First-Order Modal Logic: Frame Definability and a Lindström Theorem

Abstract

This paper involves generalizing the Goldblatt-Thomason and the Lindström characterization theorems to first-order modal logic.1 bounded morphic images, generated subframes and disjoint unions and reflects ultrafilter extensions. Our first objective here is to provide a formulation of this theorem in FML.The second aim of this paper is to study Lindström type theorems for first-order modal logic. These theorems determine the maximal expressive power of logics in terms of model theoretic concepts. In a seminal paper [15], Lindström proved that any abstract logic extending first-order logic with compactness and the Löwenheim-Skolem property is not more expressive than the first-order logic.To formulate our problems in technical terms, it is worth it to have a quick overview of basic notions of FML. Suppose τ is a language consisting of countable numbers of countably many constant and relational symbols. The usual syntactical conventions of first-order logic are assumed here. In particular, the notions of τ -terms and τ -atomic formulas are defined in the usual way. Furthermore, first-order modal τ -formulas are defined inductively in the usual pattern as follows:where P is an n-ary predicate and t 1 , . . . , t n are τ -terms. Bound and free variables in a formula are defined as in first-order logic. A τ -sentence is a formula without any free variable. The other logical connectives (∨, →, ↔, ∀), and the modal operator ◻ have their standard definitions. A constant domain Kripke model is a quadruple M = (W, R, D, I), where W ≠ ∅ is a set of possible worlds, R ⊆ W × W is an accessibility relation, and D ≠ ∅ is a domain. Moreover, i. for each w ∈ W and any n-ary predicate symbol P , I(w, P ) ⊆ D n , ii. for any w, w ′ ∈ W and each constant c, I(w, c) = I(w ′ , c) ∈ D. More generally, a varying domain Kripke model is a tuple M = (W, R, D, I, {D(w)} w∈W ), where (W, R, D, I) is a constant domain model and for each w ∈ W , D(w) ≠ ∅ is a domain of w such that D = ⋃ w∈W D(w). The pair F = (W, R) and the triple S = (W, R, D) are respectively called frame and skeleton. A Kripke model M = (W, R, D, I) is said to be based on the frame F = (W, R) or skeleton S = (W, R, D). For a given first-order Kripke model M and any w ∈ W , the pair (M, w) is called a pointed model.An assignment is a function σ, which assigns to each variable v an element σ(v) inside D. Two assignments σ and σ ′ are x-variant if σ(y) = σ ′ (y) for all variables y ≠ x. We denote this by σ ∼ x σ ′ .The notation t M,σ is used for the interpretation of t in M under assignment σ.

Probability Logic: A Model Theoretic Perspective

“…In 1969 Lindström proved that first-order logic has the maximal expressive power among the abstract logics containing it with the compactness and the Löwenheim-Skolem properties. This kind of characterization is widely studied for other logics specially for modal logics, for example [18,20,16,4,26]. In addition to the compactness, bisimulation invariance property used to prove a characterization theorem for modal logic.…”

A Lindström theorem in many-valued modal logic over a finite MTL-chain

“…The similarities mainly concern the use of Tarski Union Property, which proved to be a powerful tool in other similar contexts (see, e.g. Enqvist [16] and Zoghifard and Pourmahdian [28]). Moreover, there was also the possibility of strengthening Theorem 9 by omitting from it any mentions of this property.…”

Chain logic and Shelah’s infinitary logic

“…In spite of the abundance of abstract logics discovered in the 1960s and 1970s, rare are those that generalise first order logic and have a characterisation of this type. Search for logics with Lindström-style characterisation is an active area of research, where positive results were obtained in contexts either far from or weaker than first order logic, including various versions of modal logic [10], [4], [22], [14], [15], [36], or for fragments of first order logic [5]. Further examples and references are in [32].…”