Finding Missing Proofs with Automated Reasoning

This article features long-sought proofs with intriguing properties (such as the absence of double negation and the avoidance of lemmas that appeared to be indispensable), and it features the automated methods for finding them. The theorems of concern are taken from various areas of logic that include two-valued sentential (or propositional) calculus and infinite-valued sentential calculus. Many of the proofs (in effect) answer questions that had remained open for decades, questions focusing on axiomatic proofs. The approaches we take are of added interest in that all rely heavily on the use of a single program that offers logical reasoning, William McCune's automated reasoning program OTTER. The nature of the successes and approaches suggests that this program offers researchers a valuable automated assistant. This article has three main components. First, in view of the interdisciplinary nature of the audience, we discuss the means for using the program in question (OTTER), which flags, parameters, and lists have which effects, and how the proofs it finds are easily read. Second, because of the variety of proofs that we have found and their significance, we discuss them in a manner that permits comparison with the literature. Among those proofs, we offer a proof shorter than that given by Meredith and Prior in their treatment of ukasiewicz's shortest single axiom for the implicational fragment of two-valued sentential calculus, and we offer a proof for the ukasiewicz 23-letter single axiom for the full calculus. Third, with the intent of producing a fruitful dialogue, we pose questions concerning the properties of proofs and, even more pressing, invite questions similar to those this article answers.     

Newton''s Experimental Proofs as Eliminative Reasoning

In this paper I discuss Newton's first optical paper. My aim is to examine the type of argument which Newton uses in order to convince his readers of the truth of his theory of colors. My claim is that this argument is an induction by elimination, and that the Newtonian method of justification is a kind of generative justification, a term due to T. Nickles. To achieve my aim I analyze in some detail the arguments in Newton's first optical paper, relating the paper with Newton's other writings in optics, and especially his early correspondence in defence of his theory of colors.     

Formal Ontology and Mathematics. A Case Study on the Identity of Proofs

We propose a novel, ontological approach to studying mathematical propositions and proofs. By “ontological approach” we refer to the study of the categories of beings or concepts that, in their practice, mathematicians isolate as fruitful for the advancement of their scientific activity (like discovering and proving theorems, formulating conjectures, and providing explanations). We do so by developing what we call a “formal ontology” of proofs using semantic modeling tools (like RDF and OWL) developed by the computer science community. In this article, (i) we describe this new approach and, (ii) to provide an example, we apply it to the problem of the identity of proofs. We also describe open issues and further applications of this approach (for example, the study of purity of methods). We lay some foundations to investigate rigorously and at large scale intellectual moves and attitudes that underpin the advancement of mathematics through cognitive means (carving out investigationally valuable concepts and techniques) and social means (like communication, collaboration, revision, and criticism of specific categories, inferential patterns, and levels of analysis). Our approach complements other types of analysis of proofs such as reconstruction in a deductive system and examination through a proof-assistant.

     

Formal Operational Reasoning in African University Students

Few studies have investigated formal operational concepts among Africans. The present study examined formal operational reasoning among African university students. Both Piagetian and neo-Piagetian criteria of 75% and 50% success rate were used for determining the presence of formal operational concepts. Three formal operational concepts—propositional, proportional, and combinatorial reasoning—were assessed among African university students.     

Strategic Construction of Fitch-style Proofs

Symlog is a system for learning symbolic logic by computer that allows students to interactively construct proofs in Fitch-style natural deduction. On request, Symlog can provide guidance and advice to help a student narrow the gap between goal theorem and premises. To effectively implement this capability, the program was equipped with a theorem prover that constructs proofs using the same methods and techniques the students are being taught. This paper discusses some of the aspects of the theorem prover's design, including its set of proof-construction strategies, its unification algorithm as well as some of the tradeoffs between efficiency and pedagogy.     

Spatial Proportional Reasoning Is Associated With Formal Knowledge About Fractions

Proportional reasoning involves thinking about parts and wholes (i.e., about fractional quantities). Yet, research on proportional reasoning and fraction learning has proceeded separately. This study assessed proportional reasoning and formal fraction knowledge in 8- to 10-year-olds. Participants (N = 52) saw combinations of cherry juice and water in displays that highlighted either part–whole or part–part relations. Their task was to indicate on a continuous rating scale how much each mixture would taste of cherries. Ratings suggested the use of a proportional integration rule for both kinds of displays, although more robustly and accurately for part–whole displays. The findings indicate that children may be more likely to scale proportional components when being presented with part–whole as compared with part–part displays. Crucially, ratings for part–whole problems correlated with fraction knowledge, even after controlling for age, suggesting that a sense of spatial proportions is associated with an understanding of fractional quantities.     

The Relationships of Spatial Ability and Sex to Formal Reasoning Capabilities

The relationships of spatial ability and sex to performance on formal easoning tasks were examined for a group of 34 male and female college tudents. It was hypothesized that spatial ability is positively related to ormal reasoning task performance which is also related to male superiority on formal reasoning task performance. Results indicated that spatial ability vas unrelated to formal reasoning task performance and that sex differinces occurred with the balance task and the pendulum task which sugested that males were superior to females at manifesting the scheme of jroportionality and at being able to isolate variables.     

The Development of Proportional Reasoning: Effect of Continuous Versus Discrete Quantities

This study examines the development of children's ability to reason about proportions that involve either discrete entities or continuous amounts. Six-, 8- and 10-year olds were presented with a proportional reasoning task in the context of a game involving probability. Although all age groups failed when proportions involved discrete quantities, even the youngest age group showed some success when proportions involved continuous quantities. These findings indicate that quantity type strongly affects children's ability to make judgments of proportion. Children's greater success in judging proportions involving continuous quantities appears to be related to their use of different strategies in the presence of countable versus noncountable entities. In two discrete conditions, children—particularly 8- and 10-year-olds—adopted an erroneous counting strategy, considering the number of target elements but not the relation between target and nontarget elements, either in terms of number or amount. In contrast, in the continuous condition, when it was not possible to count, children may have relied on an early developing ability to code the relative amounts of target and nontarget regions.     

Socratic Proofs and Paraconsistency: A Case Study

This paper develops a new proof method for two propositional paraconsistent logics: the propositional part of Batens' weak paraconsistent logic CLuN and Schütte's maximally paraconsistent logic Φ_v. Proofs are de.ned as certain sequences of questions. The method is grounded in Inferential Erotetic Logic.     

Modern Infinitesimals as a Tool to Match Intuitive and Formal Reasoning in Analysis

We discuss various ways, which have been plainly justified in the secondhalf of the twentieth century, to introduce infinitesimals, and we considerthe new style of reasoning in mathematical analysis that they allow.     

Evidence for Mental-model-based Reasoning: A Comparison of Reasoning with Time and Space Concepts

Johnson-Laird (1983) has argued that spatial reasoning is based on the construction and manipulation of mental models in memory. The present article addresses the question of whether reasoning about time relations is constrained by the same factors as reasoning about spatial relations. An experiment is reported that explored the similarities and the differences in the performance of subjects in comparable spatial and temporal reasoning tasks. The results indicated that, in both the temporal and the spatial content domains, the data were in agreement with the view that subjects solved problems by constructing models in memory rather than with a logical rule conceptualisation of reasoning. An analysis of the premise-reading times on the basis of premise-linking order provided support for an on-line process of mental models construction, and offered an explanation for the finding that spatial problems that did require an inference of transitivity were easier than problems that did not. No essential differences in processing and performance were observed across the two content domains, although in the time domain the correctness data were in agreement with both the mental models theory and the logical rules view. The results are discussed with respect to the mental models theory and the structural characteristics of the problems.     

Formal paraphasias: A single case study

A case study is reported of an aphasic patient, RB, who showed frequent form-related whole-word substitutions in oral naming, writing to dictation, and reading aloud. In both written language tasks, the abstractness of the targets influenced the number of formal errors. In oral naming, a high proportion of formal paraphasias was related to the intended words in both form and meaning. A comparison between targets and formal paraphasias indicated a high agreement both in word class, number of syllables, stress pattern, and in basic (stressed) vowels. The agreement in consonants (including word-initial consonants), however, was low. It is argued that RB's formal substitutions are not caused solely by errors of lexical selection but that semantic, lexical, and segmental factors contribute to the error outcome.     

Intuition and Reasoning: A Dual-Process Perspective

A lay definition of intuition holds that it involves immediate apprehension in the absence of reasoning. From a more technical point of view, I argue also that intuition should be seen as the contrastive of reasoning, corresponding roughly to the distinction between Type 1 (intuitive) and Type 2 (reflective) processes in contemporary dual process theories of thinking. From this perspective, we already know a great deal about intuition: It is quick, provides feelings of confidence, can reflect large amounts of information processing, and is most likely to provide accurate judgments when based on relevant experiential learning. Unlike reasoning, intuition is low effort and does not compete for central working memory resources. It provides default responses which may—or often may not—be intervened upon with high effort, reflective reasoning. Intuition has, however, been blamed for a range of cognitive biases in the psychological literatures on reasoning and decision making. The evidence indicates that with novel and abstract problems, not easily linked to previous experience, intervention with effortful reasoning is often required to avoid such biases. Hence, although it seems that intuition dominates reasoning most of the time—both in the laboratory and the real world—it can indeed be a false friend.     

Practical Reasoning Arguments: A Modular Approach

This paper compares current ways of modeling the inferential structure of practical (goal-based) reasoning arguments, and proposes a new approach in which it is regarded in a modular way. Practical reasoning is not simply seen as reasoning from a goal and a means to an action using the basic argumentation scheme. Instead, it is conceived as a complex structure of classificatory, evaluative, and practical inferences, which is formalized as a cluster of three types of distinct and interlocked argumentation schemes. Using two real examples, we show how applying the three types of schemes to a cluster of practical argumentation allows an argument analyst to reconstruct the tacit premises presupposed and evaluate the argumentative reasoning steps involved. This approach will be shown to overcome the limitations of the existing models of practical reasoning arguments within the BDI and commitment theoretical frameworks, providing a useful tool for discourse analysis and other disciplines. In particular, applying this method brings to light the crucial role of classification in practical argumentation, showing how the ordering of values and preferences is only one of the possible areas of deep disagreement.