The thesis attempts to further link Cognitive Science and Computational Logic. We point to related work in both fields and give a comprehensive overview of the state-of-the-art research. The goal of this thesis, to develop a methodology that allows us to formalize episodes of human reasoning with respect to conditionals, has been carried out as follows: We investigated conditionals in human reasoning and attempted to formalize the results within a Logic Programming approach, the Weak Completion Semantics.
Background and Correspondence to other (three-valued) Semantics
• Background
We provide the first introduction to a novel cognitive theory, the Weak Completion Semantics and an implementation thereof in Prolog. Additionally, we investigate how two-valued abduction can be extended to three-valued abduction and show their correspondence. Moreover, we give a new characterization of integrity con-straints under the Weak Completion Semantics.
• Correspondence to other Semantics (Dietz, H¨olldobler and Wernhard 2014)
– We consider several well-established approaches in Logic Programming such as the Well-founded Semantics and the Stable Model Semantics and show the formal correspondence to the Weak Completion Semantics. Two main dif-ferences can be identified: One difference lies on the treatment of undefined atoms in programs, where the Well-founded Semantics and the Stable Model Semantics assume the closed-world assumption, whereas the Weak Comple-tion Semantics assumes the open-world assumpComple-tion. The second difference is about tight programs: Under certain circumstances, atoms that are involved in a positive cycle in a program, are false under the Well-founded Semantics, whereas they stay unknown under the Weak Completion Semantics.
2Wernhard (2011) and Wernhard (2012) discusses the application of different logic programming semantics to model human reasoning tasks according to the approach by Stenning and van Lambal-gen and the roles of three-valuedness in this context, within a different technical framework based on circumscription.
1.4. Contributions – We further establish an overview of related semantics and show the relations to the well-known two-valued semantics, such as the (Well)-supported Model Semantics and Clark’s Completion Semantics.
• Psychological Investigation on Cycles (Dietz, H¨olldobler and Ragni 2013)
As the Weak Completion Semantics and the Well-founded Semantics deal differ-ently with positive cycles in logic programs, we carried out a psychological study about positive cyclic conditionals. It seems that the participants understood pos-itive cyclic conditionals of length 1 differently than pospos-itive cyclic conditionals of length 2 or 3. In the first case, they understood the conditionals as facts, whereas in the second case they understood the conditionals as actual conditionals. Prelimin-ary results show that the participants’ understanding of positive cyclic conditionals seems to be in favor of the way how the Weak Completion Semantics treats positive cycles in logic programs.
Contextual Reasoning
• Contextual Programs (Dietz Saldanha, H¨olldobler and Pereira 2017)
– We extend the programs under the Weak Completion Semantics with an additional truth-functional operator, ctxt, and introduce so-called contextual programs and provide an implementation thereof in Prolog. Thectxtoperator can be seen as a mapping of three-valuedness to two-valuedness and allows us to express negation as failure under the Weak Completion Semantics.
– We reconsider former formal results of the Weak Completion Semantics and show that for contextual programs the ΦPoperator is not monotonic anymore.
Further, the ΦP operator does not necessarily have a least fixed point for this class of programs. However, we can guarantee that a least fixed point exists for the class of acyclic contextual programs.
• Contextual Abduction (Pereira, Dietz and H¨olldobler 2014a; Pereira, Dietz and H¨olldobler 2014b)
(Dietz Saldanha, H¨olldobler and Pereira 2017)
– We present a contextual abductive approach, that allows us to express a preference among explanations, where the context in which information is observed plays a central role.
– Further, this approach allows us to define more fine-grained relations between observations. We specify when an observation is a contextual (contestable) side-effect of another observation or whether they are both (jointly supported) contextual consequences of one another.
Modeling Human Reasoning Tasks
• Byrne’s Suppression Task and the Well-founded Semantics (Dietz, H¨olldobler and Ragni 2012a)
(Dietz, H¨olldobler and Wernhard 2014)
We reconsider Byrne’s suppression task and discuss open questions with respect to the formalization. Furthermore we show how the respective logic programs need to be adapted in order to adequately model the task within the Well-founded Semantics.
• Wason’s Selection Task (Dietz, H¨olldobler and Ragni 2013)
We model Wason’s selection task by taking Kowalski’s interpretation of the differ-ences between the social and the abstract case as starting point for our formaliza-tion.
• Spatial Reasoning (Dietz, H¨olldobler and H¨ops 2015)
We model a spatial reasoning task by taking the ideas of the Preferred Model Theory as starting point. For this purpose a logic program representation of the first free fit technique is provided: The idea is that a new to be included object will be either placed directly next to the already existing one, provided that there is space left, or otherwise this new object will be placed in the next available space.
As psychological experiments have shown, this technique seems to be cognitively adequate. Our results on the spatial reasoning task complies with the results of these psychological experiments.
• Modeling Quantified Statements (Dietz, H¨olldobler and Ragni 2015) (Costa, Dietz, H¨olldobler and Ragni 2016)
– We develop five principles for the representation of quantified statements mo-tivated by Logic Programming techniques and findings from Cognitive Sci-ence. We then propose a representation of the four possible quantified state-ments as logic programs.
– We predict the answers of 64 syllogistic premises and compare them to the results of psychological experiments. The Weak Completion Semantics has a matching of 85%, which is quite a good result, considering that the best of 12 other recent cognitive theories, only has a matching of 84%.
– The achievement of this contribution stands out here. For the first time we can evaluate the performance of the Weak Completion Semantics and compare our results to the results of other state-of-the-art approaches. This achievement emphasizes that the Weak Completion Semantics has to be taken seriously as a competitive cognitive theory.
• Belief-Bias Effect (Dietz 2017)
1.4. Contributions – We develop an adequate modeling approach for Evans, Barston and Pollard’s syllogistic reasoning task, taking the previous developed representations of quantified statements as starting point. We investigate the belief-bias effect and argue that the belief bias can occur either in the representational part or the reasoning part when modeling human reasoning.
– For the representational part, we model the belief bias by means of abnormal-ity predicates. For the reasoning part, we propose a new principle, in which we suggest that humans search for alternative models, when in the current one, no conclusion seems possible. For this purpose, we apply abduction, and show that by explaining the available information about the presented syllogistic premises, the belief bias can be adequately modeled.
The logic program representation of all human reasoning tasks in this thesis and their results under the Weak Completion Semantics have been implemented in Prolog or Java.
To the best of our knowledge, no one has shown a formalization of so many different human reasoning episodes within a single approach. Furthermore, we are not aware of any formalization, that has attempted to model Evans, Barston and Pollard’s syllogistic reasoning task or the belief-bias effect. We show for all tasks which steps need to be taken and motivated them by experimental findings from psychology. Summing up, the key findings are that the weak completion of a program should be favored over the completion of a program, that skeptical abduction should be favored over credulous abduction and that explanations should be minimal.
On Conditionals
• Evaluation System for Conditionals (Dietz, H¨olldobler and Pereira 2015a)
– We develop a novel system for the evaluation of conditionals in human reas-oning. For this purpose, we introduce a simple revision operator and together with abduction, we show with the help of a few examples from the literature that we can basically model any outcome on how a conditional should be eval-uated. The outcome depends mainly on the order in which we consider the conditions of a conditional. Moreover, in difference to the literature, where a conditional is either evaluated to true or false, the proposed system allows the possibility that conditionals can be evaluated to unknown.
– We additionally discuss the issue of relevance within conditionals. We identify differences between weak relevance and strong relevance and suggest a se-mantical definition of the concept of relevance.
– We conjecture that humans prefer abduction over revision, which seems to go along with what is assumed in the Cognitive Science literature.
• Correspondence to Schulz’s Approach (Dietz and H¨olldobler 2015)
We further show the correspondence of our system to another approach and show that our approach is more general.