Bealer’s Intensional Logic

Por • 31 dic, 2020 • Sección: Ambiente

Clarence Protin

Many intuitively valid arguments involving intensionality cannot be captured by first-order logic, even when extended by modal and epistemic operators. Indeed, previous attempts at providing an adequate treatment of the phenomenon of intensionality in logic and language, such as those of Frege, Church, Russell, Carnap, Quine, Montague and others are fraught with numerous philosophical and technical difficulties and shortcomings. We present Bealer’s solution to this problem which hinges on an ontological commitment to theory of Properties, Propositions and Relations (PRP). At the most basic level we can distinguish two conceptions in the theory of PRPs. An objective one tied to modality and necessary equivalence, and a mental (intentional) one tied to concepts and the requirement of non-circularity in definitions. Building on the work of Russell, Church and Quine, Bealer proposes two distinct intensional logics T1 and T2 (presented in Hilbert form) corresponding to these two conceptions, both based on the language of first-order logic extended with an intensional abstraction operator. In T1 necessitation can be directly defined and the axioms entail that we obtain standard S5 modal logic. These logics have a series of striking features and desirable aspects which set them apart from higher-order approaches. Bealer constructs a non-Tarskian algebraic semantic framework, distinct from possible worlds semantics and its problematic ontological commitments, yielding two classes of models for which T1 and T2 are both sound and complete.

arXiv:2012.09846v2 [math.LO]

Logic (math.LO)

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