In Bernard Linsky & Nicholas Griffin (eds.), The Palgrave Centenary Companion to Principia Mathematica. Palgrave-Macmillan. pp. 218-246 (2013)

Kevin Klement
University of Massachusetts, Amherst
Along with offering an historically-oriented interpretive reconstruction of the syntax of PM ( rst ed.), I argue for a certain understanding of its use of propositional function abstracts formed by placing a circum ex on a variable. I argue that this notation is used in PM only when de nitions are stated schematically in the metalanguage, and in argument-position when higher-type variables are involved. My aim throughout is to explain how the usage of function abstracts as “terms” (loosely speaking) is not inconsistent with a philosophy of types that does not think of propositional functions as mind- and language-independent objects, and adopts a nominalist/substitutional semantics instead. I contrast PM’s approach here both to function abstraction found in the typed λ-calculus, and also to Frege’s notation for functions of various levels that forgoes abstracts altogether, between which it is a kind of intermediary.
Keywords type theory  abstraction  propositional functions
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References found in this work BETA

Logic and Knowledge.BERTRAND RUSSELL - 1957 - Philosophical Quarterly 7 (29):374.
Mathematical Logic as Based on the Theory of Types.Bertrand Russell - 1908 - American Journal of Mathematics 30 (3):222-262.
On Concept and Object.Gottlob Frege - 1951 - Mind 60 (238):168-180.

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The Functions of Russell’s No Class Theory.Kevin C. Klement - 2010 - Review of Symbolic Logic 3 (4):633-664.

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