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1. Note though that in some formalisms some input may be treated in adefeasible way so that Reflexivity need not hold. For instance, incorrective approaches such as the Free Consequences contradictorypremises are not consequences.

2. We speak aboutsuper-arguments in the intuitive sense thate.g., \(A_1 \rightarrow \ldots \rightarrow A_n\) is a super-argument of\(A_1 \rightarrow \ldots \rightarrow A_m\) where \(m \lt n\).

3. Some pioneering systems in this tradition are the OSCAR system (Pollock (1995)), Defeasible Logic (Nute (1994)), or the formalisms in (Simari and Loui (1992)) and (Verheij (1996)). For a overview articles seePrakken and Vreeswijk (2002)) andPrakken (2018). More recent work in logical formal argumentation includesArieli and Straßer (2015),Besnard and Hunter (2009),Bochman (2018),Dung et al. (2009),Modgil and Prakken (2013).

4. In this and the following examples we suppose that \(\top\) is anarbitrary tautology and the formulas \(\theta , \tau\), etc. appearing inthe default rules are contingent (neither they nor their negations aretautologies).

5. One may think about a classical model M as follows: M is uniquelycharacterized by an assignment \(v\) of the truth-values 0 and 1 tothe logical atoms. Whether a formula \(\phi\) holds in M (in signs, M\(\models \phi)\) can then be determined recursively on the basis of\(v\) by means of the classical truth tables for \(\neg , \wedge\), etc.

6.We show one direction. Suppose \(\max[\phi] \models \psi\). Hence,since big-stepped probabilities are used, \(\rP(\{\max[\phi]\}) \gt\sum \{\rP(\{M\}) \mid M \prec \max[\phi]\} \ge\rP([\phi \wedge \neg \psi]) = \sum \{\rP(\{M\}) \mid M\in[\phi \wedge \neg \psi]\}\). Since max[\(\phi] \models \psi\)also \(\rP([\phi \wedge \psi]) \ge \rP(\{\max[\phi]\})\) and hence\(\rP([\phi \wedge \psi]) \gt \rP([\phi \wedge \neg \psi])\) whichimplies \(\rP(\psi \mid \phi) \gt \frac{1}{2}\).

7. For instance, “We thus argue that human rationality, and thecoherence of human thought, is definednot by logic, but byprobability.” (emphasis added,Oaksford and Chater (2009), p. 69)