[FOM] Second-order logic and neo-logicism

[Alan Weir]

I don't think Panu Raatikainen's objections to neo-logicism (FOM Vol 147 Issue 22) beg the question again the neo-logicist as suggested by Ran Lanzet (Issue 23) by simply assuming that the strong second order logic required to get Frege's theorem is part of mathematics. The key issue, it seems to me, is that the logic which the neo-logicist assumes entails strong existence claims (the instances of the comprehension axioms for example). Neo-logicists often agree that in order that  they themselves not beg the question, they should work with a logic whose first-order component is free. But when you make the same adjustment at second order level then (at least depending on how you finesse the 'freeing up' of second order logic) the  logic becomes too weak to generate infinity from Hume's Principle. See Stewart Shapiro and Alan Weir ' "Neo-logicist" Logic is not Epistemically Innocent', Philosophia Mathematica, (2000), pp. 160-189.

Best

Alan

Professor Alan Weir
Philosophy,
Sgoil nan Daonnachdan,
Oilthigh Ghlaschu/University of Glasgow
GLASGOW G12 8QQ

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