Related papers: Dismantling the Surprise Test "Paradox"
We first consider the method of scoring students' self-assessment of confidence (SAC) used by Foster in [1], and find that with it reporting their true confidence is not the optimal strategy for students. We then identify all continuously…
We present a propositional logic %which can be used to reason about the uncertainty of events, where the uncertainty is modeled by a set of probability measures assigning an interval of probability to each event. We give a sound and…
The primary objective of this paper is to revisit Simpson's paradox using a statistical misspecification perspective. It is argued that the reversal of statistical associations is sometimes spurious, stemming from invalid probabilistic…
The paper demonstrates that falsifiability is fundamental to learning. We prove the following theorem for statistical learning and sequential prediction: If a theory is falsifiable then it is learnable -- i.e. admits a strategy that…
It is well-known that Choice and Regularity are independent of each other but have important common consequences of logical character (reflection principles, representations of classes by sets, etc.). We explain this phenomenon by isolating…
The Borel--Kolmogorov paradox of conditioning with respect to events of prior probability zero has fascinated students and researchers since its discovery more than 100 years ago. Classical conditioning is only valid with respect to events…
This paper outlines a deceptively complex problem in classical mechanics which the paper names the "Falling Astronaut Problem," and it explores a method for teachers to implement this problem in an undergraduate classroom. The paper…
For half a century, authors have weakened the rule of necessitation in various more or less ad hoc ways in order to make inconsistent systems consistent. More recently, necessitation was weakened in a systematic way, not for the purpose of…
The chaotic hypothesis discussed in [GC1] is tested experimentally in a simple conduction model. Besides a confirmation of the hypothesis predictions the results suggest the validity of the hypothesis in the much wider context in which, as…
While many methods purport to explain predictions by highlighting salient features, what aims these explanations serve and how they ought to be evaluated often go unstated. In this work, we introduce a framework to quantify the value of…
We show that G\"odel's negative results concerning arithmetic, which date back to the 1930s, and the ancient "sand pile" paradox (known also as "sorites paradox") pose the questions of the use of fuzzy sets and of the effect of a measuring…
Student success models might be prone to develop weak spots, i.e., examples hard to accurately classify due to insufficient representation during model creation. This weakness is one of the main factors undermining users' trust, since model…
This article discusses the logical errors in the liar paradox, G\"odel's incompleteness theorems, Russell's paradox, and the halting problem. In order to avoid these errors, a redefinition of logic has been presented, which is concluded as…
The problem of induction has persisted since Hume exposed the logical gap between repeated observation and universal inference. Traditional attempts to resolve it have oscillated between two extremes: the probabilistic optimism of Laplace…
Breakthrough discoveries and inventions involve unexpected combinations of contents including problems, methods, and natural entities, and also diverse contexts such as journals, subfields, and conferences. Drawing on data from tens of…
An important, yet largely unstudied, problem in student data analysis is to detect misconceptions from students' responses to open-response questions. Misconception detection enables instructors to deliver more targeted feedback on the…
Common research tasks ask students to identify a correct answer and justify their answer choice. We propose expanding the array of research tasks to access different knowledge that students might have. By asking students to discuss answers…
This paper argues that, insofar as we doubt the bivalence of the Continuum Hypothesis or the truth of the Axiom of Choice, we should also doubt the consistency of third-order arithmetic, both the classical and intuitionistic versions.…
In this article we describe special type of mathematical problems that may help develop teaching methods that motivate students to explore patterns, formulate conjectures and find solutions without only memorizing formulas and procedures.…
We report an inconsistency found in probability theory (also referred to as measure-theoretic probability). For probability measures induced by real-valued random variables, we deduce an "equality" such that one side of the "equality" is a…