[FOM] Re Second-Order Logic and Logicism.

[Alan Weir]

"The excessive focus on whether second-order logic is really "logic" simply obscures what *I* take the key issue here to be, namely: Does the sort of reasoning required for the proof of Frege's Theorem (whatever 
sort that might be) preserve whatever interesting epistemic property one might think HP itself has (whatever that might be)?" Richard Heck, FOM 147.28.

I entirely agree Richard. That is exactly the sort of point Stewart and I were making. So perhaps I ought to have said that objections to the neo-logicist use of second-order logic needn't beg the question against them, even if some objections do. My own position is, indeed,  more radical than Panu's in his paper: I think that Hume's Principle can't even give us Q or the theorem of infinity without invoking  principles, whether you want to call them logical or not is irrelevant, which pose the same epistemological worries as standard mathematical theories construed platonistically. That is not just to repeat Quine's scepticism about second-order logic- as you put it: 

"the complaint that the logic has (second-order) existential implications (Quine et alia)"

Quine, remember, was very far from an ardent proponent of free logic at first-order level, his worries about second-order logic were more about the sort of existential commitments he thought it had. I have no hang-ups about the free segment of second order logic and believe in the existence of mind-independent properties, even more so than Duns Scotus. But this combination does not, I maintain, entail the epistemic innocence of the principles needed to get the infinity of the natural numbers from HP. 

I say that as a Scottish philosopher who is 100% not British, who rejects Britishness and all its works and pomps, even though, alas, I remain after last year's referendum, a UK citizen. Thus I am not one who naturally jumps in to defend 'Britain', but I do find Panu's remarks (same issue of FOM)

'However, I've had quite a lot of transaction with, e.g., British  philosophers and, believe me, these issues are not at all clear to many of them - on the contrary, what I say seems to be almost a scandal for many.'

and the contrast with 'competent people' like Burgess, Heck and Linnebo, puzzling. UK universities such as Birkbeck and St. Andrews and others have had many logicians interested in neo-logicism, (of whatever nationality) working at them, including Linnebo, also others such as Shapiro, Cook, logicians  perfectly cognizant with the complexities of second-order logic(s). And the final section criticises Hale, Wright, Boolos and Burgess for passages of which Panu says ''Whatever their actual intent, it is very easy to read such brief statements as suggesting that the full second-order arithmetic PA2 can be derived from HP alone, without any other substantial assumptions." Well, all these philosophers, not just the anti-logicists Boolos and Burgess, are well aware, even if those passages don't make it clear, that there are epistemological worries about the logic used to derive substantive mathematical results from HP.

So  whilst I am on the side of those who don't think neo-logicism yields any epistemological gains over a straightforward platonism, even leaving worries about abstraction principles aside,  I don't think only ignorance of the logical technicalities can explain taking the other view. If, for example, one had an anti-realist view of properties, but not objects, one might try to defend from the 'no better off than platonism' criticisms, the use, in proving e.g. theorems of infinity,  of predicative comprehension.

Alan Weir

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