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[ontolog-forum] FOL

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Wed, 10 Nov 2010 11:10:27 -0600
Message-id: <0A4CAB1F-9553-460F-9DC7-505043BB7AA4@xxxxxxxx>
On Nov 10, 2010, at 9:29 AM, John F. Sowa wrote:
> Patrick,
> Yes, that is the work that logicians were doing in the 1920s and '30s:
>> Presenting in a nutshell the results of our quick historical overview,
>> we can say that around 1900 logic was conceived as a theory of
>> sentences, set and relations; after World War I and as late as 1930 the
>> exemplar for modern logic was a higher-order system, simple type theory;
>> and only around 1940-1950 did the community of logicians as a whole come
>> to agree that the paradigm logical system is FOL.    (01)

Largely true, mostly because it was during this time that model theory, first 
introduced explicitly by Tarski in the early 1930s, matured into its modern 
form, as exemplified perhaps most clearly in print first by John Kemeny in his 
1956 paper "A New Approach to Semantics -- Part I" (Journal of Symbolic Logic, 
21(1)).  Prior to the development of model theory, there was no completely 
rigorous way to distinguish between first-order and higher-order systems, since 
the distinction is essentially a semantic one.  Indeed, a wholly precise 
characterization of the demarcation between first-order logic and stronger 
logics did not exist until 1969 with the publication of Lindström's theorem 
that any logic at least as strong as first-order logic that is compact* and has 
the downward Löwenheim-Skolem property** is itself first-order.    (02)

> But that is what Peirce was doing between 1897 and 1909.  He published
> much of it in 1906. but it was ignored by most logicians.  Among the few
> who read his later writings were Clarence Irving Lewis and Arthur Prior,
> both of whom made major contributions to modal logic and related topics.    (03)

Indeed, Prior essentially created modern temporal logic.    (04)

Chris Menzel    (05)

*A logic L is compact iff, for any set S of sentences of L, if every finite 
subset of S has a model, so does S.    (06)

*A logic L has the downward L-S property iff any set of sentences of L with an 
infinite model has a countable model.  (This formulation assumes that L's 
language is countable.)    (07)

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