An Algebraic Semantics for the Logic of Multiple-Source Approximation Systems

A multiple-source approximation system (MSAS) is a tuple F := (U, {Ri}i∈N ), where U is a non-empty set, N an initial segment of the set N of positive integers, and each Ri, i ∈ N, is an equivalence relation on the domain U . A quantified propositional modal logic LMSAS was defined in [1] in order to study MSAS. In this paper, we will present an algebraic semantics for LMSAS. Syntax of the logic LMSAS is given as follows: There is a (i) a non-empty countable set V ar of variables, (ii) a (possibly empty) countable set Con of constants, (iii) a non-empty countable set PV of propositional variables and (iv) the propositional constants >,⊥. The set T of terms of the language is given by V ar ∪ Con. Using the standard Boolean logical connectives ¬ (negation) and ∧ (conjunction), a unary modal connective 〈t〉 (possibility) for each term t ∈ T , and the universal quantifier ∀, well-formed formulae (wffs) of LMSAS are defined recursively as: >|⊥|p|¬α|α ∧ β|〈t〉α|∀xα, where p ∈ PV, t ∈ T, x ∈ V ar, and α, β are wffs. The set of all wffs and closed wffs of LMSAS will be denoted by F and F respectively. For a wff α of LMSAS, Con(α) will denote the set of constants used in α. Let Γ be a set of wffs of LMSAS. An interpretation for Γ is given by a triple M := (μ, V, I), where μ := (U, {Ri}i∈N ) is a MSAS, V : PV → P(U) and I : Con(Γ) → N . An assignment for an interpretation M is a map v : Term(Γ)→ N such that v(c) = I(c), for each c ∈ Con(Γ). The satisfiability in an interpretation M := (μ, V, I) of a wff α of Γ, under an assignment v, and at an object w of the domain U , denoted as M, v, w |= α is defined inductively as follows: M, v, w |= 〈t〉α, if and only if there exists w′ in U such that wRv(t)w and M, v, w′ |= α. M, v, w |= ∀xα, if and only if for every assignment v′ x-equivalent to v, M, v′, w |= α. α is valid, denoted |= α, if and only if M, v, w |= α, for every interpretation M := (μ, V, I), assignment v for M and object w in the domain of μ. The following sound and complete deductive system for LMSAS was proposed in [1]. t stands for a term in T . Axiom schema: (Ax1). All axioms of classical propositional logic. (Ax2). ∀xα→ α(t/x), where α admits the term t for the variable x. (Ax3). ∀x(α→ β)→ (α→ ∀xβ), where the variable x is not free in α. (Ax4). ∀x[t]α→ [t]∀xα, where the term t and variable x are different. (Ax5). [t](α→ β)→ ([t]α→ [t]β). (Ax6). α→ 〈t〉α. (Ax7). α→ [t]〈t〉α. (Ax8). 〈t〉〈t〉α→ 〈t〉α. Rules of inference: ∀. α MP. α N. α ∀xα α→ β [t]α β We note that this is different from both propositional quantification of modal logic, and modal predicate logic. Next, we present an algebraic semantics for LMSAS. We begin with the following definition. Definition 1 A BAOs A := (A,∩,∼, 1, fi)i∈∆ is said to be a complete BAOs (CBAOs) if it satisfies the following properties for all X ⊆ A: (A1) ⋂ X and ⋃ X exists; (A2) fk ⋂ X = ⋂ fkX, k ∈ ∆. In this paper, we are interested only in those (complete ) BAOs where ∆ = N and each fk satisfies the following additional conditions: (B1) fka ≤ fkfka; (B2) fka ≤ a and (B3) a ≤ fkgka, where gk :=∼ fk ∼.