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- M. Abraham, Dov M. Gabbay & U. Schild (2009). Analysis of the Talmudic Argumentum a Fortiori Inference Rule (Kal Vachomer) Using Matrix Abduction. Studia Logica 92 (3):281 - 364.We motivate and introduce a new method of abduction, Matrix Abduction, and apply it to modelling the use of non-deductive inferences in the Talmud such as Analogy and the rule of Argumentum A Fortiori. Given a matrix with entries in {0, 1}, we allow for one or more blank squares in the matrix, say a i , j =?. The method allows us to decide whether to declare a i , j = 0 or a i , j = 1 or a i , j =? undecided. This algorithmic method is then applied to modelling several legal and practical reasoning situations including the Talmudic rule of Kal-Vachomer. We add an Appendix showing that this new rule of Matrix Abduction, arising from the Talmud, can also be applied to the analysis of paradoxes in voting and judgement aggregation. In fact we have here a general method for executing non-deductive inferences.
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Recent developments in the cognitive sciences and artificial intelligence suggest ways of answering the most serious challenge to Peirce's notion of abduction. Either there is no such logical process as abduction or, if abduction is a form of inference, it is essentially unconscious and therefore beyond rational control so that it lacks any normative significance. Peirce himself anticipates and attempts to answer this challenge. Peirce argues that abduction is both a source of creative insight and a form of logical inference subject to a degree of conscious control. In this paper I shall sketch a developing account of abduction that is suggested by the work of Paul Churchland, Paul Thagard, Chris Eliasmith, William Wimsatt, Owen Flanagan, and others. I shall argue that a credible account of abduction will require that we approach the phenomenon from both higher and lower levels as represented by these approaches.
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Discussion of M. Abraham , Dov M. Gabbay & U. Schild, Analysis of the talmudic argumentum a fortiori inference rule (kal vachomer) using matrix abduction
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