[FOM] How much of math is logic?

Max Weiss mmweiss at sfu.ca
Tue Feb 27 21:34:16 EST 2007

On Feb 27, 2007, at 10:21 AM, Robbie Lindauer wrote:
> Frege introduces the notion of "Extension of a concept" as a Thing in
> its own right, as opposed to them (the things to which a concept
> extends) being the (several) things to which the concept refers.
> This "logical object" then is allowed to have "properties".  This is
> the essence of second-order-logic and is why, for the most part,
> logicians differentiate between second-order-logic and first-order-
> logic and why, in general, second order logic is held with suspicion
> - in particular that if SOL really is logic, then Frege's proposition
> V is apparently an obvious logical truth of it.

Historically, Lindauer's remarks are misleading.

The system of Frege's earlier book *Begriffsschrift* is a system of  
SOL, but does not contain Law V.  Likewise, a system of SOL is a  
subsystem of the system of *Principia*, but *Principia* does not  
contain Law V.

In the introduction to *Grundgesetze*, Frege writes:

``...the pronouncement is often made that arithmetic is merely a more  
highly developed logic; yet that remains disputable so long as  
transitions occur in the proofs that are not made according to  
acknowledged laws of logic, but seem rather to be based upon  
something known by intuition....  A dispute can arise, so far as I  
can see, only with regard to my Basic Law concerning courses-of- 
values (V)....  I hold that it is a law of pure logic.  In any event  
the place is pointed out where the decision must be made."

Thus, on the one hand Frege thought it obvious that his Laws I-IV are  
laws of logic.  But on the other hand, he acknowledges that the  
status of Law V is disputable, or even arbitrary.  He saw clearly  
that Law V is quite different from his Laws I-IV: that is, quite  
different from the axioms that, together with his rules of inference,  
yield a system of what is now called "second-order logic".

(For more discussion, see Tyler Burge, "Frege on Extensions of  
Concepts, 1884-1903" (Phil. Rev. 1984).)

So, I wonder what Lindauer means when he says "if SOL really is  
logic, then Frege's proposition V is apparently an obvious logical  
truth of it."  Not even Frege considered this obvious.

Max Weiss

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