"⊢" is one of the most famous symbols of modern logic. It has been introduced by Frege and for this reason is called "Frege's stroke". But it is also called by other names, in particular "turnstile", a name which has more to do with its form than its meaning. Its meaning has considerably evolved and variations of its original design have sprung, in particular its most famous double: "⊨".
In this workshop we will combine an analysis of the history of this symbol and its variations with critical reflexions about their meanings and uses. This will be a way to reflect on the evolution and central features of modern logic.

I. Origin of the symbol "⊢" and its early history
In this first session we will recall the orginal meaning of Frege's stroke, when and in which circunstances it was introduced and its reception and use or non-use by Hilbert, Whhitehead-Russell, Wittgenstein and Leśniewski. We will in particular focus on the distinction between truth and logical truth. We will furthermore discuss the symbolic dimension of "⊢" within a general discussion on symbolism, mathematics and modern logic.

II. Syntax vs Semantics, Proof Theory vs Model Theory, ⊢ vs. ⊨
In the second session we will discuss the crystallisation of the opposition in modern logic between syntax and semantic, proof-theory and model theory, typically symbolized by ⊢ vs. ⊨. An opposition which makes sense but is also overcome by the completeness theorem. We will also discuss the incompletess theorem from the perspective of these two symbols.
We will in particular emphasize the ambiguity of the use of "⊢" in sequent calculus instead of the original symbol used by Gentzen "→", explaining how this confuses one of the most important results of proof theory, the cut-elimination theorem.
We will also emphasize the ambiguity of the double use of the double "⊨", in model theory: as a symbol for a semantical consequence relation and as a symbol used for a relation between models and formulas.

III. ⊢ as an Abstract Consequence Relation
In the third session we will focus of the use of "⊢" as a symbol for an abstract consequence relation, beyond the dichotomy proof-theory / model-theory. It the denotes a fundamental relation for logical structures, slight variation of Tarski's consequence operator. We will focus in particular on the completeness theorem from this abstract perspective. We will also discuss some related notions such as logcial equivalence expressed by "⊣ ⊢" and the notion of slef-extensionality.

Bibliography

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J.-Y.Beziau, “La Puissance du Symbole”, in La Pointure du Symbole, Petra, Paris, 2014, p.9-34.

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