Beyond FOL. Where to stop?

This thesis explores extensions of FOL from a contemporary model-theoretic perspective, with particular emphasis on the criteria for the logicality of logical constants and the interaction between syntactic and semantic aspects in extended logical systems. Following a comprehensive review of the foundational concepts of FOL, including its intrinsic limitations, such as its inability to express cardinality constraints and the failure of compactness, the study examines skeptical perspectives on model theory. In this context, epistemological and doxological challenges are addressed by revisiting a refined version of the revenge argument proposed in Button and Walsh (2018) and contrasting it with the counterarguments advanced in Speitel (2024). Further analysis explores the notion of logicality from two perspectives. First, it examines the criterion introduced by Sagi (2018), which is based on set-theoretic weight measured via the Löwenheim number. Second, it investigates logicality through definability within the framework of Kennedy and Väänänen (2021), thereby highlighting distinctions from Sagi’s approach. In its final section, the study utilizes arguments grounded in metaphysical weight and the Löwenheim number to critically assess the bottom-up strategies developed by Paseau and Griffiths (2022). A refined version of McGee’s theorem for the $\Delta$-extension of $L_{\infty\omega}$ is established, revealing a fundamental asymmetry between bottom-up and top-down approaches. This imbalance not only disrupts the anticipated convergence between these methodologies but also opens the door to a non-trivial pluralism in the selection of logical constants.