Learning Logic: A Mixed Methods Study to Examine the Effects of Context Ordering on Reasoning About Conditionals

Logical statements are prevalent in mathematics, the sciences, law, and many areas of everyday life. The most common logical statements are conditionals, which have the form “If H..., then C...,” where “H” is a hypothesis (or condition) to be satisfied and “C” is a conclusion to follow. Reasoning about conditionals is a skill that is only superficially understood by most individuals and depends on four main conditional contexts (e.g., intuitive, abstract, symbolic, or counterintuitive). The purpose of this study was to test a theory about the effects of context ordering on reasoning about conditionals. To test the theory, the researcher developed, tested, and revised a virtual manipulative educational mathematics application, called the Learning Logic App. This study employed a convergent parallel mixed methods design to answer an overarching research question and two subquestions. The overarching research question was “How does the order of teaching four conditional contexts influence reasoning about conditionals?” The two subquestions examined this influence on reasoning in terms of performance and perceptions. This study involved two phases. During Phase I, 10 participants interacted with the Learning Logic App in a clinical setting. The researcher used information gathered in Phase I to revise the Learning Logic App for Phase II. During Phase II, 154 participants interacted with the Learning Logic App in a randomly assigned context ordering in an online setting. In both phases, the researcher collected quantitative and qualitative data. After independent analyses, the researcher made meta- inferences from the two data strands. The results of this study suggest that context ordering does influence learners’ reasoning. The most beneficial context ordering for learners’ performance was symbolic-intuitive-abstract-counterintuitive. The most beneficial context ordering for learners’ perceptions was intuitive-abstract-counterintuitive-symbolic. Based on these results, the researcher proposed a new context ordering: symbolic-intuitive-abstract-counterintuitive-symbolic. This progression incorporates a catalyst at the beginning (symbolic context) which aids the learner in reassessing their prior knowledge. Then, the difficulty of the contexts progresses from easiest to hardest (intuitive-abstract-counterintuitive-symbolic). These findings are important because they provide an instructional sequence for teaching and learning to reason about conditionals that is beneficial to both learners’ performance and their perceptions

Shape Analysis via Monotonic Abstraction

We propose a new formalism for reasoning about dynamic memory heaps, using monotonic abstraction and symbolic backward reachability analysis. We represent the heaps as graphs, and introduce an ordering on these graphs. This enables us to represent the violation of a given safety property as the reachability of a finitely representable set of bad graphs. We also describe how to symbolically compute the reachable states in the transition system induced by a program

Invariance properties of maximal extractable value

We develop a formalism for reasoning about trading on decentralized exchanges on blockchains and a formulation of a particular form of maximal extractable value (MEV) that represents the total arbitrage opportunity extractable from on-chain liquidity. We use this formalism to prove that for blockchains with deterministic block times whose liquidity pools satisfy some natural properties that are satisfied by pools in practice, this form of MEV is invariant under changes to the ordering mechanism of the blockchain and distribution of block times. We do this by characterizing the MEV as the profit of a particularly simple arbitrage strategy when left uncontested. These results can inform design of blockchain protocols by ruling out designs aiming to increase trading opportunity by changing the ordering mechanism or shortening block times.Comment: 39 page

COVID-19, Serendipity, and Strange Interlude: Gloria in Absurdicum with an Apology to Ovid

The absurd edicts initially issued by several governors, including those of Michigan and New York, prohibiting physicians from ordering hydroxychloroquine for patients suspected of or diagnosed with COV-19 are reprehensible. Their reasoning is beyond the pale. Hydroxychloroquine was first approved by the FDA in 1955 and has a remarkable safety record in its use as a preventative for malaria and in the treatment of several immunologic and/or light induced diseases. Its use came about serendipitously, and its mechanism of action is unclear, but the fact remains that it seems to work

Development and individual differences in transitive reasoning: A fuzzy trace theory approach

Fuzzy trace theory explains why children do not have to use rules of logic or premise information to infer transitive relationships. Instead, memory of the premises and performance on transitivity tasks is explained by a verbatim ability and a gist ability. Until recently, the processes involved in transitive reasoning and memory of the premises were studied by comparing mean performance in fixed-age groups. In this study, an individual-difference model of fuzzy trace theory for transitive reasoning was formulated and tested on a sample (N = 409) of 4- to 13-year-old children. Tasks were used which differed with respect to presentation ordering and position ordering. From this individual- difference model expectations could be derived about the individual performance on memory and transitivity test-pairs. The multilevel latent class model was used to fit the formalized individual-difference fuzzy trace theory to the sample data. The model was shown to fit the data to a large extent. The results showed that verbatim ability and gist ability drove the activation of verbatim and gist traces, respectively, and that children used combinations of these traces to solve memory tasks (testing memory of the premises) and transitivity tasks. Task format had a stronger effect on transitivity task performance than on memory of the premises. Development of gist ability was found to be faster than development of verbatim ability. Another important finding was that some children remembered the premise information correctly but were not able to infer the transitive relationship, even though the premises provided all the necessary information. This contradicts Trabasso’s linear ordering theory which posits that memory of the premises is sufficient to infer transitive relationships