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.

In this paper we contribute to bridging the gap between human reasoning as studied in Cognitive Science and commonsense reasoning based on formal logics and formal theories. In particular, the suppression task studied in Cognitive Science provides an interesting challenge problem for human reasoning based on logic. The work presented in the paper is founded on the recent approach by Stenning and van Lambalgen to model human reasoning by means of logic programs with a specific three-valued completion semantics and a semantic fixpoint operator that yields a least model, as well as abduction. Their approach has been subsequently made more precise and technically accurate by switching to three-valued Łukasiewicz logic. In this paper, we extend this refined approach by abduction. We show that the inclusion of abduction permits to adequately model additional empiric results reported from Cognitive Science. For the arising abductive reasoning tasks we give complexity results. Finally, we outline several open research issues that emerge from the application of logic to model human reasoning. 1

Abstract: Formal approaches that aim at representing human reasoning should be evaluated based on how humans actually reason. One way in doing so, is to investigate whether psychological findings of human reasoning patterns are represented in the theoretical model. The computational logic approach discussed here is the so called weak completion semantics which is based on the three-valued Lukasiewicz logic. We explain how this approach adequately models Byrne's suppression task, a psychological study where the experimental results show that participants' conclusions systematically deviate from the classical logically correct answers. As weak completion semantics is a novel technique in the field of Computational Logic, it is important to examine how it corresponds to other already established non-monotonic approaches. For this purpose we investigate the relation of weak completion with respect to completion and three-valued stable model semantics. In particular, we show that well-founded semantics, a widely accepted approach in the field of non-monotonic reasoning, corresponds to weak completion semantics for a specific class of modified programs.

The belief bias effect is a phenomenon which occurs when we think that we judge an argument based on our reasoning, but are actually influenced by our beliefs and prior knowledge. Evans, Barston and Pollard carried out a psychological syllogistic reasoning task to prove this effect. Participants were asked whether they would accept or reject a given syllogism. We discuss one specific case which is commonly assumed to be believable but which is actually not logically valid. By introducing abnormalities, abduction and background knowledge, we adequately model this case under the weak completion semantics. Our formalization reveals new questions about possible extensions in abductive reasoning. For instance, observations and their explanations might include some relevant prior abductive contextual information concerning some side-effect or leading to a contestable or refutable side-effect. A weaker notion indicates the support of some relevant consequences by a prior abductive context. Yet another definition describes jointly supported relevant consequences, which captures the idea of two observations containing mutually supportive side-effects. Though motivated with and exemplified by the running psychology application, the various new general abductive context definitions are introduced here and given a declarative semantics for the first time, and have a much wider scope of application. Inspection points, a concept introduced by Pereira and Pinto, allows us to express these definitions syntactically and intertwine them into an operational semantics.

submitted; revised; accepted Abduction has been on the back burner in logic programming, as abduction can be too difficult to implement, and costly to perform, in particular if abductive solutions are not tabled for reuse. On the other hand, current Prolog systems, with their tabling mechanisms, are mature enough to facilitate the specific introduction of tabling abductive solutions (tabled abduction) into them. Our contributions are as follows: (1) We conceptualize tabled abduction for abductive normal logic programs, permitting abductive solutions to be reused, from one abductive context to another. The approach relies on a transformation into tabled logic programs that makes use of the dual transformation, and enables efficiently handling the problem of abduction under negative goals, by introducing dual positive counterparts for them. (2) We realize tabled abduction in Tabdual, a system implemented in XSB Prolog, allowing dualization by-need only. (3) We refine the dual transformation in the context of Tabdual to permit executing programs with variables and non-ground queries. (4) We foster pragmatic approaches in Tabdual to cater to all varieties of loops in normal logic programs, now complicated by abduction. (5) We evaluate Tabdual in practice by examining five variants, according to various evaluation objectives. (6) We detail how Tabdual can be applied to declarative debugging and decision making. (7) Finally, we refer to related work, and discuss Tabdual’s correctness, complexity, and features that could migrate to the engine level, in Logic Programming systems wanting to encompass tabled abduction. Eight appendices complement and detail the main text.

The tendency to accept or reject arguments based on own beliefs or prior knowledge rather than on the rea-soning process is called the belief-bias effect. A psy-chological syllogistic reasoning task shows this phe-nomenon, wherein participants were asked whether they accept or reject a given syllogism. We discuss one case which is commonly assumed to be believable but not logically valid. By introducing abnormalities, abduction and background knowledge, we model this case under the weak completion semantics. Our formalization re-veals new questions about observations and their ex-planations which might include some relevant prior ab-ductive contextual information concerning some side-effect. Inspection points, introduced by Pereira and Pinto, allow us to express these definitions syntactically and intertwine them into an operational semantics.