Is Russell's Paradox Genuine?

Copi, Quine and van Heijenoort have each claimed that there are two fundamentally different kinds of logical paradox; namely, genuine paradoxes like Russell's and pseudo-paradoxes like the Barber of Seville. I want to contest this claim and will present my case in three stages. Firstly, I will characterize the logical paradoxes; state standard versions of three of them; and demonstrate that a symbolic formulation of each leads to a formal contradiction. Secondly, I will discuss the reasons Copi, Quine and van Heijenoort have given for the distinction between genuine and pseudo-paradoxes. Thirdly, I will attempt to explain why there is no such class as the class of all and only those classes which are not members of themselves.

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Paradoxes of set and property

Routledge Encyclopedia of Philosophy ◽

10.4324/9780415249126-y024-1 ◽

2018 ◽

Author(s):

Gregory H. Moore

Keyword(s):

Cardinal Number ◽

Liar Paradox ◽

Semantic Paradoxes ◽

Russell’S Paradox ◽

Medieval Europe ◽

Power Set ◽

Semantic Concepts ◽

Logical Paradox ◽

Russell's Paradox ◽

Logical Paradoxes

Emerging around 1900, the paradoxes of set and property have greatly influenced logic and generated a vast literature. A distinction due to Ramsey in 1926 separates them into two categories: the logical paradoxes and the semantic paradoxes. The logical paradoxes use notions such as set or cardinal number, while the semantic paradoxes employ semantic concepts such as truth or definability. Both often involve self-reference. The best known logical paradox is Russell’s paradox concerning the set S of all sets x such that x is not a member of x. Russell’s paradox asks: is S a member of itself? A moment’s reflection shows that S is a member of itself if and only if S is not a member of itself – a contradiction. Russell found this paradox by analysing the paradox of the largest cardinal. The set U of all sets has the largest cardinal number, since every set is a subset of U. But there is a cardinal number greater than that of any given set M, namely the cardinal of the power set, or set of all subsets, of M. Thus the cardinal of the power set of U is greater than that of U, a contradiction. (The paradox of the largest ordinal, discussed below, is similar in structure.) Among the semantic paradoxes, the best known is the liar paradox, found by the ancient Greeks. A man says that he is lying. Is what he says true or false? Again, either conclusion leads to its opposite. Although this paradox was debated in medieval Europe, its modern interest stems from Russell, who placed it in the context of a whole series of paradoxes, including his own.

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