Towards a Rough Mereology-Based Logic for Approximate Solution Synthesis. Part 1

We are concerned with formal models of reasoning under uncertainty. Many approaches to this problem are known in the literature e.g. Dempster-Shafer theory [29], [42], bayesian-based reasoning [21], [29], belief networks [29], many-valued logics and fuzzy logics [6], non-monotonic logics [29], neural network logics [14]. We propose rough mereology developed by the last two authors [22-25] as a foundation for approximate reasoning about complex objects. Our notion of a complex object includes, among others, proofs understood as schemes constructed in order to support within our knowledge assertions/hypotheses about reality described by our knowledge incompletely.     

Epistemic and Dialectical Models of Begging the Question

This paper addresses the problem posed by the current split between the two opposed hypotheses in the growing literature on the fallacy of begging the question the epistemic hypothesis, based on knowledge and belief, and the dialectical one, based on formal dialogue systems. In the first section, the nature of split is explained, and it is shown how each hypothesis has developed. To get the beginning reader up to speed in the literature, a number of key problematic examples are analyzed illustrating how both approaches can be applied. Useful tools are brought to bear on them, including the automated argument diagramming system Araucaria, and profiles of dialogue used to represent circular argumentation in a dialogue tableau format. These tools are used to both to model circular reasoning and to provide the contextual evidence needed to properly determine whether the circular reasoning in a given case is better judged fallacious or not. A number of technical problems that have impeded the development of both hypotheses are studied. One central problem is the distinction between argument and explanation. It is concluded that the best way to move forward and solve these problems is to reformulate the two hypotheses in such a way that they might be able to co-exist. On this basis, a unified methodology is proposed that allows each hypothesis to move forward as a legitimate avenue for research using the same tools.     

Probabilities on Sentences in an Expressive Logic

Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higher-order logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truth values. The main technical problem studied in this paper is the following: Given a set of sentences, each having some probability of being true, what probability should be ascribed to other (query) sentences? A natural wish-list, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wish-list into technical requirements for a prior probability and show that probabilities satisfying all our criteria exist. We also give explicit constructions and several general characterizations of probabilities that satisfy some or all of the criteria and various (counter)examples. We also derive necessary and sufficient conditions for extending beliefs about finitely many sentences to suitable probabilities over all sentences, and in particular least dogmatic or least biased ones. We conclude with a brief outlook on how the developed theory might be used and approximated in autonomous reasoning agents. Our theory is a step towards a globally consistent and empirically satisfactory unification of probability and logic.     

Dual Space Search During Scientific Reasoning

The purpose of the two studies reported here was to develop an integrated model of the scientific reasoning process. Subjects were placed in a simulated scientific discovery context by first teaching them how to use an electronic device and then asking them to discover how a hitherto unencountered function worked. To do this task, subjects had to formulate hypotheses based on their prior knowledge, conduct experiments, and evaluate the results of their experiments. In the first study, using 20 adult subjects, we identified two main strategies that subjects used to generate new hypotheses. One strategy was to search memory and the other was to generalize from the results of previous experiments. We described the former group as searching an hypothesis space, and the latter as searching an experiment space. In a second study, with 10 adults, we investigated how subjects search the hypothesis space by instructing them to state all the hypotheses that they could think of prior to conducting any experiments. Following this phase, subjects were then allowed to conduct experiments. Subjects who could not think of the correct rule in the hypothesis generation phase discovered the correct rule only by generalizing from the results of experiments in the experimental phase.
Both studies provide support for the view that scientific reasoning can be characterized as search in two problem spaces. By extending Simon and Lea's (1974) Generalized Rule Inducer, we present a general model of Scientific Discovery as Dual Search (SDDS) that shows how search in two problem spaces (an hypothesis space and an experiment space) shapes hypothesis generation, experimental design, and the evaluation of hypotheses. The model also shows how these processes interact with each other. Finally, we interpret earlier findings about the psychology of scientific reasoning in terms of the SDDS model.     

Memory indexing of sequential symptom processing in diagnostic reasoning

In diagnostic reasoning, knowledge about symptoms and their likely causes is retrieved to generate and update diagnostic hypotheses in memory. By letting participants learn about causes and symptoms in a spatial array, we could apply eye tracking during diagnostic reasoning to trace the activation level of hypotheses across a sequence of symptoms and to evaluate process models of diagnostic reasoning directly. Gaze allocation on former locations of symptom classes and possible causes reflected the diagnostic value of initial symptoms, the set of contending hypotheses, consistency checking, biased symptom processing in favor of the leading hypothesis, symptom rehearsal, and hypothesis change. Gaze behavior mapped the reasoning process and was not dominated by auditorily presented symptoms. Thus, memory indexing proved applicable for studying reasoning tasks involving linguistic input. Looking at nothing revealed memory activation because of a close link between conceptual and motor representations and was stable even after one week.     

Working memory limitation as a source of confusion in the abstract THOG task

Limitations of working memory are proposed as a major determinant of problem difficulty in the THOG task. This task is a logical reasoning task which uses an exclusive disjunction and requires hypothetico-deductive reasoning. Four experiments with students of mathematics or psychology were used to test the hypotheses that, first, guiding participants' attention facilitates the task and, second, the use of paper and pencil as external problem representation reliefs working memory load. Focusing participants' attention upon a critical aspect of the task does not improve solution rates. Students of mathematics were better than students of psychology, but only if they were allowed to use paper and pencil or to work on the task repeatedly. These results partially support the working memory hypothesis. They point toward the importance of training and practice in relatively simple meta-cognitive skills in logical reasoning. Received: 20 March 2000 / Accepted: 22 January 2001     

Memory activation of multiple hypotheses in sequential diagnostic reasoning

In sequential diagnostic reasoning, observed evidence activates hypotheses about possible causes in memory. These memory activations have been previously examined with a probe reaction task for problems with a single correct diagnosis. We applied this process tracing method to ambiguous problems with multiple compatible hypotheses. When participants reasoned about the causes of ambiguous symptom sequences, they were prompted to respond to probes representing hypotheses. The response time to a probe was shorter if the current support for the respective hypothesis was stronger indicating that the processing of compatible hypotheses can be traced. For sequences with two equally supported hypotheses, the initial hypothesis was more often chosen as the final diagnosis (a primacy effect). Probe reaction times suggest that the initial hypothesis has been activated more strongly already early, when it was finally chosen as the diagnosis. Nevertheless, substantial variance in response times limits the task's applicability for process tracing.     

Skepticism,Contextualism, and Discrimination*

The skeptic says that “knowledge” is an absolute term, whereas the contextualist says that ‘knowledge” is a relationally absolute term. Which is the better hypothesis about “knowledge”? And what implications do these hypotheses about “knowledge” have for knowledge? I argue that the skeptic has the better hypothesis about “knowledge”, but that both hypotheses about “knowledge” have deeply anti‐skeptical implications for knowledge, since both presuppose our capacity for epistemically salient discrimination.     

Accommodating Surprise in Taxonomic Tasks: The Role of Expertise

This paper reports a psychological study of human categorization that looked at the procedures used by expert scientists when dealing with puzzling items. Five professional botanists were asked to specify a category from a set of positive and negative instances. The target category in the study was defined by a feature that was unusual, hence situations of uncertainty and puzzlement were generated. Subjects were asked to think aloud while solving the tasks, and their verbal reports were analyzed. A number of problem solving strategies were identified, and subsequently integrated in a model of knowledge‐guided inductive categorization. Our model proposes that expert knowledge influences the subjects' reasoning in more complex ways than suggested by earlier investigations of scientific reasoning. As in previous studies, domain knowledge influenced our subjects' hypothesis generation and testing; but, additionally, it played a central role when subjects revised their hypotheses.     

A Completeness Proof for a Logic with an Alternative Necessity Operator

We show the completeness of a Hilbert-style system LK defined by M. Valiev involving the knowledge operator K dedicated to the reasoning with incomplete information. The completeness proof uses a variant of Makinson's canonical model construction. Furthermore we prove that the theoremhood problem for LK is co-NP-complete, using techniques similar to those used to prove that the satisfiability problem for propositional S5 is NP-complete.     

Multiple causes of difficulty in insight: the case of the nine-dot problem

Theories of insight problems are often tested by formulating hypotheses about the particular difficulties of individual insight problems. Such evaluations often implicitly assume that there is a single difficulty. We argue that the quantitatively small effects of many studies arise because the difficulty of many insight problems is determined by multiple factors, so the removal of 1 factor has limited effect on the solution rate. Difficulties can reside either in problem perception, in prior knowledge, or in the processing of the problem information. We support this multiple factors perspective through 3 experiments on the 9-dot problem (N.R.F. Maier, 1930). Our results lead to a significant reformulation of the classical hypothesis as to why this problem is difficult. The results have general implications for our understanding of insight problem solving and for the interpretation of data from studies that aim to evaluate hypotheses about the sources of difficulty of particular insight problems.     

Qualitative probabilistic inference under varied entropy levels

In previous work, we studied four well known systems of qualitative probabilistic inference, and presented data from computer simulations in an attempt to illustrate the performance of the systems. These simulations evaluated the four systems in terms of their tendency to license inference to accurate and informative conclusions, given incomplete information about a randomly selected probability distribution. In our earlier work, the procedure used in generating the unknown probability distribution (representing the true stochastic state of the world) tended to yield probability distributions with moderately high entropy levels. In the present article, we present data charting the performance of the four systems when reasoning in environments of various entropy levels. The results illustrate variations in the performance of the respective reasoning systems that derive from the entropy of the environment, and allow for a more inclusive assessment of the reliability and robustness of the four systems.     

Deontic Interpreted Systems

We investigate an extension of the formalism of interpreted systems by Halpern and colleagues to model the correct behaviour of agents. The semantical model allows for the representation and reasoning about states of correct and incorrect functioning behaviour of the agents, and of the system as a whole. We axiomatise this semantic class by mapping it into a suitable class of Kripke models. The resulting logic, KD45_n ^i-j, is a stronger version of KD, the system often referred to as Standard Deontic Logic. We extend this formal framework to include the standard epistemic notions defined on interpreted systems, and introduce a new doubly-indexed operator representing the knowledge that an agent would have if it operates under the assumption that a group of agents is functioning correctly. We discuss these issues both theoretically and in terms of applications, and present further directions of work.     

Solving probabilistic and statistical problems: a matter of information structure and question form

Is the human mind inherently unable to reason probabilistically, or is it able to do so only when problems tap into a module for reasoning about natural frequencies? We suggest an alternative possibility: naive individuals are able to reason probabilistically when they can rely on a representation of subsets of chances or frequencies. We predicted that naive individuals solve conditional probability problems if they can infer conditional probabilities from the subset relations in their representation of the problems, and if the question put to them makes it easy to consider the appropriate subsets. The results of seven studies corroborated these predictions: when the form of the question and the structure of the problem were framed so as to activate intuitive principles based on subset relations, naive individuals solved problems, whether they were stated in terms of probabilities or frequencies. Otherwise, they failed with both sorts of information. The results contravene the frequentist hypothesis and the evolutionary account of probabilistic reasoning.     

Knowledge of Validity

What accounts for how we know that certain rules of reasoning, such as reasoning by Modus Ponens, are valid? If our knowledge of validity must be based on some reasoning, then we seem to be committed to the legitimacy of rule‐circular arguments for validity. This paper raises a new difficulty for the rule‐circular account of our knowledge of validity. The source of the problem is that, contrary to traditional wisdom, a universal generalization cannot be inferred just on the basis of reasoning about an arbitrary object. I argue in favor of a more sophisticated constraint on reasoning by universal generalization, one which undermines a rule‐circular account of our knowledge of validity.     

Reducing structural-element salience on a source problem produces later success in analogical transfer: What role does source difficulty play?

Two experiments in reasoning by analogy were conducted to study the role of inducing source difficulty by reducing the salience of the source's structural elements. Three nonexclusive hypotheses were tested. According to the first, a difficult source problem improves analogical transfer because it increases the probability that the subject will notice the similarity between the source and the target. For example, errors made on both the source and the target can enhance the subject's awareness of the similarity between the two problems. According to the second hypothesis, a source that is difficult to solve is memorized better than an easier source. According to the third, source-problem difficulty affects the degree of abstractness in the representation of the solution elaborated by subjects. Experiment 1 showed that the higher frequency of spontaneous transfer between the source and the target when the source problem was difficult (Gick & McGarry, 1992) could be replicated in a cued-transfer situation. Experiment 2 showed that subjects given a difficult source, one in which the important element was not very salient, were better at categorizing isomorphic problems on the basis of structural features than were subjects given an easy source. The discussion deals with the implications of these results for the hypotheses tested and, more generally, for reasoning by analogy and education in general.     

Hypothetico-nomological aspects of medical diagnosis Part I: General structure of the diagnostic process and its hypothesis-directed stage

In medical diagnostic examination three main stages may be distinguished: (a) initial exploration, (b) hypothesis-directed investigation, and (c) final diagnosis making. The purpose of this work is to study some methodological problems concerning the second of the above stages of the diagnosis and to prepare a background for a mathematical model [30] of this process. In diagnostic problem solving, the reasoning proceeds along the main lines traced by some initial suggestions and passes through various intermediate elements which are connected with one another forming ramifying chains and nets of inferences and hypotheses. Such a complex mental construction is based on laws which form medical knowledge and reflect various regularities and relations, causal, structural, functional, and others. The main components of diagnostic reasoning may be divided into several classes according to their function and content: leading hypotheses, working hypotheses, main diagnostic hypotheses, statements accepted as certain, intermediary and reserve hypotheses, therapeutic suggestions of immediate consequence. In an example of diagnostic problem solving these types of propositions are defined and analysed. In diagnostic reasoning, as in every other process of rational problem solving, explanation of the observed symptoms and signs and testing of the explaining hypotheses play a predominant role. These procedures form successive, frequently numerous and diversified steps and stages of the reasoning, leading to the construction of a mental model of the patient's state. Some problems relative to the scheme of explanation, especially to that which is based on causal laws, are discussed.     

Beyond belief bias: Reasoning from conceptual structures by mental models manipulation

Mental models constitute an alternative to the rule-based systems in the explanation of human reasoning (Johnson-Laird, 1983). In this paper, we claim that the concept of believability generally used to categorize content and context effects is of little use within a semantic theory. Thus, we propose the use of categories that are directly extracted from subjective relations among concepts within the reasoning problem. We demonstrate that manipulations based on this kind of categorization produce predictable patterns of responses in reasoning problems. We present two experiments to test our predictions, using conditional and syllogistic reasoning problems, and in both cases, we demonstrate the influence of conceptual knowledge not only in natural contexts, but also in experimentally created artificial contexts.