Modern Logic: From Frege to Gödel: Ramsey

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MODERN LOGIC: FROM FREGE TO GÖDEL: RAMSEY

Frank Plumpton Ramsey (1903–1930), 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.

The second major defect that Ramsey found in Principia Mathematica concerns Russell's attempt to overcome the paradoxes, in particular his postulation of the axiom of reducibility. Ramsey accepted the simple theory of types as an unquestionably correct measure for avoiding the logical contradictions, such as Russell's paradox and the Burali-Forti paradox, but he claimed that the contradictions that the hierarchy of orders had been introduced to avoid are of no concern either to logic or to mathematics. These contradictions—for instance, the Richard paradox and Weyl's contradiction concerning the word heterological —cannot be stated in logical terms alone but contain some further reference to thought, language, or symbolism. Rejecting Russell's conception of orders, Ramsey put forward a less restrictive theory based on his extensional view of propositional functions. Just as "(x ). ϕx " represents an infinite conjunction of atomic propositions "ϕa. ϕb. · · · " so "(ϕ )ϕa " expands to "ϕ ₁a. ϕ ₂a. · · · " and similarly with disjunctions replacing conjunctions for existential quantifiers. Accordingly, if we start with truth-functions of atomic formulas, then no matter how often or in what respect we generalize upon them, we shall never pass to propositions significantly different from these elementary truth-functions; the only difference will lie in the notation introduced with the quantifiers. There is consequently no need for the axiom of reducibility—which, Ramsey claimed, could anyhow be false—and although the resultant theory countenances definitions of propositions in terms of totalities to which they belong, such definitions are in Ramsey's eyes no more vicious than an identification of a man as the tallest in a group of which he is a member.

The third great defect in Principia Mathematica which Ramsey proposed to rectify concerns Russell's definition of identity, according to which it is impossible for two objects to have all their properties in common. Ramsey held that this consequence shows that identity has been wrongly defined, and he advanced a definition of "x = y " designed to render the phrase tautological when x and y have the same value and contradictory otherwise.