Non-Arbitrary Termination: Recursive Base Cases and the Third Horn of the Agrippan Trilemma

Day and Athos (2026) identified a structural amphiboly in the third horn of the Agrippan Trilemma: the conflation of “terminates” with “terminates arbitrarily.” This paper demonstrates that the class of non-arbitrary termination revealed by the amphiboly is not merely a theoretical possibility but a ubiquitous feature of computational and mathematical practice. Recursive algorithms terminate at base cases whose warrant derives from the mathematical structure of the problem, not from further chain-extension. These base cases satisfy Reading A (the process terminates) without satisfying Reading B (the termination is arbitrary). Recursive computation has operated outside the scope of the Trilemma’s third horn since the foundational work of Church and Turing in the 1930s, providing nearly a century of empirical evidence that non-arbitrary termination is achievable. The Trilemma’s failure to account for structurally warranted termination is therefore not a narrow philosophical technicality but a defect that rendered an entire domain of human knowledge invisible to epistemological analysis. This paper examines the structural parallel between recursive base cases, transcendental arguments, and the gap in the Trilemma’s trichotomy, and argues that the identification of the amphiboly retroactively explains why computational epistemology and classical epistemology have failed to make contact for nearly a century.