Negative Results in Proof Theory Workshop 2018 The Proof Society - PowerPoint PPT Presentation

method For a class of proof systems PS and a regular property RP of logics establish theorems of the form: If a logic has a proof system in PS , then it has regular property RP . Or, equivalently, If a logic does not have RP , then it does not have a proof system in PS . The strength of the method depends on the size of the class PS and the frequency with which RP occurs among the considered logics. In this talk: ◦ the logics are intermediate, modal, and intuitionistic modal logics; ◦ the proof systems are abstract versions of sequent calculi. ◦ the regular property is uniform interpolation. 6 / 21

method For a class of proof systems PS and a regular property RP of logics establish theorems of the form: If a logic has a proof system in PS , then it has regular property RP . Or, equivalently, If a logic does not have RP , then it does not have a proof system in PS . The strength of the method depends on the size of the class PS and the frequency with which RP occurs among the considered logics. In this talk: ◦ the logics are intermediate, modal, and intuitionistic modal logics; ◦ the proof systems are abstract versions of sequent calculi. ◦ the regular property is uniform interpolation. Side benefit: Uniform interpolation in a uniform, modular way, and for new logics. 6 / 21

uniform interpolation Dfn A logic L has (Craig) interpolation if whenever ⊢ ϕ → ψ there is a χ in the common language L ( ϕ ) ∩ L ( ψ ) such that ⊢ ϕ → χ and ⊢ χ → ψ . 7 / 21

uniform interpolation Dfn A logic L has (Craig) interpolation if whenever ⊢ ϕ → ψ there is a χ in the common language L ( ϕ ) ∩ L ( ψ ) such that ⊢ ϕ → χ and ⊢ χ → ψ . A propositional (modal) logic has uniform interpolation if the interpolant depends only on the premiss or the conclusion: For all ϕ there are formulas ∃ p ϕ and ∀ p ϕ not containing p and no atoms not in ϕ such that for all ψ not containing p: ⊢ ψ → ϕ iff ⊢ ψ → ∀ p ϕ ⊢ ϕ → ψ iff ⊢ ∃ p ϕ → ψ. 7 / 21

uniform interpolation Dfn A logic L has (Craig) interpolation if whenever ⊢ ϕ → ψ there is a χ in the common language L ( ϕ ) ∩ L ( ψ ) such that ⊢ ϕ → χ and ⊢ χ → ψ . A propositional (modal) logic has uniform interpolation if the interpolant depends only on the premiss or the conclusion: For all ϕ there are formulas ∃ p ϕ and ∀ p ϕ not containing p and no atoms not in ϕ such that for all ψ not containing p: ⊢ ψ → ϕ iff ⊢ ψ → ∀ p ϕ ⊢ ϕ → ψ iff ⊢ ∃ p ϕ → ψ. ∃ p ϕ is the right interpolant and ∀ p ϕ the left interpolant: ⊢ ϕ → ∃ p ϕ ⊢ ∀ p ϕ → ϕ. 7 / 21

uniform interpolation Dfn A logic L has (Craig) interpolation if whenever ⊢ ϕ → ψ there is a χ in the common language L ( ϕ ) ∩ L ( ψ ) such that ⊢ ϕ → χ and ⊢ χ → ψ . A propositional (modal) logic has uniform interpolation if the interpolant depends only on the premiss or the conclusion: For all ϕ there are formulas ∃ p ϕ and ∀ p ϕ not containing p and no atoms not in ϕ such that for all ψ not containing p: ⊢ ψ → ϕ iff ⊢ ψ → ∀ p ϕ ⊢ ϕ → ψ iff ⊢ ∃ p ϕ → ψ. ∃ p ϕ is the right interpolant and ∀ p ϕ the left interpolant: ⊢ ϕ → ∃ p ϕ ⊢ ∀ p ϕ → ϕ. Note Uniform interpolation implies interpolation: the interpolant is ∃ p 1 . . . p n ϕ , where p 1 , . . . , p n are the atoms that occur in ϕ but not in ψ . 7 / 21

uniform interpolation in modal and intermediate logics Theorem (Pitts ’92) IPC has uniform interpolation. 8 / 21

uniform interpolation in modal and intermediate logics Theorem (Pitts ’92) IPC has uniform interpolation. (this was the inspiration for our approach) 8 / 21

uniform interpolation in modal and intermediate logics Theorem (Pitts ’92) IPC has uniform interpolation. (this was the inspiration for our approach) Theorem (Shavrukov ’94) GL has uniform interpolation. 8 / 21

uniform interpolation in modal and intermediate logics Theorem (Pitts ’92) IPC has uniform interpolation. (this was the inspiration for our approach) Theorem (Shavrukov ’94) GL has uniform interpolation. Theorem (Ghilardi & Zawadowski ’95) K has uniform interpolation. S 4 does not. 8 / 21

uniform interpolation in modal and intermediate logics Theorem (Pitts ’92) IPC has uniform interpolation. (this was the inspiration for our approach) Theorem (Shavrukov ’94) GL has uniform interpolation. Theorem (Ghilardi & Zawadowski ’95) K has uniform interpolation. S 4 does not. Theorem (Bilkova ’06) KT and Grz have uniform interpolation. K 4 does not. 8 / 21

uniform interpolation in modal and intermediate logics Theorem (Pitts ’92) IPC has uniform interpolation. (this was the inspiration for our approach) Theorem (Shavrukov ’94) GL has uniform interpolation. Theorem (Ghilardi & Zawadowski ’95) K has uniform interpolation. S 4 does not. Theorem (Bilkova ’06) KT and Grz have uniform interpolation. K 4 does not. Theorem (Maxsimova ’77, Ghilardi & Zawadowski ’02) There are exactly seven intermediate logics with (uniform) interpolation: IPC , Sm, GSc, LC, KC, Bd 2 , CPC . 8 / 21

uniform interpolation in modal and intermediate logics Theorem (Pitts ’92) IPC has uniform interpolation. (this was the inspiration for our approach) Theorem (Shavrukov ’94) GL has uniform interpolation. Theorem (Ghilardi & Zawadowski ’95) K has uniform interpolation. S 4 does not. Theorem (Bilkova ’06) KT and Grz have uniform interpolation. K 4 does not. Theorem (Maxsimova ’77, Ghilardi & Zawadowski ’02) There are exactly seven intermediate logics with (uniform) interpolation: IPC , Sm, GSc, LC, KC, Bd 2 , CPC . Theorem (Maxsimova ’79) Among the normal extensions of S 4 there are at least 31 and at most 37 logics with interpolation. 8 / 21

uniform interpolation in modal and intermediate logics Theorem (Pitts ’92) IPC has uniform interpolation. (this was the inspiration for our approach) Theorem (Shavrukov ’94) GL has uniform interpolation. Theorem (Ghilardi & Zawadowski ’95) K has uniform interpolation. S 4 does not. Theorem (Bilkova ’06) KT and Grz have uniform interpolation. K 4 does not. Theorem (Maxsimova ’77, Ghilardi & Zawadowski ’02) There are exactly seven intermediate logics with (uniform) interpolation: IPC , Sm, GSc, LC, KC, Bd 2 , CPC . Theorem (Maxsimova ’79) Among the normal extensions of S 4 there are at least 31 and at most 37 logics with interpolation. Pitts uses a terminating sequent calculus for IPC . (developed independently by Dyckhoff and Hudelmaier in ’92) 8 / 21

aim In the case of (intuitionistic) modal and intermediate logic, isolate a (large) class of proof systems and prove that any logic with a proof system in that class has uniform interpolation. 9 / 21

aim In the case of (intuitionistic) modal and intermediate logic, isolate a (large) class of proof systems and prove that any logic with a proof system in that class has uniform interpolation. Since uniform interpolation is rare among modal and intermediate logics, this establishes the negative result (not having a proof system in that class) for many such logics. 9 / 21

aim In the case of (intuitionistic) modal and intermediate logic, isolate a (large) class of proof systems and prove that any logic with a proof system in that class has uniform interpolation. Since uniform interpolation is rare among modal and intermediate logics, this establishes the negative result (not having a proof system in that class) for many such logics. The method also provide a uniform and modular way to prove uniform interpolation for classes of logics, including some logics for which this was unknown, such as KD . 9 / 21

aim In the case of (intuitionistic) modal and intermediate logic, isolate a (large) class of proof systems and prove that any logic with a proof system in that class has uniform interpolation. Since uniform interpolation is rare among modal and intermediate logics, this establishes the negative result (not having a proof system in that class) for many such logics. The method also provide a uniform and modular way to prove uniform interpolation for classes of logics, including some logics for which this was unknown, such as KD . The class of proof systems is defined not in terms of concrete rules but in terms of the structural properties of rules. 9 / 21

aim In the case of (intuitionistic) modal and intermediate logic, isolate a (large) class of proof systems and prove that any logic with a proof system in that class has uniform interpolation. Since uniform interpolation is rare among modal and intermediate logics, this establishes the negative result (not having a proof system in that class) for many such logics. The method also provide a uniform and modular way to prove uniform interpolation for classes of logics, including some logics for which this was unknown, such as KD . The class of proof systems is defined not in terms of concrete rules but in terms of the structural properties of rules. In this talk: classical modal logic with one modal operator. 9 / 21

the proof systems Dfn The language consists of ⊥ , ∧ , ∨ , → , ✷ , p 1 , p 2 , . . . . 10 / 21

the proof systems Dfn The language consists of ⊥ , ∧ , ∨ , → , ✷ , p 1 , p 2 , . . . . A sequent is an expression (Γ ⇒ ∆) , where Γ and ∆ are multisets. 10 / 21

the proof systems Dfn The language consists of ⊥ , ∧ , ∨ , → , ✷ , p 1 , p 2 , . . . . A sequent is an expression (Γ ⇒ ∆) , where Γ and ∆ are multisets. ✷ Γ ≡ df { ✷ ϕ | ϕ ∈ Γ } ✷ (Γ ⇒ ∆) ≡ df ( ✷ Γ ⇒ ✷ ∆) (Γ ⇒ ∆) · (Π ⇒ Σ) ≡ df (Γ , Π ⇒ ∆ , Σ) . 10 / 21

the proof systems Dfn The language consists of ⊥ , ∧ , ∨ , → , ✷ , p 1 , p 2 , . . . . A sequent is an expression (Γ ⇒ ∆) , where Γ and ∆ are multisets. ✷ Γ ≡ df { ✷ ϕ | ϕ ∈ Γ } ✷ (Γ ⇒ ∆) ≡ df ( ✷ Γ ⇒ ✷ ∆) (Γ ⇒ ∆) · (Π ⇒ Σ) ≡ df (Γ , Π ⇒ ∆ , Σ) . Dfn A sequent calculus is a set of rules, where a rule R is an expression of the form S 1 . . . S n R S 0 ( rl) for certain sequents S 0 , . . . , S n (that may be empty). An instance R of a rule is of the form σ S 1 . . . σ S n R σ S 0 where σ is a substitution for the modal language. 10 / 21

the proof systems Dfn The language consists of ⊥ , ∧ , ∨ , → , ✷ , p 1 , p 2 , . . . . A sequent is an expression (Γ ⇒ ∆) , where Γ and ∆ are multisets. ✷ Γ ≡ df { ✷ ϕ | ϕ ∈ Γ } ✷ (Γ ⇒ ∆) ≡ df ( ✷ Γ ⇒ ✷ ∆) (Γ ⇒ ∆) · (Π ⇒ Σ) ≡ df (Γ , Π ⇒ ∆ , Σ) . Dfn A sequent calculus is a set of rules, where a rule R is an expression of the form S 1 . . . S n R S 0 ( rl) for certain sequents S 0 , . . . , S n (that may be empty). An instance R of a rule is of the form σ S 1 . . . σ S n R σ S 0 where σ is a substitution for the modal language. Dfn Rule (rl) is unary if S 0 contains a single nonboxed formula and all atoms in the premisses occur in S 0 , and thinnable (closed under weakening) if for every instance R = ( S ′ 1 . . . S ′ n / S ′ 0 ) and sequent S the following is an instance of R : S · S ′ . . . S · S ′ 1 n R ( S ) S · S ′ 0 10 / 21

the proof system G 3 Dfn In a unary thinnable (ut) rule the conclusion consists of a single, nonboxed formula and for every instance R and sequent S, R ( S ) is an instance of the rule. 11 / 21

the proof system G 3 Dfn In a unary thinnable (ut) rule the conclusion consists of a single, nonboxed formula and for every instance R and sequent S, R ( S ) is an instance of the rule. Dfn All rules in G 3 that are not axioms are thinnable and unary: Γ ⇒ ϕ, ∆ Γ , ϕ ⇒ ∆ Γ , ¬ ϕ ⇒ ∆ Γ ⇒ ¬ ϕ, ∆ Γ , ϕ ⇒ ∆ Γ , ψ ⇒ ∆ Γ ⇒ ϕ, ψ, ∆ Γ , ϕ ∨ ψ ⇒ ∆ Γ ⇒ ϕ ∨ ψ, ∆ . . . 11 / 21

the proof system G 3 Dfn In a unary thinnable (ut) rule the conclusion consists of a single, nonboxed formula and for every instance R and sequent S, R ( S ) is an instance of the rule. Dfn All rules in G 3 that are not axioms are thinnable and unary: Γ ⇒ ϕ, ∆ Γ , ϕ ⇒ ∆ Γ , ¬ ϕ ⇒ ∆ Γ ⇒ ¬ ϕ, ∆ Γ , ϕ ⇒ ∆ Γ , ψ ⇒ ∆ Γ ⇒ ϕ, ψ, ∆ Γ , ϕ ∨ ψ ⇒ ∆ Γ ⇒ ϕ ∨ ψ, ∆ . . . The canonical rules of (Avron ’08) are instances of unary thinnable rules. 11 / 21

the proof system G 3 Dfn In a unary thinnable (ut) rule the conclusion consists of a single, nonboxed formula and for every instance R and sequent S, R ( S ) is an instance of the rule. Dfn All rules in G 3 that are not axioms are thinnable and unary: Γ ⇒ ϕ, ∆ Γ , ϕ ⇒ ∆ Γ , ¬ ϕ ⇒ ∆ Γ ⇒ ¬ ϕ, ∆ Γ , ϕ ⇒ ∆ Γ , ψ ⇒ ∆ Γ ⇒ ϕ, ψ, ∆ Γ , ϕ ∨ ψ ⇒ ∆ Γ ⇒ ϕ ∨ ψ, ∆ . . . The canonical rules of (Avron ’08) are instances of unary thinnable rules. Dfn A calculus is terminating if there is a well-founded order on sequents such that in every rule the premisses come before the conclusion, and . . . 11 / 21

the proof system G 3 Dfn In a unary thinnable (ut) rule the conclusion consists of a single, nonboxed formula and for every instance R and sequent S, R ( S ) is an instance of the rule. Dfn All rules in G 3 that are not axioms are thinnable and unary: Γ ⇒ ϕ, ∆ Γ , ϕ ⇒ ∆ Γ , ¬ ϕ ⇒ ∆ Γ ⇒ ¬ ϕ, ∆ Γ , ϕ ⇒ ∆ Γ , ψ ⇒ ∆ Γ ⇒ ϕ, ψ, ∆ Γ , ϕ ∨ ψ ⇒ ∆ Γ ⇒ ϕ ∨ ψ, ∆ . . . The canonical rules of (Avron ’08) are instances of unary thinnable rules. Dfn A calculus is terminating if there is a well-founded order on sequents such that in every rule the premisses come before the conclusion, and . . . In general, the cut rule does not belong to a terminating calculus: Γ ⇒ ϕ, ∆ Γ , ϕ ⇒ ∆ Γ ⇒ ∆ 11 / 21

the proof systems for modal logic Dfn A nonaxiom rule R = ( S 1 . . . S n / S 0 ) is unary and thinnable if S 0 contains a single, nonboxed formula and for every instance R and sequent S, R ( S ) is an instance of the rule. 12 / 21

the proof systems for modal logic Dfn A nonaxiom rule R = ( S 1 . . . S n / S 0 ) is unary and thinnable if S 0 contains a single, nonboxed formula and for every instance R and sequent S, R ( S ) is an instance of the rule. Unary thinnable axioms are: (Γ , p ⇒ p , ∆) (Γ , ⊥ ⇒ ∆) (Γ ⇒ ⊤ , ∆) . 12 / 21

the proof systems for modal logic Dfn A nonaxiom rule R = ( S 1 . . . S n / S 0 ) is unary and thinnable if S 0 contains a single, nonboxed formula and for every instance R and sequent S, R ( S ) is an instance of the rule. Unary thinnable axioms are: (Γ , p ⇒ p , ∆) (Γ , ⊥ ⇒ ∆) (Γ ⇒ ⊤ , ∆) . A unary thinnable modal rule is of the form ◦ S 1 · S 0 S 2 · ✷ S 1 · ✷ S 0 R where S 0 contains a single formula, that is boxed, S 2 is of the form (Π ⇒ ∆) , S 1 contains only multisets, and ◦ S 1 denotes S 1 or � S 1 . 12 / 21

the proof systems for modal logic Dfn A nonaxiom rule R = ( S 1 . . . S n / S 0 ) is unary and thinnable if S 0 contains a single, nonboxed formula and for every instance R and sequent S, R ( S ) is an instance of the rule. Unary thinnable axioms are: (Γ , p ⇒ p , ∆) (Γ , ⊥ ⇒ ∆) (Γ ⇒ ⊤ , ∆) . A unary thinnable modal rule is of the form ◦ S 1 · S 0 S 2 · ✷ S 1 · ✷ S 0 R where S 0 contains a single formula, that is boxed, S 2 is of the form (Π ⇒ ∆) , S 1 contains only multisets, and ◦ S 1 denotes S 1 or � S 1 . Example Unary thinnable (modal) rules: Γ ⇒ ϕ, ψ, ∆ Γ ⇒ ϕ Γ , ϕ ⇒ Π , ✷ Γ ⇒ ✷ ϕ, ∆ R K Π , ✷ Γ , ✷ ϕ ⇒ ∆ R D Γ ⇒ ϕ ∨ ψ, ∆ 12 / 21

the proof systems for modal logic Dfn A nonaxiom rule R = ( S 1 . . . S n / S 0 ) is unary and thinnable if S 0 contains a single, nonboxed formula and for every instance R and sequent S, R ( S ) is an instance of the rule. Unary thinnable axioms are: (Γ , p ⇒ p , ∆) (Γ , ⊥ ⇒ ∆) (Γ ⇒ ⊤ , ∆) . A unary thinnable modal rule is of the form ◦ S 1 · S 0 S 2 · ✷ S 1 · ✷ S 0 R where S 0 contains a single formula, that is boxed, S 2 is of the form (Π ⇒ ∆) , S 1 contains only multisets, and ◦ S 1 denotes S 1 or � S 1 . Example Unary thinnable (modal) rules: Γ ⇒ ϕ, ψ, ∆ Γ ⇒ ϕ Γ , ϕ ⇒ Π , ✷ Γ ⇒ ✷ ϕ, ∆ R K Π , ✷ Γ , ✷ ϕ ⇒ ∆ R D Γ ⇒ ϕ ∨ ψ, ∆ Example Rules that are not unary (modal): Γ , ψ → χ ⇒ ϕ → ψ Γ , χ ⇒ ∆ � Γ , ✷ ϕ ⇒ ϕ Π , ✷ Γ ⇒ ✷ ϕ, ∆ R GL Γ , ( ϕ → ψ ) → χ ⇒ ∆ 12 / 21

results for modal logic Theorem (Iemhoff 2016) A logic with a terminating calculus that consists of unary thinnable (modal) rules has uniform interpolation. 13 / 21

results for modal logic Theorem (Iemhoff 2016) A logic with a terminating calculus that consists of unary thinnable (modal) rules has uniform interpolation. Corollary (well-known) Classical propositional logic CPC has uniform interpolation. 13 / 21

results for modal logic Theorem (Iemhoff 2016) A logic with a terminating calculus that consists of unary thinnable (modal) rules has uniform interpolation. Corollary (well-known) Classical propositional logic CPC has uniform interpolation. Proof 13 / 21

results for modal logic Theorem (Iemhoff 2016) A logic with a terminating calculus that consists of unary thinnable (modal) rules has uniform interpolation. Corollary (well-known) Classical propositional logic CPC has uniform interpolation. Proof All rules in the sequent calculus G 3 are unary and thinnable. ⊣ 13 / 21

results for modal logic Theorem (Iemhoff 2016) A logic with a terminating calculus that consists of unary thinnable (modal) rules has uniform interpolation. Corollary (well-known) Classical propositional logic CPC has uniform interpolation. Proof All rules in the sequent calculus G 3 are unary and thinnable. ⊣ Corollary The modal logics K (Ghilardi) and KD (Iemhoff) have uniform interpolation. 13 / 21

results for modal logic Theorem (Iemhoff 2016) A logic with a terminating calculus that consists of unary thinnable (modal) rules has uniform interpolation. Corollary (well-known) Classical propositional logic CPC has uniform interpolation. Proof All rules in the sequent calculus G 3 are unary and thinnable. ⊣ Corollary The modal logics K (Ghilardi) and KD (Iemhoff) have uniform interpolation. A promised negative result: 13 / 21

results for modal logic Theorem (Iemhoff 2016) A logic with a terminating calculus that consists of unary thinnable (modal) rules has uniform interpolation. Corollary (well-known) Classical propositional logic CPC has uniform interpolation. Proof All rules in the sequent calculus G 3 are unary and thinnable. ⊣ Corollary The modal logics K (Ghilardi) and KD (Iemhoff) have uniform interpolation. A promised negative result: Corollary If a modal logic does not have uniform interpolation, then it does not have a terminating calculus that consists of unary thinnable (modal) rules. 13 / 21

results for modal logic Theorem (Iemhoff 2016) A logic with a terminating calculus that consists of unary thinnable (modal) rules has uniform interpolation. Corollary (well-known) Classical propositional logic CPC has uniform interpolation. Proof All rules in the sequent calculus G 3 are unary and thinnable. ⊣ Corollary The modal logics K (Ghilardi) and KD (Iemhoff) have uniform interpolation. A promised negative result: Corollary If a modal logic does not have uniform interpolation, then it does not have a terminating calculus that consists of unary thinnable (modal) rules. Examples are K 4 and S 4. 13 / 21

results for modal logic Theorem (Iemhoff 2016) A logic with a terminating calculus that consists of unary thinnable (modal) rules has uniform interpolation. Corollary (well-known) Classical propositional logic CPC has uniform interpolation. Proof All rules in the sequent calculus G 3 are unary and thinnable. ⊣ Corollary The modal logics K (Ghilardi) and KD (Iemhoff) have uniform interpolation. A promised negative result: Corollary If a modal logic does not have uniform interpolation, then it does not have a terminating calculus that consists of unary thinnable (modal) rules. Examples are K 4 and S 4. Interplay: Semantics (algebraic logic) and proof theory. 13 / 21

so far Aim: Isolate a (large) class of proof systems and prove that any (intuitionistic) modal and intermediate logic with a proof system in that class has uniform interpolation. 14 / 21

so far Aim: Isolate a (large) class of proof systems and prove that any (intuitionistic) modal and intermediate logic with a proof system in that class has uniform interpolation. Side benefit: Establishing uniform interpolation in a uniform, modular way, and for new logics. 14 / 21

so far Aim: Isolate a (large) class of proof systems and prove that any (intuitionistic) modal and intermediate logic with a proof system in that class has uniform interpolation. Side benefit: Establishing uniform interpolation in a uniform, modular way, and for new logics. So far: a uniform way to prove uniform interpolation for modal logics, where the proof systems consist of unary thinnable modal rules. 14 / 21

so far Aim: Isolate a (large) class of proof systems and prove that any (intuitionistic) modal and intermediate logic with a proof system in that class has uniform interpolation. Side benefit: Establishing uniform interpolation in a uniform, modular way, and for new logics. So far: a uniform way to prove uniform interpolation for modal logics, where the proof systems consist of unary thinnable modal rules. To come: ◦ extend the method to intermediate and intuitionistic modal logics, ◦ explain the proof method, in particular its modularity. 14 / 21

proof method Theorem A modal logic with a terminating calculus that consists of unary thinnable modal rules has uniform interpolation. 15 / 21

proof method Theorem A modal logic with a terminating calculus that consists of unary thinnable modal rules has uniform interpolation. Proof 15 / 21

proof method Theorem A modal logic with a terminating calculus that consists of unary thinnable modal rules has uniform interpolation. Proof idea 15 / 21

proof method Theorem A modal logic with a terminating calculus that consists of unary thinnable modal rules has uniform interpolation. Proof idea Define for each instance R of a rule in the calculus and each sequent S an expression ∀ R pS. E.g, if R is an instance of a unary thinnable rule: R R = ( S 1 . . . S n / S 0 ) ∀ pS 0 ≡ df ∀ pS 1 ∧ . . . ∧ ∀ pS n . 15 / 21

proof method Theorem A modal logic with a terminating calculus that consists of unary thinnable modal rules has uniform interpolation. Proof idea Define for each instance R of a rule in the calculus and each sequent S an expression ∀ R pS. E.g, if R is an instance of a unary thinnable rule: R R = ( S 1 . . . S n / S 0 ) ∀ pS 0 ≡ df ∀ pS 1 ∧ . . . ∧ ∀ pS n . � {∀ R Inductively define ∀ pS ≡ df pS | R a rule instance with conclusion S } . 15 / 21

proof method Theorem A modal logic with a terminating calculus that consists of unary thinnable modal rules has uniform interpolation. Proof idea Define for each instance R of a rule in the calculus and each sequent S an expression ∀ R pS. E.g, if R is an instance of a unary thinnable rule: R R = ( S 1 . . . S n / S 0 ) ∀ pS 0 ≡ df ∀ pS 1 ∧ . . . ∧ ∀ pS n . � {∀ R Inductively define ∀ pS ≡ df pS | R a rule instance with conclusion S } . For free sequents S, ∀ pS is defined separately. 15 / 21

proof method Theorem A modal logic with a terminating calculus that consists of unary thinnable modal rules has uniform interpolation. Proof idea Define for each instance R of a rule in the calculus and each sequent S an expression ∀ R pS. E.g, if R is an instance of a unary thinnable rule: R R = ( S 1 . . . S n / S 0 ) ∀ pS 0 ≡ df ∀ pS 1 ∧ . . . ∧ ∀ pS n . � {∀ R Inductively define ∀ pS ≡ df pS | R a rule instance with conclusion S } . For free sequents S, ∀ pS is defined separately. Prove with induction along the order that for any rule in the calculus, if the premisses of a rule have a uniform interpolant, then so does the conclusion. 15 / 21

proof method Theorem A modal logic with a terminating calculus that consists of unary thinnable modal rules has uniform interpolation. Proof idea Define for each instance R of a rule in the calculus and each sequent S an expression ∀ R pS. E.g, if R is an instance of a unary thinnable rule: R R = ( S 1 . . . S n / S 0 ) ∀ pS 0 ≡ df ∀ pS 1 ∧ . . . ∧ ∀ pS n . � {∀ R Inductively define ∀ pS ≡ df pS | R a rule instance with conclusion S } . For free sequents S, ∀ pS is defined separately. Prove with induction along the order that for any rule in the calculus, if the premisses of a rule have a uniform interpolant, then so does the conclusion. Some details are omitted . . . ⊣ 15 / 21

proof method Theorem A modal logic with a terminating calculus that consists of unary thinnable modal rules has uniform interpolation. Proof idea Define for each instance R of a rule in the calculus and each sequent S an expression ∀ R pS. E.g, if R is an instance of a unary thinnable rule: R R = ( S 1 . . . S n / S 0 ) ∀ pS 0 ≡ df ∀ pS 1 ∧ . . . ∧ ∀ pS n . � {∀ R Inductively define ∀ pS ≡ df pS | R a rule instance with conclusion S } . For free sequents S, ∀ pS is defined separately. Prove with induction along the order that for any rule in the calculus, if the premisses of a rule have a uniform interpolant, then so does the conclusion. Some details are omitted . . . ⊣ Uniform and modular proof. 15 / 21

intermediate logic Similar to the classical case, but far more complicated: ∃ p and ∀ p. 16 / 21

intermediate logic Similar to the classical case, but far more complicated: ∃ p and ∀ p. One needs a terminating calculus for IPC . 16 / 21

intermediate logic Similar to the classical case, but far more complicated: ∃ p and ∀ p. One needs a terminating calculus for IPC . Use G 4 i by Dyckhoff and Hudelmaier. Not all rules of G 4 i are focussed. 16 / 21

intermediate logic Similar to the classical case, but far more complicated: ∃ p and ∀ p. One needs a terminating calculus for IPC . Use G 4 i by Dyckhoff and Hudelmaier. Not all rules of G 4 i are focussed. Theorem (Iemhoff 2017) Any calculus that is an extension of G 4 i with unary one-sided thinnable modal rules has uniform interpolation. 16 / 21

intermediate logic Similar to the classical case, but far more complicated: ∃ p and ∀ p. One needs a terminating calculus for IPC . Use G 4 i by Dyckhoff and Hudelmaier. Not all rules of G 4 i are focussed. Theorem (Iemhoff 2017) Any calculus that is an extension of G 4 i with unary one-sided thinnable modal rules has uniform interpolation. Proof 16 / 21

intermediate logic Similar to the classical case, but far more complicated: ∃ p and ∀ p. One needs a terminating calculus for IPC . Use G 4 i by Dyckhoff and Hudelmaier. Not all rules of G 4 i are focussed. Theorem (Iemhoff 2017) Any calculus that is an extension of G 4 i with unary one-sided thinnable modal rules has uniform interpolation. Proof For rules in G 4 i that are nonunary or not one-sided, prove that if the premisses have a uniform interpolant, then so does the conclusion. Further proceed as in the classical case. ⊣ 16 / 21

intermediate logic Similar to the classical case, but far more complicated: ∃ p and ∀ p. One needs a terminating calculus for IPC . Use G 4 i by Dyckhoff and Hudelmaier. Not all rules of G 4 i are focussed. Theorem (Iemhoff 2017) Any calculus that is an extension of G 4 i with unary one-sided thinnable modal rules has uniform interpolation. Proof For rules in G 4 i that are nonunary or not one-sided, prove that if the premisses have a uniform interpolant, then so does the conclusion. Further proceed as in the classical case. ⊣ Corollary No intermediate logic except the 7 with uniform interpolation has such a calculus. 16 / 21

intermediate logic Similar to the classical case, but far more complicated: ∃ p and ∀ p. One needs a terminating calculus for IPC . Use G 4 i by Dyckhoff and Hudelmaier. Not all rules of G 4 i are focussed. Theorem (Iemhoff 2017) Any calculus that is an extension of G 4 i with unary one-sided thinnable modal rules has uniform interpolation. Proof For rules in G 4 i that are nonunary or not one-sided, prove that if the premisses have a uniform interpolant, then so does the conclusion. Further proceed as in the classical case. ⊣ Corollary No intermediate logic except the 7 with uniform interpolation has such a calculus. Corollary When developing a calculus based on G 4 i for an intermediate logic without uniform interpolation, then some of the rules cannot be unary, thinnable, and one-sided. 16 / 21

intuitionistic modal logic Work in progress. 17 / 21

intuitionistic modal logic Work in progress. The logics are extensions of iK (only ✷ , no diamond ✸ ). 17 / 21

intuitionistic modal logic Work in progress. The logics are extensions of iK (only ✷ , no diamond ✸ ). The sequent calculi are extensions of G 4 iK , which is G 4 i plus the rules Γ ⇒ ϕ Π , ✷ Γ ⇒ ✷ ϕ, ∆ R K 17 / 21

intuitionistic modal logic Work in progress. The logics are extensions of iK (only ✷ , no diamond ✸ ). The sequent calculi are extensions of G 4 iK , which is G 4 i plus the rules Γ ⇒ ϕ Γ ⇒ ϕ Π , ✷ Γ , ψ ⇒ ∆ L ✷ Π , ✷ Γ ⇒ ✷ ϕ, ∆ R K Π , ✷ Γ , ✷ ϕ → ψ ⇒ ∆ → 17 / 21

intuitionistic modal logic Work in progress. The logics are extensions of iK (only ✷ , no diamond ✸ ). The sequent calculi are extensions of G 4 iK , which is G 4 i plus the rules Γ ⇒ ϕ Γ ⇒ ϕ Π , ✷ Γ , ψ ⇒ ∆ L ✷ Π , ✷ Γ ⇒ ✷ ϕ, ∆ R K Π , ✷ Γ , ✷ ϕ → ψ ⇒ ∆ → Lemma G 4 iK is terminating. 17 / 21

intuitionistic modal logic Work in progress. The logics are extensions of iK (only ✷ , no diamond ✸ ). The sequent calculi are extensions of G 4 iK , which is G 4 i plus the rules Γ ⇒ ϕ Γ ⇒ ϕ Π , ✷ Γ , ψ ⇒ ∆ L ✷ Π , ✷ Γ ⇒ ✷ ϕ, ∆ R K Π , ✷ Γ , ✷ ϕ → ψ ⇒ ∆ → Lemma G 4 iK is terminating. Theorem Any logic with a calculus that is an extension of G 4 iK with unary one-sided thinnable (modal) rules has uniform interpolation. This holds in particular for iK and iKD . 17 / 21

intuitionistic modal logic Work in progress. The logics are extensions of iK (only ✷ , no diamond ✸ ). The sequent calculi are extensions of G 4 iK , which is G 4 i plus the rules Γ ⇒ ϕ Γ ⇒ ϕ Π , ✷ Γ , ψ ⇒ ∆ L ✷ Π , ✷ Γ ⇒ ✷ ϕ, ∆ R K Π , ✷ Γ , ✷ ϕ → ψ ⇒ ∆ → Lemma G 4 iK is terminating. Theorem Any logic with a calculus that is an extension of G 4 iK with unary one-sided thinnable (modal) rules has uniform interpolation. This holds in particular for iK and iKD . Modularity of the proof: Six properties of rules are isolated such that: 17 / 21