[FOM] Origins of the double turnstile

[WILLIAM TAIT]

Again, notice that Lyndon used \vDash as a relation between sentences.  Bill

> On Jan 7, 2017, at 12:23 PM, Christopher Menzel <cmenzel at tamu.edu> wrote:
> 
> Thanks to Guillermo, Richard, and Bill for their references and comments — super useful! I of course agree with Richard and Bill that the discussion in _Theory of Models_ suggests that the various symbols noted were in use prior to 1963, so it's curious that they seem so difficult to find. As Bill notes, Lyndon (1967) uses \vDash but Lyndon mentions that the book is basically his notes from courses taught in 1962 and 1963, so that seems to be the earliest I've been able to trace its use.
> 
> For FOM's convenience, the various LaTeX symbols and formulas mentioned in this exchange can be seen here:
> 
> http://cmenzel.org/|=.pdf
> 
> -chris
> 
> ps: I am embarrassed to admit that for the past 25+ years that I've been using LaTeX, until today, I've thought that "\Vdash" and "\models" were synonyms like "\to" and "\rightarrow" but now see that "\models" yields a somewhat stubbier and less pleasing symbol.
> 
> 
>> On 7 Jan 2017, at 9:25 AM, WILLIAM TAIT <williamtait at mac.com> wrote:
>> 
>> It is indeed listed in the “forward on terminology” in *Theory of Models: Proceedings of the 1963 InternationalSymposium at Berkeley* (1965) and, as Richard suggested, that seems to indicate a prior usage. But I looked (roughly) through the proceedings of the Summer Institute for Symbolic Logic (Cornell. 1957) and couldn’t find that notation. Feferman used the notation \vdash_{M} A to mean the sentence A is true in the structure M. Roger Lyndon (*Notes on Logic* 1967) used A \v= B with sentences A and B to mean that B is true when A is. 
>> 
>> Bill
> 
>> On 6 Jan 2017, at 9:28 PM, Richard Heck <richard_heck at brown.edu> wrote:
>> 
>> 
>> The book in question is on Google Books here:
>> 
>>    https://books.google.com/books?id=Tb_SBQAAQBAJ
>> 
>> See p. xiv, perhaps, for what Hodges had in mind. This occurs in a "Foreword on Terminology" in which Addison, Henkin, and Tarski ("the editors") attempt to establish, or at least encourage, some uniformity in terminology and symbolism. They explicitly recommend \vDash over \Vdash and \varVdash, which is similar to \Vdash except that the horizontal line cuts across both verticals. (This is apparently provided by stix. See the Comprehensive LaTeX Symbol List of the stix docs. This is apparently also U+2AE6.) 
>> 
>> This suggests to me, though of course I'm no expert, that all of these symbols were already in use by 1965 (when the book was published). Maybe a look at some of the papers to which the authors in this volume refer would help untangle the history.
>> 
>> Richard
> 
>> On 6 Jan 2017, at 4:41 PM, Guillermo Badia <guillebadia89 at gmail.com> wrote:
>> 
>> According to Hodges (Model Theory, p. 83), the notation goes back to the foreword of "The theory of models" edited  by Henkin and Tarski in 1965. Hope this is useful :-).
>> 
>> Cheers,
>> Guillermo
> 
>>> On Jan 6, 2017, at 11:47 AM, Christopher Menzel <cmenzel at tamu.edu> wrote:
>>> 
>>> FOM folk,
>>> 
>>> The turnstile (\vdash) is typically traced back to Frege and was picked up by Whitehead and Russell, who used it more or less to indicate provability, and of course this largely continues to the present day. But whence the double turnstile (\models) to indicate logical consequence? I thought perhaps I'd find it in Carnap or, at least, in Kemeny's famous 1956 JSL articles, but it's not there; they just adopt ordinary language expressions to indicate semantic relations. From the very limited bit of searching I've done, the double turnstile appears to be of fairly recent vintage. Does anyone know its origins or, at least, can anyone point to an earlyish (even pre-1960) use of the notation?
>>> 
>>> Thanks.
>>> 
>>> Chris Menzel
> 
> 
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