Knowing-How and the Deduction Theorem

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Andrei Rodin, Vladimir Krupski
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preprint description
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 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. Finally we extend our analysis to the case of extra-logical knowledge-how representable with Gentzen-style formal systems, which admit constructive meaning explanations. For this end we build a typed sequential calculus and prove for it a ``constructive'' Deduction Theorem interpretable in extra-logical terms. We conclude with a number of open questions, which concern translations between knowledge-how and knowledge-that in this more general semantic setting.

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