# Propositional discourse logic

- 222 Downloads
- 5 Citations

## Abstract

A novel normal form for propositional theories underlies the logic pdl, which captures some essential features of natural discourse, independent from any particular subject matter and related only to its referential structure. In particular, pdlallows to distinguish vicious circularity from the innocent one, and to reason in the presence of inconsistency using a minimal number of extraneous assumptions, beyond the classical ones. Several, formally equivalent decision problems are identified as potential applications: non-paradoxical character of discourses, admissibility of arguments in argumentation networks, propositional satisfiability, and the existence of kernels of directed graphs. Directed graphs provide the basis for the semantics of pdl and the paper concludes by an overview of relevant graph-theoretical results and their applications in diagnosing paradoxical character of natural discourses.

## Keywords

Semantic paradox Circularity Kernels of digraphs Classical logic Graph normal form## References

- Baroni, P., & Giacomin, M. (2003). Solving semantic problems with odd-length cycles in argumentation. In T. D. Nielsen & N. L. Zhang (Eds.),
*Symbolic and quantitative approaches to reasoning with uncertainty. Lecture notes in computer science*(Vol. 2711, pp. 440–451). Berlin: Springer.CrossRefGoogle Scholar - Baroni, P., Giacomin, M., & Guida, G. (2005). Scc-recursiveness: A general schema for argumentation semantics.
*Artificial Intelligence*,*168*(1), 162–210.CrossRefGoogle Scholar - Barwise, J., & Moss, L. (1996).
*Vicious circles: On the mathematics of non-wellfounded phenomena*. Stanford: CSLI.Google Scholar - Bezem, M., Grabmayer, C., & Walicki, M. (2012). Expressive power of digraph solvability.
*Annals of Pure and Applied Logic*,*162*(3), 200–212.Google Scholar - Béziau, J.-Y. (1998). A sequent calculus for Łukasiewicz’s three-valued logic based on suszko’s bivalent semantics.
*Bulletin of the Section of Logic*,*28*(2), 89–97.Google Scholar - Boros, E., & Gurvich, V. (2006). Perfect graphs, kernels and cooperative games.
*Discrete Mathematics*,*306*, 2336–2354.CrossRefGoogle Scholar - Chvátal, V. (1973). On the computational complexity of finding a kernel. Technical Report CRM-300, Centre de Recherches Mathématiques, Univeristé de Montréal. http://users.encs.concordia.ca/~chvatal
- Cook, R. (2004). Patterns of paradox.
*The Journal of Symbolic Logic*,*69*(3), 767–774.CrossRefGoogle Scholar - Cook, R. (2006). There are non-circular paradoxes (but Yablo’s isn’t one of them).
*The Monist*,*89*, 118–149.CrossRefGoogle Scholar - Creignou, N. (1995). The class of problems that are linearly equivalent to satisfiability or a uniform method for proving np-completeness.
*Theoretical Computer Science*,*145*, 111–145.CrossRefGoogle Scholar - Dimopoulos, Y., & Magirou, V. (1994). A graph theoretic approach to default logic.
*Information and Computation*,*112*, 239–256.CrossRefGoogle Scholar - Dimopoulos, Y., Magirou, V., & Papadimitriou, C. H. (1997). On kernels, defaults and even graphs.
*Annals of Mathematics and Artificial Intelligence*,*20*, 1–12.CrossRefGoogle Scholar - Dimopoulos, Y., & Torres, A. (1996). Graph theoretical structures in logic programs and default theories.
*Theoretical Computer Science*,*170*(1–2), 209–244.CrossRefGoogle Scholar - Doutre, S. (2002). Autour de la sématique préférée des systèmes d’argumentation. PhD thesis, Université Paul Sabatier, Toulouse.Google Scholar
- Duchet, P. (1980). Graphes noyau-parfaits, II.
*Annals of Discrete Mathematics*,*9*, 93–101.CrossRefGoogle Scholar - Duchet, P., & Meyniel, H. (1983). Une généralisation du théorème de Richardson sur l’existence de noyaux dans les graphes orientés.
*Discrete Mathematics*,*43*(1), 21–27.CrossRefGoogle Scholar - Dung, P. M. (1995). On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and \(n\)-person games.
*Artificial Intelligence*,*77*, 321–357.CrossRefGoogle Scholar - Gabbay, D. (2009). Modal provability foundations for argumentation networks.
*Studia Logica*,*93*(2–3), 181–198.CrossRefGoogle Scholar - Gabbay, D., & van der Torre, L. (Eds.) (2009). New ideas in argumentation theory [Special issue].
*Studia Logica, 93*(2–3) 357–381.Google Scholar - Gaifman, H. (1988). Operational pointer semantics: Solution to self-referential puzzles. In M. Vardi (Ed.),
*Theoretical aspects of reasoning about knowledge*(pp. 43–59). Los Atlos, CA: Morgan Kaufman.Google Scholar - Gaifman, H. (1992). Pointers to truth.
*The Journal of Philosophy*,*89*(5), 223–261.CrossRefGoogle Scholar - Gaifman, H. (2000). Pointers to propositions. In A. Chapuis & A. Gupta (Eds.),
*Circularity, definition and truth*(pp. 79–121). New Delhi: Indian Council of Philosophical Research.Google Scholar - Galeana-Sánchez, H., & Neumann-Lara, V. (1984). On kernels and semikernels of digraphs.
*Discrete Mathematics*,*48*(1), 67–76.CrossRefGoogle Scholar - Grossi, D. (2010a). Argumentation in the view of modal logic. In P. McBurney, I. Rahwan, & S. Parsons (Eds.),
*ArgMAS. Lecture notes in computer science*(Vol. 6614, pp. 190–208). Berlin-Heidelberg: Springer.Google Scholar - Grossi, D. (2010b). On the logic of argumentation theory. In W. van der Hoek, G. A. Kaminka, Y. Lespérance, M. Luck, & S. Sen (Eds.), 9th
*AAMAS*(pp. 409–416). Toronto, Canada.Google Scholar - Kripke, S. (1975). Outline of a theory of truth.
*The Journal of Philosophy*,*72*(19), 690–716.CrossRefGoogle Scholar - Neumann-Lara, V. (1971).
*Seminúcleos de una digráfica*. Technical report Anales del Instituto de Matemáticas II, Universidad Nacional Autónoma México.Google Scholar - Prakken, H., & Vreeswijk, G. (2002). Logics for deafeasible argumentation. In D. Gabbay & F. Guenthner (Eds.),
*Handbook of philosophical logic*(Vol. 4, pp. 219–318). Dordrecht: Kluwer Academic Publishers.Google Scholar - Richardson, M. (1953). Solutions of irreflexive relations.
*The Annals of Mathematics, Second Series*,*58*(3), 573–590.CrossRefGoogle Scholar - Sorensen, R. (1998). Yablo’s paradox and kindred infinite liars.
*Mind*,*107*, 137–155.CrossRefGoogle Scholar - von Neumann, J., & Morgenstern, O. (1947).
*Theory of games and economic behavior*. Princeton: Princeton University Press.Google Scholar - Walicki, M., & Dyrkolbotn, S. (2012). Finding kernels or solving SAT.
*Journal of Discrete Algorithms*,*10*, 146–164.Google Scholar