[FOM] First- Vs Second-Order Logic: Origins of the Distinction?

Robert Black mongre at gmx.de
Tue May 31 11:04:01 EDT 2016

On 31/05/2016 00:46, Joseph Shipman wrote:
> Are you maintaining that prior to 1929 there was not a clear enough understanding of the concept "every model of S is also a model of x" that the question of whether that concept implied x was deducible from S was considered an actual open question? In other words, that Godel's paper on the completeness theorem decisively solved for FOL a problem that it also was the first to clearly formulate?
> If so, Godel  is even more impressive than I had realized.
I think the first reasonably clear presentation of the completeness of 
first-order logic as an open problem is at the end of chapter 3 section 
9 of the Hilbert/Ackermann book of 1928 (the logic text Gödel quotes in 
his completeness paper), and even there a little charity is needed since 
their statement of completeness inadvertently uses the expression 
'logische Formel' in a sense other than that which they had defined for 
it. We are today so used to clear separation between syntax and 
semantics that it's perhaps hard for us to realize how unclear the 
pioneers were. Well before Gödel Skolem had all the materials for a 
completeness proof but failed to put them together because he didn't 
realize what had to be proved.

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