Who needs (to assume) Hume's principle? July 2006


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In the Foundations of Arithmetic, Frege famously developed a theory which today goes by the name of logicism - that it is possible to prove the truths of arithmetic using only logical principles and definitions. Logicism fell out of favor for various reasons, most spectacular of which was that the system, which Frege thought would definitively prove his thesis, turned out to be inconsistent. In the early 1980s a movement called neo-logicism was begun by Crispin Wright. Neo-logicism holds that Frege was almost right, in that arithmetic can be proven in second-order logic using only definitions and one quasi-logical proposition, called Hume's Principle, which says that the number of Ps equals the number of Qs if and only if they can be put into one-to-one correspondence. There has been some controversy about the status of Hume’s Principle - for instance, whether it counts as a logical or analytic proposition. (See e.g. the similarly titled, “Is Hume’s Principle Analytic?, by Crispin Wright and George Boolos.) In this paper a different tack will be tried. Indeed Frege is almost right. He is almost right because a large part of arithmetic and number theory, or at the least a large part of something which looks like them, can indeed be generated using only logical principles and definitions, without the assumption of any quasi-logical assertion and in particular without Hume’s Principle. Specifically, logic will be taken as second-order logic with full comprehension and the addition of one distinguished 2-ary predicate “!”. A large amount of arithmetic and number theory will then be developed, using only (second-order) logical principles and definitions. It can thus be seen that the epistemological status of this large part of arithmetic is independent of the question of the status of Hume’s Principle.
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