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Projective unification in modal logic II

Projective unification in modal logic II

Piotr Wojtylak

Institute of Mathematics, Silesian University, Katowice, Poland

Projective unification in modal logic II

Piotr Wojtylak

Institute of Mathematics, Silesian University, Katowice, Poland This talk is based on a paper by Wojciech Dzik and Piotr Wojtylak, Projective Unification in Modal Logic, submitted to the Logic Journal of the IGPL.

Projective unification in modal logic II

Piotr Wojtylak

Institute of Mathematics, Silesian University, Katowice, Poland This talk is based on a paper by Wojciech Dzik and Piotr Wojtylak, Projective Unification in Modal Logic, submitted to the Logic Journal of the IGPL.

Utrecht, 26-28th May 2011

Modal Logic

Modal Logic

Let Var = {x, y, z, . . . } be the set of propositional variables and Fm be the set of modal formulas. For each formula A, let Var(A) denote the (finite) set of variables occurring in A.

Modal Logic

Let Var = {x, y, z, . . . } be the set of propositional variables and Fm be the set of modal formulas. For each formula A, let Var(A) denote the (finite) set of variables occurring in A. By a modal logic we mean any consistent and normal extension of

• S4. More specifically, a modal logic is a proper subset of Fm

closed under substitutions, closed under MP : A → B, A B and RG : A A, containing all classical tautologies, and A → A A → A (A → B) → (A → B).

Modal Logic

Let Var = {x, y, z, . . . } be the set of propositional variables and Fm be the set of modal formulas. For each formula A, let Var(A) denote the (finite) set of variables occurring in A. By a modal logic we mean any consistent and normal extension of

• S4. More specifically, a modal logic is a proper subset of Fm

closed under substitutions, closed under MP : A → B, A B and RG : A A, containing all classical tautologies, and A → A A → A (A → B) → (A → B). Given a modal logic L, we define its global entailment relation ⊢L. Thus, X ⊢L A means that A can be derived from X ∪ L using the rules MP and RG.

Unifiers

By a substitution we mean any finite mapping ε: Var → Fm: ε := x1/B1 · · · xn/Bn and A[ε] or A[x1/B1 · · · xn/Bn] is the result of the substitution.

Unifiers

By a substitution we mean any finite mapping ε: Var → Fm: ε := x1/B1 · · · xn/Bn and A[ε] or A[x1/B1 · · · xn/Bn] is the result of the substitution. A substitution ε is called a unifier for a formula A in L if ⊢L A[ε]. If ε is an L-unifier for A, then εδ is also an L-unifier for A for each substitution δ.

Unifiers

By a substitution we mean any finite mapping ε: Var → Fm: ε := x1/B1 · · · xn/Bn and A[ε] or A[x1/B1 · · · xn/Bn] is the result of the substitution. A substitution ε is called a unifier for a formula A in L if ⊢L A[ε]. If ε is an L-unifier for A, then εδ is also an L-unifier for A for each substitution δ. Unifiers of the form v : Var(A) → {⊥, ⊤} are called ground unifiers for A. They can be identified with valuations in the two-element topological Boolean algebra 2 which satisfy the formula A. Given a unifier ε for A and any substitution δ: Var → {⊥, ⊤} we get the ground unifier εδ for the formula A.

Projectivity

A substitution ε is said to be a projective unifier of a formula A if (i) ⊢L A[ε]; (ii) A ⊢L x[ε] ↔ x, for each variable x.

Projectivity

A substitution ε is said to be a projective unifier of a formula A if (i) ⊢L A[ε]; (ii) A ⊢L x[ε] ↔ x, for each variable x. The concept of a projective unifier (formula, substitution) is due to S.Ghilardi. He used them extensively throughout his papers though the term ‘projective unifier’ did not appear until

• Baader, F.,Ghilardi, S., Unification in modal and description

logics, Logic Journal of the IGPL.

Projectivity

A substitution ε is said to be a projective unifier of a formula A if (i) ⊢L A[ε]; (ii) A ⊢L x[ε] ↔ x, for each variable x. The concept of a projective unifier (formula, substitution) is due to S.Ghilardi. He used them extensively throughout his papers though the term ‘projective unifier’ did not appear until

• Baader, F.,Ghilardi, S., Unification in modal and description

logics, Logic Journal of the IGPL. Let ε0 be a unifier for A in L. Then ε0 is said to be a most general unifier (or, in short, an MGU) for A, on the ground of L, if each L-unifier is an instantiation of ε0, that is if for each L-unifier ε there is a substitution δ such that ε =L ε0δ.

Projectivity

A substitution ε is said to be a projective unifier of a formula A if (i) ⊢L A[ε]; (ii) A ⊢L x[ε] ↔ x, for each variable x. The concept of a projective unifier (formula, substitution) is due to S.Ghilardi. He used them extensively throughout his papers though the term ‘projective unifier’ did not appear until

• Baader, F.,Ghilardi, S., Unification in modal and description

logics, Logic Journal of the IGPL. Let ε0 be a unifier for A in L. Then ε0 is said to be a most general unifier (or, in short, an MGU) for A, on the ground of L, if each L-unifier is an instantiation of ε0, that is if for each L-unifier ε there is a substitution δ such that ε =L ε0δ.

Theorem

Each projective unifier for A is an MGU for A.

Intuitionistic logic

The problem of projective unification for intermediate logics was solved by A.Wro´ nski

• Wro´

nski A., Transparent verifiers in intermediate logics, Abstracts of the 54-th Conference in History of Mathematics, Cracow (2008).

Theorem

An intermediate logic LINT has projective unification iff LC ⊆ LINT.

Intuitionistic logic

The problem of projective unification for intermediate logics was solved by A.Wro´ nski

• Wro´

nski A., Transparent verifiers in intermediate logics, Abstracts of the 54-th Conference in History of Mathematics, Cracow (2008).

Theorem

An intermediate logic LINT has projective unification iff LC ⊆ LINT. One implication of the above theorem follows from

• Minari P. , Wro´

nski A., The property (HD) in intuitionistic

• Logic. A Partial Solution of a Problem of H. Ono, Reports on

mathematical logic 22 (1988), 21–25. They defined a substitution ε (in {⇒, ∧, ¬}) by putting B[ε] = A ⇒ B if B[v] = ⊤ ¬¬A ∧ (A ⇒ B) if B[v] = ⊥ .

Intutionistic Logic

Now, to prove that LC has projective unification, it suffices to know that the disjunction q ∨ r is defined in LC as ((q → r) → r) ∧ ((r → q) → q).

Intutionistic Logic

Now, to prove that LC has projective unification, it suffices to know that the disjunction q ∨ r is defined in LC as ((q → r) → r) ∧ ((r → q) → q). To prove the reverse implication, let us assume that ε is a projective unifier for (q ⇒ r) ∨ (r ⇒ q), denoted by A, which is an axiom for LC. Then (by A ⊢L x[ε] ↔ x) A ∧ x ⊢ x[ε] and x[ε] ⊢ A ⇒ x, for each variable x.

Intutionistic Logic

Now, to prove that LC has projective unification, it suffices to know that the disjunction q ∨ r is defined in LC as ((q → r) → r) ∧ ((r → q) → q). To prove the reverse implication, let us assume that ε is a projective unifier for (q ⇒ r) ∨ (r ⇒ q), denoted by A, which is an axiom for LC. Then (by A ⊢L x[ε] ↔ x) A ∧ x ⊢ x[ε] and x[ε] ⊢ A ⇒ x, for each variable x. Now, we need to know that (in intuitionistic logic) (A ⇒ q) = q = (A ∧ q) and (A ⇒ r) = r = (A ∧ r) which is sufficient to show that x[ε] = x for each variable x ∈ {q, r}. Thus, we get A[ε] = A. Since ε is an LINT-unifier for A, we conclude that LINT ⊢ A and hence LC ⊆ LINT.

Problem

The question arises if the same characterization of logics with projective unification, as for intermediate logics, can be proved for modal systems.

Problem

The question arises if the same characterization of logics with projective unification, as for intermediate logics, can be proved for modal systems.

Theorem

If a modal logic L has projective unification, then (y → z) ∨ (z → y) ∈ L. Thus, to characterize all modal logics with projective unification,

• ne should consider the modal logic S4.3, which is obtained by

extending S4 with (A → B) ∨ (B → A).

Problem

The question arises if the same characterization of logics with projective unification, as for intermediate logics, can be proved for modal systems.

Theorem

If a modal logic L has projective unification, then (y → z) ∨ (z → y) ∈ L. Thus, to characterize all modal logics with projective unification,

• ne should consider the modal logic S4.3, which is obtained by

extending S4 with (A → B) ∨ (B → A). However, it is not an easy matter to show that S4.3 enjoys projective unification. It turns out, in particular, that the method

• f ground unifiers, as used for intermediate logics, does not work

for modal systems.

Example

Example

Suppose that a formula A is unifiable, let v : Var(A) → {⊤, ⊥} be a ground unifier for A. Let us define a substitution ε as follows: x[ε] = A → x if x[v] = ⊤ A ∧ x if x[v] = ⊥ for each variable x ∈ Var(A).

Example

Example

Suppose that a formula A is unifiable, let v : Var(A) → {⊤, ⊥} be a ground unifier for A. Let us define a substitution ε as follows: x[ε] = A → x if x[v] = ⊤ A ∧ x if x[v] = ⊥ for each variable x ∈ Var(A). One could try to prove that B[ε] = A → B if B[v] = ⊤ A ∧ B if B[v] = ⊥

Example

Example

Suppose that a formula A is unifiable, let v : Var(A) → {⊤, ⊥} be a ground unifier for A. Let us define a substitution ε as follows: x[ε] = A → x if x[v] = ⊤ A ∧ x if x[v] = ⊥ for each variable x ∈ Var(A). One could try to prove that B[ε] = A → B if B[v] = ⊤ A ∧ B if B[v] = ⊥ The problem rises when B is B1 and B[v] = ⊤. Then B[ε] = (A → B1) = (A → B). Since (A → B) is equivalent to A → B only if A = ♦A (which is not S4.3 valid), we only prove that S5 enjoys projective unification, see

• Dzik, W., Unitary unification of S5 logic and its extensions,

Bulletin of the Section of Logic 32(1–2) (2003), 19–26.

Example

Example

Let us try a modified approach. Suppose that v : Var(A) → {⊤, ⊥} is a ground unifier for A, and define x[ε] = ♦A → x if x[v] = ⊤ ♦A ∧ x if x[v] = ⊥ for each variable x ∈ Var(A).

Example

Example

Let us try a modified approach. Suppose that v : Var(A) → {⊤, ⊥} is a ground unifier for A, and define x[ε] = ♦A → x if x[v] = ⊤ ♦A ∧ x if x[v] = ⊥ for each variable x ∈ Var(A). By induction with respect to the length of a formula B in {→, ⊥, }, with Var(B) ⊆ Var(A), one proves without problems B[ε] = ♦A → B if B[v] = ⊤ ♦A ∧ B if B[v] = ⊥

Example

Example

Let us try a modified approach. Suppose that v : Var(A) → {⊤, ⊥} is a ground unifier for A, and define x[ε] = ♦A → x if x[v] = ⊤ ♦A ∧ x if x[v] = ⊥ for each variable x ∈ Var(A). By induction with respect to the length of a formula B in {→, ⊥, }, with Var(B) ⊆ Var(A), one proves without problems B[ε] = ♦A → B if B[v] = ⊤ ♦A ∧ B if B[v] = ⊥ Thus, A[ε] = ♦A → A which means we get, again, a projective unifier for A in S5.

Example

Example

Let us consider a ‘modal’ version of the projective unifier in LC x[ε] = (A → x) if x[v] = ⊤ ♦A ∧ (A → x) if x[v] = ⊥ assuming that v is a ground unifier for A and A = A.

Example

Example

Let us consider a ‘modal’ version of the projective unifier in LC x[ε] = (A → x) if x[v] = ⊤ ♦A ∧ (A → x) if x[v] = ⊥ assuming that v is a ground unifier for A and A = A. Then B[ε] = (A → B) if B[v] = ⊤ ♦A ∧ (A → B) if B[v] = ⊥ for each intuitionistic formula B.

Example

Example

Let us consider a ‘modal’ version of the projective unifier in LC x[ε] = (A → x) if x[v] = ⊤ ♦A ∧ (A → x) if x[v] = ⊥ assuming that v is a ground unifier for A and A = A. Then B[ε] = (A → B) if B[v] = ⊤ ♦A ∧ (A → B) if B[v] = ⊥ for each intuitionistic formula B. However, the above fails for formulas B with ‘unboxed’ variables, e.g. for x → y.

Example

Example

Let us consider a ‘modal’ version of the projective unifier in LC x[ε] = (A → x) if x[v] = ⊤ ♦A ∧ (A → x) if x[v] = ⊥ assuming that v is a ground unifier for A and A = A. Then B[ε] = (A → B) if B[v] = ⊤ ♦A ∧ (A → B) if B[v] = ⊥ for each intuitionistic formula B. However, the above fails for formulas B with ‘unboxed’ variables, e.g. for x → y. We could

• nly settle the problem of projective unification for formulas with

‘boxed’ variables on the ground of S4.3Grz, the extension of S4.3 with ((A → A) → A) → A (Grzegorczyk’s formula). It is known that each ’boxed’ formula with ’boxed’ variables is, in S4.3Grz, an intuitionistic formula.

Example

Example

Let A be x ∨ ∼x.

Example

Example

Let A be x ∨ ∼x. We have two ground unifiers for A and hence we get two projective substitutions for A ε0 : x/A ∧ x, where A ∧ x = (x ∨ ∼x) ∧ x = x ε1 : x/A → x, where A → x = (x ∨ ∼x) → x = ♦x

Example

Example

Let A be x ∨ ∼x. We have two ground unifiers for A and hence we get two projective substitutions for A ε0 : x/A ∧ x, where A ∧ x = (x ∨ ∼x) ∧ x = x ε1 : x/A → x, where A → x = (x ∨ ∼x) → x = ♦x Note that neither ε0, nor ε1 is a unifier for A. Indeed, we have A[ε0] = x ∨ ∼x = ♦x → x A[ε1] = ♦x ∨ ∼♦x = ♦x → ♦x.

Example

Example

Let A be x ∨ ∼x. We have two ground unifiers for A and hence we get two projective substitutions for A ε0 : x/A ∧ x, where A ∧ x = (x ∨ ∼x) ∧ x = x ε1 : x/A → x, where A → x = (x ∨ ∼x) → x = ♦x Note that neither ε0, nor ε1 is a unifier for A. Indeed, we have A[ε0] = x ∨ ∼x = ♦x → x A[ε1] = ♦x ∨ ∼♦x = ♦x → ♦x. However, if one takes the substitution ε0ε1, or ε1ε0, then A[ε0ε1] = ♦♦x → ♦x = ⊤ A[ε1ε0] = ♦x → ♦x = ⊤.

Example

Thus, both substitutions ε0ε1 and ε0ε1 are projective unifiers for the formula x ∨ ∼x.

Example

Thus, both substitutions ε0ε1 and ε0ε1 are projective unifiers for the formula x ∨ ∼x. Note that x[ε0ε1] = ♦x and x[ε1ε0] = ♦x which means that ε1 and ε0 are not equivalent.

Example

Thus, both substitutions ε0ε1 and ε0ε1 are projective unifiers for the formula x ∨ ∼x. Note that x[ε0ε1] = ♦x and x[ε1ε0] = ♦x which means that ε1 and ε0 are not equivalent. This shows that projective unifiers, similarly as MGU’s, for a given formula A, may be not unique. A unifiable formula may have, if any, several non-equivalent projective unifiers.

Main idea

The above example suggests the way in which one should try to get a projective unifier for a given unifiable formula A. To this aim

• ne should define a sequence ε1, . . . , εn of projective substitutions

for A in L, and take their composition ε = ε1 · · · εn.

Main idea

The above example suggests the way in which one should try to get a projective unifier for a given unifiable formula A. To this aim

• ne should define a sequence ε1, . . . , εn of projective substitutions

for A in L, and take their composition ε = ε1 · · · εn. If one takes sufficiently many projective substitutions in the sequence ε1, . . . , εn, one may expect to get a unifier for A. However, it is not clear what ‘sufficiently many’ means here and, besides, the

• rder of substitutions in the sequence may also be important.

Main idea

The above example suggests the way in which one should try to get a projective unifier for a given unifiable formula A. To this aim

• ne should define a sequence ε1, . . . , εn of projective substitutions

for A in L, and take their composition ε = ε1 · · · εn. If one takes sufficiently many projective substitutions in the sequence ε1, . . . , εn, one may expect to get a unifier for A. However, it is not clear what ‘sufficiently many’ means here and, besides, the

• rder of substitutions in the sequence may also be important.

The above described idea is due to Silvio Ghilardi who used it to settle the unification problem for a number of intermediate and modal logics, including INT, K4, S4 and S4.2, see

• Ghilardi S., Unification through projectivity, Journal of

Symbolic Computation 7 (1997), 733–752.

• Ghilardi S., Unification in intuitionistic logic, Journal of

Symbolic Logic 64(2) (1999), 859–880.

• Ghilardi S.,Best solving modal equations, Annals of Pure and

Applied Logic 102 (2000), 183–198.

Deviations

The substitutions such that x[εi] is either A → x or A ∧ x, for each x ∈ Var(A) are taken as ε1, . . . , εn. They are called simplified L¨

• wenheim substitutions for A.

Deviations

The substitutions such that x[εi] is either A → x or A ∧ x, for each x ∈ Var(A) are taken as ε1, . . . , εn. They are called simplified L¨

• wenheim substitutions for A. In the case of modal systems, a

formula A is projective iff a suitable composition of simplified L¨

• wenheim substitutions is a unifier for A. Some bounds on the

number of substitutions in the compositions are also given.

Deviations

The substitutions such that x[εi] is either A → x or A ∧ x, for each x ∈ Var(A) are taken as ε1, . . . , εn. They are called simplified L¨

• wenheim substitutions for A. In the case of modal systems, a

formula A is projective iff a suitable composition of simplified L¨

• wenheim substitutions is a unifier for A. Some bounds on the

number of substitutions in the compositions are also given. To settle the problem of unification in S4.3 we follow Ghilardi’s ideas with some modifications. First of all, we extend the class of simplified L¨

• wenheim substitutions by allowing x[εi] to be A → x,
• r A ∧ x (in addition to A → x or A ∧ x) and even allowing

x[εi] = x.

Deviations

The substitutions such that x[εi] is either A → x or A ∧ x, for each x ∈ Var(A) are taken as ε1, . . . , εn. They are called simplified L¨

• wenheim substitutions for A. In the case of modal systems, a

formula A is projective iff a suitable composition of simplified L¨

• wenheim substitutions is a unifier for A. Some bounds on the

number of substitutions in the compositions are also given. To settle the problem of unification in S4.3 we follow Ghilardi’s ideas with some modifications. First of all, we extend the class of simplified L¨

• wenheim substitutions by allowing x[εi] to be A → x,
• r A ∧ x (in addition to A → x or A ∧ x) and even allowing

x[εi] = x. Secondly, the substitutions εi, from which the projective unifier is composed, are generalized, in the just mentioned sense, L¨

• wenheim substitutions, but not for the master formula A, only

for its certain fragments.

CNF

Our approach to the problem of unification in S4.3 is based on the reduction of any formula to its conjunctive normal form (CNF-formula) in which only variables are ‘boxed’. This reduction is possible at the cost of introducing (fresh) variables.

CNF

Our approach to the problem of unification in S4.3 is based on the reduction of any formula to its conjunctive normal form (CNF-formula) in which only variables are ‘boxed’. This reduction is possible at the cost of introducing (fresh) variables.

Theorem

For every formula A one can find two disjoint sets of variables X and Y and a CNF formula A⋆ which is a conjunction of the following clauses: (⋆) x1 ∨ · · · ∨ xn∨ ∼ y1 ∨ · · · ∨ ∼ ym ∨ l1 ∨ · · · ∨ lk where xi ∈ X for 0 ≤ i ≤ n, and yi ∈ Y for 0 ≤ i ≤ m, and li is a variable – or its negation – from the set X ∪ Y for 0 ≤ i ≤ k, and (i) A is unifiable iff A⋆ is unifiable; (ii) if A⋆ has a projective unifier, then A has a projective unifier, as well.

Proof

Example

Let A = z ∨ C where z is a variable. If does not occur in C, one may simply take z/A → z as a projective unifier for A: A[z/A → z] = (A → z) ∨ C = (z ∨ C) → (z ∨ C) = ⊤. If we allow z to appear in C, we take z/A → z.

Proof

Example

Let A = z ∨ C where z is a variable. If does not occur in C, one may simply take z/A → z as a projective unifier for A: A[z/A → z] = (A → z) ∨ C = (z ∨ C) → (z ∨ C) = ⊤. If we allow z to appear in C, we take z/A → z.

Example

Similarly, z/A ∧ z is a projective unifier for A if A =∼ z ∨ C.

Proof

Example

Let A = z ∨ C where z is a variable. If does not occur in C, one may simply take z/A → z as a projective unifier for A: A[z/A → z] = (A → z) ∨ C = (z ∨ C) → (z ∨ C) = ⊤. If we allow z to appear in C, we take z/A → z.

Example

Similarly, z/A ∧ z is a projective unifier for A if A =∼ z ∨ C.

Example

Let A = x1 ∨ · · · ∨ xn. Then x1/A → x1 · · · xn/A → xn. is a projective unifier for A. Note it is evident for n = 2

Proof

Example

The worst case consists of U-clauses x1 ∨ · · · ∨ xn∨ ∼ y1 ∨ · · · ∨ ∼ ym, where n ≥ 1. Let {y1, . . . , ym} = U and write down the above clause as U → x1 ∨ · · · ∨ xn.

Proof

Example

The worst case consists of U-clauses x1 ∨ · · · ∨ xn∨ ∼ y1 ∨ · · · ∨ ∼ ym, where n ≥ 1. Let {y1, . . . , ym} = U and write down the above clause as U → x1 ∨ · · · ∨ xn. Let ε be the substitution x1/A → x1 · · · xn/A → xn. It is clear that ε is projective for A. Note that ε is a substitution for the variables X and hence

Proof

Example

The worst case consists of U-clauses x1 ∨ · · · ∨ xn∨ ∼ y1 ∨ · · · ∨ ∼ ym, where n ≥ 1. Let {y1, . . . , ym} = U and write down the above clause as U → x1 ∨ · · · ∨ xn. Let ε be the substitution x1/A → x1 · · · xn/A → xn. It is clear that ε is projective for A. Note that ε is a substitution for the variables X and hence A[ε] = U → (A → x1) ∨ · · · ∨ (A → xn) = = U →

• U ∧ (A → x1)
• ∨ · · · ∨
• U ∧ (A → xn)
• =

= U →

• (x1∨· · ·∨xn) → x1
• ∨· · ·∨
• (x1∨· · ·∨xn) → xn

Theorem

Theorem

Each unifiable formula has a projective unifier in S4.3.

Theorem

Theorem

Each unifiable formula has a projective unifier in S4.3. A projective unifier for a unifiable formula A is defined as the composition ε1 · · · εn of the just described ‘partial’ unifiers for A.

Theorem

Theorem

Each unifiable formula has a projective unifier in S4.3. A projective unifier for a unifiable formula A is defined as the composition ε1 · · · εn of the just described ‘partial’ unifiers for A. The decisive step of our reduction procedure is the removal of all U-clauses, for any non-empty U ⊆ Y . This is done (similarly as in Ghilardi’s papers) in accordance with a linearization of the inclusion relation on subsets of Y .

Theorem

Theorem

Each unifiable formula has a projective unifier in S4.3. A projective unifier for a unifiable formula A is defined as the composition ε1 · · · εn of the just described ‘partial’ unifiers for A. The decisive step of our reduction procedure is the removal of all U-clauses, for any non-empty U ⊆ Y . This is done (similarly as in Ghilardi’s papers) in accordance with a linearization of the inclusion relation on subsets of Y .

Corollary

A modal logic L containing S4 has projective unification if and

• nly if S4.3 ⊆ L.

S.Ghilardi

• Ghilardi S.,Best solving modal equations, Annals of Pure and

Applied Logic 102 (2000), 183–198.

Theorem

The following conditions are equivalent: (i) A is projective; (iii) ModL(A) has the extension property.

S.Ghilardi

• Ghilardi S.,Best solving modal equations, Annals of Pure and

Applied Logic 102 (2000), 183–198.

Theorem

The following conditions are equivalent: (i) A is projective; (iii) ModL(A) has the extension property. A variant of a Kripke model < F, u >, where F =< F, R, ρ > is a finite rooted frame and u : F → P(Var) is a Kripke model < F, u0 > such that u(p) = u0(p) holds for all p ∈ cl(ρ). A class K ⊆ ModL of Kripke models over is said to have the extension property if for every Kripke model < F, u >∈ ModL, if < Fp, up >∈ K holds for every p ∈ cl(ρ), then there is a variant < F, u0 > of < F, u >, such that < F, u0 >∈ K.

Relative unification

A formula A(X, Y ) is X-unifiable (on the ground of L) if there is a substitution ε: X → Fm such that L ⊢ A[ε].

Relative unification

A formula A(X, Y ) is X-unifiable (on the ground of L) if there is a substitution ε: X → Fm such that L ⊢ A[ε].Under certain assumptions one can prove Each X-unifiable formula has a projective X-unifier in S4.3.

Relative unification

A formula A(X, Y ) is X-unifiable (on the ground of L) if there is a substitution ε: X → Fm such that L ⊢ A[ε].Under certain assumptions one can prove Each X-unifiable formula has a projective X-unifier in S4.3.

Corollary

Let A, F be modal formulas and S4.3 ⊆ L. If A is unifiable, then there exists a substitution ε such that (i) F ⊢L A[ε]; (ii) F → A ⊢L x[ε] ↔ x, for each variable x.

Extensions of S4.3

All normal extensions of S4.3 have the finite model property and they are all finitely axiomatizable.

Extensions of S4.3

All normal extensions of S4.3 have the finite model property and they are all finitely axiomatizable. However, the full lattice of normal extensions of S4.3 is ‘one of great complexity’, as Kit Fine puts it in

• Fine K., The logics containing S4.3, Zeitschrift fur Math.

Logik und Grundlagen der Mathematik 17 (1971), 371–376.

Extensions of S4.3

All normal extensions of S4.3 have the finite model property and they are all finitely axiomatizable. However, the full lattice of normal extensions of S4.3 is ‘one of great complexity’, as Kit Fine puts it in

• Fine K., The logics containing S4.3, Zeitschrift fur Math.

Logik und Grundlagen der Mathematik 17 (1971), 371–376.

Theorem

Every extension (with inferential rules) of S4.3 is finitely axiomatizable and has the strong finite model property.

SCPL

We consider structural rules of the form A/B.

SCPL

We consider structural rules of the form A/B. The rule is passive, if A is not unifiable in L.

SCPL

We consider structural rules of the form A/B. The rule is passive, if A is not unifiable in L. The logic S5 is not structurally complete, since the following rule is admissible but not derivable in S5: P2 : ♦x ∧ ♦∼x ⊥

SCPL

We consider structural rules of the form A/B. The rule is passive, if A is not unifiable in L. The logic S5 is not structurally complete, since the following rule is admissible but not derivable in S5: P2 : ♦x ∧ ♦∼x ⊥ We say that a logic L is almost structurally complete, L ∈ ASCpl, if every admissible rule in L, which is not passive, is derivable in L.

Theorem

Every modal logic extending S4.3 is almost structurally complete.

Topological Boolean Algebras

Let K be a class of finite subdirectly irreducible TBA regarded as matrix models for S4.3.

Topological Boolean Algebras

Let K be a class of finite subdirectly irreducible TBA regarded as matrix models for S4.3. Let L be the modal logic determined by K.

Topological Boolean Algebras

Let K be a class of finite subdirectly irreducible TBA regarded as matrix models for S4.3. Let L be the modal logic determined by

• K. Suppose that ⊢ is a consequence relation extending L (regarded

as a proof system) with some passive rules.

Topological Boolean Algebras

Let K be a class of finite subdirectly irreducible TBA regarded as matrix models for S4.3. Let L be the modal logic determined by

• K. Suppose that ⊢ is a consequence relation extending L (regarded

as a proof system) with some passive rules.

Theorem

The class {B × Hn : B ∈ K and B × Hn is a model for ⊢ } is strongly complete for Cn; where Hn denotes the Henle algebra with n-atoms and n ≥ 1.

Topological Boolean Algebras

Let K be a class of finite subdirectly irreducible TBA regarded as matrix models for S4.3. Let L be the modal logic determined by

• K. Suppose that ⊢ is a consequence relation extending L (regarded

as a proof system) with some passive rules.

Theorem

The class {B × Hn : B ∈ K and B × Hn is a model for ⊢ } is strongly complete for Cn; where Hn denotes the Henle algebra with n-atoms and n ≥ 1.

Theorem

The class {B × H1 : B ∈ K} determines the structurally complete extension of L.

K4.3

The modal logic K4.3 is axiomatized by the formulas A → A (A → B) → (A → B) (A → B) ∨ (B → A).

K4.3

The modal logic K4.3 is axiomatized by the formulas A → A (A → B) → (A → B) (A → B) ∨ (B → A). The axiom (of S4) A → A is lacking

K4.3

The modal logic K4.3 is axiomatized by the formulas A → A (A → B) → (A → B) (A → B) ∨ (B → A). The axiom (of S4) A → A is lacking and, in result, the formula x has no projective unifier.

K4.3

The modal logic K4.3 is axiomatized by the formulas A → A (A → B) → (A → B) (A → B) ∨ (B → A). The axiom (of S4) A → A is lacking and, in result, the formula x has no projective unifier. Indeed, the only unifier of x is x/T,

K4.3

The modal logic K4.3 is axiomatized by the formulas A → A (A → B) → (A → B) (A → B) ∨ (B → A). The axiom (of S4) A → A is lacking and, in result, the formula x has no projective unifier. Indeed, the only unifier of x is x/T, but x ⊢K4.3 x ↔ T is not valid as it would lead to K4.3 ⊢ x → x.