Products of infinitely many modal logics

Yasusi Hasimoto

                          hasimoto@tokiwa.ac.jp

Department of Management Information, Tokiwa Junior College, Mito, Ibaraki, 310-8585, Japan

Abstract The definition of normal products introduced in [3] is based on the standard way of introducing “products” in measure theory and topology. Here, we will show that the same idea is also applicable to infinitely many products.

1 Products of infinitely many modal general frames

We assume the logics are extensions of K ⊕ ◇⊤ in order to define products of infinitely many modal logics.

For modal algebras P_{λ} on (W_{λ}, R_{λ}), define × ×_{λ∈Λ}P_{λ} to be the modal algebra on ( ∏_{λ∈Λ}W_{λ}, ∏_{λ∈Λ}R_{λ})

generated by {π_{λ}^{−1}(X_{λ}) | X_{λ} ∈ P_{λ} and λ ∈ Λ}. We say π_{{λ_{1},...,λ_{n}}}^{−1}(X_{λ_{1}} × ⋯ × X_{λ_{n}}) to be cylindric set of {P_{λ} | λ ∈ Λ} for a finite subset {λ_{1}, …, λ_{n}} of Λ and X_{λ_{i}} ∈ P_{λ_{i}}. Note that π_{{λ_{1},...,λ_{n}}}^{−1}(X_{λ_{1}} ×⋯×X_{λ_{n}}) =

π_{λ_{1}}^{−1}(X_{λ_{1}}) ∩ ⋯ ∩ π_{λ_{1}}^{−1}(X_{λ_{n}}) and that C is a cylindric set of {P_{λ} | λ ∈ Λ} iff C = ∏_{λ∈Λ}X_{λ}, where X_{λ} ∈ P_{λ}

for every λ ∈ Λ and X_{λ} = W_{λ} finitely except λ ∈ Λ.

Then we can also define as follows.

× ×_{λ∈Λ} P_{λ} = {⋃_{i∈I} C_{i} | C_{i} is a cylindric set of {P_{λ} | λ ∈ Λ} for a finite I }.

For given modal general frames F_{λ} = (W_{λ}, R_{λ}, P_{λ}) (λ ∈ Λ), the normal product × ×_{λ∈Λ}F_{λ} of F_{λ}’s is a

modal general frame ( ∏_{λ∈Λ}W_{λ}, ∏_{λ∈Λ}R_{λ}, × ×_{λ∈Λ}P_{λ}).

Theorem 1 A projection π_{λ} : ∏_{λ∈Λ}W_{λ} → W_{λ} is always a reduction of × ×_{λ∈Λ}F_{λ} to F_{λ} .

The next theorem gathers together several essential properties of normal products.

Theorem 2 1. If f_{λ} is a reduction of F_{λ} to G_{λ} for λ ∈ Λ, then the product map × ×_{λ∈Λ}f_{λ} is a reduction of × ×_{λ∈Λ}F_{λ}

to × ×_{λ∈Λ}G_{λ}, where × ×_{λ∈Λ}f_{λ} is defined by ((× ×_{λ∈Λ}f_{λ})(w))(λ) = f_{λ}(w_{λ}) for w ∈ ∏_{λ∈Λ}W_{λ}.

2. If G_{λ} is a generated subframe of F_{λ} for λ ∈ Λ, then × ×_{λ∈Λ}G_{λ} is also a generated subframe of × ×_{λ∈Λ}F_{λ}.

3. The identity map id : ∑ ; λ∈∏_{μ∈M} Λ_{μ} ∏_{μ∈M}W_{λ_{μ}}^{μ} → ∏_{μ∈M} ∑_{λ_{μ}∈Λ_{μ}}W_{λ_{μ}}^{μ} is a reduction of ∑ ; λ∈∏_{μ∈M} Λ_{μ} × ×_{μ∈M}F_{λ_{μ}}^{μ}

to × ×_{μ∈M} ∑_{λ_{μ}∈Λ_{μ}}F_{λ_{μ}}^{μ}.

For modal algebras A_{λ}’s, we define their normal product × ×_{λ∈Λ}A_{λ} as (× ×_{λ∈Λ}(A_{λ})_{+})^{+}.

Corollary 3 For modal general frames F_{λ} for λ ∈ Λ,

(× ×_{λ∈Λ}F_{λ})^{+} ≅ × ×_{λ∈Λ}(F_{λ})^{+}.  (1)

Theorem 4 If modal general frame F_{λ} is descriptive for λ ∈ Λ, then their normal product × ×_{λ∈Λ}F_{λ} is also descriptive. Corollary 5 For modal algebras A_{λ}’s,

(× ×_{λ∈Λ}A_{λ})_{+} ≅ × ×_{λ∈Λ}(A_{λ})_{+}.  (2)

We can show the duality of Theorem 2 as follows. Theorem 6

1. If B_{λ} is a homomorphic image of A_{λ}, then × ×_{λ∈Λ}B_{λ} is also a homomorphic image of × ×_{λ∈Λ}A_{λ}.

2. If B_{λ} is a subalgebra of A_{λ}, then × ×_{λ∈Λ}B_{λ} is isomorphic to a subalgebra of × ×_{λ∈Λ}A_{λ}.

3. × ×_{μ∈M} ∏_{λ_{μ}∈Λ_{μ}}A_{λ_{μ}}^{μ} is isomorphic to a subalgebra of ∏ ; λ∈∏_{μ∈M} Λ_{μ} × ×_{μ∈M}A_{λ_{μ}}^{μ}.

By Theorem 6, we have the following. Theorem 7 For classes K_{λ} of modal algebras for λ ∈ Λ,

1. × ×_{λ∈Λ}H(K_{λ}) ⊆ H(× ×_{λ∈Λ}K_{λ}),

2. × ×_{λ∈Λ}S(K_{λ}) ⊆ IS(× ×_{λ∈Λ}K_{λ}),

3. × ×_{λ∈Λ}P (K_{λ}) ⊆ ISP (× ×_{λ∈Λ}K_{λ}).

Hence,

4. V (× ×_{λ∈Λ}V (K_{λ})) = V (× ×_{λ∈Λ}K_{λ}).

2 Products of infinitely many modal logics

For modal logics L_{λ}’s, define the normal product × ×_{λ∈Λ}L_{λ} to be × ×_{λ∈Λ}L(F (L_{λ}). By (1) and (2), × ×_{λ∈Λ}L_{λ} = L(× ×_{λ∈Λ}V (L_{λ})) holds. Thus, we may also take the latter as the definition of × ×_{λ∈Λ}L_{λ}. By Theorem 7(4), we have the following. Theorem 8 Let K_{λ} be classes of modal algebras, and let C_{λ} be classes of modal general frames for λ ∈ Λ. Then

1. × ×_{λ∈Λ}L(K_{λ}) = L(× ×_{λ∈Λ}K_{λ}),

2. × ×_{λ∈Λ}L(C_{λ}) = L(× ×_{λ∈Λ}C_{λ}).

Theorem 9 Suppose that L_{λ} is modal logic for λ ∈ Λ. Then

⋂_{λ∈Λ} L_{λ} ⊇ × ×_{λ∈Λ} L_{λ}.

Theorem 10 Let φ be a formula which is equivalent to a conjunction of formulas of the form □^{m}(ψ → χ), where m ≥ 0, χ is constructed by ∧, □, ◇, ⊤, ⊥ from propositional variables and ψ is obtained from propositional variables applying ∧, ∨, □, ◇, ⊤, ⊥ in such a way that no subformula of the form either ψ_{1} ∨ ψ_{2} or ◇ψ_{1} occurs in the scope of any □ and that ψ_{1} and ψ_{2} have no common propositional variable in a subformula of the form ψ_{1} ∧ ψ_{2}.Then φ ∈ ⋂_{λ∈Λ} L_{λ} ⇒ φ ∈ × ×_{λ∈Λ} L_{λ}.

3 Shifted products of infinitely many modal logics

Suppose that A_{λ} = (A_{λ}, ∧, ∨, →, 0, 1, □_{λ}) is mono-modal algebra, for λ ∈ Λ. Define multi-modal algebra A_{λ}^{∘} as A_{λ}^{∘} = (A_{λ}, ∧, ∨, →, 0, 1, {□_{μ} | μ ∈ Λ}) with □_{μ}a = a for all μ ≠ λ and all a ∈ A. For modal general frame F_{λ} for λ ∈ Λ, define modal general frame F_{λ}^{∘}, with binary relations, as F_{λ}^{∘} = (W_{λ}, {R_{μ} | μ ∈ Λ}, P), where R_{μ} = Δ for all μ ≠ λ. For a class K of mono-modal algebras and a class C of modal general frames, K^{∘} and C^{∘} are defined in a natural way. Now, define the shifted product ⨂_{λ∈Λ}A_{λ} of A_{λ}’s,

(the shifted product ⨂_{λ∈Λ}F_{λ} of F_{λ}’s ) by ⨂_{λ∈Λ}A_{λ} = × ×_{λ∈Λ}A_{λ}^{∘} (⨂_{λ∈Λ}F_{λ} = × ×_{λ∈Λ}F_{λ}^{∘}). We can define shifted

product of classes of modal algebras and those of classes of modal general frames in an obvious way. Then, we define the shifted product ⨂_{λ∈Λ}L_{λ} of modal logics L_{λ}’s by ⨂_{λ∈Λ}L_{λ} = L(⨂_{λ∈Λ}F (L_{λ})).

Theorem 11 Let K_{λ} be classes of modal algebras, and let C_{λ} be classes of modal general frames for λ ∈ Λ. Then

1. ⨂_{λ∈Λ}L(K_{λ}) = L(⨂_{λ∈Λ}K_{λ}),

2. ⨂_{λ∈Λ}L(C_{λ}) = L(⨂_{λ∈Λ}C_{λ}).

Suppose that L_{λ} is a modal logic. Define multi-modal logics L_{λ}^{∘} as L_{λ}^{∘} = ∗_{μ∈Λ}L_{μ}, where L_{μ} =

K ⊕ □p ↔ p for μ ≠ λ and ∗ denotes the fusion of logics.

Theorem 12 For modal logics L_{λ}’s,

⨂_{λ∈Λ} L_{λ} = × ×_{λ∈Λ} L_{λ}^{∘}.

We define the multi-modal logic [L_{λ} | λ ∈ Λ] to be ∗_{λ∈Λ}L_{λ} ⊕ □_{λ_{i}}□_{λ_{j}}p ↔ □_{λ_{j}}□_{λ_{i}}p ⊕ ◇_{λ_{i}}□_{λ_{j}}p →

□_{λ_{j}}◇_{λ_{i}}p, for i ≠ j. The following theorem lists some properties of ⊗.

Theorem 13 Let L_{λ} be modal logic for λ ∈ Λ. Then

1. If L_{λ}’s are consistent logics, then ⨂_{λ∈Λ} L_{λ} is a conservative extension of L_{λ},

2. ⨂_{λ∈Λ} L_{λ} ⊇ [L_{λ} | λ ∈ Λ].

References

[1] V. Shehtman, Two-dimensional modal logics. Math. Zametki, vol. 5, pp. 759 – 772, (In Russian), 1978.

[2] D. Gabbay and V. Shehtman, Products of modal logics, Part I . L. J. of IGPL, vol. 6 No. 1, pp. 73 – 146, 1998.

[3] Y. Hasimoto, Normal products of modal logics. to appear in Advances in Modal Logic, Volume 3, CSLI Publications.