Consider propositional logic with primitive connectives $\{{\to},{\land},{\lor},{\bot}\}$. We view $\neg \varphi$ as an abbreviation of $\varphi\to\bot$ and $\varphi\leftrightarrow\psi$ as an abbreviation of $(\varphi\to\psi)\land(\psi\to\varphi)$, etc.

The *classical sequent calculus* LK has rules such as

$$\frac{\Gamma, \varphi\vdash \Pi \qquad \Sigma, \psi\vdash \Pi} {\Gamma, \Sigma, \varphi\lor\psi \vdash \Pi}{\lor}L \qquad \frac{\Gamma\vdash\varphi,\Delta}{\Gamma\vdash\varphi\lor\psi,\Delta}{\lor}R_1 \qquad \frac{\Gamma \vdash \psi, \Delta}{\Gamma\vdash\varphi\lor\psi,\Delta}{\lor}R_2 $$ $$ \frac{\Gamma\vdash\varphi,\Delta \qquad \Sigma,\psi\vdash\Pi} {\Gamma, \Sigma, \varphi\to\psi \vdash \Delta,\Pi}{\to}L \qquad \frac{\Gamma,\varphi\vdash \psi,\Delta}{\Gamma\vdash\varphi\to\psi,\Delta}{\to}R $$ and so forth, where $\Gamma$, $\Sigma$, $\Pi$, and $\Delta$ are finite multisets of formulae.

It is well known that if we restrict the shape of all sequents to have exactly one formula to the right of the $\vdash$, we get a proof system LJ for *intuitionistic* propositional logic. This corresponds to requiring that every $\Delta$ is empty and every $\Pi$ is a singleton in the formulations of the rules above.

**Do we still get intuitionistic logic if the only rule we make this change to is ${\to}R$?** In other words, we have
$$\frac{\Gamma,\varphi\vdash\psi}{\Gamma\vdash\varphi\to\psi}{\to}R'$$
together with all of the other rules in their classical formulation, including structural rules with long $\Delta$s and $\Pi$s.

(Motivation: I'm trying, yet again, to wrap my head around what the *essential* difference between intuitionistic and classical logic is. It's often said that intuitionistic logic grew out of Brouwer's stricter demands on how a *disjunction* can be proved, but that can't be the whole story because there's a difference even for the implicational fragment (with no disjunctions). Now I'm wondering whether disjunction is actually part of the story at all. Kripke semantics treats it completely classically, and intuitionistic logic becomes fully classical if we add Peirce's law as an axiom (which again does not mention disjunction). The above conjecture is inspired by the Curry-Howard isomorphism where Peirce's law maps to `call/cc`

, more or less. Therefore it would make sense that we can get intuitionistic logic simply by forbidding lambda abstractions from capturing continuation variables.)