[FOM] computable numbers, independent freindly logic

[praatika@mappi.helsinki.fi]

George Kapoulas <gkapou at yahoo.gr> wrote:

> 2) There is an approach for logic 
> called independent friendly logic.
> Does anyone know  if there is 
> proof that the approach used by 
> this logic  cannot be expressed 
> in 1st order logic, 
> and references for this?

This logic, isolated by Hintikka and Sandu, essentially 
uses "partially ordered" or "branching" quantifiers 
(also known as "Henkin quantifiers"). It has been known 
for some time that the resulting logic is equivalent to
Sigma-1-1 fragment of second order logic (Enderton 1970; 
Walkoe 1970). Hence it is much stronger than the standard 
first order logic, and also essentially incomplete. 
Hintikka and Sandu have also included partially ordered 
connectives to their IF logic, but this does not make it 
really stronger. 

References:

Enderton, H.B. (1970) 'Finite partially ordered quantifiers', 
Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik 16,  
393-397.
Walkoe, W. (1970) 'Finite partially ordered quantification'. JSL 35, 
535-555.


Best

Panu

Panu Raatikainen
Ph.D., Docent in Theoretical Philosophy
Fellow, Helsinki Collegium for Advanced Studies
P.O. Box 4
FIN-00014 University of Helsinki
Finland

E-mail: panu.raatikainen at helsinki.fi

http://www.helsinki.fi/collegium/eng/Raatikainen/raatikainen.htm