Infinite Reasoning

Philosophy and Phenomenological Research 103 (2):385-407 (2021)
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Our relationship to the infinite is controversial. But it is widely agreed that our powers of reasoning are finite. I disagree with this consensus; I think that we can, and perhaps do, engage in infinite reasoning. Many think it is just obvious that we can't reason infinitely. This is mistaken. Infinite reasoning does not require constructing infinitely long proofs, nor would it gift us with non-recursive mental powers. To reason infinitely we only need an ability to perform infinite inferences. I argue that we have this ability. My argument looks to our best current theories of inference and considers examples of apparent infinite reasoning. My position is controversial, but if I'm right, our theories of truth, mathematics, and beyond could be transformed. And even if I'm wrong, a more careful consideration of infinite reasoning can only deepen our understanding of thinking and reasoning. (Note for readers: the paper's brief discussion of uniform reflection and omega inconsistency is misleading. The imagined interlocutor's argument makes an assumption about the PA-provability of provability generalizations that, while true for the Godel sentence's instances, is unjustified, in general. This means my position is stronger against this objection than the paper suggests, since omega inconsistent theories are not automatically inconsistent with their uniform reflection principles, you also need to assume the arithmetically true Pi-2 sentences.)



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Jared Warren
Stanford University

Citations of this work

Functionalism About Inference.Jared Warren - forthcoming - Inquiry: An Interdisciplinary Journal of Philosophy.
Carnap's philosophy of mathematics.Benjamin Marschall - 2022 - Philosophy Compass 17 (11):e12884.
Carnap and Beth on the Limits of Tolerance.Benjamin Marschall - 2021 - Canadian Journal of Philosophy 51 (4):282–300.

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Counterfactuals.David Lewis - 1973 - Philosophy of Science 42 (3):341-344.
What is inference?Paul Boghossian - 2014 - Philosophical Studies 169 (1):1-18.
Wittgenstein on rules and private language.Saul A. Kripke - 1982 - Revue Philosophique de la France Et de l'Etranger 173 (4):496-499.

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