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A definition of abstract logic is presented. This is used to explore and compare some abstract logics, such as logics with generalised quantifiers and infinitary logics, and their properties. Special focus is given to the properties of completeness, compactness, and the Löwenheim-Skolem property. A method of comparing different logics is presented and the concept of equivalent logics introduced. Lastly a proof is given for Lindström's theorem, which provides a characterization of elementary logic, also known as first-order logic, as the strongest logic for which both the compactness property and the Löwenheim-Skolem property, holds.

Learning to write formal mathematical proofs presents a major challenge to undergraduates. Students who have succeeded in algorithm-intensive courses such as calculus often find the abstract logic and nonprocedural nature of proof writing to be technically difficult, ambiguous and filled with potential errors and misconceptions. This mixed-methods study examines 23 undergraduate students’ attempts to write one-to-one and onto proofs in an introductory abstract algebra course. Data collected consisted of six rounds of assessments on one-to-one and onto proofs, including homework, quizzes and exams. Using an existing framework of undergraduate proof writing, the researchers found that students’ misconceptions and errors varied substantially by student and task, with one-to-one proofs presenting unique challenges. Implications for teaching and research include emphasis on the logic of proof approaches and providing structured proof frameworks to assist undergraduates with the procedural and conceptual challe...

The Craig Interpolation Theorem is intimately connected with the emergence of abstract logic and continues to be the driving force of the field. I will argue in this paper that the interpolation property is an important litmus test in abstract model theory for identifying "natural," robust extensions of first order logic. My argument is supported by the observation that logics which satisfy the interpolation property usually also satisfy a Lindström type maximality theorem. Admittedly, the range of such logics is small.