J. R. Lucas argues in “Minds, Machines, and Gödel” that his potential output of truths of arithmetic cannot be duplicated by any Turing machine, and a fortiori cannot be duplicated by any machine. Given any Turing machine that generates a sequence of truths of arithmetic, Lucas can produce as true some sentence of arithmetic that the machine will never generate. Therefore Lucas is no machine.

I believe Lucas's critics have missed something true and important in his argument. I shall restate the argument in order to show this. Then I shall try to show how we may avoid the anti-mechanistic conclusion of the restated argument.

As I read Lucas, he is rightly defending the soundness of a certain infinitary rule of inference. Let L be some adequate formalization of the language of arithmetic; henceforth when I speak of sentences, I mean sentences of L, and when I call them true, I mean that they are true on the standard interpretation of L. We can define a certain effective function Con from machine tables to sentences, such that we can prove the following by metalinguistic reasoning about L.

1. C1. Whenever M specifies a machine whose potential output is a set S of sentences, Con (M) is true if and only if S is consistent.

2. C2. Whenever M specifies a machine whose potential output is a set S of true sentences, Con (M) is true.

3. C3. Whenever M specifies a machine whose potential output is a set S of sentences including the Peano axioms, Con (M) is provable from S only if S is inconsistent.