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

[WILLIAM TAIT]

> On May 27, 2016, at 9:39 AM, Joseph Shipman <JoeShipman at aol.com> wrote:
> 
> II claim that before 1929, the former was generally held, because there was a notion of semantic entailment distinct from deductibility, so that soundness and completeness were clearly distinct properties of logical deductive calculi (soundness meaning that only validities were derivable, where a validity is an open statement whose universalized form is true in all models or interpretations, and completeness meaning that all validities are derivable).

That is surely wrong. In 1929 one of the most influential people thinking about logic in mathematics was Hilbert. It was his view that, to establish the existence of, say, the system of natural numbers, i.e. to give any semantic content to PA, one had to establish its SYNTACTIC consistency. (In a large part, the development of first and higher order logic (in Hilbert’s lectures of 1917-8) was motivated by the need to give a precise sense to the notion of a syntactic consistency proof.

Bill