Frege, Boolos, and Logical Objects


David J. Anderson and Edward N. Zalta


Journal of Philosophical Logic, 33/1 (February 2004): 1-26.


In this paper, the authors discuss "logical objects" (extensions, numbers, truth-values, etc.) and the recent attempts to rehabilitate Frege's strategy for introducing such objects. We focus on George Boolos's work in the mid-80s and early 90s, and establish that the `eta' relation that he deployed on Frege's behalf is similar, if not identical, to the encoding mode of predication that underlies Zalta's theory of abstract objects. Boolos accepted unrestricted comprehension over Properties and used the `eta' relation to assert the existence of logical objects under certain highly restricted conditions. By contrast, Zalta accepts unrestricted comprehension over logical objects and banish encoding formulas from comprehension over Properties. Boolos' proposes 3 different systems, all of which have the requisite mathematical power to derive the natural numbers and 2 of which yield something like sets, in the first instance. But none of these systems can extend Frege's conception to cover directions, shapes, truth values, and other objects abstracted from equivalence relations The system Zalta advocates has the philosophical power for the latter, but requires additional (though plausible) assumptions to attain the natural mathematical power Frege desired. Some new results in object theory, concerning natural sets, truth-values, and other abstract, logical objects, are developed in detail.

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