[FOM] Fwd: First- Vs Second-Order Logic: Origins of the Distinction?
WILLIAM TAIT
williamtait at mac.com
Tue May 31 10:53:23 EDT 2016
By mistake, I sent this response to Joseph only to him and not to FOM. It may be of interest to others.
Bill
> Begin forwarded message:
>
> From: WILLIAM TAIT <williamtait at mac.com>
> Subject: Re: [FOM] First- Vs Second-Order Logic: Origins of the Distinction?
> Date: May 30, 2016 at 8:01:28 PM CDT
> To: Joseph Shipman <JoeShipman at aol.com>
>
> At root is a misunderstanding, signaled by calling Hilbert a 'formalist' as opposed to a 'methodological formalist'. Hilbert's conception, from before the turn of the century was that, e.g., the system of natural numbers exists if its (categorical) theory is consistent. His problem, not really solved until 1917-8, was to explain what he meant by 'consistent' without reference to semantic content. To prove consistency syntactically, one has to be a 'formalist' and more: one has to prove propositions about symbols without using methods that presuppose the same mathematics one is trying to prove consistency. This project failed, of course, but we are talking about what conception of logic prevailed prior to 1929---the year of the split among the post-Kantians at Davos, publication of John Dewey's Experience and Nature, Wittgenstein's 'return' to philosophy, the Great Depression---and my birth. (So I obviously know a great deal about it!) it was also the year after the publication of Hilbert's 1917-8 lectures under the names of Hilbert and Ackerman. It was this book, as I understand it, that posed to Goedel the problem of syntactical completeness.
>
> Bill
>
> Sent from my iPad
>
>> On May 30, 2016, at 6:46 PM, Joseph Shipman <JoeShipman at aol.com> wrote:
>>
>> 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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