Some properties on orthologics

Yutaka Miyazaki

                        y-miya@logic.jaist.ac.jp

School of Information Science, JAIST

We have only little knowledge about the class of orthologics now. We intend this study to be a small step to throw light on the problems of these area of mathematical logics. Here we discuss some logical properties, and the structure of the lattice of modular orthologics.

1 Preliminaries

The notion of orthologics stems from an algebraic structure called ortholattice, which we will define first.

Definition 1 (Ortholattice) An ortholattice is an algebraic structure 𝔄 = ⟨A, ∩, ∪, ′ , 0.1⟩, which is a bounded lattice with a unary operation (·)′ that satisfies the following: for any x, y ∈ A,

(a) x′′ = x.  (b) x ∩ x′ = 0.  (c) x ≤ y implies y′ ≤ x′.

Our propositional language consists of the following primitive symbols: a collection of propositional variables p, q, r, p_{0}, p_{1}, … etc., a propositional constant ⊥ (falsity), a unary connective ¬ (negation), a binary connective ∧ (conjunction), and a pair of parentheses ( , ). The set Φ of all formulas of this language is defined in a usual way, and the notion of orthologics is defined in the following way.

Definition 2 (Orthologic) ([1]) An orthologic L on the set Φ of formulas is a subset of the product Φ × Φ (write α ⊢_{L} β, instead of ⟨α, β⟩ ∈ L) which includes the following axiom schemes and is closed under the following inference rules: Axiom schemes:

(Ax1)

α ⊢_{L} α

(Ax2)

¬¬α ⊢_{L} α

(Ax3)

α ⊢_{L} ¬¬α

(Ax4)

α ∧ β ⊢_{L} α

(Ax5)  α ∧ β ⊢_{L} β (Ax6)  α ∧ ¬α ⊢_{L} β (Ax7)  ⊥ ⊢_{L} α

Inference Rules:

(R1)

\begin{matrix}α ⊢_{L} β & β ⊢_{L} γ \\ α ⊢_{L} γ \end{matrix}

(R2)

\begin{matrix}α ⊢_{L} β & α ⊢_{L} γ \\ α ⊢_{L} β ∧ γ \end{matrix}

(R3)

{α ⊢_{L} β \atop ¬β ⊢_{L} ¬α}

The intersection of all orthologics on Φ, that is called the (smallest) orthologic, is denoted by O.

The class of all ortholattices is defined by a set of identities, which means that the class forms a variety. This variety is denoted by O. The orthologics O is complete with respect to the variety O. The most frequently considered subvarieties of O, the variety of orthomodular lattices (OML), the variety of modular ortholattices (MOL), and the variety of Boolean algebras(BA) are obtained by adding the following identities to the defining identities of O.

For OML · · · · · · x ∩ {(x ∩ y) ∪ x′} = x ∩ y (orthomodular law)
For MOL · · · · · · x ∩ {(x ∩ y) ∪ z} = (x ∩ y) ∪ (x ∩ z) (modular law)
For BA · · · · · · · · · x ∩ (y ∪ z) = (x ∩ y) ∪ (x ∩ z) (distributive law)

Of course, by adding suitable axiom schemes to O, we consider the (smallest) orthomodular logic OML, the (smallest) modular orthologic MOL, and the classical logic CL, which correspond to OML, MOL, and BA respectively.

2 Variable separation, variable sharing, and interpolation

For a formula α, the set of all propositional variables appeared in α is denoted by V (α). For an orthologic L, the binary relation of logically equivalence in L is denoted by ≡_{L}. That is, for formulas α, β, α ≡_{L} β if and only if ⟨α, β⟩ ∈ L and ⟨β, α⟩ ∈ L. Note that V (α) = ∅ means that α is ⊥ or α is ⊤(= ¬⊥).

Definition 3 Let L be an orthologic.

(1) L has the variable separation property (VSEP) if and only if for α, β ∈ Φ such that V (α)∩V (β) = ∅, ⟨α, β⟩ ∈ L implies that α ≡_{L} ⊤ or else β ≡_{L} ⊥.

(2) L has the variable sharing property (VSHP) if and only if for α_{1}, α_{2}, β_{1}, β_{2} ∈ Φ, such that V (α_{1}, α_{2})∩ V (β_{1}, β_{2}) = ∅, ⟨α_{1} ∧ β_{1}, α_{2} ∨ β_{2}⟩ ∈ L implies that ⟨α_{1}, α_{2}⟩ ∈ L or else ⟨β_{1}, β_{2}⟩ ∈ L.

(3) L has the interpolation property (IP) if and only if for α, β ∈ Φ, if ⟨α, β⟩ ∈ L, then there exists a formula γ which satisfies the following:

(a) ⟨α, γ⟩ ∈ L and ⟨γ, β⟩ ∈ L.

(b) V (γ) ⊆ V (α) ∩ V (β).

It is easily seen that IP implies VSEP. A sufficient algebraic condition for an orthologic to have IP was obtained in [2], and only O and CL are orthologics that are known to have IP at present. On VSEP and VSHP of orthologics, we consider the following constructions of ortholattices.

1. reversing: For an ortholattice 𝔄 = ⟨A, ∩, ∪, (·)′, 0, 1⟩, a reverse of 𝔄 is an ortholattice \overline{𝔄} = ⟨A, ∪, ∩, (·)′, 1, 0⟩. That is, only the order relation in \overline{𝔄} is redefined as the converse of the order relation in 𝔄.

2. {0, 1}-pasting: For ortholattices 𝔄 = ⟨A, ∩, ∪, (·)′, 0_{A}, 1_{A}⟩ and 𝔅 = ⟨B, ∩, ∪, (·)′, 0_{B}, 1_{B}⟩, {0, 1}- pasting of 𝔄 and 𝔅 is 𝔄+𝔅 = ⟨(A∪B)/≈, ∩, ∪, (·)′, 0, 1⟩, where ≈ is an equivalence relation defined ≈:= {(x, x) | x ∈ A ∪ B} ∪ {(0_{A}, 0_{B}), (0_{B}, 0_{A}), (1_{A}, 1_{B}), (1_{B}, 1_{A})}. That is, 𝔄 + 𝔅 is an ortholattice in which 0_{A} and 0_{B} are identical to the new smallest element 0, and 1_{A} and 1_{B} are identical to the new largest element 1 and others are inherited from the original 𝔄 and 𝔅.

Indeed, the reverse of an ortholattice is proved to be an ortholattice, and moreover, the varieties OML, MOL, and BA are closed under reversing. On the other hand, O and OML are closed under {0, 1}- pasting, whereas MOL and BA are not. Using the construction of {0, 1}-pasting, the next theorem on VSEP can be proved.

Theorem 4 Let C be a class of ortholattices closed under {0, 1}-pasting, and L = L(C) the orthologic characterized by C. Then L has the variable separation property. □ By the above theorem, we now recognize that the orthologic OML has VSEP. About VSHP, we can show only an algebraic equivalent conditions below at present.

Theorem 5 Let C be a class of ortholattices closed under reversing and {0, 1}-pasting, and L = L(C) the orthologic characterized by C. Then the following 3 conditions are equivalent.

(1) L has the variable sharing property.

(2) Suppose α, β ∈ Φ, V (α) ∩ V (β) = ∅. Then α ∨ β ≡_{L} ⊤ implies α ≡_{L} ⊤ or β ≡_{L} ⊤.

(3) Suppose α, β ∈ Φ, V (α) ∩ V (β) = ∅. Then α ∧ β ≡_{L} ⊥ implies α ≡_{L} ⊥ or β ≡_{L} ⊥. □

It is easily seen that CL has VSHP.

Question: Is CL the only orthologic that has VSHP?

3 A chain of modular orthologics

Finally, we consider extremely simple ortholattices and their corresponding logics. An ortholattice is lantern-type if and only if it satisfies that for any x, y, 0 < x ≤ y < 1 implies x = y. An element a in a lantern-type ortholattice is a middle element if and only if 0 < a < 1. A lantern-type ortholattice that has k middle elements is denoted by 𝔄_{k}. Note that k is always even because of the operation (·)′. Also, the orthologic characterized by an ortholattice 𝔄 is denoted by L(𝔄). It is proved easily that for any k ∈ ω, L(𝔄_{k}) is a modular orthologic since 𝔄_{k} is a modular ortholattice. On the structure of modular orthologics characterized by lantern-type ortholattices, the next lemma holds. Lemma 6

(1) For all n ∈ ω, L(𝔄_{2n}) ⊇ L(𝔄_{2n+2}).

(2) For all n ≥ 1, L(𝔄_{2n}) ≠ L(𝔄_{2n+2}). □

Proof of (1) is straightforward because 𝔄_{2n} can be embedded into 𝔄_{2n+2}. To show (2) above, employ 2n + 1 propositional variables p_{0}, p_{1}, p_{2}, …, p_{2n}, and build up the following formulas:

α_{2n} := p_{0} ∧ ⋀_{1≤i<j≤2n}(p_{i} ∨ p_{j}). β_{2n} := ⋁_{i=1}^{2n}(p_{0} ∧ p_{i})

Then the following proposition can be proved, which leads to Lemma 6 (2). Proposition 7

(1) For n ≥ 1, ⟨α_{2n}, β_{2n}⟩ ∉ L(𝔄_{2n+2}).

(2) For n ∈ ω, ⟨α_{2n}, β_{2n}⟩ ∈ L(𝔄_{2n}). □

By Lemma 6, we have the following theorem on the lattice of modular orthologics. Theorem 8

(1) L(𝔄_{0}) = L(𝔄_{2}) ⊃ L(𝔄_{4}) ⊃ · · · · · · ⊃ L(𝔄_{2n}) ⊃ L(𝔄_{2n+2}) ⊃ · · ·.

(2) Orthologics which are characterized by lantern-type ortholattices form an infinite descending chain.

(3) There exist at least countable modular orthologics. □

These analysis highly depends on their simple form of lantern-type ortholattices. It is unknown at least today whether there exist other classes of ortholattices in which each propositional logic determined by its single member can be distinguished from each other by a special form of pairs of formulas.

Question: Can we axiomatize the orthologic L := ⋂{L(𝔄) | 𝔄 is a lantern-type ortholattice} ?

References

[1] Goldblatt R.I., Semantic analysis of orthologic, Journal of Philosophical Logic 3, 19–35 (1974).

[2] Miyazaki Y., The super-amalgamation property of the variety of ortholattices, Reports on Mathematical Logic 33, 45–63 (1999).