On Argumentation Logic and Propositional Logic

Abstract

Abstract.This paper studies the relationship between Argumentation Logic (AL), a recently defined logic based on the study of argumentation in AI, and classical Propositional Logic (PL). In particular, it shows that AL and PL are logically equivalent in that they have the same entailment relation from any given classically consistent theory. This equivalence follows from a correspondence between the non-acceptability of (arguments for) sentences in AL and Natural Deduction (ND) proofs of the complement of these sentences. The proof of this equivalence uses a restricted form of ND proofs, where hypotheses in the application of the Reductio of Absurdum inference rule are required to be "relevant" to the absurdity derived in the rule. The paper also discusses how the argumentative re-interpretation of PL could help control the application of ex-falso quodlibet in the presence of inconsistencies.

Informalizing Formal Logic

“…Let us first consider how arguments would be constructed in an argumentation-based reformulation of formal logical reasoning. As mentioned above, we will follow the approach of argumentationbased logic (Kakas et al 2014;Kakas et al 2018), adopting the proof theory formulation of formal logic and building on the connection of proof theory and argumentation. The construction of arguments in informal logic typically follows some accepted argument schemes (Pollock 1995;Walton 1995) that would link premises to a conclusion, often called the claim or position, supported by the argument.…”

“…For example, in legal argumentation this could reflect legal precedence such as lex-posterior where later laws are generally stronger than earlier laws. For the general framework of argumentation-based logical reasoning as applied for the case of propositional logic (Kakas et al 2014;Kakas et al 2018) the preference and ensuing defense relation is minimal and consists of two elements:…”

“…In this way, argumentation re-covers semantically the effects of proof by contradiction in logical entailment and gives us a formal derivation for classical logic that is logically sound and complete with respect to the semantics of classical logic, when the given theory, T, is classically consistent. The interested reader can see, in (Kakas et al 2018), the technical details of this correspondence with natural deduction derivations and how this gives a way to understand the precise equivalence of argumentation logic to formal classical propositional logic.…”

“…This paper does not aim to describe in detail the technical reformulation of formal logic in argumentation. This can be found in the development of argumentation-based logic in the articles Kakas et al 2018) which give a precise technical reformulation of classical deductive reasoning in propositional logic (PL) in terms of argumentation. Rather, we will adopt an informal presentation of these technical results, concentrating in exposing the essential elements of this reformulation, as the salient and universal elements of logic through argumentation, across its whole spectrum from formal to informal reasoning.…”

“…We will therefore avoid the heavy use of formal terminology and notation, confining ourselves to some parts of the standard language of propositional and first order predicate logic. As mentioned above, readers who are interested in the technical and formal details can consult the articles (Kakas et al 2014;Kakas et al 2018).…”