The Consistency of Arithmetic by Storrs McCall
Page last updated 10 Oct 2020
While doing some research recently, I came across an article MS Word doc The Consistency of Arithmetic by Storrs McCall, a professor of philosophy at McGill University, and which is included in his book The Consistency of Arithmetic which is claimed be a new “semantic” proof of the consistency of Peano arithmetic. While there have been a few comments on it on the New York University FOM forum as listed at FOM March 2015 Archives, they were not very informative, so I had a quick look at the article and soon discovered the rather obvious flaw in McCall’s argument, and which had not been mentioned on the FOM forum. I have summarized McCall’s error here as it may save others from wasting time on it.
Essentially what McCall does is this:
He starts off by referring to a specific formal system, in this case Peano arithmetic, but as we shall see, the actual system chosen is immaterial, as the basis of McCall’s argument is fundamentally flawed.
Rather than using the chosen system itself, he uses a modified form of his chosen formal system. In this modified system, all meaningful propositions of the system can be distinguished from all other sequences of the symbols of the system.
But, crucially, McCall simply assumes that in this modified system, every such meaningful proposition must have either a “true” or a “false” value, stating:
As it turns out, the semantics for [the modified system] rather than the traditional axioms … are bivalent, meaning that every [well-formed formula] in every model is either true or false.
But we know that the truth value of a sequence of symbols depends completely on the axioms and rules of the system for which the sequences of symbols are to be associated with; a sequence of symbols that is true in one system can be false in another system. And if the axioms and rules of the system are inconsistent, then a sequence of symbols can be both “true” and “false” at the same time.
By the naive assumption that every meaningful proposition has either a true or a false value, McCall is simply assuming that the axioms and rules of his modified system cannot generate a sequence of symbols where both that sequence and its negation are true. In other words, McCall is assuming consistency in order to prove consistency, which is obviously an absurdly circular argument. (Footnote: Note that even if one were to make the bizarre suggestion that the truth or falsity of a proposition of a system is in some way determined independently of the axioms and rules of that system, that would also mean that one could not then assert that the system could not prove, by those rules and axioms, both a proposition of the system and its negation, see also The notion of ‘truth’.)
It can be noted that McCall muddles the waters considerably by his notion of associating every symbol with a physical cube, and also associating every sequence of such symbols with a physical array of these cubes. This is a completely specious complication which only serves to obfuscate the crucial fallacious assumption.
Finally, it might be noted that the naive assumption that there is a difference between a meaningful proposition of the system being true and being provable also serves to create further obfuscation, see The notion of ‘truth’.
Rationale: Every logical argument must be defined in some language, and every language has limitations. Attempting to construct a logical argument while ignoring how the limitations of language might affect that argument is a bizarre approach. The correct acknowledgment of the interactions of logic and language explains almost all of the paradoxes, and resolves almost all of the contradictions, conundrums, and contentious issues in modern philosophy and mathematics.
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