Knowing-How and the Deduction Theorem Vladimir Krupski ∗1 and Andrei Rodin y2 1Department of Mechanics and Mathematics, Moscow State University 2Institute of Philosophy, Russian Academy of Sciences and Department of Liberal Arts and Sciences, Saint-Petersburg State University July 25, 2017 Abstract: In his seminal address delivered in 1945 to the Royal Society Gilbert Ryle considers a special case of knowing-how, viz., knowing how to reason according to logical rules. He argues that knowing how to use logical rules cannot be reduced to a propositional knowledge. We evaluate this argument in the context of two different types of formal systems capable to represent knowledge and support logical reasoning: Hilbert-style systems, which mainly rely on axioms, and Gentzen-style systems, which mainly rely on rules. We build a canonical syntactic translation between appropriate classes of such systems and demonstrate the crucial role of Deduction Theorem in this construction. This analysis suggests that one’s knowledge of axioms and one’s knowledge of rules ∗
[email protected] [email protected] ; the author thanks Jean Paul Van Bendegem, Bart Van Kerkhove, Brendan Larvor, Juha Raikka and Colin Rittberg for their valuable comments and discussions and the Russian Foundation for Basic Research for a financial support (research grant 16-03-00364). 1 under appropriate conditions are also mutually translatable. However our further analysis shows that the epistemic status of logical knowing- how ultimately depends on one’s conception of logical consequence: if one construes the logical consequence after Tarski in model-theoretic terms then the reduction of knowing-how to knowing-that is in a certain sense possible but if one thinks about the logical consequence after Prawitz in proof-theoretic terms then the logical knowledge- how gets an independent status.