[FOM] Origins of the double turnstile

Richard Heck richard_heck at brown.edu
Fri Jan 6 21:28:44 EST 2017

The book in question is on Google Books here:


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.


On 01/06/2017 04:41 PM, Guillermo Badia 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 Fri, Jan 6, 2017 at 6:47 PM, Christopher Menzel <cmenzel at tamu.edu
> <mailto: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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Richard G Heck Jr
Professor of Philosophy
Brown University

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