A Computational Logic Approach to the Suppression Task

A Computational Logic Approach to the Suppression Task Emmanuelle-Anna Dietz and Steffen H¨olldobler ({dietz,sh}@iccl.tu-dresden.de) International Center for Computation Logic, TU Dresden D-01062 Dresden, Germany Marco Ragni (ragni@cognition.uni-freiburg.de) Center for Cognitive Science, Friedrichstrase 50 D-79098 Freiburg, Germany Table 1: The suppression task (Byrne, 1989) and used abbre- viations. Subjects received conditionals A, B or C and facts E, E,L or L and had to draw inferences. Abstract A novel approach to human conditional reasoning based on the three-valued Łukasiewicz logic is presented. We will demon- strate that the Łukasiewicz logic overcomes problems the so- far proposed Fitting logic has in reasoning with the suppres- sion task. While adequately solving the suppression task, the approach gives rise to a number of open questions concerning the use of Łukasiewicz logic, unique fixed points, completion versus weak completion, explanations, negation, and sceptical versus credulous approaches in human reasoning. A B C E E L L Keywords: Łukasiewicz logic; computational logic; suppres- sion task; human reasoning. Introduction An interesting study is the suppression task, in which Byrne (1989) has shown that graduate students with no previous ex- posure to formal logic did suppress previously drawn conclu- sions when additional information became available. Inter- estingly, in some instances the previously drawn conclusions were valid whereas in other instances the conclusions were invalid with respect to classical two-valued logic. Consider the following example: If she has an essay to finish then she will study late in the library and She has an essay to finish. Then most subjects (96%) conclude: She will study late in the library. If subjects, however, receive an additional condi- tional: If the library stays open she will study late in the li- brary then only 38% of the subjects conclude: She will study late in the library. This shows, that, although the conclusion is still correct, the conclusion is suppressed by an additional conditional. This is an excellent example for human capabil- ity to draw non-monotonic inferences. Table 1 shows the abbreviations that will be used through- out the paper, whereas Table 2 gives an account of the find- ings of Byrne (1989). As we are using a formal language, propositions like “She will go to the library” (abbreviated L) will be represented by propositional variables like l, with the intended interpretation that if l is true (>), then “She will go to the library”. Taking a naive propositional approach, we can represent A by the implication e ← l, where the propositional variables e and l represent the facts E and L, respectively, and so on. It is straightforward to see that classical two-valued logic cannot model the suppression task adequately: Applying the classical logical consequence operator to some instances of the suppression task (like A, C, E) yields qualitatively wrong answers, due to the monotonic nature of the classical logic. If she has an essay to finish then she will study late in the library. If she has a textbook to read then she will study late in the library. If the library stays open she will study late in the library. She has an essay to finish. She does not have an essay to finish. She will study late in the library. She will not study late in the library. Table 2: The drawn conclusions in the experiment of Byrne. Conditional(s) Fact Experimental Findings A A, B A, C E E E 96% of subjects conclude L. 96% of subjects conclude L. 38% of subjects conclude L. A A, B A, C E E E 46% of subjects conclude L. 4% of subjects conclude L. 63% of subjects conclude L. A A, B A, C L L L 53% of subjects conclude E. 16% of subjects conclude E. 55% of subjects conclude E. A A, B A, C L L L 69% of subjects conclude E. 69% of subjects conclude E. 44% of subjects conclude E. Consequently, at least a non-monotonic operator is needed. As argued by Stenning and van Lambalgen (2008) 1 human reasoning should be modeled by, first, reasoning towards an appropriate representation and, second, by reasoning with re- spect to this representation. As appropriate representation Stenning and van Lambalgen propose logic programs under completion semantics based on the three-valued logic used by Fitting (1985), which itself is based on the three-valued Kleene (1952) logic. Unfortunately, some technical claims made by Stenning and van Lambalgen (2008) are wrong concerning their sec- ond step. H¨olldobler and Kencana Ramli (2009b) have shown that the three-valued logic proposed by Fitting is inadequate for the suppression task. Somewhat surprisingly, the sup- pression task can be adequately modeled if the three-valued 1 There is an earlier publication (Stenning & van Lambalgen, 2005), but Michiel van Lambalgen advised us to refer to their text- book.