 It has been a considerable challenge to come up with analytic proof systems for fuzzy logics. These are systems that share important features, like the eliminability of cuts and the subformula property, with Gentzen’s sequent calculi for classical and intuitionistic logic (see entry on  the development of proof theory).  A major breakthrough has been achieved with the introduction of a hypersequent calculus for Gödel–Dummett logic by Arnon Avron (1991). Hypersequent calculi arise from sequent calculi by defining inference rules that refer to finite multisets of sequents, rather than to single sequents. Hypersequents are interpreted, at the meta-level, as disjunctions of sequents. This interpretation implies that it is sound to add additional sequents (external weakening) or to replace multiple copies of the same sequent within a given hypersequent by a single copy of that sequent (external contraction). In the case of Gödel–Dummett logic, the rules for introducing logical connectives arise from the rules of Gentzen’s intuitionistic sequent calculus by simply adding side-hypersequents the upper and lower sequents of the original rule. For example, the sequent rule for introducing conjunction on the right-hand side   \[\frac{\Gamma_1 \Rightarrow \phi \hspace{3ex} \Gamma_2 \Rightarrow \psi}{\Gamma_1,\Gamma_2 \Rightarrow \phi \wedge \psi}\]   where \(\Gamma_1\) and \(\Gamma_2\) are finite sequences of formulas, is turned into the following hypersequent rule:   \[\frac{H \mid \Gamma_1 \Rightarrow \phi \hspace{3ex} H' \mid \Gamma_2 \Rightarrow \psi}{H \mid H' \mid \Gamma_1,\Gamma_2 \Rightarrow \phi \wedge \psi }\]   where \(H\) and \(H'\) denote the side-hypersequents, i.e., finite multisets of sequents added disjunctively to the exhibited sequents. This by itself does not change the corresponding logic (intuitionistic logic, in this case). The crucial additional structural rule is the so-called communication rule:   \[\frac{H \mid \Gamma_1,\Pi_1 \Rightarrow \Delta_1 \hspace{3ex} H' \mid \Gamma_2,\Pi_2 \Rightarrow \Delta_2}{H \mid H' \mid \Gamma_1,\Gamma_2 \Rightarrow \Delta_1 \mid \Pi_2,\Pi_2 \Rightarrow \Delta_2}\]   Here \(\Gamma_1, \Gamma_2,\Pi_1, \Pi_2\) are finite sequences of formulas; \(\Delta_1\) and \(\Delta_2\) are either single formulas or remain empty; \(H\) and \(H'\) denote the side-hypersequents, like above. To understand the impact of the communication rule, we present a derivation of the crucial axiom \((\varphi \to \psi) \lor (\psi \to \varphi)\), which features an application of this rule immediately below the intial (hyper)sequents:  \[ \frac{\displaystyle\frac{\displaystyle\frac{\displaystyle\frac{\displaystyle\frac{\displaystyle\frac{\varphi \Rightarrow \varphi \hspace{5ex} \psi \Rightarrow \psi}{\varphi \Rightarrow \psi \mid \psi \Rightarrow \varphi}}{\Rightarrow \varphi \to \psi \mid \psi \Rightarrow \varphi}}{\Rightarrow \varphi \to \psi \mid\, \Rightarrow  \psi \to \varphi}}{\Rightarrow (\varphi \to \psi) \vee (\psi \to \varphi) \mid\, \Rightarrow \psi \to \varphi}}{\Rightarrow (\varphi \to \psi) \vee (\psi \to \varphi) \mid\, \Rightarrow  (\varphi \to \psi) \vee (\psi \to \varphi)}}{\Rightarrow (\varphi \to \psi) \lor (\psi \to \varphi)} \]   