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# MODERN LOGIC: FROM FREGE TO GÖDEL: RAMSEY

Frank Plumpton Ramsey (19031930), a brilliant Cambridge philosopher and logician, attempted to give a satisfactory account of the foundations of mathematics in accordance with the method of Frege, Russell, and Whitehead, defending their view that mathematics is logic while proposing revisions in the system of Principia Mathematica suggested by the work of Wittgenstein.

According to Russell, pure mathematics consists of "the class of all propositions of the form 'p implies q ' where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants" (The Principles of Mathematics, p. 3). Ramsey agreed with this definition insofar as it characterizes the generality that is a feature of pure mathematics, but he claimed that it takes no account of an equally important mark of mathematics, its tautological character. The term tautological in the relevant sense derives from Wittgenstein, who applied it to formulas of the propositional calculus which come out true no matter what combinations of the values true and false are assigned to the component propositions. Ramsey extended the term to apply to valid formulas of the predicate calculus. Thus, the formula "(x ). ϕx : : ϕa " is tautological, since "ϕa " expresses one of the possibilities which go to make up the possibly infinite conjunction abbreviated by "(x ). ϕx."

Admittedly we cannot write down the fully expanded versions of quantified formulas, but this inability does not affect the tautological character of truths formulated in the compressed notation. Similarly, Ramsey maintained, the inability of human beings to list the members of an infinite class is no bar to our conceiving of classes whose members could be indicated only in this way and not via the specification of a defining predicate. Indeed, the possibility of such indefinable classes is an essential part of the extensional attitude of modern mathematics, and Ramsey regarded the neglect of this possibility in Principia Mathematica as one of the work's three major defects. Thus, as interpreted in the system of Principia Mathematica the multiplicative axiom (axiom of choice) is logically doubtful, but on an extensional view of classes it is, according to Ramsey, an evident tautology.