Logic & Philosophy of Science


Crispin Wright
University of St. Andrews

Basic Law V and Indefinite Extensibility


Many diagnoses remain on offer in recent philosophy of why Frege’s system of Grundgesetze collapsed in contradiction. They include his recourse to unrestricted objectual quantification and — not the same thing — to impredicative first-order quantification, the simultaneous impredicativity of the higher-order logic of Begriffschrift, and his oversight of the ontologically  ‘inflationary’ character of the course-of-values operator as characterized by Law V. There are issues about whether (and which of) these diagnoses are fundamentally in tension. In any case, the focus of the present paper will be on another one, less clear and even harder to assess: that Frege’s problems were owing to his having not even a “glimmering of a suspicion of the existence of indefinitely extensible concepts” (Dummett, my italics).

The very idea that there is any coherent notion of “indefinite extensibility” has been received skeptically by some authorities (Boolos, Burgess). My purpose here is to address this skepticism.  I will offer a new characterization of indefinite extensibility, outline its connection with paradox, and try to explain wherein Frege’s oversight of it consisted and the obstacle posed to anything worth regarding as a logicist foundation for the mathematics of indefinitely extensible domains.  

Saturday, February 11, 2006
SST 777
4 pm

Refreshments will be served