[FOM] Logicism: a survey

[Joseph Shipman]

If Logicism is the philosophy that all or most mathematics is reducible to "logic", it seems to me that critics of Logicism do a poor job of stating their objections in a way that allows any nontrivial mathematics whatsoever to count as "reducible to logic". Surely Russell and other logicists cannot have been so fundamentally misguided that their program fails even to establish a very easy theorem such as "for any integer n, there is a prime larger than n" as following from logical considerations alone.

On the one hand, if your criterion for a mathematical proposition being "logical" depends only on its form and not on its being derivable in some logical deductive calculus, and if you allow Logicism to proceed as far as Euclid's theorem, you have to allow arbitrary true AE arithmetic sentences to be equivalent to logical validities, which means you can't have a recursively enumerable set of validities, and should therefore not balk at set-theoretic semantics for SOL making the validity of some sentences be undecidable.

On the other hand, if you think that a necessary condition for a formal system to qualify as a "logic" is that there be a recursively enumerable set of validities, and if you allow Logicism to proceed as far as Euclid's theorem, 
you need to state how much farther you think Logicism may be taken, by identifying the most powerful deductive system to which you are willing to grant the epistemologically preferred adjective "logical" rather than the putatively less well-supported epithet "mathematical".

I'll grant you an upper bound, namely "the set of sentences of SOL which, under set-theoretic semantics, are consequences of ZFC". Any anti-logicist ought to either specify a weaker deductive calculus than this that he is willing to call a "logic", or declare that even Euclid's theorem is merely mathematical and doesn't count as following from logic, or allow for a non-r.e. set of validities and give a *different* argument against set-theoretic semantics for Second Order Logic.

Please reply with your preferred alternative and explain why!

-- JS

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