This paper presents a comprehensive study of argumentation frameworks with necessities (AFNs), a bipolar extension of Dung Abstract argumentation frameworks (AFs) where the support relation captures a positive interaction between arguments having the meaning of necessity: the acceptance of an argument may require the acceptance of other argument(s). The paper discusses new main acceptability semantics for AFNs and their characterization both by a direct approach and a labelling approach. It examines the relationship between AFNs and Dung AFs and shows the (...) gain provided by the former in terms of concision. Finally, the paper shows how to represent an AFN as a normal logic program (LP) and vice versa and in both cases establishes a one-to-one correspondence between extensions under the main acceptability semantics (except for semi-stable semantics where the correspondence is not completely full) of an AFN and particular cases of 3-valued stable models of normal LPs. (shrink)

This paper explores the relationship between informal reasoning, creativity in mathematics, and problem solving. It underscores the importance of environments that promote interaction, hypothesis generation, examination, refutation, derivation of new solutions, drawing conclusions, and reasoning with others, as key factors in enhancing mathematical creativity. Drawing on argumentation logic, the paper proposes a novel approach to uncover specific characteristics in the development of formalized proving using “proof-events.” Argumentation logic can offer reasoning mechanisms that facilitate these environments. This paper proposes how argumentation (...) can be implemented to discover certain characteristics in the development of formalized proving with “proof-events”. The concept of a proof-event was introduced by Goguen who described mathematical proof as a multi-agent social event involving not only “classical” formal proofs, but also other informal proving actions such as deficient or alleged proofs. Argumentation is an integral component of the discovery process for a mathematical proof since a proof necessitates a dialogue between provers and interpreters to clarify and resolve gaps or assumptions. By formalizing proof-events through argumentation, this paper demonstrates how informal reasoning and conflicts arising during the proving process can be effectively simulated. The paper presents an extended version of the proof-events calculus, rooted in argumentation theories, and highlights the intricate relationships among proof, human reasoning, cognitive processes, creativity, and mathematical arguments. (shrink)