The glory of classes, over against properties, is their clear individuation : they are identical if and only if they have the same members(flpovix)

The fact is that, if there are classes, classes are universals. This is sometimes obscured by speaking of classes as mere aggregates or collections(flpov114)

From Georg Cantor's theorem it follows that there must be classes, in particular classes of linguistic forms, having no linguistic forms corresponding to them(flpov121)

When classes are treated as constructions rather than discoveries it is a position of conceptualism as opposed to Platonic realism(flpov125)

There is little reason to distinguish classes from properties. if there is any difference between classes and properties, it is merely this: classes are the same when members are the same, whereas it is not universally conceded properties are the same when possessed by the same objects. Classes may be thought of as properties if the latter notion is so qualified that properties become identical when their instances are identical. For mathematics certainly, there is no need of countenancing properties in any other sense(ml120)

The usual way of specifying a class is by citing a necessary and sufficient condition for membership in it(ml128)

Individuals are distinguished from other classes by the peculiar circumstance of being their own sole members(ml135)

I(Willard Quine) have exploited the set-theoretic notation of class abstraction without assuming classes. Abstracts emerge with all the innocence of relative clauses. The device pays its way in the treatment of substitution, it clarifies the pronominal character of bound variables, and it eventuates in a philosophically agreeable perspective on classes(molvii)

The logical theory of classes, or set theory, proves to be the basic discipline of pure mathematics(mol5)

This dissociation from truth-function logic is accentuated by further departures in notation; thus 'F arrow G' and 'F double-headed arrow G' are not written, and instead of the alternation 'F v G' and conjunction 'FG' of terms we get 'F u G' and 'F n G', said to designate the union and intersection of classes(mol130)

There is philosophical comfort in the assurance that we can talk of logical validity and consistency without appealing to a limitless realm of abstract objects called classes(mol212)

A trend of the system is that a class exists unless it is too big. Consequently Zermelo cannot use the Fregean version of number; each number so construed, except 0, would be too big a class to exist. But the von Neumann version works very well(mol302)

There is no denying the access of power that accrues to our conceptual scheme through the positing of abstract objects. Most of what is gained by positing attributes, however, is gained equally by positing classes. Classes are on a par with attributes on the score of abstractness or universality, and they serve the purposes of attributes so far as mathematics and certainly most of science are concerned; and they enjoy, unlike attributes, a crystal-clear Identity concept. No wonder that in mathematics the murky intensionality of attributes tends to give way to the limpid extensionality of classes; and likewise in other sciences, roughly in proportion to the rigor and austerity of their systematization(or21)

Any such language is extensional; that is, coextensive predicates are interchangeable salva veritate. No true sentence in the language can become false when an occurrence of a predicate is supplanted by a coextensive predicate. The fictitious objects named by the predicates thus behave as classes rather than attributes(pol74)

The only difference between classes and attributes is that classes are taken always to be identical if they have the same members, whereas attributes are not always taken to be identical when they hold of exactly the same objects. A trouble with attributes is that we are never told, or anyway not in clear enough terms, what the further conditions of their identity might be. Talk of attributes does fit ordinary language more closely than talk of classes and I(Quine) think I know why. I suspect, as usual, a lingering tendency to confuse use and mention(rr102)

The old copula of predication, 'is a', has become 'is a member of' or epsilon(rr106)

A class is an abstract entity, a universal, even if it happens to be a class of concrete things. To the nominalist temper, accordingly, the elimination of classes in favor of expressions is a congenial objective; and this is what Russell is sometimes believed to have done in 1908 when he showed how contexts ostensibly treating of classes could be construed as abbreviations of other expressions wherein reference is made only to "propositional functions."(slp21)

Classes are clearer than attributes because they have relatively definite principle of individuation: they differ from one another just in case their members differ, whereas attributes (if they diverge from classes at all) differ from one another also under additional circumstances whose nature is left, in Principia, quite unspecified. Once classes have been introduced, attributes are scarcely mentioned again in the course of the three volumes. The clear course would have been to introduce the membership notation 'x E a' at an earlier point, then the notion of attribute, insofar as it diverges from that of class, would not occur(slp22)

The null class presupposes a distinction between classes and individuals which is inexpressible in terms of membership, since the null class is like an individual in lacking members(slp85)

It becomes imperative to distinguish between a Unit class and its member when the member is itself a class. For instance the empty class has no members whereas its unit class has one(slp269)

Classes are values of variables(slp275)

Virtual classes is a dummy set theory to begin with, dummy in that there are as yet no existence assumptions. There is just this axiom of extensionality(slp282)

The virtual theory of classes is a partial counterfeit of set theory fashioned purely of logic(stlix)

The virtual theory of classes eventually gets merged with the real theory in such a way as to produce a combination which, though not strictly more powerful than the real theory alone would be, is smoother in its running(stlix)

There are advantages (and disadvantages) in holding with John von Neumann and perhaps Cantor that not all classes are capable of being members of classes. In theories that hold this, the excess vocabulary has come in handy for marking the distinction; classes capable of being members are called sets(stl3)

Ultimate classes are not being members in turn of further classes. We can know this technical sense of 'set' and still use the terms 'set' and 'class' almost interchangeably. For the distinction emerges only in systems that admit ultimate classes, and even in such systems the classes we have to do with tend to be sets rather than ultimate classes until we get pretty far out(stl3)

The natural attitude on the question what classes exist is that any open sentence determines a class. Since this is discredited, we have to be deliberate about our axioms of class existence and explicit about our reasoning from them; Intuition is not in general to be trusted here(stl5)

The distinction between set and ultimate class is not to be confused with that between real and virtual class. Ultimate classes, for theories that admit them, are real: they belong to the universe of discourse, they are values of quantifiable variables. The virtual theory of classes, on the other hand, does not invoke classes as values of variables; it talks much as if there were classes, but explains this talk without assuming them(stl20)

Willard Quine's virtual classes are precisely Kurt Godel's "notions", apart from a couple points of style. One point is that unlike Gödel Quine talks of them as if of classes, subjecting them to class operations and the like(stl21)

A motive for talking thus ostensibly and eliminably of classes and relations is compactness of expression(stl26)

The acceptance of classes as members of further classes adds very significantly to what can be said about numbers and other mathematical objects(stl29)

What constitutes them individuals is not inclassitude, but identity with their unit classes (or, what comes to the same thing, identity with their own sole members)(stl32)

Even the many theorems which, because they are governed by universal rather than existential quantifiers, we can prove without axioms or premises of existence, would lose all potential content and interest if classes were excluded(stl38)

We are committed to there being at least one class. For we are assuming the classical logic of quantification, which, as is well known, assumes there to be at least one thing(stl38)

The notation of the theory of classes suffices for the arithmetic of natural numbers, and also suffices for further sorts of numbers(stl119)

A certain axiom schema may not imply there are Infinite classes; however, a certain axiom schema may state that given certain ones, other infinite classes must accompany them(stl164)

For belonging to a class, fulfilling the membership conditions is not enough; existence is required(stl170)

A class is a set, that is, if no bigger than some set; so John von Neumann. But in the end he settles on a stronger axiom of replacement, whereby a class is a set if and only if it is not as big as the class of all sets(stl311)

In the bipartite ontology of Principles of Mathematics, classes counted as things rather than as concepts; classes existed. Russell observed against Peano that "we must not identify the class with the class-concept," because of extensionality: classes with the same members are the same. Since the class was not the class-concept, Russell took it not to be a concept at all; hence it had to be a thing. But then, he felt, it ought to be no more than the sum of the things in it; and here was his problem of the one and the many. We saw that in 1905 Russell freed himself of Meinong's impossibles and the like by a doctrine of incomplete symbols. Classes were next(tt76)

There seems to be an unarticulated feeling that a class is best given by enumeration. For large classes this is impossible, granted; but still it is felt that you know a class only insofar as you know its members(tt106)

You specify a class not by its members but by its membership condition, its open sentence(tt107)

An ultimate class is a class that does not belong to any classes. In Mathematical Logic Quine called them nonelements. Often in the literature they are called proper classes. This perverse terminology has its etiology: they are classes proper as opposed to sets. But it is perverse; 'ultimate' is more suggestive(tt108)

Classes raise no perplexities over identity, being identical if and only if their members are identical. Not only is the use of classes instead of attributes, where possible, to be desired on account of the identity question. It is also important in that intensional abstraction is opaque whereas class abstraction is transparent(wo209)

Thus what disqualifies classes as objects of propositional attitudes is laxity of the condition of class identity: for two open sentences to determine the same class they need only be coextensive, i.e., fulfilled by the same values of the variable(wo211)

The efficacy of classes becomes more impressive when we find that they can be made to serve the purposes also of a great lot of further abstract objects of undeniable utility: relations, functions, numbers themselves(wo237)

Classes, if uncritically accepted, lead to absurdities. Yet the admission of classes as values of variables of quantification brings power that is not lightly to be surrendered(wo266)