. Towards the end of the last century interest revived in the logical PARADOXES, from which the semantic ones were not yet distinguished. To cope with RUSSELL’S PARADOX and others, Russell enunciated the vicious circle principle: ‘If, provided a certain collection had a total, it would have members only definable in terms of that total, then the said collection has no total’, i.e. we cannot talk of the totality of its members. Classes form such a collection, for him. There are, he said, first-type, or first-level, classes whose members are ordinary objects, second-type classes whose members are first-type classes, and so on. The class of cats and the class of dogs are animal classes. They are first-type classes themselves, and are members of the second-type class of animal classes. There is a class of all classes of a given type which will itself be one type higher, but no class of all classes (see also CATEGORIES). Ordinary objects are of type zero. The hierarchy of types also applies to properties: A property of properties of objects belongs to the second type. Black is a property of some cats. It has the property of applying to some cats. Applying to some cats is therefore a second-type property of the first-type property black.
In ‘Napoleon had all the properties of a great general’, having all the properties of a great general is a property of Napoleon, and so is of the first type. But it refers to properties, so is said to be of the second order. It attributes to Napoleon only the relevant first-order properties (on this theory). The addition of the hierarchy of orders to that of types gives us the ramified as against simple theory of types. (But words like ‘second-order’ are normally used more loosely.) Propositions too are distinguished into orders. A proposition referring to no other propositions is of the first order. One referring to propositions of the first order (e.g. ‘Some first-order propositions are false’) is of the second order, and so on. The ramified theory was used to solve the semantic PARADOXES, e.g. the LIAR.
Since the ramified theory invalidates certain mathematical procedures, Russell introduced the controversial axiom of reducibility,saying that any higher-order property or proposition could be replaced by some first-order one.
Classes, etc., defined in ways violating the vicious circle principle are said to have impredicative definitions (for an example see Carnap, p. 37–8).
One disadvantage of the theory is that many words, e.g. ‘class’, ‘proposition’, ‘true’, become systematically, or ‘typically’, AMBIGUOUS, with different senses for each type or order.
R.Carnap, ‘The logicist foundations of mathematics’ in P.Benacerraf and H.Putnam (eds), Philosophy of Mathematics, Blackwell, 1964, pp. 31–41. (Elementary but illuminating.)
I.M.Copi, The Theory of Logical Types, RKP, 1971. (Fuller treatment.)
A.Whitehead and B.Russell, Principia Mathematica, Cambridge UP, vol. 1, 1910. (See Introduction, chapter 2, § 1, for vicious circle principle, repeated (though not there so called) in Russell’s Logic and Knowledge, Allen and Unwin, 1956, p. 63.)
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