An analytic tableau calculus for a temporalised belief logic

A tableau is a refutation-based decision procedure for a related logic, and is among the most popular proof procedures for modal logics. In this paper, we present a labelled tableau calculus for a temporalised belief logic called TML⁺, which is obtained by adding a linear-time temporal logic onto a belief logic by the temporalisation method of Finger and Gabbay. We first establish the soundness and the completeness of the labelled tableau calculus based on the soundness and completeness results of its constituent logics. We then sketch a resolution-type proof procedure that complements the tableau calculus and also propose a model checking algorithm for TML⁺ based on the recent results for model checking procedures for temporalised logics. TML⁺ is suitable for formalising trust and agent beliefs and reasoning about their evolution for agent-based systems. Based on the logic TML⁺, the proposed labelled tableau calculus could be used for analysis, design and verification of agent-based systems operating in dynamic environments.     

The computational value of debate in defeasible reasoning

Defeasible reasoning is concerned with the logics of non-deductive argument. As is described in the literature, the study of this type of reasoning is considerably more involved than the study of deductive argument, even so that, in realistic applications, there is often a lack of resources to perform an exhaustive analysis. It follows that, in a theory of defeasible reasoning, the order and direction in which arguments are developed, i.e. theprocedure, is important. The aim of this article is to show that debate is the most efficient procedure to argue in the presence of limited resources. To do so, there is first some general theory on defeasible argumentation, which is followed by an introduction to the problem of dialectical search. The problem of dialectical search is (or at least, should be) the essential issue in every theory on argumentation, and emerges at every occasion that involves adjudication on competing arguments. Starting with an example, it is explained that dialectical search can be best scheduled according to classical debating techniques, that work along well-tried methods. These methods (which include various forms of curtailment, interruption, and interpretation) have proven their value in keeping debating efforts within reasonable bounds. How they apply in a theory of formal argument, will be shown in this article.This research was made possible by SION, and is financed by NWO under contract number 612-316-019. Part of this research has been conducted at the Vrije Universiteit Amsterdam. This article contains fragments of Chapter 6 and Chapter 7 of the author's dissertation. Studies in Defeasible Argumentation (1993).     

A Sound and Complete Tableau Calculus for Reasoning about only Knowing and Knowing at Most

We define a tableau calculus for the logic of only knowing and knowing at most ON, which is an extension of Levesque's logic of only knowing O. The method is based on the possible-world semantics of the logic ON, and can be considered as an extension of known tableau calculi for modal logic K45. From the technical viewpoint, the main features of such an extension are the explicit representation of "unreachable" worlds in the tableau, and an additional branch closure condition implementing the property that each world must be either reachable or unreachable. The calculus allows for establishing the computational complexity of reasoning about only knowing and knowing at most. Moreover, we prove that the method matches the worst-case complexity lower bound of the satisfiability problem for both ON and O. With respect to [22], in which the tableau calculus was originally presented, in this paper we both provide a formal proof of soundness and completeness of the calculus, and prove the complexity results for the logic ON.     

Logic programming,probability, and two-system accounts of reasoning: a rejoinder to Oaksford and Chater (2014)

This reply to Oaksford and Chater’s (O&C)’s critical discussion of our use of logic programming (LP) to model and predict patterns of conditional reasoning will frame the dispute in terms of the semantics of the conditional. We begin by outlining some common features of LP and probabilistic conditionals in knowledge-rich reasoning over long-term memory knowledge bases. For both, context determines causal strength; there are inferences from the absence of certain evidence; and both have analogues of the Ramsey test. Some current work shows how a combination of counting defeaters and statistics from network monitoring can provide the information for graded responses from LP reasoning. With this much introduction, we then respond to O&C’s specific criticisms and misunderstandings.     

The distinct contributions of cause–effect order and reasoning type in judgments of causality

ABSTRACT

People generally perceive a stronger link between smoking and cancer than between cancer and smoking. Generally, prior research on asymmetrical causal reasoning has not distinguished predictive (searching for effects) and diagnostic reasoning (searching for causes) from the order in which causes and effects are presented. Across 6 studies (overall N = 627), we show that order and reasoning have an additive influence on the causality perception: causes, spatially or temporally presented before the effect, strengthen the causality attribution associated to predictive (vs. diagnostic) frames. Moreover, we show that order and reasoning frame are bi-directionally related, as the cause-first order triggers predictive reasoning and vice versa, and people mentally maintain the cause-first order when envisaging a causal relation. Besides its methodological contribution to the causal reasoning literature, this research demonstrates the powerful role of word order in causal reasoning. Implications for the role of word order in communication and risk prevention are discussed.     

An epistemic and dynamic approach to abductive reasoning: Abductive problem and abductive solution

We propose a study of abductive reasoning addressing it as an epistemic process that involves both an agent?s information and the actions that modify this information. More precisely, we present and discuss definitions of an abductive problem and an abductive solution in terms of an agent?s information, that is, in terms of knowledge and beliefs. The discussion is then formalised by ‘implementing’ our definitions in a dynamic epistemic logic framework, where the properties of these definitions are studied, an epistemic action that represents the application of an abductive step is introduced, and an illustrative example is provided. A number of the most interesting properties of abductive reasoning (those highlighted by Peirce) are shown to be better modelled within this approach.     

Probabilistic single function dual process theory and logic programming as approaches to non-monotonicity in human vs. artificial reasoning

In this paper, it is argued that single function dual process theory is a more credible psychological account of non-monotonicity in human conditional reasoning than recent attempts to apply logic programming (LP) approaches in artificial intelligence to these data. LP is introduced and among other critiques, it is argued that it is psychologically unrealistic in a similar way to hash coding in the classicism vs. connectionism debate. Second, it is argued that causal Bayes nets provide a framework for modelling probabilistic conditional inference in System 2 that can deal with patterns of inference LP cannot. Third, we offer some speculations on how the cognitive system may avoid problems for System 1 identified by Fodor in 1983. We conclude that while many problems remain, the probabilistic single function dual processing theory is to be preferred over LP as an account of the non-monotonicity of human reasoning.     

Disjunctive logic programming with types and objects: The DLV system

The paper presents DLV⁺, a Disjunctive Logic Programming (DLP) system with object-oriented constructs, including classes, objects, (multiple) inheritance, and types. DLV⁺ is built on top of DLV (a state-of-the art DLP system), and provides a graphical user interface that allows one to specify, update, browse, query, and reason on knowledge bases. Two strong points of the system are the powerful type-checking mechanism and the advanced interface for visual querying.DLV⁺ is already used for the development of knowledge based applications for information extraction and text classification.     

Bridging learning theory and dynamic epistemic logic

This paper discusses the possibility of modelling inductive inference (Gold 1967) in dynamic epistemic logic (see e.g. van Ditmarsch et al. 2007). The general purpose is to propose a semantic basis for designing a modal logic for learning in the limit. First, we analyze a variety of epistemological notions involved in identification in the limit and match it with traditional epistemic and doxastic logic approaches. Then, we provide a comparison of learning by erasing (Lange et al. 1996) and iterated epistemic update (Baltag and Moss 2004) as analyzed in dynamic epistemic logic. We show that finite identification can be modelled in dynamic epistemic logic, and that the elimination process of learning by erasing can be seen as iterated belief-revision modelled in dynamic doxastic logic. Finally, we propose viewing hypothesis spaces as temporal frames and discuss possible advantages of that perspective.     

Children's reasoning about the temporal order of past and future events

Four- and five-year-olds completed two sets of tasks that involved reasoning about the temporal order in which events had occurred in the past or were to occur in the future. Four-year-olds succeeded on the tasks that involved reasoning about the order of past events but not those that involved reasoning about the order of future events, whereas 5-year-olds passed both types of tasks. Individual children who failed the past-event tasks were not particularly likely to fail the more difficult future-event tasks. However, children's performance on the reasoning tasks was predictive of their performance on a task assessing their comprehension of the terms “before” and “after.” Our results suggest that there may be a developmental change over this age range in the ability to flexibly represent and reason about the before-and-after relationships between events.     

The link between deductive reasoning and mathematics

ABSTRACT

Recent studies have shown that deductive reasoning skills (including transitive and conditional inferences) are related to mathematical abilities. Nevertheless, so far the links between mathematical abilities and these two forms of deductive inference have not been investigated in a single study. It is also unclear whether these inference forms are related to both basic maths skills and mathematical reasoning, and whether these relationships still hold if the effects of fluid intelligence are controlled. We conducted a study with 87 adult participants. The results showed that transitive reasoning skills were related to performance on a number line task, and conditional inferences were related to arithmetic skills. Additionally, both types of deductive inference were related to mathematical reasoning skills, although transitive and conditional reasoning ability were unrelated. Our results also highlighted the important role that ordering abilities play in mathematical reasoning, extending findings regarding the role of ordering abilities in basic maths skills. These results have implications for the theories of mathematical and deductive reasoning, and they could inspire the development of novel educational interventions.     

A cirquent calculus system with clustering and ranking

Cirquent calculus is a new proof-theoretic and semantic approach introduced by G. Japaridze for the needs of his theory of computability logic (CoL). The earlier article “From formulas to cirquents in computability logic” by Japaridze generalized formulas in CoL to circuit-style structures termed cirquents. It showed that, through cirquents with what are termed clustering and ranking, one can capture, refine and generalize independence-friendly (IF) logic. Specifically, the approach allows us to account for independence from propositional connectives in the same spirit as IF logic accounts for independence from quantifiers. Japaridze's treatment of IF logic, however, was purely semantical, and no deductive system was proposed. The present paper syntactically constructs a cirquent calculus system with clustering and ranking, sound and complete w.r.t. the propositional fragment of cirquent-based semantics. Such a system captures the propositional version of what is called extended IF logic, thus being an axiomatization of a nontrivial fragment of that logic.     

Pandora: A Reasoning Toolbox using Natural Deduction Style

Pandora is a tool for supporting the learning of first ordernatural deduction. It includes a help window, an interactivecontext sensitive tutorial known as the "e-tutor" and facilitiesto save, reload and export to LATEX. Every attempt to applya natural deduction rule is met with either success or a helpfulerror message, providing the student with instant feedback.Detailed electronic logs of student usage are recorded for evaluationpurposes. This paper describes the basic functionality, thee-tutor, our experiences of using the tool in teaching and ourfuture plans.