I believe I once had a proof of this proposition, but it's been lost to the mists of time and old hard drives, so who knows if it was correct, and try as I might I can't seem to reproduce it.

Is it possible, in Melliès' tensorial logic, to give a proof of ¬(¬1 ⊗ ¬1) (a.k.a. 1 ⅋ 1)? Equivalently, is there an arrow in a dialogue category (with monoidal unit 1 and negation functor ¬) from ¬1 to 1?


Your notation is a bit confusing because usually in linear logic and related systems $\top$ denotes the unit of additive conjuction (i.e., categorically, the terminal object), but then when you formulate your question in categorical terms you clearly say that by $\top$ you mean the monoidal unit, which instead is traditionally denoted by $1$ (this is the notation used by Melliès, for instance in his habilitation thesis, see p.83-84).

So, the answer depends on what is $\top$. If it is the terminal object, then $\lnot(\lnot\top\otimes\lnot\top)$ is indeed provable; if it is the monoidal unit (which I am going to denote by $1$ here), then $\lnot(\lnot 1\otimes\lnot 1)$ is not provable.

In the former case, here's a proof: $$ \begin{array}{rcl} \lnot\top & \vdash & \top \\ \hline \lnot\top,\lnot\top & \vdash & \bot \\ \hline \lnot\top\otimes\lnot\top & \vdash & \bot \\ \hline & \vdash & \lnot(\lnot\top\otimes\lnot\top) \\ \end{array} $$

In the latter case, non-provability comes from the non-provability of $1⅋1$ in linear logic without mix. If you want to see it directly in tensorial logic, simply observe that, by cut-elimination (which holds in tensorial logic) and reversibility of the $\lnot$-right and $\otimes$-left rules, derivability of $\vdash\lnot(\lnot 1\otimes\lnot 1)$ is equivalent to derivability of $\lnot 1,\lnot 1\vdash\bot$, which may only come from a proof of $\lnot 1\vdash 1$, which is unprovable because no rule (except cut) admits such a sequent as its conclusion, as shown by a straightforward inspection of the various rules (cf. for instance p.84 of Melliès habilitation thesis).

  • $\begingroup$ Apologies for the notation. I should have said 1 — I've edited the question. $\endgroup$ – Twey Oct 19 '18 at 20:24
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    $\begingroup$ I have to say, this is very surprising to me in a game-semantic sense. Surely it's always possible to win the dialogue game represented by ¬(¬1 ⊗ ¬1). $\endgroup$ – Twey Oct 19 '18 at 20:51
  • $\begingroup$ I honestly don't know the exact definition of innocent sequential strategy (the one corresponding to tensorial logic) so I can't tell you what goes wrong. I'm pretty sure that it has to do with the fact that the two copies of $1$ are in parallel here, which is precisely the intuition behind the mix rule... but I don't know, it's a good question. $\endgroup$ – Damiano Mazza Oct 20 '18 at 19:11

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