The \(\lambda \) -Calculus as a Solution to a Problem of Philosophical Analysis

In his remarkable but scarcely noticed paper “Logic and Analysis” ( \(=\) L&A), Church explains what counts as a formally correct and materially adequate solution to a problem of philosophical analysis. In this chapter it is shown that according to the L&A methodology, the \(\lambda \) -calculus may be considered as Church’s solution to the problem of a philosophical analysis of functions. L&A establishes that problems of philosophical analysis are often problems of applied logic in the sense of formulating some system of ideas in coherent logical form. Usually, the ideas in question are already known to common sense. However, the common-sense formulation requires amendment and supplementation, in order to remove uncertainties and perplexities. In the case of the (non-extensional) concept of function-as-rules, the common-sense formulation corresponds to the traditional (or Euler’s) notation for functions. And there is a double ambiguity found in this traditional approach, one related to first-order functions (the Church-Kleene examples) and one related to higher-order functions (the Curry-Feys example) as well as Wittgenstein’s question on “ \(f(f)\) ”. In the chapter it is shown that the \(\lambda \) -calculus qualifies as an adequate solution to the problem of a philosophical analysis of functions according to L&A and Quine’s conceptions of philosophical analysis.