[FOM] Weak system for SO logic

[praatika@mappi.helsinki.fi]

Richard Heck wrote:

> I think nowadays "the more usual rules" aren't really very usual.

I based calling them "more usual" simply to my quick check in few  
books and papers I was more familiar with: e.g. van Dalen, Logic and  
Structure; Troelstra & Schwichtenberg, Basic Proof Theory; Prawitz,  
"Ideas and results in proof theory"; and Leivant, "Higher order logic"  
- all (if I read them correctly) present the rules amounting to the  
full impredicative comprehension, and only them.

In addition to Takeuti, at least Manzano, in Extensions of First-Order  
Logic, mentions other options. She refers back to the classic (Henkin  
1953).

But it seems to me that at least in the sources with deal with natural  
deduction, these really are the "more usual" ones. However, I would be  
very interested in learning about contrary evidence...

> That is how Frege formulates the rules in _Begriffsschrift_, but
> even in _Grundgesetze_ he shows some awareness that there are really two
> things going on. It's worth disentangling these, which of course is what
> happens when we formulate the rules as simply as possible and add
> comprehension axioms.

I whole-heartedly agree!


All the Best, Panu



Panu Raatikainen, Ph.D.

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