p$ occurring in τ _i . If p is a τ _i -free premiss or the conclusion of a τ _i -valid argument, then q is not because τ _i is in equilibrium and thence $\mathcal{Q}(p)={\bf t}, \mathcal{Q}(q)={\bf f}$. If, in contrast, both p and q are neither τ _i -free premisses nor conclusions of τ _i -valid arguments, then both are conclusions of τ _i -invalid arguments only. Yet as τ _i is complete, there are no pairs of contradictory sentences which are but conclusions τ _i -invalid arguments. So the second case does not arise. We still have to show that $\mathcal{Q}|_{\tau_i}$ assigns conclusions the value t if the corresponding premisses are true (constraint 3 in Definition 5): If for all premisses $p_1\ldots p_n$ of some $a \in T_i$ it holds that $\mathcal{Q}(p_1)=\ldots=\mathcal{Q}(p_n)={\bf t}$, then a is by construction not attacked by any τ _i -valid argument—τ _i would not be in equilibrium otherwise—, and therefore $\mathcal{Q}(C(a))={\bf t}$.

Step 2: We extend $\mathcal{Q}$ to $\tau \setminus \tau_i$ as follows (note that we consider sentences that do occur in τ but not in τ _i ): Every sentence p whose negation occurs in τ _i is assigned the complementary truth value to $\mathcal{Q}(\lnot p)$. Every remaining sentence is set to f. Now, let us complete the check for dialectical coherency. Let p, q be two contradictory sentences, not both in τ _i (if one of both is in τ _i the construction ensures that they are assigned complementary truth values). But by completeness of S, there is a thesis in τ _i that states either p or q, and therefore the construction guarantees $\mathcal{Q}(p)$ and $\mathcal{Q}(q)$ are complementary. Next, does $\mathcal{Q}$ satisfy the ‘deduction constraint’ (constraint 3 in Definition 5)? The first thing to note is that every argument $a \in \tau \backslash \tau_i$ contains at least one premiss which is false. For otherwise, every premiss p of a were either (1) a τ _i -free premiss in τ _i , (2) a conclusion of a τ _i -valid argument, or (3) the negation of a sentence in τ _i that is neither (1) nor (2). Yet, since by completeness of S the only sentences in τ _i that are neither (1) nor (2) are negations of conclusions of τ _i -valid arguments, (3) amounts to being the conclusion of a τ _i -valid argument, i.e. (2). Hence τ _i would not be a closed subdebate. Now because every argument a in $\tau\setminus\tau_i$ has at least one false premiss, $\mathcal{Q}$ satisfies the deduction constraint. Also, this fact guarantees that v(a) = 0. $\square$

The final proposition tells us how to construct a closed subdebate in equilibrium which corresponds to a given dialectically coherent, complete position.

Proposition 10

(Construction of stance-attribution) Let $\tau=\langle T,A,U \rangle$ be a dialectical structure and v a v-function that is generated by a dialectically coherent, complete position $\mathcal{Q}$ on τ. There exists a stance-attribution $S:\{o\}\rightarrow {\bf P}(T)$ inducing the subdebate τ _o such that

• 1. v is a validity function on τ _o , 

• 2. τ _o is in equilibrium,

• 3. τ _o is closed.

Proof

First, we construct τ _o iteratively. Let $T_0=\emptyset$ and apply the following rule provided T _n is given

• (R) Let T ^* be the set of all arguments $a \in T \backslash T_n$ such that for every premiss p of a: $\mathcal{Q}(p)={\bf t}$ or p negates the conclusion of an argument $b\in T_n$ with v(b) = 1. If T ^* = $\emptyset$ then T _o  = T _n , STOP. Otherwise T _n+1 = T _n ∪ T ^*.

Ad 1): We show that $\vartheta:T_o \rightarrow \{0,1\}$ with $a \mapsto v(a)$ is a validity function on τ _o . By construction an argument $a \in T_o$ has a premiss p with $\mathcal{Q}(p)={\bf f}$ if and only if there is an argument b which is τ _o -valid and attacks a. Thus, ϑ does satisfy the recursive definition of a validity function.

Ad 2): Assume that τ _o were not in equilibrium, that is there were a sentence p such that both p and $\lnot p$ are (1) a τ _o -free premiss or (2) a conclusion of a τ _o -valid argument. If p were a τ _o -free premiss in argument a, then (by definition of “free premiss”) $\lnot p$ couldn’t be the conclusion of a τ _o -valid argument. So $\lnot p$ would be a τ _o -free premiss in some argument b, too. Because of dialectical coherency, $\mathcal{Q}(p)$ is complementary to $\mathcal{Q}(\lnot p)$, and thence the algorithm would not have picked a and b. Yet if p were the conclusion of a τ _o -valid argument, $\lnot p$ would not be τ _o -free and would thus be the conclusion of a τ _o -valid argument, too. Still, this contradicts the assumption that $\mathcal{Q}$ is dialectically coherent.

Ad 3): Assume there were an argument $a \in T\setminus T_o$ such that adding a to τ _o would not increase the set of τ _o -free premisses. Then every premiss of a would either be (1) a τ _o -free premiss of some argument in T _o (and thus be true), (2) the conclusion of a τ _o -valid argument’s conclusion (and thus be true), or (3) the negation of a τ _o -valid argument’s conclusion. Therefore, the rule (R) would have picked a and would not have stopped. $\square$

So not only can we construct dialectically coherent, complete positions from stance-attributions, but, inversely, every coherent position corresponds to a closed subdebate in equilibrium. Note that such a subdebate is not necessarily complete since the dialectical structure τ might simply not contain enough theses.