WAYNE AITKEN and JEFFREY A. BARRETT COMPUTER IMPLICATION AND THE CURRY PARADOX Received 26 August 2003; received in revised version 4 April 2004 ABSTRACT. There are theoretical limitations to what can be implemented by a computer program. In this paper we are concerned with a limitation on the strength of computer implemented deduction. We use a version of the Curry paradox to arrive at this limitation. KEY WORDS: Curry paradox, implication, modus ponens, theory of algorithms 1. I NTRODUCTION Computers can be used to check properties. Indeed, properties of com- puter programs might themselves be checked by appropriately designed programs. As is well-known there are limits on what can be done. Testing for the halting property is one of these. In this paper we are concerned with another limitation, one concerning computer-implemented deduction. An implication program is a program that takes as input two statements concerning the behavior of programs, then tries to deduce, by way of a specified library of rules, the second statement from the first. It keeps looking until it finds a deduction, thus it may never halt. If it does find a deduction, it halts and outputs 1 to signal that a proof has been found. An implication program can be used to prove statements involving the im- plication program itself. Since recursive and partially recursive functions can be implemented as computer programs, an implication program might also be used to prove relationships between such functions. Throughout we assume that programs are written in a fixed language for a computer with unlimited memory. In this paper we point out a limitation on implication programs, namely that no sufficiently powerful implication program can incorporate an unre- stricted form of modus ponens. So modus ponens is an example of a valid rule of inference that can be defined algorithmically, but cannot be used by the implication program. The structure of the argument follows the Curry paradox. The algorith- mic version of the paradox parallels the classical version which proceeds as follows (see [3] and [1] for discussions of the classical paradox which was first stated in [2]). Consider the property C(X) defined to hold if and Journal of Philosophical Logic 33: 631–637, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.