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Logic Journal of IGPL Advance Access originally published online on November 9, 2008
Logic Journal of IGPL 2008 16(6):585-590; doi:10.1093/jigpal/jzn023
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Vol. 16 No. 6, © The Author 2008. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

The Ricean Objection: An Analogue of Rice's Theorem for First-order Theories

Igor Carboni Oliveira and Walter Carnielli

Centre for Logic, Epistemology and the History of Science (CLE), and Department of Philosophy (IFCH), State University of Campinas - UNICAMP, C.P. 6133 - 13083-970 Campinas, SP, Brazil.


  Abstract

We propose here an extension of Rice's Theorem to first-order logic, proven by totally elementary means. If P is any property defined over the collection of all first-order theories and P is non-trivial over the set of finitely axiomatizable theories (i.e., P holds for some, but not all theories), then P is undecidable. This not only means that the problem of deciding properties of first-order theories is as hard as the problem of deciding properties about languages accepted by Turing machines, but also offers a general setting for proving several undecidability results in first-order theories.


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