[FOM] How much of math is logic?
Robbie Lindauer
rlindauer at gmail.com
Tue Feb 27 13:21:44 EST 2007
The program of logicism was simultaneously to extend what is called
logic and to attach it to mathematics.
Thus the notion of "logical facts" and "logical objects" were
introduced as "things that are true because there is logic".
Then the number and character of these logical objects and logical
facts were extended at the same time.
Russel says, roughly, "we don't accept these axioms because they're
true or obvious, but because we can reconstruct lots of mathematics
from them".
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.
Robbie
On Feb 27, 2007, at 9:38 AM, joeshipman at aol.com wrote:
> In that case "logicism" is trivially false, so why was there such a
> debate about it in the first few decades of the 20th century?
>
>
>
>> The existence of the empty set is not a logical truth either.
>
>> Robbie Lindauer
>
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