Abstract Logics as classifications of abstract structures
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Abstract
Abstract logics are developed within a structuralist approach to the subject and disciplinary boundaries of logic. Nevertheless, within algebraic, proof-theoretic, and model-theoretic traditions, diverse interpretations of abstract logics are possible. Focusing on the model-theoretic tradition, this paper presents an alternative viewpoint that challenges the standard conception of abstract logics as structures. Instead, it argues for interpreting abstract logics as classifications of abstract structures (i.e., isomorphism types). From this perspective, the property of invariance under isomorphism ceases to function as a traditional criterion for distinguishing logical from non-logical terms—a criterion that has faced legitimate criticism in contemporary philosophy of logic. Invariance is rather treated as meta-constraints imposed on classes of structures, rendering it impossible to distinguish among structures belonging to the same isomorphism type using the tools of abstract model theory. The variability of such meta-constraints in early and modern model theories is examined. Two distinct ontological perspectives are compared: one perspective regards an abstract structure as a form that is common to all members of an isomorphism type, while the other perspective sees it as the isomorphism type itself, which can be represented by any of its tokens. The roles played by Edmund Husserl’s concept of definite manifold and Rudolf Carnap’s notion of model structure in shaping the metatheoretical apparatus of model theory are elucidated. Supporting evidence for interpreting definitive manifolds as isomorphism types is presented, alongside counterarguments highlighting specific shortcomings. Furthermore, the paper investigates a trichotomy within early model theory involving monomorphicity, deductive completeness, and non-forkability. Finally, prospects for advancing the proposed interpretation of abstract logics are explored, motivated by the diversity of criteria for structural similarity currently employed in model-theoretic research.
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