Warning: This post will not make sense if you do not have the original article Wang’s Paradox by Michael Dummett. You also need to know what is a valid argument (an argument whose conclusion cannot be false if the premises are true) and what is a sound argument (a valid argument with true premises).
After some preliminary comments on strict finitism as a philosophy of mathematics, Dummett introduces Wang’s Paradox:
Premise 1:0 is small;
Premise 2 (induction step):If n is small, n+1 is small:
Conclusion:Therefore, every number is small.
This is a paradox because the premises are true, the argument is valid but the conclusion patently false since there are numbers that are not small. According to Dummett, “The paradox is evidently due to the vagueness of the predicate “small”“. But how exactly? Someone who wants to reject the conclusion of this argument has two options, to either show the argument is not valid or one of the premises is not true.
ARGUMENT IS INVALID
Suppose the argument is invalid. Consider these two explanations.
First explanation: the invalidity comes from wrongly using universal generalisation on vague predicates. In other words, we cannot use universal generalisation on premise 2 (which contains the vague predicate “smallâ€) to arrive at the conclusion that all numbers are small.
Response to first explanation: Even so, we still have a problem since we can arrive at the conclusion that k is small, for some k is not small. Furthermore, to declare that universal generalisation is invalid in this context is to violate the meaning of “allâ€.
Second explanation: the invalidity comes from wrongly applying induction to vague predicates, i.e. inductive arguments are only valid when applied to arguments containing non-vague predicates.
Response to second explanation: Even so, we still have a problem since we can arrive at the conclusion that k is small, for some k that is not small. Just use a series of conditionals and modus ponens [see footnote 1 below]. If we go with the second explanation, we would also have to reject modus ponens as a valid way of arguing and surely, we don’t want to do that. [Comment: Actually, one option is to say that modus ponens is invalid when applied to vague predicates.]
Ok, so both explanations above are unsatisfactory. How about this third explanation? Third explanation: Deny that “in the presence of vague predicates, an argument each step of which is valid is necessarily validâ€.
Response to third explanation: This violates our whole concept of what a constitutes a valid argument!
Counter-response: But hang on, a strict finitist (this is where Dummett’s preliminary comments on strict finitism comes in useful) would accept this explanation since for the strict finitist, “a proof is valid just in case it can in practice be recognised by us as valid; and, when it exceeds a certain length and complexity, that capacity failsâ€.
Counter-counter response: But Wang’s Paradox can also be a paradox for the strict finitist since “it will always be possible so to interpret “small†that we can find a number which is not small for which there apparently exists a proof, in the strict finitist’s sense of “proofâ€, that it is small, a proof not expressly appealing to inductionâ€. For the full argument, see footnote 2 below.
This concludes my first post on Wang’s Paradox.
Footnote 1: First find some particular n that is not small, then apply premise 2 as often as you need with actual numbers to conclude that that the particular n is small, (i.e. start with these premises: 0 is small; If 0 is small, then 1 is small; to conclude that 1 is small. And then use “1 is small†and “If 1 is small, then 2 is small†to conclude that 2 is small etc…). The upshot is that we can still use modus ponens to generate a paradox for some particular n that is not small. It would seem like madness to reject modus ponens.
Footnote 2: My simplified version of Dummett’s argument follows: Call n an apodictic number if it is possible for a proof (in the strict finitist’s sense of “proofâ€) to contain as many as n steps. First premise: 1 is apodictic. Second premise: If m is apodictic, then so is m+1. Conclusion: All numbers are apodictic.