Abstract
In abstract argumentation arguments are just points in a network of attacks: they do not hold premisses, conclusions or internal structure. So is there a meaningful way in which two arguments, belonging possibly to different attack graphs, can be said to be equivalent? The paper argues for a positive answer and, interfacing methods from modal logic, the theory of argument games and the equational approach to argumentation, puts forth and explores a formal theory of equivalence for abstract argumentation.
[...] actual reasoning may be more like weaving a piece of cloth from many threads than forging a chain with links in linear mathematical proof style [...]
van Benthem [7, p. 83]
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Notes
- 1.
- 2.
Cf. [24] for a recent overview of argument games.
- 3.
This property is known in modal logic as image-finiteness of the accessibility relation of a Kripke frame [9, Chap. 2].
- 4.
The reader is referred to [12] for a detailed presentation of this result.
- 5.
A proof of this statement in the general setting of complete partial orders can be found in [31, Corollary 3.7].
- 6.
- 7.
More generally, the claim is a direct consequence of the existence of a homomorphism from the term algebra \(\mathsf {Term} = \langle \fancyscript{L}, \wedge , \lnot , \bot , \lozenge \rangle \) of language \(\fancyscript{L}\) (without universal modality) to the complex algebra \(\mathsf {Set}_{\fancyscript{A}} = \langle 2^A, \cap , -, \emptyset , f \rangle \) where \(f: \wp (A) \longrightarrow \wp (A)\) such that
[9, Chap. 5]. - 8.
See Blackburn et al. [9, Chap. 3.1].
- 9.
It is worth observing that this three-set partition corresponds to the labeling of arguments as “in” (i.e., belonging to the extension at issue), “out” (i.e, being attacked by the extension at issue), and “undecided” (i.e., neither of the above) of the labeling-based semantics of argumentation [2, 10].
- 10.
It is worth stressing that this is a refinement of the common understanding of ‘status of an argument’ in the literature on argumentation theory.
- 11.
See [9, Chap. 2].
- 12.
- 13.
The function is partial because only sequences compatible with the move function \(\mathtt{m}\) below need to be considered.
- 14.
These games are determined by the Gale-Stewart theorem since it can always be decided whether a dialogue is winning for \(\fancyscript{P}\), i.e., the winning positions for \(\fancyscript{P}\) are an open set (cf. [23, Chap. 6]).
- 15.
- 16.
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Acknowledgments
We would like to thank Johan van Benthem for the many useful suggestions that helped us shape this last version of the paper.
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Gabbay, D., Grossi, D. (2014). When are Two Arguments the Same? Equivalence in Abstract Argumentation. In: Baltag, A., Smets, S. (eds) Johan van Benthem on Logic and Information Dynamics. Outstanding Contributions to Logic, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-06025-5_25
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