Download Admissible Infinitary Rules in Modal Logic. Part II

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.