Classical logic, argument and dialectic

Abstract

A well studied instantiation of Dung's abstract theory of argumentation yields argumentationbased characterisations of non-monotonic inference over possibly inconsistent sets of classical formulae. This provides for single-agent reasoning in terms of argument and counter-argument, and distributed non-monotonic reasoning in the form of dialogues between computational and/or human agents. However, features of existing formalisations of classical logic argumentation (Cl-Arg) that ensure satisfaction of rationality postulates, preclude applications of Cl-Arg that account for real-world dialectical uses of arguments by resource-bounded agents. This paper formalises dialectical classical logic argumentation that both satisfies these practical desiderata and is provably rational. In contrast to standard approaches to Cl-Arg we: 1) draw an epistemic distinction between an argument's premises accepted as true, and those assumed true for the sake of argument, so formalising the dialectical move whereby arguments' premises are shown to be inconsistent, and avoiding the foreign commitment problem that arises in dialogical applications; 2) provide an account of Cl-Arg suitable for real-world use by eschewing the need to check that an argument's premises are subset minimal and consistent, and identifying a minimal set of assumptions as to the arguments that must be constructed from a set of formulae in order to ensure that the outcome of evaluation is rational. We then illustrate our approach with a natural deduction proof theory for propositional classical logic that allows measurement of the 'depth' of an argument, such that the construction of depth-bounded arguments is a tractable problem, and each increase in depth naturally equates with an increase in the inferential capabilities of real-world agents. We also provide a resource-bounded argumentative characterisation of non-monotonic inference as defined by Brewka's Preferred Subtheories.

A Study of Argumentative Characterisations of Preferred Subtheories

“…This result is shown for Dung AF s in[Dung, 1995], and for dialectical AF s in[D'Agostino and Modgil, 2018].Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence …”

“…We now show that given a dialectical AF (A, D) defined by (Γ, ≤) in Section 2.3, where A satisfies P1-P3, then the preferred and stable extensions coincide (it is straightforward to show that stable implies preferred for dialectical AF s). We make use of Lemma 9 in [D'Agostino and Modgil, 2018]:…”

“…Moreover, the inconsistency of arguments' premises is typically demonstrated dialectically (as in the classic Socratic move whereby an interlocutor is shown to commit to inconsistent premises). Hence [D'Agostino and Modgil, 2016;2018] develop a dialectical account of Cl-Arg that is provably rational under minimal assumptions as to the arguments that need be constructed, and that eschews the need to check for consistency of premises; inconsistency is demonstrated dialectically. The key insight is to account for a distinction ubiquitous in dialectical practice, whereby an argument entails a conclusion from assumptions regarded as premises assumed to be true, and assumptions that are supposed true for the 'sake of argument' (i.e., those premises that an interlocutor commits to).…”

“…A (i.e, if one can construct arguments with conflicting conclusions then one can combine their assumptions to yield an argument concluding ⊥). Then [D'Agostino and Modgil, 2018] show satisfaction of the consistency and closure postulates assuming any preference relation. The crash resistance and non-interference rationality postulates [Caminada et al, 2012] are satisfied if: 1) whenever X in A includes premises redundantly used to entail the conclusion (since subset minimality is not enforced) by virtue of these premises being syntactically disjoint from the remaining premises and the conclusion, then resources suffice to construct the non-redundant argument excluding these premises; 2) ≺ is such that adding redundant premises does not strengthen an argument (this is satisfied by ≺ as defined in Definition 2).…”

“…Moreover, they enforce checking that the arguments' premises are consistent and 'subset minimal' (no proper subset of the premises entail the conclusion). Hence, [D'Agostino and Modgil, 2016;2018] provide a fully rational dialectical account of Cl-Arg that makes minimal assumptions as to the arguments that need to be constructed from B, and eschews the need to check for subset minimality and consistency of premises. They then give an argumentative characterisation (again by reference to stable extensions) of Preferred Subtheories inference defined under the assumption that agents have limited inferential capabilities.…”