[FOM] How much of math is logic?

praatika@mappi.helsinki.fi praatika at mappi.helsinki.fi
Wed Feb 28 07:43:02 EST 2007

Quoting joeshipman at aol.com:

> This is a silly objection, the point is that the mapping **can be 
> seen** to connect arithmetical statements to logical ones in a 
> truth-preserving way. What do you think Russell thought he was doing?

I don't think this is silly. For every consistent recursively
axiomatizable theory S there is an effective translation of S into the weak
Robinson Arithmetic Q which preserves theoremhood,and more (Pour-El and
Kripke,1967). We can take S to be arbitrarily strong, e.g., ZFC+
supercompact cardinals exist. Yet, we would not say that such set theory
really is just elementary arithmetic. One should be careful in not
concluding too much from such translations. 

I don't think it matters much what Russell thought he was doing. What is
important is what he in fact did, and did not do. He managed to develop 
mathematics in his type theory only by postulating further non-logical
axioms, namely, AC, reducibility and infinity.      

> This is exactly what I was trying to say -- I was careful to 
> distinguish the ZFC axiom of Infinity as something that was NOT 
> "logical", in order to argue that those parts of math which do not need 
> it CAN be thought of as simply logical.

I am afraid I fail to see any clear motivation for drawing the line between
logical and non-logical to exactly that point. Moreover, Shipman wrote earlier: 

> (By the way, I use "logical truth" and "logical validity"
> interchangeably to mean sentences true in all models; 

Certainly the axioms of ZFC without the axiom of infinity (call it ZFC-) are
not together logically true in this sense, for they are false in many models. 

> Why not? In my opinion, many people who say that second-order logic is 
> not logic are prejudiced because they find the view that statements 
> like CH do not have an absolute truth value congenial; that view allows 
> one to maintain the Tennantian anti-realist position that there do not 
> exist truths which are in principle unknowable.

I do allow, tentatively at least, that CH and such have a truth value. This
issue has nothing to do with my dislike of SO "logic" as logic. I think the
same is true of many others here in FOM who also dislike SO logic. 



Panu Raatikainen

Academy Research Fellow, The Academy of Finland
Docent in Theoretical Philosophy, University of Helsinki

Department of Philosophy
P.O.Box 9
FIN-00014 University of Helsinki

e-mail: panu.raatikainen at helsinki.fi


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