# A conditional logic for abduction

- 385 Downloads
- 7 Citations

## Abstract

We propose a logic of abduction that (i) provides an appropriate formalization of the explanatory conditional, and that (ii) captures the defeasible nature of abductive inference. For (i), we argue that explanatory conditionals are non-classical, and rely on Brian Chellas’s work on conditional logics for providing an alternative formalization of the explanatory conditional. For (ii), we make use of the adaptive logics framework for modeling defeasible reasoning. We show how our proposal allows for a more natural reading of explanatory relations, and how it overcomes problems faced by other systems in the literature.

## Keywords

Abduction Adaptive logics Conditional logic Non-monotonic logic## Notes

### Acknowledgments

Research for this article was partially supported by the project “Logics of discovery, heuristics and creativity in the sciences” (PAPIIT, IN400514-3) granted by the National Autonomous University of Mexico (UNAM). We are greatly indebted to the *Dirección General de Asuntos del Personal Académico (UNAM) and to the Programa de Becas Posdoctorales de la Coordinación de Humanidades* (UNAM). We also thank Laura Leonides and two anonymous referees for their many helpful comments and suggestions regarding this paper.

## References

- Aliseda, A. (2006).
*Abductive reasoning logical investigations into discovery and explanations*. Berlin: Springer, Synthese Library.Google Scholar - Aliseda, A., & Leonides, L. (2013). Hypotheses testing in adaptive logics: An application to medical diagnosis.
*Logic Journal of the IGPL*,*21*, 915–930.CrossRefGoogle Scholar - Batens, D. (2000). A survey of inconsistency-adaptive logics. In D. Batens, G. Priest, & J-Pl van Bendegem (Eds.),
*Frontiers of paraconsistent logic*(pp. 49–73). Baldock: Research Studies Press, Kings College Publication.Google Scholar - Batens, D. (2007). A universal logic approach to adaptive logics.
*Logica Universalis*,*1*, 221–242.CrossRefGoogle Scholar - Ben-David, S., & Ben-Eliyahu-Zohary, R. (2000). A modal logic for subjective default reasoning.
*Artificial Intelligence*,*116*, 217–236.CrossRefGoogle Scholar - Boutilier, C., & Becher, V. (1995). Abduction as belief revision.
*Artificial Intelligence*,*77*(1), 43–94.CrossRefGoogle Scholar - Campos, D. (2011). On the distinction between Peirce’s abduction and Lipton’s inference to the best explanation.
*Synthese*,*180*, 419–442.CrossRefGoogle Scholar - Carnielli, W. (2006). Surviving abduction.
*Logic Journal of IGPL*,*14*(2), 237–256.CrossRefGoogle Scholar - Chellas, B. F. (1975). Basic conditional logic.
*Journal of Philosophical Logic*,*4*, 133–153.CrossRefGoogle Scholar - Ciampolini, A., & Torroni, P. (2004). Using abductive logic agents for modeling the judicial evaluation of criminal evidence.
*Applied Artificial Intelligence*,*18*(3–4), 251–275.CrossRefGoogle Scholar - Console, L., & Torasso, P. (1991). On the co-operation between abductive and temporal reasoning in medical diagnosis.
*Artificial Intelligence in Medicine*,*3*(6), 291–311.CrossRefGoogle Scholar - Douven, I. (2011). Abduction. In E. N. Zalta (Ed),
*The Stanford Encyclopedia of Philosophy*(Spring 2011 ed.). http://plato.stanford.edu/cgi-bin/encyclopedia/archinfo.cgi?entry=abduction. - Fann, K. T. (1970).
*Peirce’s theory of abdcution*. The Hague: Martinus Nijhoff.CrossRefGoogle Scholar - Frankfurt, H. G. (1958). Peirce’s notion of abduction.
*The Journal of Philosophy*,*55*(14), 593–597.CrossRefGoogle Scholar - Gauderis, T. (2013). Modelling abduction in science by means of a modal adaptive logic.
*Foundations of Science*,*18*(4), 611–624.Google Scholar - Harman, G. (1965). The inference to the best explanation.
*Philosophical Review*,*74*, 88–95.CrossRefGoogle Scholar - Hintikka, J. (1998). What is abduction? the fundamental problem of contemporary epistemology. Transactions of the Charles S.
*Peirce Society*,*34*, 503–533.Google Scholar - Hobbs, Jerry R. (2008). Abduction in natural language understanding. In L. Horn & G. Ward (Eds.),
*The handbook of pragmatics*(pp. 724–741). Oxford: Blackwell Publishing Ltd.Google Scholar - Hoffmann, M. H. G. (2011). ‘Theoric transformations’ and a new classification of abductive inferences. Transactions of the Charles S.
*Peirce Society*,*46*, 570–590.CrossRefGoogle Scholar - Horty, J. F. (2002). Skepticism and floating conclusions.
*Artificial Intelligence*,*135*, 55–72.CrossRefGoogle Scholar - Horty, J. (2012).
*Reasons as defaults*. Oxford: Oxford University Press.CrossRefGoogle Scholar - Kakas, A. C., Kowalski, R. A., & Toni, F. (1995). Abductive logic programming.
*Journal of Logic and Computation*,*2*(6), 719–770.CrossRefGoogle Scholar - Lycke, H. (2012). A formal explication of the search for explanations: the adaptive logics approach to abductive reasoning.
*Logic Journal of IGPL*,*20*(2), 497–516.CrossRefGoogle Scholar - Mackonis, A. (2013). Inference to the best explanation, coherence and other explanatory virtues.
*Synthese*,*190*, 975–995.CrossRefGoogle Scholar - Magnani, L. (Ed.) (2013). Special issue on formal representations in model-based reasoning and abduction.
*Logic Journal of the IGPL*,*21*(6), 931–942Google Scholar - Magnani, L. (2001).
*Abduction, reason, and science: Processes of discovery and explanation*. New York: Kluwer-Plenum.CrossRefGoogle Scholar - Makinson, D., & Schlechta, K. (1991). Floating conclusions and zombie paths: two deep difficulties in the “directly skeptical” approach to defeasible inheritance nets.
*Artificial Intelligence*,*48*, 199–209.CrossRefGoogle Scholar - Marquis, P. (1991). Extending abduction from propositional to first-order logic. In
*Fundamentals of Artificial Intelligence Research (Lecture Notes in Computer Science*, Vol. 535), (pp. 141–155). Berlin: Springer-Verlag.Google Scholar - Mayer, M. C., & Pirri, F. (1993). First order abduction via tableau and sequent calculi.
*Bulletin of the IGPL*,*1*, 99–117.CrossRefGoogle Scholar - Mayer, M. C., & Pirri, F. (1996). Abduction is not deduction-in-reverse.
*Logic Journal of the IGPL*,*4*(1), 95–108.CrossRefGoogle Scholar - Meheus, J., & Batens, D. (2006). A formal logic for abductive reasoning.
*Logic Journal of The IGPL*,*14*, 221–236.CrossRefGoogle Scholar - Meheus, J. (2011). A formal logic for the abduction of singular hypotheses. In D. Dieks, W. Gonzalez, S. Hartmann, T. Uebel, & M. Weber (Eds.),
*Explanation, prediction, and confirmation. New trends and old ones reconsidered*(pp. 93–108). Berlin: Springer.CrossRefGoogle Scholar - Nepomuceno-Fernández, A., Soler-Toscano, F., & Velázquez-Quesada, F. (2013). An epistemic and dynamic approach to abductive reasoning: Selecting the best explanation.
*Logic Journal of the IGPL*,*21*(6), 943–961.CrossRefGoogle Scholar - Niiniluoto, I. (2000). Hempel’s theory of statistical explanation. In J. H. Fetzer (Ed.),
*Science, explanation, and rationality: The philosophy of Carl G. Hempel*(pp. 138–163). Oxford: Oxford University Press.Google Scholar - Peirce, C. S. (1932–1958). In P. Weiss, C. Hartshorne, & A. W. Burk (Eds.).
*Collected papers of Charles Sanders Peirce*(Vols. 1–8). Cambridge, MA: Harvard University Press. (Abbreviated CP).Google Scholar - Prendinger, H., & Ishizuka, M. (2005). A creative abduction approach to scientific and knowledge discovery.
*Knowledge-Based Systems*,*18*(7), 321–326.CrossRefGoogle Scholar - Priest, G. (2008).
*An introduction to non-classical logic*(2nd ed.). Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Psillos, S. (2002).
*Causation and explanation*. Stocksfield: Acumen Publishing Limited.Google Scholar - Psillos, S. (2002). Simply the best: a case for abduction. In A. C. Kakas & F. Sadri (Eds.),
*Computational logic: Logic programming and beyond*(pp. 605–625). Berlin: Springer-Verlag.Google Scholar - Psillos, S. (2007). Past and contemporary perspectives on explanation. In T. A. F. Kuipers (Ed.),
*General philosophy of science. Focal issues*(pp. 97–173). Amsterdam: North-Holland Publishers.CrossRefGoogle Scholar - Ruben, D.-H. (1990).
*Explaining explanation*. London/New York: Routledge.CrossRefGoogle Scholar - Salmon, W. (1990).
*Four decades of scientific explanation*. Minneapolis, MN: University of Minnesota Press.Google Scholar - Straßer, C. (2014). Trends in Logic.
*Adaptive Logics for Defeasible Reasoning*, Vol. 38. Berlin: Springer.Google Scholar - Straßer, C. (2011). A deontic logic framework allowing for factual detachment.
*Journal of Applied Logic*,*9*(1), 61–80.CrossRefGoogle Scholar - Straßer, C. (2012). Adaptively applying modus ponens in conditional logics of normality.
*Journal of Applied Non-Classical Logic*,*22*, 125–148.CrossRefGoogle Scholar - Thagard, P. (1988).
*Computational Philosophy of Science*. Cambridge, MA: MIT Press.Google Scholar - Touretzky, D. S., Horty, F. J., & Thomason, R. H., (1987). A clash of intuitions: The current state of nonmonotonic multiple inheritance systems’. In
*Proceedings of the IJCAl-87*(pp. 476–482). Burlington, MA: Morgan Kaufmann.Google Scholar - Van De Putte, F., & Straßer, C. (2013). Three formats of prioritized adaptive logics: A comparative study.
*Logic Journal of the IGPL*,*22*, 127–159.Google Scholar - van Fraassen, B. (1980).
*The scientific image*. Oxford: Clarendon Press.Google Scholar - Verdée, P. (2009). Adaptive logics using the minimal abnormality strategy are \(\pi ^1_1\)-complex.
*Synthese*,*167*, 93–104.Google Scholar - Wilson, I. (1979). Explanatory and inferential conditionals.
*Philosophical Studies*,*35*, 269–278.CrossRefGoogle Scholar - Woods, J. (2012). Cognitive economics and the logic of abduction.
*The Review of Symbolic Logic*,*5*, 148–161.CrossRefGoogle Scholar