Axiom of Reducibility .sx For every concept F i ( x ) of objects of type n , for i> n , there exists an extensionally equivalent concept formula of type n+1 , i.e. a concept formula such that formula .sx This axiom asserts that increasingly complex definitions do not give any new powers of classification - all classes of individuals can be marked out with basic concepts of type 1 , and all classes of objects of type n can be marked out by concepts of type n+1 .sx This means that when using a version of Frege's definition of 'natural number' but with its quantifier limited to ranging only over concepts whose type is next above that of the numbers being defined one will have achieved almost the same effect as quantifying over all concepts , of whatever type , since one will have quantified over all the classes that they could delimit .sx This is again a principle which has to be admitted not to be a purely logical principle .sx It is not something which can be proved , or even , strictly speaking , formulated as a logically legitimate statement , for it involves generalizing over concepts of more than one type and is thus not itself logically correct .sx Russell is by this means able to avoid the inconsistencies of Frege's system and retain many of the essentials of his account of arithmetic but only at a price and only in a way which must lead one to ask whether he has not demonstrated that and why arithmetic is not reducible to logic , rather than that it is .sx To evaluate the position further it is necessary to inquire into Russell's justification for adopting the Vicious Circle Principle .sx EMPIRICISM , LOGICAL POSITIVISM AND THE STERILITY OF REASON .sx If one were to adopt a strongly Platonist position , saying that numbers , classes , concepts and functions have an existence which is independent of us and our mathematical activities , then the Vicious Circle Principle could not be justified as a general logical principle .sx A definition such as .sx Ben Cullin unch the highest mountain in the Hebrides .sx which specifies the reference of a name by quantifying over a class in order to select an individual member from that same class , violates the Vicious Circle Principle .sx Nonetheless , provided that we are confident that there must be a highest mountain in the Hebrides ( there are mountains there , all mountains are comparable with respect to height and it is extremely unlikely that there are two of exactly the same height ) , the definition would normally be regarded as legitimate , even though it does not necessarily provide sufficient information to enable someone to identify the mountain in question .sx This definition would be regarded as legitimate because the totality by reference to which the name is defined exists as a determinate collection of objects prior to and independently of the definition .sx The definition does not introduce a new object to add to the totality in question , but names an already existing member .sx For the Platonist all definitions of ( words referring to ) numbers , classes , concepts and functions will be of this kind ; they are definitions whose function is to link linguistic expressions with pre-existing entities .sx So for the Platonist , violation of the Vicious Circle Principle cannot , in and of itself , invalidate a definition .sx This means that he must look elsewhere for a solution to Russell's paradox .sx One route available to him is to treat Russell's paradox as proving two truths about classes :sx ( i ) no class belongs to itself , and ( ii ) there is no class of all classes .sx Alternatively he could treat it as indication of a need to distinghish between classes and sets ; mathematicians deal with sets and the nature of the set-theoretic universe is captured by the axioms of set theory whose truth is recognized by set-theoretic intuition .sx These routes are available , but they represent a departure from the logicist programme .sx Once appeal to set theoretic intuition is allowed it is not clear why intuition of numbers and geometric intuition should not also be allowed .sx Russell's advocacy of logicism and his reasons for thinking the Vicious Circle Principle to be a general logical principle are , on the other hand , both grounded in his empiricism .sx His early , more rationalist , philosophy gave way to empiricism at the same time that his account of mathematics became more closely tied to formal logic .sx For empiricists the only reality is the empirical world , the world with which we are acquainted through sense experience .sx Abstract objects , such as numbers or classes , have no independent existence but must be a product of our linguistic or mental constructions .sx Since there is no realm of mathematical reality of which to have mathematical intuitions , true mathematical statements , if they can be known to be true independently of experience , must be analytic truths , having their origin in the way in which the abstract objects are constructed .sx As Russell concurred with Frege's desire to separate sharply between the logical and the psychological , he too shied away from any appeal to mental constructions .sx The objectivity of mathematics requires that the meaning of mathematical language cannot be given by reference to ideas or mental constructions .sx Rather the meaning of all expressions which apparently refer to abstract objects must be shown , by the provision of suitable definitions , to be logical constructs ( fictions ) built up from constituents of the empirical world .sx The assignment of meaning to names can then take one of two forms .sx Either the name is simply a label for an empirically given object , in which case Russell called it a logically proper name , one which has a reference but no sense , or it is a descriptive expression ( a definite description ) , which identifies an object via its relations to other given objects , or via its mode of construction out of given objects .sx The only kind of definition which can be provided for a logically proper name is an ostensive one - the label is attached by pointing to the object and uttering the name .sx Thus only empirically given entities can have logically proper names .sx Here Russell disregards Frege's injunction to treat a name as having meaning only in the context of a sentence ; logically proper names can function in isolation .sx Definite descriptions , on the other hand , are always defined in a sentential context .sx Such expressions are required either for picking out empirical individuals via their relation to other empirical individuals , or for showing how new entities can be constructed out of those which are empirically given or have previously been defined .sx In the former case one will be using a description to pick out an object from a given class , but in the latter one will not be picking out an already existing entity since one will be introducing a logical fiction .sx The two cases need to be treated differently if the Vicious Circle Principle is not to seem unduly and unjustifiably restrictive .sx Definite descriptions are treated as having meaning only in the context of a sentence .sx According to Russell's theory of definite descriptions .sx Ben Cullin is the highest mountain in the Hebrides unch .sx There is a highest mountain in the Hebrides , there is at most one and it is Ben Cullin .sx Here there is quantification over the domain to which the descriptively identified object belongs , but the definition makes it quite clear that the function of the descriptive phrase is to pick out an object from that domain and that no object is being added to it .sx Moreover , the descriptive phrase does not have to have a reference for the sentence containing it to make sense .sx The sentence is simply false if the descriptive identification fails .sx Logical fictions are constructed by collecting already defined objects into classes which are themselves regarded as mere fictions .sx To this end Russell gives contextual definitions showing how apparent reference to classes can be eliminated .sx For example , the simplest eliminations would be .sx 1 a belongs to the class of Fs unch F(a) .sx 2 the class of Fs = the class of Gs unch formula .sx In ( 2 ) the Vicious Circle Principle requires that the domain of quantification on the right hand side not include the class of Fs or of Gs .sx Moreover , it requires that the concepts F(x ) and G(x ) not themselves contain any quantification over a domain which includes themselves or a domain which includes entities of higher type than their arguments .sx This means that the actual contextual definitions used in Principia have to be somewhat more complicated .sx The basic idea behind the remaining clauses , which cover the cases where one wants to say things about classes or to form classes into further classes , is that every statement about a class is really a statement about its members ( since by ( 1 ) and ( 2 ) classes are identical when they have the same members) .sx If this is so , then an apparently simple statement about a class can , in principle , be written as a logically complex statement concerning its members .sx So , for example , .sx ' n = the number of Fs' becomes .sx 'the class of Fs belongs to the class of all n-membered classes' .sx which in turn reduces to .sx formula .sx It is this view of classes as in principle eliminable logical constructs which justifies the Vicious Circle Principle .sx An entity ( whether object or concept ) cannot be constructed out of itself , but only out of entities previously given or constructed .sx Similarly a verbal expression cannot be defined in terms of itself , but only from expressions which are given as primitive or have previously been defined .sx But if this is the justification for the Vicious Circle Principle , then the introduction of the Axiom of Reducibility looks even more embarrassing than it seemd at first sight .sx Not only is it a non-logical assumption needed for the derivation of arithmetic , and hence an admission of failure in the logicist programme , but , being an existential axiom , it suggests a return to some form of Platonism .sx This would undercut the Vicious Circle Principle which formed the foundation of Russell's theories of logical types which in turn represented his solution to the paradoxes .sx This is essentially the argument presented by G o del ( 1944) .sx If we are entitled to assume the existence of concepts independently of their definition , as the Axiom of Reducibility seems to assert , then , as outlined above , there is no good reason for thinking the Vicious Circle Principle to be a general logical principle .sx However , it is also possible to read the Axiom of Reducibility as an assertion of faith in the logicist programme in mathematics , and more generally of the philosophical position of the logical positivists .sx This does not wholly exonerate Russell .sx There is a curious circularity in having the success of a position rest on an assertion to the effect that it can be successfully carried out .sx But , in the first place , it is not clear that any philosophy can ever avoid this kind of circularity and , in the second , it will at least defend Russell against the charge of having introduced an element which renders his position philosophically incoherent .sx The position of the logical positivists , unlike that of some earlier empiricists , such as Berkeley and Hume , is not phenomenalist .sx The logical positivists had , as Russell ( 1919 p.170 ) said , " a robust sense of reality " , i.e. a belief that the empirical world exists independently of us and our experiences - we do not live only in a world of ideas .sx There is a sharp distinction to be drawn between Hamlet and Napoleon , between unicorns and lions .sx The former are fictions , existing only as ideas , whereas the latter are empirically real and have an existence beyond our ideas .sx In this respect their position has more in common with Locke's empiricism than with Hume's .sx Sense experience affords genuine knowledge of the empirical world and is our only route to knowledge of it .sx Since this is the only reality of which we have an experience , empirical reality is the only reality which we have any basis for supposing to exist independently of us .sx ( For further discussion and detailed arguments concerning the grounds for belief in the existence of an external world see , for example , Russell ( 1912) .sx ) The only objective knowledge there can be is factual knowledge of this independently existing reality .sx