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

Joseph Shipman JoeShipman at aol.com
Mon May 30 19:46:56 EDT 2016

But not everyone was a formalist like Hilbert. It is not wrong to say, as I said, that there was a "notion of semantic entailment", even though Hilbert had rejected this notion.

After all, Godel's completeness theorem is frequently described as having shown that any sentence x that was entailed by a set of sentences S in FOL is derivable from them by the standard deductive rules, or equivalently that for any sentence x not so derivable, so that ~x was consistent with S, was false in some model of S so that it was not semantically entailed.

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.

-- JS

Sent from my iPhone

> On May 29, 2016, at 4:38 PM, WILLIAM TAIT <williamtait at mac.com> wrote:
>> 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
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