The Implicit Commitment of Arithmetical Theories and Its Semantic Core

Erkenntnis 84 (4):913-937 (2019)
  Copy   BIBTEX


According to the implicit commitment thesis, once accepting a mathematical formal system S, one is implicitly committed to additional resources not immediately available in S. Traditionally, this thesis has been understood as entailing that, in accepting S, we are bound to accept reflection principles for S and therefore claims in the language of S that are not derivable in S itself. It has recently become clear, however, that such reading of the implicit commitment thesis cannot be compatible with well-established positions in the foundations of mathematics which consider a specific theory S as self-justifying and doubt the legitimacy of any principle that is not derivable in S: examples are Tait’s finitism and the role played in it by Primitive Recursive Arithmetic, Isaacson’s thesis and Peano Arithmetic, Nelson’s ultrafinitism and sub-exponential arithmetical systems. This casts doubts on the very adequacy of the implicit commitment thesis for arithmetical theories. In the paper we show that such foundational standpoints are nonetheless compatible with the implicit commitment thesis. We also show that they can even be compatible with genuine soundness extensions of S with suitable form of reflection. The analysis we propose is as follows: when accepting a system S, we are bound to accept a fixed set of principles extending S and expressing minimal soundness requirements for S, such as the fact that the non-logical axioms of S are true. We call this invariant component the semantic core of implicit commitment. But there is also a variable component of implicit commitment that crucially depends on the justification given for our acceptance of S in which, for instance, may or may not appear reflection principles for S. We claim that the proposed framework regulates in a natural and uniform way our acceptance of different arithmetical theories.



    Upload a copy of this work     Papers currently archived: 93,296

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library


Added to PP

37 (#445,119)

6 months
9 (#355,374)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Carlo Nicolai
King's College London

References found in this work

Outline of a theory of truth.Saul Kripke - 1975 - Journal of Philosophy 72 (19):690-716.
Basic proof theory.A. S. Troelstra - 1996 - New York: Cambridge University Press. Edited by Helmut Schwichtenberg.
Axiomatic Theories of Truth.Volker Halbach - 2010 - Cambridge, England: Cambridge University Press.
Finitism.W. W. Tait - 1981 - Journal of Philosophy 78 (9):524-546.
Reflecting on incompleteness.Solomon Feferman - 1991 - Journal of Symbolic Logic 56 (1):1-49.

View all 54 references / Add more references