Inference, Belief, and Understanding

Abstract

Someone might accept the hypothetical proposition that if A and B are true then Z is true without accepting either A or B (although Achilles is surely right that such a person would be wise to abandon Euclid and take to football). Both the tortoise and Achilles also agree that someone could accept A and B without accepting the conditional 'If A and B are true then Z is true'. The tortoise challenges Achilles to consider him as such a person and to 'force him, logically, to accept Z as true' (ibid.). The story suggests that he can never succeed. The tortoise claims at the outset not to accept the conditional (call it C) that if A and B are true then Z is true,2 but he gladly accepts it as soon as Achilles asks him to. Since he insists that every proposition he accepts should be written down, C is duly added to A, B, and Z in Achilles's notebook. When the tortoise points out, as before, that it is possible to accept A, B, and C without accepting the conditional that if A and B and C are true then Z is true, Achilles asks him to accept one more conditional. He aggrees, and that conditional in turn is added to the notebook and called D. Before the same kind of interchange can be repeated the narrator, having pressing business at the bank, is obliged to leave. We are left to draw our own moral. It seems to me that there is no sound moral to be drawn from the story about the nature of validity or logical consequence as such. Lewis Carroll does not succeed in showing that Z does not