Understanding and Inductive Inference

The problem of justifying induction has vexed philosophers for centuries. It has been entangled with issues concerning logic and probability and the philosophy of science. This article proposes a new approach to untangling these issues and resolving the overall problem. This new approach is by way of the perspective of realist ontology. Induction should not be seen as a debased form of logic and the search for inflexible rules for justifying particular sorts of inductive inference should be abandoned. Rather, induction can be justified pragmatically as a general practice because reality possesses ordered patterns. Induction, being dependent on repetition, can sometimes identify aspects of those patterns simply because the instances of repetition really do occur. Thus, confidence in conclusions based on inductive inferences is not properly to be derived from unchanging logical rules governing when and where inductive inferences may be successfully applied, but from the involvement of such inferences with a wider, deeper process in which not only deductive inferences and other inductive inferences are involved, but also a set of relations to broader bodies of relevant knowledge. That is, unlike particular deductive inferences, inductive inferences may not be engaged in isolation from all other information about the world and completely in the abstract. Induction is always connected with a process of retroduction.

PATTERN RECOGNITION AS AN INDUCTIVE PROCESS

Children and adults rated their own certainty following inductive inferences, deductive inferences, and guesses. Beginning in kindergarten, participants rated deductions as more certain than weak inductions or guesses. Deductions were rated as more certain than strong inductions beginning in Grade 3, and fourth-grade children and adults differentiated strong inductions, weak inductions, and informed guesses from pure guesses. By Grade 3, participants also gave different types of explanations for their deductions and inductions. These results are discussed in relation to children's concepts of cognitive processes, logical reasoning, and epistemological development.

It is argued that inductive inferences from behaviors to traits are performed more frequently than deductive inferences from traits to behaviors-a phenomenon referred to as the induction-deduction asymmetry. Two experiments are reported in which behavior-to-trait inferences and trait-to-behavior inferences were compared within the same paradigm: Participants learned a series of descriptions of a target person, half of which were presented in trait form, half in behavior form. A subsequent recognition task was constructed so that some of the items (traits and behaviors) had actually been seen, some were entirely new, and some were new but had been implied by the information given. The 2 experiments provide clear evidence for the hypothesis that traits implied by a behavior are more frequently misidentified as already seen than behaviors implied by a trait. Response-time data in Experiment 2 further suggest that inferences from behaviors to traits are made on-line, whereas inferences from traits to behaviors appear to be memory based.

Most attribution theories focus on inductive inferences and abstract causal categories. By contrast, goal-based and knowledge structure theories focus on people's perceptions of intentional actions, their deductive inferences, and their concrete explanations of actions. Goal-based theories have demonstrated the importance of goals and intentions as explanations. However, research shows that explanations of goal-based actions reflect the extremity and controllability of the actions, the presence of obstructions, and the type of causal question eliciting the explanation. These factors determine whether people prefer to invoke goals or enabling conditions (or conjunctions) as explanations. Judgments of explanations also reflect communicative principles or informativeness and relevance, more than logical judgments of probability and necessity. The goal-based approach is often seen as an alternative to the inductive covariational approach, but the two lines of research can be integrated.

Parametric logic is a framework that generalises classical first-order logic. A generalised notion of logical consequence—a form of preferential entailment based on a closed world assumption—is defined as a function of some parameters. A concept of possible knowledge base—the counterpart to the consistent theories of first-order logic—is introduced. The notion of compactness is weakened. The degree of weakening is quantified by a nonnull ordinal—the larger the ordinal, the more significant the weakening. For every possible knowledge base T, a hierarchy of sentences that are generalised logical consequences of T is built. The first layer of the hierarchies corresponds to sentences that can be obtained by a deductive inference, characterised by the compactness property. The second layer of the hierarchies corresponds to sentences that can be obtained by an inductive inference, characterised by the property of weak compactness quantified by 1. Weaker forms of compactness—quantified by nonnull ordinals—determine higher layers in the hierarchies, corresponding to more complex inferences. The naturalness of the hierarchies built over the possible knowledge bases is attested by fundamental connections with notions from Learning theory and from topology. The naturalness of the hierarchies built over the possible knowledge bases is attested by fundamental connections with notions from Learning theory—classification in the limit, with or without a bounded number of mind changes—and from topology—in reference to the Borel and the difference hierarchies. In this paper, we introduce the key model-theoretic aspects of Parametric logic, justify the concept of the knowledge base, define the hierarchies of generalised logical consequences and illustrate their relevance to Nonmonotonic reasoning. More specifically, we show that the degree of nonmonotonicity that is required to infer a sentence can be characterised by the least nonnull ordinal that quantifies the weakening of compactness used to locate the inferred sentence in the hierarchies.

The theory of Boolean functions, especially in respect to representing these functions in the disjunctive or conjunctive normal forms, is extended in this chapter onto the case of finite predicates. Finite predicates are decomposed by that into some binary units, which will correspond to components of Boolean vectors and matrices and are represented as combinations of these units. Further, the main concepts used for solving pattern recognition problems are defined, namely world model, data, and knowledge. The data presenting information about the existence of some objects with definite combinations of properties is considered, as well as the knowledge presenting information about the existence of regular relationships between attributes. These relationships prohibit some combinations of properties. In this way, the knowledge gives the information about the non-existence of objects with some definite (prohibited) combinations of attribute values. A special form of regularity representation, called implicative regularities, is introduced. Any implicative regularity generates an empty interval in the Boolean space of object descriptions, which do not contradict the data. The problem of plausibility evaluation of induced implicative regularities should be solved by that. The pattern recognition problem is solved by two steps. First, regularities are extracted from the database (inductive inference); second, the obtained knowledge is used for the object recognition (deductive inference).

An ongoing debate in education has centered on the concept of domain-general reasoning practices and whether such a position can explain how student learning in one discipline can influence their practices in another. For example, student learning in science impacts their learning in mathematics. Based on a model highlighting a significant relationship between reasoning practices used across science and mathematics, this study explored the lasting impact of learning science through the Science Writing Heuristics (SWH) approach on inductive and deductive reasoning in mathematics as well as deductive reasoning in science. Utilizing fourth-grade student data and follow-up data collected over three subsequent years, we applied the generalized DINA (deterministic inputs, noisy “and” gate) and linear regression models for analysis. The findings provide evidence of the immeote utilization of inductive reasoning in mathematics and a delayed advantage of deductive reasoning in both mathematics and science. These results support the implementation of argument-based approaches, like the SWH approach, on student development and utilization of domain-general reasoning. The findings of this study highlight the need for further research on the development of such reasoning resources when students are immersed in epistemically rich learning environments as proposed by national curricula. Additionally, further investigation is needed to determine whether these reasoning skills are applied across different disciplines, regardless of context.

Alexandre Koyré praised Gassendi for having published the first correct formulation of the principle of inertia thanks to his “deep understanding” of Galileo’s science, but also criticized him for his distrust of mathematics and his physics made of “bits and pieces.” Taking its cue from Koyré’s ambivalent judgment, this chapter provides a reconstruction of Gassendi’s engagement with Galileo’s new science, starting with the De motu impresso a motore translato epistolae duae (1642) and ending with the Syntagma philosophicum. The chapter focuses on three themes: Gassendi’s account of free fall and projectile motion; his formulation of the principle of inertia; and his reinterpretation of Galileo’s theory of tides. More specifically, it deals with Gassendi’s project to provide Galileo’s laws of motion with an ontological foundation; analyzes his attempt to bridge the gap between the principle of inertia, valid at the level of the res concretae, and the theory of the self-motion of atoms; and tries to make sense of Gassendi’s simultaneous endorsement of Galileo’s theory of tides, on the one hand, and of Kepler’s laws of planetary motion, on the other. The chapter also shows that Gassendi was aware of some conceptual tensions in his theories, which he partially modified over the course of his career.

An outline is presented of the line of reasoning which Newton used in his Principia to obtain Kepler’s three laws of planetary motion from his own laws of motion and of universal gravitation. His combination of deductive and retroductive reasoning is related to his general philosophical outlook on the nature of science. The actual structure of the argument differs significantly from that often found in physics texts. A straightforward presentation is developed which allows the modern reader to see the type of derivation Newton employed to deduce Kepler’s three laws. The discussion preserves the essentially geometric character of Newton’s constructions in the Principia. It can even be presented to students in a noncalculus-based introductory physics course as a case study showing how one of the great achievements of modern science was produced.

The goal of this chapter is to derive Kepler's laws of planetary motion from Newton's laws of motion and gravitation. We will analyze the motion of two massive particles (such as the sun and Mars), assuming that all forces other than mutual gravitational attraction are negligible. We want to find explicit formulas allowing us to predict motions. Will Mars crash into the sun? Will comet Hale-Bopp return? If so, when? Our formulas will be of use to engineers, who may want to maneuver satellites or send a space probe that is “parked,” i.e., orbiting a planet, off to orbit a different planet. During this chapter we will pretend to be physicists. We will make savvy use of conserved quantities and particular functional forms to pick coordinates that simplify our calculations.KeywordsAngular MomentumOrdinary Differential EquaInertial SystemAstronomical UnitPlanetary MotionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Kepler’s three laws of planetary motion are derived algebraically from conservation principles without introducing Newton’s law of force explicitly. This procedure can be presented to students who have had no more than high school algebra.

Three-experiments examined the number of qualitatively different processing dimensions needed to account for inductive and deductive reasoning. In each study, participants were presented with arguments that varied in logical validity and consistency with background knowledge (believability), and evaluated them according to deductive criteria (whether the conclusion was necessarily true given the premises) or inductive criteria (whether the conclusion was plausible given the premises). We examined factors including working memory load (Experiments 1 and 2), individual working memory capacity (Experiments 1 and 2), and decision time (Experiment 3), which according to dual-processing theories, modulate the contribution of heuristic and analytic processes to reasoning. A number of empirical dissociations were found. Argument validity affected deduction more than induction. Argument believability affected induction more than deduction. Lower working memory capacity reduced sensitivity to argument validity and increased sensitivity to argument believability, especially under induction instructions. Reduced decision time led to decreased sensitivity to argument validity. State-trace analyses of each experiment, however, found that only a single underlying dimension was required to explain patterns of inductive and deductive judgments. These results show that the dissociations, which have traditionally been seen as supporting dual-processing models of reasoning, are consistent with a single-process model that assumes a common evidentiary scale for induction and deduction. (PsycINFO Database Record

Knowledge underlies rational choice between options. Predictive optimization is the prioritization of options according to their predicted outcomes based on available knowledge. The epistemological justification of predictive optimization is based on positivism, which asserts that facts and laws about the world exist and are discoverable. However, knowledge of human affairs in strategic adversarial interactions is often severely limited and erroneous: residual uncertainty is often vast. This results especially from deception and innovation by the adversary which introduce deep Knightian uncertainty. Consequently, predictive optimization is unreliable: outcomes may differ substantially from predictions. An alternative strategy for prioritization of options is info-gap robust satisficing: achieve critical goals (that are adequate but perhaps suboptimal) over a wide range of deviation of reality from current knowledge. The epistemological justification of robust satisficing is based on extending positivism to acknowledge and manage the unknown. Prioritization of options by robust satisficing manages both the limitations of knowledge and the need for achieving critical goals. This critique of positivism is not constructivist. Rather, we extend positivism to account for highly deficient knowledge. We present several examples and conclude by discussing the relation between inductive, abductive and deductive inference.

In this paper, we propose a knowledge processing neural network which is capable of inductive and deductive inference. The proposed network looks up relations between words in a concept dictionary and co-occurrence dictionary. First, the proposed network divides sentences into the subject words and the other words. Then these words are input into two-layer network. Second, hierarchical structure is composed using concept dictionary. Third, the network induces general knowledge from individual knowledge. We added a function to respond to questions in natural language with ”Yes/No/I don’t know” in order to confirm the validity of proposed network by evaluating the quantity of correct answers.