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1. The role of unitizing predicates in the construction of logic

   Dawkins, Paul Christian; Roh, Kyeong Hah (January 2022, Proceedings of the 12th Congress of the European Society for Research in Mathematics Education)

   Based on data from a teaching experiment with two undergraduate students, we propose the unitizing of predicates as a construct to describe how students render various mathematical conditions as predicates such that various theorems have the same logical structure. This may be a challenge when conditions are conjunctions, negative, involve auxiliary objects, or are quantified. We observe that unitizing predicates in theorems and proofs seemed necessary for students in our study to see various theorems as having the same structure. Once they had done so, they reiterated an argument for why contrapositive proofs proved their associated theorems, showing the emergence of logical structure. 

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2. The role of unitizing predicates in the construction of logic

   Dawkins, Paul Christian; Roh, Kyeong Hah (January 2022, Proceedings of the 12th Congress of the European Society for Research in Mathematics Education)

   Based on data from a teaching experiment with two undergraduate students, we propose the unitizing of predicates as a construct to describe how students render various mathematical conditions as predicates such that various theorems have the same logical structure. This may be a challenge when conditions are conjunctions, negative, involve auxiliary objects, or are quantified. We observe that unitizing predicates in theorems and proofs seemed necessary for students in our study to see various theorems as having the same structure. Once they had done so, they reiterated an argument for why contrapositive proofs proved their associated theorems, showing the emergence of logical structure. 

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3. Asynchronous Probabilistic Couplings in Higher-Order Separation Logic

   https://doi.org/10.1145/3632868  

   Gregersen, Simon Oddershede; Aguirre, Alejandro; Haselwarter, Philipp G.; Tassarotti, Joseph; Birkedal, Lars (January 2024, Proceedings of the ACM on Programming Languages)

   Probabilistic couplings are the foundation for many probabilistic relational program logics and arise when relating random sampling statements across two programs. In relational program logics, this manifests as dedicated coupling rules that, e.g., say we may reason as if two sampling statements return the same value. However, this approach fundamentally requires aligning or synchronizing the sampling statements of the two programs which is not always possible. In this paper, we develop Clutch, a higher-order probabilistic relational separation logic that addresses this issue by supporting asynchronous probabilistic couplings. We use Clutch to develop a logical step-indexed logical relation to reason about contextual refinement and equivalence of higher-order programs written in a rich language with a probabilistic choice operator, higher-order local state, and impredicative polymorphism. Finally, we demonstrate our approach on a number of case studies. All the results that appear in the paper have been formalized in the Coq proof assistant using the Coquelicot library and the Iris separation logic framework. 

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4. LINKING STRUCTURES ACROSS LOGIC AND SPACE: THE ROLE OF EULER DIAGRAMS

   Antonides, J; Norton, A; Arnold, R (July 2024, For the learning of mathematics)

   This theoretical article explores the affordances and challenges of Euler diagrams as tools for supporting undergraduate introduction-to-proof students to make sense of, and reason about, logical implications. To theoretically frame students’ meaning making with Euler diagrams, we introduce the notion of logico-spatial linked structuring (or LSLS). We argue that students’ use of Euler diagrams as representations of logical statements entails a conceptual linking between spatial and non-spatial representations, and the LSLS framework provides a tool for modeling this conceptual linking. Moreover, from our Piagetian epistemological perspective, reasoning with Euler diagrams entails engaging in spatial mental operations and making a logical conclusion from the result. We illustrate the utility of the LSLS framework through examples with two undergraduate students as they reasoned about the truth of the converse and contrapositive of a given logical implication, and we identify specific spatial operations that they used and coordinated in their problem solving. 

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5. Secondary Prospective Teachers’ Strategies to Determine Equivalence of Conditional Statements

   Buchbinder, O.; McCrone, S. (January 2020, Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)

   Karunakaran, S. S.; Reed, Z.; Higgins, A. (Ed.)

   Future mathematics teachers must be able to interpret a wide range of mathematical statements, in particular conditional statements. Literature shows that even when students are familiar with conditional statements and equivalence to the contrapositive, identifying other equivalent and non-equivalent forms can be challenging. As a part of a larger grant to enhance and study prospective secondary teachers’ (PSTs’) mathematical knowledge for teaching proof, we analyzed data from 26 PSTs working on a task that required rewriting a conditional statement in several different forms and then determining those that were equivalent to the original statement. We identified three key strategies used to make sense of the various forms of conditional statements and to identify equivalent and non-equivalent forms: meaning making, comparing truth-values and comparing to known syntactic forms. The PSTs relied both on semantic meaning of the statements and on their formal logical knowledge to make their judgments. 

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