Version: v1, Published online: 1998
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8. Problems with ramified types
Russell’s reluctance to embrace the ramified theory of types was well-motivated, for it raised serious obstacles to logicism. In the first place, Russell’s definition of a cardinal number as the class of all similar classes is no longer admissible. There is no longer a class of all similar classes, but only a class of all similar first-order classes, a class of all similar second-order classes, and so on. (Strictly, of course, the definition will proceed in terms of the hierarchy of functions, but the class terminology is more familiar and admissible as a façon de parler.) There then ceases to be a unique cardinal 2, for example, but a different 2 at each level of the hierarchy; ordinary talk about 2 is typically ambiguous between these different cardinals. Similar problems affect the statement of logical principles which, strictly, would need to be stated for each order. In Principia this is avoided by the use of free variables which are typically ambiguous. Thus ⊢. φx ∨∼φx asserts the law of excluded middle for monadic propositions of undetermined order; whereas ⊢. (φ). φx ∨∼φx asserts it for those of some particular order. These results are untoward but not intolerable.
A similar, but more troubling difficulty threatened Russell’s treatment of the continuum, for on the ramified theory the least upper bound of a bounded set of real numbers is a number of a higher level. The most serious difficulty, however, concerns the principle of mathematical induction: namely, that any function satisfied by 0, and satisfied by x + 1 if satisfied by x, is satisfied by all natural numbers. These two difficulties threaten the logicist programme for transfinite and finite arithmetic respectively. Russell surmounted them by making the assumption that for every nonpredicative function there is an equivalent predicative function in the same variables. This is his ‘axiom of reducibility’: (∃f) (φx ≡x f ! x) (1910–13: 12.1).
It has often been objected that the axiom of reducibility undoes all the good that ramifying type theory achieved, on the grounds that it renders the ramified theory equivalent to the simple theory, thereby readmitting all the paradoxes ramification was designed to prevent (Ramsey 1925; Copi 1950). Myhill (1979) shows that this objection is ill-founded.
Frank Ramsey (1925) proposed that the paradoxes should be divided into those which belong to logic and set theory and those which are semantical or linguistic and that logic need solve only the former. Since the simple theory of types solves the logical paradoxes, the ramified theory is not needed in logic. This suggestion has been very influential. However, it still leaves the semantic paradoxes to be dealt with and Ramsey’s demarcation between the two kinds is hardly adequate. His suggestion that the semantic paradoxes are in some way empirical is neither clear nor plausible. Moreover, the fact that both types of paradox arise from self-reference suggests a uniform treatment. Above all, Ramsey’s distinction does little justice to Russell’s appealing intuition that the same underlying logic should be applied universally. It is clear now that semantics is capable of just as rigorous, formal treatment as set theory and the suggestion that the proper formal treatment of each requires different logical principles lacks prima facie plausibility (see Semantic paradoxes and theories of truth §2).
Under the influence of Wittgenstein, Russell came to accept the view that logical propositions were all tautologies. From this point of view the axiom of reducibility is clearly unsatisfactory (Wittgenstein 1922). From much the same point of view, Ramsey argued that it was not only contingent but a contingent falsehood – though it is not clear that his counterexample is a good one (Ramsey 1925). In the second edition of Principia, therefore, Russell tried to dispense with the axiom. He conceded that much of transfinite arithmetic would be lost as a result, but hoped none the less to save the theory of natural numbers by proving a version of the principle of mathematical induction within the constraints of the ramified theory and without the axiom of reducibility. Gödel (1944) noted the flaw in the proof and Myhill (1974) has since shown it to be irremediable. It follows, therefore, that even finite arithmetic cannot be logicized within the constraints of the ramified theory without the axiom of reducibility.
Griffin, Nicholas. Problems with ramified types. Russell, Bertrand Arthur William (1872–1970), 1998, doi:10.4324/9780415249126-DD059-1. Routledge Encyclopedia of Philosophy, Taylor and Francis, https://www.rep.routledge.com/articles/biographical/russell-bertrand-arthur-william-1872-1970/v-1/sections/problems-with-ramified-types.
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