Are there objects that are “thin” in the sense that their existence does not make a substantial demand on the world? Frege famously thought so. He claimed that the equinumerosity of the knives and the forks suffices for there to be objects such as the number of knives and the number of forks, and for these objects to be identical. The idea of thin objects holds great philosophical promise but has proved hard to explicate. This book attempts to develop the needed explanations by drawing on some Fregean ideas. First, to be an object is to be a possible referent of a singular term. Second, singular reference can be achieved by providing a criterion of identity for the would-be referent. The second idea enables a form of easy reference and thus, via the first idea, also a form of easy being. Paradox is avoided by imposing a predicativity restriction on the criteria of identity. But the abstraction based on a criterion of identity may result in an expanded domain. By iterating such expansions, a powerful account of dynamic abstraction is developed.

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Some reasons to regard the cumulative hierarchy of sets as potential rather than actual are discussed. Motivated by this, a modal set theory is developed which encapsulates this potentialist conception. The resulting theory is equi-interpretable with Zermelo Fraenkel set theory but sheds new light on the set-theoretic paradoxes and the foundations of set theory.

English contains two sorts of object quantifiers. In addition to ordinary singular quantifiers, as in the sentence`There is a Cheerio in the bowl', there are plural quantifiers, as in`There are some Cheerios in the bowl'. In the 1980s, George Boolos showed how these plural quantifiers can be used to interpret monadic second-order logic (henceforth MSOL). 1 This is an indisputable technical result. However, as with most technical results, it is not obvious what its philosophical cash value is.According to Boolos, its cash value is that MSOL really is logic. In particular, the result is said to show that MSOL introduces no new ontological commitments: that no sets or classes or Fregean concepts are needed as values of the second-order variables, but that the entities already in the first-order domain suffice. Despite some isolated voices of dissent, 2 this philosophical interpretation of Boolos' technical result has become enormously popular. MSOL is now widely regarded as an important part of the philosopher's logical tool kit, of great value not only in the philosophy of mathematics but also in analytic metaphysics more generally. Most of all, Boolos' interpretation of MSOL has been hailed as a way of making great ontological bargains. According to its supporters, it allows us to pay the ontological price of a mere first-order theory and get the corresponding monadic second-order theory for free. 3 In this paper I will challenge this wide-spread view of the philosophical cash value of Boolos' technical result. I will reject Boolos' claim that this result shows MSOL to be pure logic. And I will express serious misgivings about his claim that MSOL in this way is shown to be ontologically innocent.To assess what is achieved by Boolos' result, we need to get clear on the logical status of the theory of plural quantification. Only if this theory deserves the honorific``logic'' will Boolos be justified in claiming that his # 2003 Blackwell Publishing Inc.,