## Frege, Boolos, and Logical Objects

### Authors

David J. Anderson and Edward N. Zalta
### Reference

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

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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