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Admissible Infinitary Rules in Modal Logic. Part II Admissible Infinitary Rules in Modal Logic. Part II Piotr Wojtylak University of Opole, Opole, Poland Admissible Infinitary Rules in Modal Logic. Part II Piotr Wojtylak University of Opole, Opole, Poland Wojciech Dzik Silesian University, Katowice, Poland Admissible Infinitary Rules in Modal Logic. Part II Piotr Wojtylak University of Opole, Opole, Poland Wojciech Dzik Silesian University, Katowice, Poland key words: consequence operations, projective unification, admissible rules, structural completeness Admissible Infinitary Rules in Modal Logic. Part II Piotr Wojtylak University of Opole, Opole, Poland Wojciech Dzik Silesian University, Katowice, Poland key words: consequence operations, projective unification, admissible rules, structural completeness Admissible Infinitary Rules in Modal Logic. Part II Piotr Wojtylak University of Opole, Opole, Poland Wojciech Dzik Silesian University, Katowice, Poland key words: consequence operations, projective unification, admissible rules, structural completeness Les Diablerets, 1 February, 2015 Abstract Abstract • Projective Unification in Modal Logic, The Logic Journal of the IGPL 20(2012) No.1, 121–153. • Modal Consequence Relations Extending S4.3, Notre Dame Journal of Formal Logic, (to appear) • Almost structurally complete consequence operations extending S4.3, (in preparation). Abstract • Projective Unification in Modal Logic, The Logic Journal of the IGPL 20(2012) No.1, 121–153. • Modal Consequence Relations Extending S4.3, Notre Dame Journal of Formal Logic, (to appear) • Almost structurally complete consequence operations extending S4.3, (in preparation). Theorem Each unifiable formula has a projective unifier in S4.3. Consequently, each modal consequence operation extending S4.3 is (structurally)-complete with respect finitary non-passive rules. Abstract • Projective Unification in Modal Logic, The Logic Journal of the IGPL 20(2012) No.1, 121–153. • Modal Consequence Relations Extending S4.3, Notre Dame Journal of Formal Logic, (to appear) • Almost structurally complete consequence operations extending S4.3, (in preparation). Theorem Each unifiable formula has a projective unifier in S4.3. Consequently, each modal consequence operation extending S4.3 is (structurally)-complete with respect finitary non-passive rules. Theorem A modal logic L containing S4 enjoys projective unification if and only if S4.3 ⊆ L. Abstract Abstract Theorem (i) Each finitary consequence operation Cn extending S4.3 has a finite basis; (ii) Each consequence operation Cn extending S4.3 coincide on finite sets with a FA modal consequence operation. Abstract Theorem (i) Each finitary consequence operation Cn extending S4.3 has a finite basis; (ii) Each consequence operation Cn extending S4.3 coincide on finite sets with a FA modal consequence operation. A consequence operation Cn is finitely approximable (Cn ∈ FA) if − → Cn = K for some class K of finite matrices. Abstract Theorem (i) Each finitary consequence operation Cn extending S4.3 has a finite basis; (ii) Each consequence operation Cn extending S4.3 coincide on finite sets with a FA modal consequence operation. A consequence operation Cn is finitely approximable (Cn ∈ FA) if − → Cn = K for some class K of finite matrices. Theorem A modal consequence operation Cn extending S4.3 is structurally complete with respect to infinitary non-passive rules iff Cn is FA. Abstract Theorem (i) Each finitary consequence operation Cn extending S4.3 has a finite basis; (ii) Each consequence operation Cn extending S4.3 coincide on finite sets with a FA modal consequence operation. A consequence operation Cn is finitely approximable (Cn ∈ FA) if − → Cn = K for some class K of finite matrices. Theorem A modal consequence operation Cn extending S4.3 is structurally complete with respect to infinitary non-passive rules iff Cn is FA. We provide an uniform basis for all non-passive admissible rules of any L ∈ NExt(S4.3) consisting of infinitary rules like {(αi ↔ αj ) → α0 : 0 < i < j} α0 Modal Logic Modal Logic Let Var be the set of propositional variables and Fm be the set of modal formulas in {→, ⊥, }. For each formula α, let Var (α) denote the (finite) set of variables occurring in α. Modal Logic Let Var be the set of propositional variables and Fm be the set of modal formulas in {→, ⊥, }. For each formula α, let Var (α) denote the (finite) set of variables occurring in α. By a modal logic we mean any proper subset of Fm closed under substitutions, closed under MP : α → β, α β and RG : containing all classical tautologies, and (α → β) → (α → β). α , α Modal Logic Let Var be the set of propositional variables and Fm be the set of modal formulas in {→, ⊥, }. For each formula α, let Var (α) denote the (finite) set of variables occurring in α. By a modal logic we mean any proper subset of Fm closed under substitutions, closed under MP : α → β, α β and RG : α , α containing all classical tautologies, and (α → β) → (α → β). Each modal logic is an extension of the logic K with axiom schemata. S4 extends K with (T ) : α → α oraz (4) : α → α. The logic S4.3 contains additionally (α → β) ∨ (β → α); and S5 is axiomatized with (5) : ♦α → α. Modal consequences Given a modal logic L, we define its global entailment relation CnL . Thus, α ∈ CnL (X ) means that α can be derived from X ∪ L using the rules MP and RG . Modal consequences Given a modal logic L, we define its global entailment relation CnL . Thus, α ∈ CnL (X ) means that α can be derived from X ∪ L using the rules MP and RG . Theorem If S4 ⊆ L, then α ∈ CnL (X , β) iff β → α ∈ CnL (X ). Modal consequences Given a modal logic L, we define its global entailment relation CnL . Thus, α ∈ CnL (X ) means that α can be derived from X ∪ L using the rules MP and RG . Theorem If S4 ⊆ L, then α ∈ CnL (X , β) iff β → α ∈ CnL (X ). By a modal consequence operation we mean any structural consequence operation Cn which extends CnK . Thus, Cn may be given by extending a modal logic with some inferential rules. Modal consequences Given a modal logic L, we define its global entailment relation CnL . Thus, α ∈ CnL (X ) means that α can be derived from X ∪ L using the rules MP and RG . Theorem If S4 ⊆ L, then α ∈ CnL (X , β) iff β → α ∈ CnL (X ). By a modal consequence operation we mean any structural consequence operation Cn which extends CnK . Thus, Cn may be given by extending a modal logic with some inferential rules. Let us note that such extensions may violate the above deduction theorem. Modal consequences Given a modal logic L, we define its global entailment relation CnL . Thus, α ∈ CnL (X ) means that α can be derived from X ∪ L using the rules MP and RG . Theorem If S4 ⊆ L, then α ∈ CnL (X , β) iff β → α ∈ CnL (X ). By a modal consequence operation we mean any structural consequence operation Cn which extends CnK . Thus, Cn may be given by extending a modal logic with some inferential rules. Let us note that such extensions may violate the above deduction theorem. Let NExt(L) denote the lattice of all normal extensions of L. Modal consequences Given a modal logic L, we define its global entailment relation CnL . Thus, α ∈ CnL (X ) means that α can be derived from X ∪ L using the rules MP and RG . Theorem If S4 ⊆ L, then α ∈ CnL (X , β) iff β → α ∈ CnL (X ). By a modal consequence operation we mean any structural consequence operation Cn which extends CnK . Thus, Cn may be given by extending a modal logic with some inferential rules. Let us note that such extensions may violate the above deduction theorem. Let NExt(L) denote the lattice of all normal extensions of L. EXTfin (L) be the lattice of finitary consequence extensions of CnL Modal consequences Given a modal logic L, we define its global entailment relation CnL . Thus, α ∈ CnL (X ) means that α can be derived from X ∪ L using the rules MP and RG . Theorem If S4 ⊆ L, then α ∈ CnL (X , β) iff β → α ∈ CnL (X ). By a modal consequence operation we mean any structural consequence operation Cn which extends CnK . Thus, Cn may be given by extending a modal logic with some inferential rules. Let us note that such extensions may violate the above deduction theorem. Let NExt(L) denote the lattice of all normal extensions of L. EXTfin (L) be the lattice of finitary consequence extensions of CnL and EXT(L) is the lattice of all extensions of CnL . Modal consequences Given a modal logic L, we define its global entailment relation CnL . Thus, α ∈ CnL (X ) means that α can be derived from X ∪ L using the rules MP and RG . Theorem If S4 ⊆ L, then α ∈ CnL (X , β) iff β → α ∈ CnL (X ). By a modal consequence operation we mean any structural consequence operation Cn which extends CnK . Thus, Cn may be given by extending a modal logic with some inferential rules. Let us note that such extensions may violate the above deduction theorem. Let NExt(L) denote the lattice of all normal extensions of L. EXTfin (L) be the lattice of finitary consequence extensions of CnL and EXT(L) is the lattice of all extensions of CnL . The lattice ordering in each case is determined by inclusion. Modal algebras Modal algebras Modal algebra A = (A, →, ⊥, ) is an extension of a Boolean algebra (A, →, ⊥) with a monadic operator operator (1) > = >; (2) (a ∧ b) = a ∧ b, for a, b ∈ A. Modal algebras Modal algebra A = (A, →, ⊥, ) is an extension of a Boolean algebra (A, →, ⊥) with a monadic operator operator (1) > = >; (2) (a ∧ b) = a ∧ b, for a, b ∈ A. Modal algebras will be regarded as logical matrices with a designated element >. Modal algebras Modal algebra A = (A, →, ⊥, ) is an extension of a Boolean algebra (A, →, ⊥) with a monadic operator operator (1) > = >; (2) (a ∧ b) = a ∧ b, for a, b ∈ A. Modal algebras will be regarded as logical matrices with a designated element >. Each valuation v : Var → A extends to a homomorphism v : For → A. If v (α) = > for some v , then α is said to be satisfiable in A; in symbols α ∈ Sat(A). Modal algebras Modal algebra A = (A, →, ⊥, ) is an extension of a Boolean algebra (A, →, ⊥) with a monadic operator operator (1) > = >; (2) (a ∧ b) = a ∧ b, for a, b ∈ A. Modal algebras will be regarded as logical matrices with a designated element >. Each valuation v : Var → A extends to a homomorphism v : For → A. If v (α) = > for some v , then α is said to be satisfiable in A; in symbols α ∈ Sat(A). Let Log(A) be the set of formulas valid in A. Modal algebras Modal algebra A = (A, →, ⊥, ) is an extension of a Boolean algebra (A, →, ⊥) with a monadic operator operator (1) > = >; (2) (a ∧ b) = a ∧ b, for a, b ∈ A. Modal algebras will be regarded as logical matrices with a designated element >. Each valuation v : Var → A extends to a homomorphism v : For → A. If v (α) = > for some v , then α is said to be satisfiable in A; in symbols α ∈ Sat(A). Let Log(A) be the set of formulasTvalid in A. Given a class K of modal algebras, we put Log(K) = {Log(A) : A ∈ K}. Modal algebras Modal algebra A = (A, →, ⊥, ) is an extension of a Boolean algebra (A, →, ⊥) with a monadic operator operator (1) > = >; (2) (a ∧ b) = a ∧ b, for a, b ∈ A. Modal algebras will be regarded as logical matrices with a designated element >. Each valuation v : Var → A extends to a homomorphism v : For → A. If v (α) = > for some v , then α is said to be satisfiable in A; in symbols α ∈ Sat(A). Let Log(A) be the set of formulasTvalid in A. Given a class K of modal algebras, we put Log(K) = {Log(A) : A ∈ K}. Theorem For each modal logic L there is a class K of modal algebras such that L = Log(K). Modal algebras Modal algebra A = (A, →, ⊥, ) is an extension of a Boolean algebra (A, →, ⊥) with a monadic operator operator (1) > = >; (2) (a ∧ b) = a ∧ b, for a, b ∈ A. Modal algebras will be regarded as logical matrices with a designated element >. Each valuation v : Var → A extends to a homomorphism v : For → A. If v (α) = > for some v , then α is said to be satisfiable in A; in symbols α ∈ Sat(A). Let Log(A) be the set of formulasTvalid in A. Given a class K of modal algebras, we put Log(K) = {Log(A) : A ∈ K}. Theorem For each modal logic L there is a class K of modal algebras such that L = Log(K). It means each modal logic has na adequate class of modal algebras. Matrix consequences → − Each modal algebra A generates modal consequence operation A : → − α ∈ A (X ) ⇔ v [X ] ⊆ {>} ⇒ v (α) = >, dla v : Var → A . Matrix consequences → − Each modal algebra A generates modal consequence operation A : → − α ∈ A (X ) ⇔ v [X ] ⊆ {>} ⇒ v (α) = >, dla v : Var → A . V → → − − For each class K of modal algebras, we put K = { A : A ∈ K}. Matrix consequences → − Each modal algebra A generates modal consequence operation A : → − α ∈ A (X ) ⇔ v [X ] ⊆ {>} ⇒ v (α) = >, dla v : Var → A . V → → − − For each class K of modal algebras, we put K = { A : A ∈ K}. Theorem For each modal consequence operation Cn there exists a class K of → − modal algebras such that Cn = K . Matrix consequences → − Each modal algebra A generates modal consequence operation A : → − α ∈ A (X ) ⇔ v [X ] ⊆ {>} ⇒ v (α) = >, dla v : Var → A . V → → − − For each class K of modal algebras, we put K = { A : A ∈ K}. Theorem For each modal consequence operation Cn there exists a class K of → − modal algebras such that Cn = K . L ∈ FMP means that L has an adequate family of finite algebras. Matrix consequences → − Each modal algebra A generates modal consequence operation A : → − α ∈ A (X ) ⇔ v [X ] ⊆ {>} ⇒ v (α) = >, dla v : Var → A . V → → − − For each class K of modal algebras, we put K = { A : A ∈ K}. Theorem For each modal consequence operation Cn there exists a class K of → − modal algebras such that Cn = K . L ∈ FMP means that L has an adequate family of finite algebras. A consequence operation Cn is finitely approximatizable (Cn ∈ FA) → − if Cn = K where each A ∈ K is finite. Matrix consequences → − Each modal algebra A generates modal consequence operation A : → − α ∈ A (X ) ⇔ v [X ] ⊆ {>} ⇒ v (α) = >, dla v : Var → A . V → → − − For each class K of modal algebras, we put K = { A : A ∈ K}. Theorem For each modal consequence operation Cn there exists a class K of → − modal algebras such that Cn = K . L ∈ FMP means that L has an adequate family of finite algebras. A consequence operation Cn is finitely approximatizable (Cn ∈ FA) → − if Cn = K where each A ∈ K is finite. If, additionally, K is finite, then Cn is said to be strongly finite (Cn ∈ SF). Matrix consequences → − Each modal algebra A generates modal consequence operation A : → − α ∈ A (X ) ⇔ v [X ] ⊆ {>} ⇒ v (α) = >, dla v : Var → A . V → → − − For each class K of modal algebras, we put K = { A : A ∈ K}. Theorem For each modal consequence operation Cn there exists a class K of → − modal algebras such that Cn = K . L ∈ FMP means that L has an adequate family of finite algebras. A consequence operation Cn is finitely approximatizable (Cn ∈ FA) → − if Cn = K where each A ∈ K is finite. If, additionally, K is finite, then Cn is said to be strongly finite (Cn ∈ SF). Theorem If Cn is strongly finite (Cn ∈ SF ), then Cn is finitary (Cn ∈Fin); Matrix consequences → − Each modal algebra A generates modal consequence operation A : → − α ∈ A (X ) ⇔ v [X ] ⊆ {>} ⇒ v (α) = >, dla v : Var → A . V → → − − For each class K of modal algebras, we put K = { A : A ∈ K}. Theorem For each modal consequence operation Cn there exists a class K of → − modal algebras such that Cn = K . L ∈ FMP means that L has an adequate family of finite algebras. A consequence operation Cn is finitely approximatizable (Cn ∈ FA) → − if Cn = K where each A ∈ K is finite. If, additionally, K is finite, then Cn is said to be strongly finite (Cn ∈ SF). Theorem If Cn is strongly finite (Cn ∈ SF ), then Cn is finitary (Cn ∈Fin); For consequence operations over (S4.3): FA∩ Fin = SF. S4.3-models Topological Boolean Algebras (TBA’s) are modal algebras which are models for S4; i.e., they fulfill (3) a ≤ a; (4) a = a, for each a ∈ A. S4.3-models Topological Boolean Algebras (TBA’s) are modal algebras which are models for S4; i.e., they fulfill (3) a ≤ a; (4) a = a, for each a ∈ A. Any (Kripke) structure F = (V , R) consists of a non-empty set V and a binary relation R on V . Each F = (V , R) determines a modal algebra F+ , on the power set A = P(V ), with a = {x ∈ V : R(x) ⊆ a}, where R(x) = {y ∈ V : xRy }. S4.3-models Topological Boolean Algebras (TBA’s) are modal algebras which are models for S4; i.e., they fulfill (3) a ≤ a; (4) a = a, for each a ∈ A. Any (Kripke) structure F = (V , R) consists of a non-empty set V and a binary relation R on V . Each F = (V , R) determines a modal algebra F+ , on the power set A = P(V ), with a = {x ∈ V : R(x) ⊆ a}, where R(x) = {y ∈ V : xRy }. A subset C of V is called a cluster if xRy and yRx, for each x, y ∈ C . Symbols 1, 2 and 3, etc., stand for 1- , 2- and 3-element clusters, which are S5 models. Then n+ is called a Henle Algebra. S4.3-models Topological Boolean Algebras (TBA’s) are modal algebras which are models for S4; i.e., they fulfill (3) a ≤ a; (4) a = a, for each a ∈ A. Any (Kripke) structure F = (V , R) consists of a non-empty set V and a binary relation R on V . Each F = (V , R) determines a modal algebra F+ , on the power set A = P(V ), with a = {x ∈ V : R(x) ⊆ a}, where R(x) = {y ∈ V : xRy }. A subset C of V is called a cluster if xRy and yRx, for each x, y ∈ C . Symbols 1, 2 and 3, etc., stand for 1- , 2- and 3-element clusters, which are S5 models. Then n+ is called a Henle Algebra. Finite subdirectly irreducible S4.3-algebras are TBA’s in which all open elements (i.e. such a’s that a = a) form a chain. They coincide with algebras F+ for finite quasi chains F which are, in turn, determined by lists, i.e. finite sequences of positive integers. Projective unification Projective unification A substitution ε is a unifier for a formula α (a set X ) in logic L if ε(α) ∈ L (if ε[X ] ⊆ L). Projective unification A substitution ε is a unifier for a formula α (a set X ) in logic L if ε(α) ∈ L (if ε[X ] ⊆ L). Logical constants {⊥, >} form a 2-element subalgebra, say 2, of Lindenbauma-Tarski’s algebra for L (if L ⊇ S4). Unifiers v : Var → {⊥, >} will be called ground unifiers. Projective unification A substitution ε is a unifier for a formula α (a set X ) in logic L if ε(α) ∈ L (if ε[X ] ⊆ L). Logical constants {⊥, >} form a 2-element subalgebra, say 2, of Lindenbauma-Tarski’s algebra for L (if L ⊇ S4). Unifiers v : Var → {⊥, >} will be called ground unifiers. Theorem The following conditions are equivalent for each L ⊇ S4: (i) α is L-unifiable; (ii) α has a ground unifier w L; (iii) α is satisfiable in 2; (iv) ∼ α 6∈ Tr = Log (2). Projective unification A substitution ε is a unifier for a formula α (a set X ) in logic L if ε(α) ∈ L (if ε[X ] ⊆ L). Logical constants {⊥, >} form a 2-element subalgebra, say 2, of Lindenbauma-Tarski’s algebra for L (if L ⊇ S4). Unifiers v : Var → {⊥, >} will be called ground unifiers. Theorem The following conditions are equivalent for each L ⊇ S4: (i) α is L-unifiable; (ii) α has a ground unifier w L; (iii) α is satisfiable in 2; (iv) ∼ α 6∈ Tr = Log (2). A unifier ε is said to be projective for X if ε(β) ↔ β ∈ CnL (X ), for each formula β; we say that L has projective unification, if each L-unifiable formula has a projective unifier is L. Projective unification A substitution ε is a unifier for a formula α (a set X ) in logic L if ε(α) ∈ L (if ε[X ] ⊆ L). Logical constants {⊥, >} form a 2-element subalgebra, say 2, of Lindenbauma-Tarski’s algebra for L (if L ⊇ S4). Unifiers v : Var → {⊥, >} will be called ground unifiers. Theorem The following conditions are equivalent for each L ⊇ S4: (i) α is L-unifiable; (ii) α has a ground unifier w L; (iii) α is satisfiable in 2; (iv) ∼ α 6∈ Tr = Log (2). A unifier ε is said to be projective for X if ε(β) ↔ β ∈ CnL (X ), for each formula β; we say that L has projective unification, if each L-unifiable formula has a projective unifier is L. Structural completeness Structural completeness We consider (infinitary) inferential rules X /β. Structural completeness We consider (infinitary) inferential rules X /β. Finitary rules may be reduced to the form α/β. Structural completeness We consider (infinitary) inferential rules X /β. Finitary rules may be reduced to the form α/β. Given a set R of rules and a modal logic L, we denote the modal consequence operation extending CnL with the rules R by CnLR . If Cn = CnLR , then R is said to be a (rule-)basis of the consequence Cn on the ground of L. Structural completeness We consider (infinitary) inferential rules X /β. Finitary rules may be reduced to the form α/β. Given a set R of rules and a modal logic L, we denote the modal consequence operation extending CnL with the rules R by CnLR . If Cn = CnLR , then R is said to be a (rule-)basis of the consequence Cn on the ground of L. The consequence CnLR is finitary if the rules R are finitary. Structural completeness We consider (infinitary) inferential rules X /β. Finitary rules may be reduced to the form α/β. Given a set R of rules and a modal logic L, we denote the modal consequence operation extending CnL with the rules R by CnLR . If Cn = CnLR , then R is said to be a (rule-)basis of the consequence Cn on the ground of L. The consequence CnLR is finitary if the rules R are finitary. The rule X /β is said to be admissible for a modal consequence operation Cn, if ε[X ] ⊆ Cn(∅) implies ε(β) ∈ Cn(∅), for each substitution ε. Structural completeness We consider (infinitary) inferential rules X /β. Finitary rules may be reduced to the form α/β. Given a set R of rules and a modal logic L, we denote the modal consequence operation extending CnL with the rules R by CnLR . If Cn = CnLR , then R is said to be a (rule-)basis of the consequence Cn on the ground of L. The consequence CnLR is finitary if the rules R are finitary. The rule X /β is said to be admissible for a modal consequence operation Cn, if ε[X ] ⊆ Cn(∅) implies ε(β) ∈ Cn(∅), for each substitution ε. The rule is derivable for Cn (is Cn–derivable), if β ∈ Cn(X ). Structural completeness We consider (infinitary) inferential rules X /β. Finitary rules may be reduced to the form α/β. Given a set R of rules and a modal logic L, we denote the modal consequence operation extending CnL with the rules R by CnLR . If Cn = CnLR , then R is said to be a (rule-)basis of the consequence Cn on the ground of L. The consequence CnLR is finitary if the rules R are finitary. The rule X /β is said to be admissible for a modal consequence operation Cn, if ε[X ] ⊆ Cn(∅) implies ε(β) ∈ Cn(∅), for each substitution ε. The rule is derivable for Cn (is Cn–derivable), if β ∈ Cn(X ). The consequence operation Cn is structurally complete (Cn ∈ SCpl) if every admissible rule for Cn is Cn-derivable. Structural completeness We consider (infinitary) inferential rules X /β. Finitary rules may be reduced to the form α/β. Given a set R of rules and a modal logic L, we denote the modal consequence operation extending CnL with the rules R by CnLR . If Cn = CnLR , then R is said to be a (rule-)basis of the consequence Cn on the ground of L. The consequence CnLR is finitary if the rules R are finitary. The rule X /β is said to be admissible for a modal consequence operation Cn, if ε[X ] ⊆ Cn(∅) implies ε(β) ∈ Cn(∅), for each substitution ε. The rule is derivable for Cn (is Cn–derivable), if β ∈ Cn(X ). The consequence operation Cn is structurally complete (Cn ∈ SCpl) if every admissible rule for Cn is Cn-derivable.Cn ∈ SCplfin means that each finitary rule which is admissible for Cn is Cn–derivable. Almost structural completeness Almost structural completeness The rule X /β is passive if X is not unifiable, Almost structural completeness The rule X /β is passive if X is not unifiable, e.g. P2 : ♦α ∧ ♦ ∼ α β Almost structural completeness The rule X /β is passive if X is not unifiable, e.g. P2 : ♦α ∧ ♦ ∼ α β Cn is almost structurally complete, Cn ∈ ASCpl, if each admissible non-passive rule is derivable (for Cn). Almost structural completeness The rule X /β is passive if X is not unifiable, e.g. P2 : ♦α ∧ ♦ ∼ α β Cn is almost structurally complete, Cn ∈ ASCpl, if each admissible non-passive rule is derivable (for Cn). Corollary Cn ∈ ASCplfin for each Cn ∈ EXT(S4.3). Almost structural completeness The rule X /β is passive if X is not unifiable, e.g. P2 : ♦α ∧ ♦ ∼ α β Cn is almost structurally complete, Cn ∈ ASCpl, if each admissible non-passive rule is derivable (for Cn). Corollary Cn ∈ ASCplfin for each Cn ∈ EXT(S4.3). Corollary If Cn ∈ EXT(S4.3M), where (M) is McKinsey’s axiom ♦α → ♦α, then Cn ∈ ASCpl(fin) ⇔ Cn ∈ SCpl(fin) . Almost structural completeness The rule X /β is passive if X is not unifiable, e.g. P2 : ♦α ∧ ♦ ∼ α β Cn is almost structurally complete, Cn ∈ ASCpl, if each admissible non-passive rule is derivable (for Cn). Corollary Cn ∈ ASCplfin for each Cn ∈ EXT(S4.3). Corollary If Cn ∈ EXT(S4.3M), where (M) is McKinsey’s axiom ♦α → ♦α, then Cn ∈ ASCpl(fin) ⇔ Cn ∈ SCpl(fin) . Passive rules over S4.3 Passive rules over S4.3 Let n ≥ 0 and let us consider the sublanguage of Fm spanned on p1 , . . . , pn . There are 2n (Boolean) atoms there and each of them can be represented by the formula σ(1) p1 σ(n) ∧ · · · ∧ pn where σ : {1, . . . , n} → {0, 1}, and p 0 = p, and p 1 =∼ p. Suppose we denote these atoms by: θ1 , . . . , θ2n Passive rules over S4.3 Let n ≥ 0 and let us consider the sublanguage of Fm spanned on p1 , . . . , pn . There are 2n (Boolean) atoms there and each of them can be represented by the formula σ(1) p1 σ(n) ∧ · · · ∧ pn where σ : {1, . . . , n} → {0, 1}, and p 0 = p, and p 1 =∼ p. Suppose we denote these atoms by: θ1 , . . . , θ2n Theorem Each finitary consequence operation Cn ∈ EXT(S4.3) is an extension of some CnL (for L ⊇ S4.3 and some n ≥ 0) with a finite number of passive rules ♦θ1 ∧ · · · ∧ ♦θs α where 2 ≤ s ≤ 2n and Var (α) ∩ {p1 , . . . , pn } = ∅. Finitary extensions of S4.3 Finitary extensions of S4.3 Theorem Let Cn ≥ CnL and K be a class of finite subdirectly irreducible S4.3-algebras such that L = Cn(∅) = Log(K). Finitary extensions of S4.3 Theorem Let Cn ≥ CnL and K be a class of finite subdirectly irreducible S4.3-algebras such that L = Cn(∅) = Log(K). Then one can find K1 , · · · , Km such that S(K) = K1 ⊇ · · · ⊇ Km and → − Cn(X ) = L (X ), for each finite sets X , where L = Km ∪ (Km−1 \ Km ) × (m − 1)+ ∪ · · · ∪ (K1 \ K2 ) × 1+ Finitary extensions of S4.3 Theorem Let Cn ≥ CnL and K be a class of finite subdirectly irreducible S4.3-algebras such that L = Cn(∅) = Log(K). Then one can find K1 , · · · , Km such that S(K) = K1 ⊇ · · · ⊇ Km and → − Cn(X ) = L (X ), for each finite sets X , where L = Km ∪ (Km−1 \ Km ) × (m − 1)+ ∪ · · · ∪ (K1 \ K2 ) × 1+ Corollary Let L ∈ NExtS4.3 and let K an adequate for L class of finite subdirectly irreducible S4.3-algebras. Then Finitary extensions of S4.3 Theorem Let Cn ≥ CnL and K be a class of finite subdirectly irreducible S4.3-algebras such that L = Cn(∅) = Log(K). Then one can find K1 , · · · , Km such that S(K) = K1 ⊇ · · · ⊇ Km and → − Cn(X ) = L (X ), for each finite sets X , where L = Km ∪ (Km−1 \ Km ) × (m − 1)+ ∪ · · · ∪ (K1 \ K2 ) × 1+ Corollary Let L ∈ NExtS4.3 and let K an adequate for L class of finite subdirectly irreducible S4.3-algebras. Then (i) − → β ∈ K (X , α) iff − → α → β ∈ K (X ). Finitary extensions of S4.3 Theorem Let Cn ≥ CnL and K be a class of finite subdirectly irreducible S4.3-algebras such that L = Cn(∅) = Log(K). Then one can find K1 , · · · , Km such that S(K) = K1 ⊇ · · · ⊇ Km and → − Cn(X ) = L (X ), for each finite sets X , where L = Km ∪ (Km−1 \ Km ) × (m − 1)+ ∪ · · · ∪ (K1 \ K2 ) × 1+ Corollary Let L ∈ NExtS4.3 and let K an adequate for L class of finite subdirectly irreducible S4.3-algebras. Then (i) − → β ∈ K (X , α) iff − → α → β ∈ K (X ). (ii)The structurally complete extension of CnL is determined by K × 1+ = {B × 1+ : B ∈ K}. Strongly finite logics Strongly finite logics By use of the above theorem we can describe the lattice EXTfin (L) of all finitary consequence extensions of any logic L over S4.3. Strongly finite logics By use of the above theorem we can describe the lattice EXTfin (L) of all finitary consequence extensions of any logic L over S4.3. The results, however, would be insufficient for EXT(L). Strongly finite logics By use of the above theorem we can describe the lattice EXTfin (L) of all finitary consequence extensions of any logic L over S4.3. The results, however, would be insufficient for EXT(L). Exceptions are tabular (and strongly finite) logics Strongly finite logics By use of the above theorem we can describe the lattice EXTfin (L) of all finitary consequence extensions of any logic L over S4.3. The results, however, would be insufficient for EXT(L). Exceptions are tabular (and strongly finite) logics Lemma Suppose that L ∈ NExt(S4.3). Then L is tabular iff CnL is strongly finite (i.e. CnL ∈ SF ). Strongly finite logics By use of the above theorem we can describe the lattice EXTfin (L) of all finitary consequence extensions of any logic L over S4.3. The results, however, would be insufficient for EXT(L). Exceptions are tabular (and strongly finite) logics Lemma Suppose that L ∈ NExt(S4.3). Then L is tabular iff CnL is strongly finite (i.e. CnL ∈ SF ). Theorem If a logic L ∈ NExt(S4.3) is tabular, then EXT(L) ⊆ SF and consequently EXT(L) = EXTfin (L). Strongly finite logics By use of the above theorem we can describe the lattice EXTfin (L) of all finitary consequence extensions of any logic L over S4.3. The results, however, would be insufficient for EXT(L). Exceptions are tabular (and strongly finite) logics Lemma Suppose that L ∈ NExt(S4.3). Then L is tabular iff CnL is strongly finite (i.e. CnL ∈ SF ). Theorem If a logic L ∈ NExt(S4.3) is tabular, then EXT(L) ⊆ SF and consequently EXT(L) = EXTfin (L). We show EXT(Cn) ⊆ FA and EXT(Cn) ≡ EXTfin (L) (where ≡ is a lattice isomorphism and L = Cn(∅)) for every consequence operation Cn ∈FA extending (S4.3). Strongly finite logics By use of the above theorem we can describe the lattice EXTfin (L) of all finitary consequence extensions of any logic L over S4.3. The results, however, would be insufficient for EXT(L). Exceptions are tabular (and strongly finite) logics Lemma Suppose that L ∈ NExt(S4.3). Then L is tabular iff CnL is strongly finite (i.e. CnL ∈ SF ). Theorem If a logic L ∈ NExt(S4.3) is tabular, then EXT(L) ⊆ SF and consequently EXT(L) = EXTfin (L). We show EXT(Cn) ⊆ FA and EXT(Cn) ≡ EXTfin (L) (where ≡ is a lattice isomorphism and L = Cn(∅)) for every consequence operation Cn ∈FA extending (S4.3). FA logics FA logics We should stress that, in contrast to the finitary case, infinitary modal consequence operations over S4.3 are not characterized by finite algebras. We can prove FA logics We should stress that, in contrast to the finitary case, infinitary modal consequence operations over S4.3 are not characterized by finite algebras. We can prove Theorem For consequence operations over S4.3: FA = ASCpl. FA logics We should stress that, in contrast to the finitary case, infinitary modal consequence operations over S4.3 are not characterized by finite algebras. We can prove Theorem For consequence operations over S4.3: FA = ASCpl. Corollary For consequence operations over S4.3: ASCpl ∩ Fin =SF . FA logics We should stress that, in contrast to the finitary case, infinitary modal consequence operations over S4.3 are not characterized by finite algebras. We can prove Theorem For consequence operations over S4.3: FA = ASCpl. Corollary For consequence operations over S4.3: ASCpl ∩ Fin =SF . Variants of structural completeness Variants of structural completeness We have distinguished four variants of structural completeness: SCpl, SCplfin , ASCpl and ASCplfin . Variants of structural completeness We have distinguished four variants of structural completeness: SCpl, SCplfin , ASCpl and ASCplfin . To show that SCpl 6= SCplfin (or ASCpl6= ASCplfin ) it suffices to consider the rule : (%) {(αi ↔ αj ) → α0 : 0 < i < j} α0 Variants of structural completeness We have distinguished four variants of structural completeness: SCpl, SCplfin , ASCpl and ASCplfin . To show that SCpl 6= SCplfin (or ASCpl6= ASCplfin ) it suffices to consider the rule : (%) {(αi ↔ αj ) → α0 : 0 < i < j} α0 It can be easily seen that the rule is valid for each finite modal algebra which means it is derivable for the modal consequence − → operation A determined by a finite A. Variants of structural completeness We have distinguished four variants of structural completeness: SCpl, SCplfin , ASCpl and ASCplfin . To show that SCpl 6= SCplfin (or ASCpl6= ASCplfin ) it suffices to consider the rule : (%) {(αi ↔ αj ) → α0 : 0 < i < j} α0 It can be easily seen that the rule is valid for each finite modal algebra which means it is derivable for the modal consequence − → operation A determined by a finite A. Since each modal logic extending S4.3 is know to have the finite model property, we obtain admissibility of the rule for any extension of S4.3. Variants of structural completeness We have distinguished four variants of structural completeness: SCpl, SCplfin , ASCpl and ASCplfin . To show that SCpl 6= SCplfin (or ASCpl6= ASCplfin ) it suffices to consider the rule : (%) {(αi ↔ αj ) → α0 : 0 < i < j} α0 It can be easily seen that the rule is valid for each finite modal algebra which means it is derivable for the modal consequence − → operation A determined by a finite A. Since each modal logic extending S4.3 is know to have the finite model property, we obtain admissibility of the rule for any extension of S4.3.On the other hand , % is not derivable for S4.3 (nor S4.3M) as p0 cannot be deduced, in S4.3, from any finite subset of {(pi ↔ pj ) → p0 : 0 < i < j}. Variants of structural completeness We have distinguished four variants of structural completeness: SCpl, SCplfin , ASCpl and ASCplfin . To show that SCpl 6= SCplfin (or ASCpl6= ASCplfin ) it suffices to consider the rule : (%) {(αi ↔ αj ) → α0 : 0 < i < j} α0 It can be easily seen that the rule is valid for each finite modal algebra which means it is derivable for the modal consequence − → operation A determined by a finite A. Since each modal logic extending S4.3 is know to have the finite model property, we obtain admissibility of the rule for any extension of S4.3.On the other hand , % is not derivable for S4.3 (nor S4.3M) as p0 cannot be deduced, in S4.3, from any finite subset of {(pi ↔ pj ) → p0 : 0 < i < j}.So, S4.3 ∈ASCplfin \ ASCpl (and S4.3M ∈ SCplfin \ SCpl). Variants of structural completeness We have distinguished four variants of structural completeness: SCpl, SCplfin , ASCpl and ASCplfin . To show that SCpl 6= SCplfin (or ASCpl6= ASCplfin ) it suffices to consider the rule : (%) {(αi ↔ αj ) → α0 : 0 < i < j} α0 It can be easily seen that the rule is valid for each finite modal algebra which means it is derivable for the modal consequence − → operation A determined by a finite A. Since each modal logic extending S4.3 is know to have the finite model property, we obtain admissibility of the rule for any extension of S4.3.On the other hand , % is not derivable for S4.3 (nor S4.3M) as p0 cannot be deduced, in S4.3, from any finite subset of {(pi ↔ pj ) → p0 : 0 < i < j}.So, S4.3 ∈ASCplfin \ ASCpl (and S4.3M ∈ SCplfin \ SCpl). Projective unification of infinite sets of formulas Projective unification of infinite sets of formulas The projective unification of S4.3 will not be preserved if we extend the concept of projective unifier on infinite sets of formulas. Projective unification of infinite sets of formulas The projective unification of S4.3 will not be preserved if we extend the concept of projective unifier on infinite sets of formulas. Namely, let us suppose that ε is a unifier for the set {(pi ↔ pj ) → p0 : 0 < i < j} in S4.3 (notice that the set is unifiable) such that (ε(β) ↔ β) ∈ CnS4.3 ({(pi ↔ pj ) → p0 : 0 < i < j}), for each β Projective unification of infinite sets of formulas The projective unification of S4.3 will not be preserved if we extend the concept of projective unifier on infinite sets of formulas. Namely, let us suppose that ε is a unifier for the set {(pi ↔ pj ) → p0 : 0 < i < j} in S4.3 (notice that the set is unifiable) such that (ε(β) ↔ β) ∈ CnS4.3 ({(pi ↔ pj ) → p0 : 0 < i < j}), for each β Since % is S4.3-admissible and ε is a unifier for any (pi ↔ pj ) → p0 with 0 < i < j, we get ε(p0 ) ∈ S4.3. Thus, we obtain (for β = p0 ) p0 ∈ CnS4.3 ({(pi ↔ pj ) → p0 : 0 < i < j}) which is impossible. Consequently, the set {(pi ↔ pj ) → p0 : 0 < i < j} cannot have a projective unifier in S4.3 (and any weaker logic). FA extensions FA extensions Let Cn ∈ EXT(S4.3) and K be a class of finite subdirectly irreducible S4.3-algebras such that Log (K) = Cn(∅) = L. FA extensions Let Cn ∈ EXT(S4.3) and K be a class of finite subdirectly irreducible S4.3-algebras such that Log (K) = Cn(∅) = L. Then one can find K1 ⊇ · · · ⊇ Km such that S(K) = K1 , and → − Ki = S(Ki ) for i = 1, . . . , m, and Cn =fin L , where L = Km ∪ (Km−1 \ Km ) × (m − 1)+ ∪ · · · ∪ (K1 \ K2 ) × 1+ FA extensions Let Cn ∈ EXT(S4.3) and K be a class of finite subdirectly irreducible S4.3-algebras such that Log (K) = Cn(∅) = L. Then one can find K1 ⊇ · · · ⊇ Km such that S(K) = K1 , and → − Ki = S(Ki ) for i = 1, . . . , m, and Cn =fin L , where L = Km ∪ (Km−1 \ Km ) × (m − 1)+ ∪ · · · ∪ (K1 \ K2 ) × 1+ Theorem −→ (i) L = CnASCpl ; FA extensions Let Cn ∈ EXT(S4.3) and K be a class of finite subdirectly irreducible S4.3-algebras such that Log (K) = Cn(∅) = L. Then one can find K1 ⊇ · · · ⊇ Km such that S(K) = K1 , and → − Ki = S(Ki ) for i = 1, . . . , m, and Cn =fin L , where L = Km ∪ (Km−1 \ Km ) × (m − 1)+ ∪ · · · ∪ (K1 \ K2 ) × 1+ Theorem −→ (i) L = CnASCpl ; → − → − (ii) If K ≤ Cn, then Cn = L ; FA extensions Let Cn ∈ EXT(S4.3) and K be a class of finite subdirectly irreducible S4.3-algebras such that Log (K) = Cn(∅) = L. Then one can find K1 ⊇ · · · ⊇ Km such that S(K) = K1 , and → − Ki = S(Ki ) for i = 1, . . . , m, and Cn =fin L , where L = Km ∪ (Km−1 \ Km ) × (m − 1)+ ∪ · · · ∪ (K1 \ K2 ) × 1+ Theorem −→ (i) L = CnASCpl ; → − → − (ii) If K ≤ Cn, then Cn = L ; Corollary −−−−→ K × 1+ = CnLSCpl and −→ K = CnLASCpl . Infinitary Admissible Rules Infinitary Admissible Rules Let Σ be the family of all strictly increasing number-theoretic functions. For each function f ∈ Σ, let rf be ^ _ {[ (pi ↔ pj )] → p0 : n > 0} n<j≤f (n) 0<i≤n p0 Example Let f (n) = n + 1 for each n. Then rf is equivalent to the rule %: {(pi ↔ pj ) → p0 : 0 < i < j} p0 Example Let f (n) = n + 2 for each n. Then rf is equivalent to {(pi ↔ pn+1 ) ∧ (pj ↔ pn+2 )) → p0 : 0 < i, j ≤ n} p0 Basis for admissible rules Basis for admissible rules Note that the rules rf are unifiable, i.e. they are non-passive as p0 /> is a unifier for each of the premises. Basis for admissible rules Note that the rules rf are unifiable, i.e. they are non-passive as p0 /> is a unifier for each of the premises. Lemma The rule rf , for any f ∈ Σ, is valid in any finite algebra A, i.e. ^ _ − → p0 ∈ A ({[ (pi ↔ pj )] → p0 : n > 0}) n<j≤f (n) 0<i≤n Basis for admissible rules Note that the rules rf are unifiable, i.e. they are non-passive as p0 /> is a unifier for each of the premises. Lemma The rule rf , for any f ∈ Σ, is valid in any finite algebra A, i.e. ^ _ − → p0 ∈ A ({[ (pi ↔ pj )] → p0 : n > 0}) n<j≤f (n) 0<i≤n Thus, similarly as % each rule rf is admissible for any extension of S4.3. Basis for admissible rules Note that the rules rf are unifiable, i.e. they are non-passive as p0 /> is a unifier for each of the premises. Lemma The rule rf , for any f ∈ Σ, is valid in any finite algebra A, i.e. ^ _ − → p0 ∈ A ({[ (pi ↔ pj )] → p0 : n > 0}) n<j≤f (n) 0<i≤n Thus, similarly as % each rule rf is admissible for any extension of S4.3. For any consequence operation Cn, let CnΣ be the extension of Cn with the rules {rf }f ∈Σ . We prove that {rf }f ∈Σ is a rule basis for all admissible non-passive (infinitary) rules of any Cn ∈ EXT(S4.3): Basis for admissible rules Note that the rules rf are unifiable, i.e. they are non-passive as p0 /> is a unifier for each of the premises. Lemma The rule rf , for any f ∈ Σ, is valid in any finite algebra A, i.e. ^ _ − → p0 ∈ A ({[ (pi ↔ pj )] → p0 : n > 0}) n<j≤f (n) 0<i≤n Thus, similarly as % each rule rf is admissible for any extension of S4.3. For any consequence operation Cn, let CnΣ be the extension of Cn with the rules {rf }f ∈Σ . We prove that {rf }f ∈Σ is a rule basis for all admissible non-passive (infinitary) rules of any Cn ∈ EXT(S4.3): Theorem Cnfin Σ = CnΣ = CnASCpl =fin Cn, for any Cn ∈ EXT(S4.3). Corollaries Corollaries Corollary Let Cn ∈ EXT(S4.3). Then Cn is almost structurally complete (Cn ∈ ASCpl) iff the rules {rf }f ∈Σ are derivable for Cn (i.e. Cn = CnΣ ). Corollaries Corollary Let Cn ∈ EXT(S4.3). Then Cn is almost structurally complete (Cn ∈ ASCpl) iff the rules {rf }f ∈Σ are derivable for Cn (i.e. Cn = CnΣ ). It also follows from the above result that almost structurally complete consequence operations are hereditary almost structurally complete, i.e. Corollary If Cn, Cn0 ∈ EXT(S4.3), then Cn ∈ ASCpl ∧ Cn ≤ Cn0 ⇒ Cn0 ∈ ASCpl. Cn ∈ FA ∧ Cn ≤ Cn0 ⇒ Cn0 ∈ FA. Corollaries Corollary Let Cn ∈ EXT(S4.3). Then Cn is almost structurally complete (Cn ∈ ASCpl) iff the rules {rf }f ∈Σ are derivable for Cn (i.e. Cn = CnΣ ). It also follows from the above result that almost structurally complete consequence operations are hereditary almost structurally complete, i.e. Corollary If Cn, Cn0 ∈ EXT(S4.3), then Cn ∈ ASCpl ∧ Cn ≤ Cn0 ⇒ Cn0 ∈ ASCpl. Cn ∈ FA ∧ Cn ≤ Cn0 ⇒ Cn0 ∈ FA. The lattice ASCpl(S4.3) The lattice ASCpl(S4.3) Corollary ASCpl(S4.3) is a sublattice of EXT(S4.3). The lattices ASCpl(S4.3) and EXTfin (S4.3) are isomorphic and as the lattice isomorphisms one can take the mappings Cn 7→ CnΣ and Cn 7→ Cnfin . The lattice ASCpl(S4.3) Corollary ASCpl(S4.3) is a sublattice of EXT(S4.3). The lattices ASCpl(S4.3) and EXTfin (S4.3) are isomorphic and as the lattice isomorphisms one can take the mappings Cn 7→ CnΣ and Cn 7→ Cnfin . → − The consequence operation K , where K = Alg(S4.3)si fin , is the least element of the lattice ASCpl(S4.3). This consequence operation can also be given as the ASCpl extension of S4.3, that is as the extension of CnS4.3 with the rules {rf }f ∈Σ . The lattice ASCpl(S4.3) Corollary ASCpl(S4.3) is a sublattice of EXT(S4.3). The lattices ASCpl(S4.3) and EXTfin (S4.3) are isomorphic and as the lattice isomorphisms one can take the mappings Cn 7→ CnΣ and Cn 7→ Cnfin . → − The consequence operation K , where K = Alg(S4.3)si fin , is the least element of the lattice ASCpl(S4.3). This consequence operation can also be given as the ASCpl extension of S4.3, that is as the extension of CnS4.3 with the rules {rf }f ∈Σ .Each element of → − ASCpl(S4.3) has a finite rule basis, over the above defined K , consisting of passive rules just as each element of EXTfin (S4.3)) over S4.3. Infinitary proof theory Infinitary proof theory Corollary For each formula α ∈ CnΣ (X ) we have: 1. either a finitary proof (of α) in which we apply MP, RG and some finitary passive rules valid for Cn (with respect to some formulas in L ∪ X ); 2. or α is given by a single application of one of the rules rf with respect to formulas which have finitary proofs as above. Infinitary proof theory Corollary For each formula α ∈ CnΣ (X ) we have: 1. either a finitary proof (of α) in which we apply MP, RG and some finitary passive rules valid for Cn (with respect to some formulas in L ∪ X ); 2. or α is given by a single application of one of the rules rf with respect to formulas which have finitary proofs as above. In other words each formula α ∈ CnΣ (X ) has a syntactic proof of the type ≤ ω + 1. Infinitary proof theory Corollary For each formula α ∈ CnΣ (X ) we have: 1. either a finitary proof (of α) in which we apply MP, RG and some finitary passive rules valid for Cn (with respect to some formulas in L ∪ X ); 2. or α is given by a single application of one of the rules rf with respect to formulas which have finitary proofs as above. In other words each formula α ∈ CnΣ (X ) has a syntactic proof of the type ≤ ω + 1. Although the rules rf , for f ∈ Σ, look artificial they generate a natural (and simple) proof system. Infinitary proof theory Corollary For each formula α ∈ CnΣ (X ) we have: 1. either a finitary proof (of α) in which we apply MP, RG and some finitary passive rules valid for Cn (with respect to some formulas in L ∪ X ); 2. or α is given by a single application of one of the rules rf with respect to formulas which have finitary proofs as above. In other words each formula α ∈ CnΣ (X ) has a syntactic proof of the type ≤ ω + 1. Although the rules rf , for f ∈ Σ, look artificial they generate a natural (and simple) proof system. The only problem is that the rule basis {rf : f ∈ Σ} has the cardinality c. Σ We suspect that CnS4.3 has no countable rule basis though we failed to prove it and left this problem open.