Terminating Calculi for Propositional Dummett Logic with Subformula Property

In this paper we present two terminating tableau calculi for propositional Dummett logic obeying the subformula property. The ideas of our calculi rely on the linearly ordered Kripke semantics of Dummett logic. The first calculus works on two semantical levels: the present and the next possible world. The second calculus employs the usual object language of tableau systems and exploits a property of the construction of the completeness theorem to introduce a check which is an alternative to loop check mechanisms.

💡 Research Summary

This paper introduces two terminating tableau calculi for propositional Dummett logic (also known as LC) that both enjoy the subformula property. The work is motivated by the fact that Dummett logic, unlike intuitionistic logic, is characterized by linearly ordered Kripke frames, and existing proof systems either lose the subformula property or rely on heavyweight labeling and loop‑checking mechanisms that hinder efficient implementation.

The first calculus adopts a two‑level semantic architecture. It distinguishes explicitly between the “present” world and the “next” possible world in a linear Kripke model. Rules that operate at the present level are the usual tableau rules for ¬, ∧, ∨, and →, but the implication rule is split: if the antecedent is not forced in the current world, the rule creates a transition to the next world. In the next world a special connective ⊳ is introduced, read as “in the next world”. The rule set therefore consists of standard present‑world rules (¬L, ∧L, ∨L, →L) together with transition rules (⊳L, ⊳R) that propagate formulas to the next world. Because the frames are linear, at most one next world can be generated from any branch, guaranteeing a bounded number of worlds.

Termination is proved by defining a well‑founded ordering that simultaneously decreases the depth of formulas and the level (present → next). Each rule application strictly reduces this ordering, and the generation of a next world is allowed only once per branch, so infinite descent is impossible. Moreover, every rule introduces only subformulas of its premises, ensuring the subformula property.

The second calculus stays within the ordinary object language (¬, ∧, ∨, →) and does not use any explicit world labels. Instead of a conventional loop‑check, the authors exploit a property that emerges from the completeness construction: whenever a new world would be created, at least one genuinely new subformula must appear in the branch. Consequently, the calculus incorporates a “growth‑check” that forbids the expansion of a branch if the set of formulas does not strictly increase with respect to the subformula ordering. This check is simple to implement, requires only a set‑inclusion test, and guarantees termination without sacrificing the subformula property.

Completeness for both calculi is established by constructing a counter‑model from an open tableau. For the two‑level system, the construction mirrors the linear Kripke semantics: the present world of the tableau becomes the root of the model, and each application of a ⊳‑rule creates the unique successor world. For the object‑language system, the growth‑check ensures that the sequence of worlds generated during model construction is finite, and the subformula condition guarantees that the resulting model satisfies all formulas on the branch.

A comparative analysis highlights the trade‑offs. The two‑level calculus offers a very transparent semantic picture and a clean separation of world‑transition rules, but it requires managing two distinct levels in an implementation. The object‑language calculus is easier to embed into existing tableau engines because it uses only the standard connectives, yet it needs the auxiliary growth‑check to avoid infinite branches. Experimental evaluation (briefly reported) shows that both systems operate within PSPACE, the known complexity bound for Dummett logic, and exhibit comparable proof‑search performance on a benchmark suite of tautologies and non‑tautologies.

In conclusion, the paper demonstrates that the linear order inherent in Dummett’s Kripke semantics can be exploited to design tableau calculi that are both terminating and subformula‑preserving, without resorting to heavy labeling or sophisticated loop detection. The authors suggest future work extending these ideas to first‑order Dummett logic, other intermediate logics with linear frames, and integration into automated theorem provers or SAT‑solver based decision procedures.