By Irving M. Copi

This reissue, first released in 1971, presents a quick ancient account of the speculation of Logical forms; and describes the issues that gave upward thrust to it, its a variety of diverse formulations (Simple and Ramified), the problems attached with each one, and the criticisms which have been directed opposed to it. Professor Copi seeks to make the topic available to the non-specialist and but offer a sufficiently rigorous exposition for the intense scholar to work out precisely what the idea is and the way it really works.

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Thus the order can be ‘recaptured’ from the abstract by our defining convention that the first element of the ordered pair is the element which belongs to both classes contained in the class {{x}, {x, y}}. The ordered pair is {{y}, {y, x}}, clearly different from {{x}, {x, y}}, which is the ordered pair The ordered n-tuple were it needed, could be defined analogously as {{x1}, {x1, x2}, …, {x1, x2, …, xn}}. The usual class operations of ordinary Boolean Algebra are easily defined as follows: 28â•… The Theory of Logical Types And the empty classes for types n=1, 2, 3, …are defined by: DΛ: Λn for {xn−1:~(xn−1=xn−1)}.

03) will be adopted for as € This is a single axiom rather than an axiom schema. No typical ambiguity occurs here, nor is any needed, because if type 0 contains infinitely many individuals, type (0) will contain infinitely many classes of individuals, and so on. Hence the full array of natural numbers 0, 1, 2, 3, …exist as distinct classes in every type from ((0)) on up. There have been many attempts to prove, rather than to postulate, the existence of infinite classes. Several such putative proofs were accepted as valid by Russell in 1903 (Russell 1903, 357–8), but were rejected by him in Principia Mathematica (Whitehead and Russell, 1910, II, 183).

A separate axiom is required to guarantee the existence of a class which will contain exactly one member from each of the classes F1, F2, F3, …whose cardinal numbers are to be multiplied. Because of its connection with the multiplication of cardinal numbers, such an axiom is sometimes called the ‘multiplicative principle’. But because it specifies a class which in effect is produced by choosing one element from each of a set of classes, the axiom is more often called the ‘axiom of choice’. There are other propositions to which the axiom is logically equivalent, for example, that any two cardinal numbers are either equal or one is larger than the other, or the proposition that any class can be well-ordered (Lemmon, 1968, 111–20).

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