We develop a general criterion for cut elimination in sequent calculi for propositional modal logics, which rests on absorption of cut, contraction, weakening and inversion by the purely modal part of the rule system. Our criterion applies also to a wide variety of logics outside the realm of normal modal logic. We give extensive example instantiations of our framework to various conditional logics. For these, we obtain fully internalised calculi which are substantially simpler than those known in the literature, along with leaner proofs of cut elimination and complexity. In one case, conditional logic with modus ponens and conditional excluded middle, cut elimination and complexity were explicitly stated as open in the literature.
Cut elimination, originally conceived by Gentzen [7], is one of the core concepts of proof theory and plays a major role in particular for algorithmic aspects of logic, including the complexity of automated reasoning and, via interpolation, modularity issues. The large number of logical calculi that are currently in use, in particular in various areas of computer science, motivates efforts to define families of sequent calculi that cover a variety of logics and admit uniform proofs of cut elimination, enabled by suitable sufficient conditions. Here, we present such a method for modal sequent calculi that applies to possibly non-normal normal modal logics, which appear, e.g. in concurrency and knowledge representation. We use a separation of the modal calculi into a fixed underlying propositional part and a modal part. The core of our criterion, that we call the absorption of cut, stipulates that an application of the cut rule to conclusions of modal rules can be replaced by a single rule application. This concept generalises the notion of resolution closed rule set [14,18], dropping the assumption that the logic at hand is rank-1, i.e. axiomatised by formulas in which the nesting depth of modal operators is uniformly equal to 1 (such as K). where p ∈ V and ♥ ∈ Λ is n-ary. We use standard abbreviations of the other propositional connectives ⊤, ∨ and →. A Λ-sequent is a finite multiset of Λ-formulas, and the set of Λsequents is denoted by S(Λ). We write the multiset union of Γ and ∆ as Γ, ∆ and identify a formula A ∈ F(Λ) with the singleton sequent containing only A. If S ⊆ F(Λ) is a set of formulas, then an S-substitution is a mapping σ : V → S. We denote the result of uniformly substituting σ(p) for p in a formula A by Aσ. This extends pointwise to Λ-sequents so that Γσ = A 1 σ, . . . , A n σ if Γ = A 1 , . . . , A n . If S ⊆ F(Λ) is a set of Λ-formulas and A ∈ F(Λ), we say that A is a propositional consequence of S if there exist A 1 , . . . , A n ∈ S such that
A is a substitution instance of a propositional tautology. We write S ⊢ PL A if A is a propositional consequence of S and A ⊢ PL B for {A} ⊢ PL B for the case of single formulas.
To facilitate the task of comparing the notion of provability in both Hilbert and Gentzen type proof systems, we introduce the following notion of a proof rule that can be used, without any modifications, in both systems. Definition 3.1. A Λ-rule is of the form Γ 1 …Γn Γ 0
where n ≥ 0 and Γ 0 , . . . , Γ n are Λ-sequents. The sequents Γ 1 , . . . , Γ n are the premises of the rule and Γ 0 its conclusion. A rule Γ 0 without premises is called a Λ-axiom, which we denote by just its conclusion, Γ 0 . A rule set is just a set of Λ-rules, and we say that a rule set R is substitution closed, if Γ 1 σ . . . Γ n σ/Γ 0 σ ∈ R whenever Γ 1 . . . Γ n /Γ 0 ∈ R and σ : V → F(Λ) is a substitution.
In view of the sequent calculi that we introduce later, we read sequents disjunctively. Consequently, a rule Γ 1 . . . , Γ n /Γ 0 can be used to prove the disjunction of Γ 0 , provided that Γ i is provable, for all 1 ≤ i ≤ n. We emphasise that a rule is an expression of the object language, i.e. it does not contain meta-linguistic variables. As such, it represents a specific deduction step rather than a family of possible deductions, which helps to economise on syntactic categories. In our examples, concrete rule sets are presented as instances of rule schemas.
Example 3.2. For the modal logics K, K4 and T , we fix the modal signature Λ = { } consisting of a single modal operator with arity one. The language of conditional logic is given by the similarity type Λ = {⇒} where the conditional arrow ⇒ has arity 2. We use infix notation and write A ⇒ B instead of ⇒ (A, B) for A, B ∈ F(Λ). Formulas A ⇒ B are interpreted as various forms of conditionals, e.g. default implication ‘if A then normally B’, relevant implication and others, depending on the choice of semantics and imposed logical principles. Deduction over modal and conditional logics are governed by the following rule sets:
Figure 2: Axioms and Rules of conditional Hilbert Systems (1) The rule set K associated to the modal logic K consists of all instances of the necessitation rule (N) and the distribution axiom (D) in Figure 1. The rule sets that axiomatise the logics T and K4 arise by extending this set with the reflexivity axiom (R) and the (4)-axiom, respectively. We reserve the name (T) for the reflexivity rule in a cut-free system. (2) The basic conditional logic is the system CK of [3], axiomatised by the rule set that consists of all instances of (RCEA) and (RCK) in Figure 2. The system CK constitutes a minimal set of properties to be reasonably expected of any conditional, however nonstandard: replacement of equivalents in the left hand argument, and compatibility with conjunction in the right-hand argument. Additional properties are typically imposed when more specific interpretations of ⇒ are intended. E.g. the basic prop
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