We define analogues of modal Sahlqvist formulas for the modal mu-calculus, and prove a correspondence theorem for them.

In the paper we present a technique for eliminating quantifiers of arbitrary order, in particular of first-order. Such a uniform treatment of the elimination problem has been problematic up to now, since techniques for eliminating first-order quantifiers do not scale up to higher-order contexts and those for eliminating higher-order quantifiers are usually based on a form of monotonicity w.r.t implication (set inclusion) and are not applicable to the first-order case. We make a shift to arbitrary relations “ordering” the underlying universe. (...) This allows us to incorporate background theories into higher-order quantifier elimination methods which, up to now, has not been achieved. The technique we propose subsumes many other results, including the Ackermann's lemma and various forms of fixpoint approaches when the “ordering” relations are interpreted as implication and reveals the common principle behind these approaches. (shrink)

In this paper, we extend the canonicity methodology in Ghilardi & Meloni (1997) to arbitrary lattice expansions, and syntactically describe canonical inequalities for lattice expansions consisting of -meet preserving operations, -multiplicative operations, adjoint pairs, and constants. This approach gives us a uniform account of canonicity for substructural and lattice-based logics. Our method not only covers existing results, but also systematically accounts for many canonical inequalities containing nonsmooth additive and multiplicative uniform operations. Furthermore, we compare our technique with the approach in (...) Dunn et al. (2005) and Gehrke et al. (2005). (shrink)

In this paper, we establish the first-order definability of sequents with consistent variable occurrence on bi-approximation semantics by means of the Sahlqvist–van Benthem algorithm. Then together with the canonicity results in Suzuki (2011), this allows us to establish a Sahlqvist theorem for substructural logic. Our result is not limited to substructural logic but is also easily applicable to other lattice-based logics.

Modal logic is concerned with the analysis of sentential operators in the widest sense. Originally invented to analyse the notion of necessity applications have been found in many areas of philosophy, logic, linguistics and computer science. This in turn has led to an increased interest in the technical development of modal logic.

Sahlqvist formulas are a syntactically specified class of modal formulas proposed by Hendrik Sahlqvist in 1975. They are important because of their first-order definability and canonicity, and hence axiomatize complete modal logics. The first-order properties definable by Sahlqvist formulas were syntactically characterized by Marcus Kracht in 1993. The present paper extends Kracht's theorem to the class of ‘generalized Sahlqvist formulas' introduced by Goranko and Vakarelov and describes an appropriate generalization of Kracht formulas.

In this paper we consider the normal modal logics of elementary classes defined by first-order formulas of the form \. We prove that many properties of these logics, such as finite axiomatisability, elementarity, axiomatisability by a set of canonical formulas or by a single generalised Sahlqvist formula, together with modal definability of the initial formula, either simultaneously hold or simultaneously do not hold.

Minimal predicates P satisfying a given first-order description φ(P) occur widely in mathematical logic and computer science. We give an explicit first-order syntax for special first-order ‘PIA conditions’ φ(P) which guarantees unique existence of such minimal predicates. Our main technical result is a preservation theorem showing PIA-conditions to be expressively complete for all those first-order formulas that are preserved under a natural model-theoretic operation of ‘predicate intersection’. Next, we show how iterated predicate minimization on PIA-conditions yields a language MIN(FO) equal (...) in expressive power to LFP(FO), first-order logic closed under smallest fixed-points for monotone operations. As a concrete illustration of these notions, we show how our sort of predicate minimization extends the usual frame correspondence theory of modal logic, leading to a proper hierarchy of modal axioms: first-order-definable, first-order fixed-point definable, and beyond. (shrink)

We elaborate on semantically labelled syntax trees that provide a method of proving the non-existence of modal formulae satisfying certain syntactic properties and defining a given class of frames and use them to show that there are classes of Kripke frames that are definable by both non-Sahlqvist and Sahlqvist formulae, but the latter requires more propositional variables.

We characterize the modal logics of elementary classes of Kripke frames as precisely those modal logics that are axiomatized by modal axioms synthesized in a certain effective way from "quasi-positive" sentences of hybrid logic. These are pure positive hybrid sentences with arbitrary existential and relativized universal quantification over nominals. The proof has three steps. The first step is to use the known result that the modal logic of any elementary class of Kripke frames is also the modal logic of the (...) closure of this class under disjoint unions, generated subframes, bounded morphic images, and ultraroots. This latter class can be defined by the first-order sentences of a special syntactic form (called pseudo-equations by Goldblatt) that are valid in the former class. The second step is to translate these pseudo-equations into equivalent quasi-positive hybrid sentences. In the third and main step, we show that any quasi-positive sentence S generates an infinite set of modal formulas called "approximants," which together axiomatize a canonical modal logic that is sound and complete for the class of frames validating S. The proof is analogous to standard proofs of Sahlqvist's theorem. It generalizes to sets of quasi-positive sentences. The main result now follows. (shrink)

Our concern is the axiomatisation problem for modal and algebraic logics that correspond to various fragments of two-variable first-order logic with counting quantifiers. In particular, we consider modal products with Diff, the propositional unimodal logic of the difference operator. We show that the two-dimensional product logic $Diff \times Diff$ is non-finitely axiomatisable, but can be axiomatised by infinitely many Sahlqvist axioms. We also show that its ‘square’ version (the modal counterpart of the substitution and equality free fragment of two-variable first-order (...) logic with counting to two) is non-finitely axiomatisable over $Diff \times Diff$ , but can be axiomatised by adding infinitely many Sahlqvist axioms. These are the first examples of products of finitely axiomatisable modal logics that are not finitely axiomatisable, but axiomatisable by explicit infinite sets of canonical axioms. (shrink)

We provide syntactic necessary and sufficient conditions on the formulae reducible by the second-order quantifier elimination algorithm DLS. It is shown that DLS is compete for all modal Sahlqvist and Inductive formulae, and that all modal formulae in a single propositional variable on which DLS succeeds are canonical.

In a previous work we introduced the algorithm \SQEMA\ for computing first-order equivalents and proving canonicity of modal formulae, and thus established a very general correspondence and canonical completeness result. \SQEMA\ is based on transformation rules, the most important of which employs a modal version of a result by Ackermann that enables elimination of an existentially quantified predicate variable in a formula, provided a certain negative polarity condition on that variable is satisfied. In this paper we develop several extensions of (...) \SQEMA\ where that syntactic condition is replaced by a semantic one, viz. downward monotonicity. For the first, and most general, extension \SSQEMA\ we prove correctness for a large class of modal formulae containing an extension of the Sahlqvist formulae, defined by replacing polarity with monotonicity. By employing a special modal version of Lyndon's monotonicity theorem and imposing additional requirements on the Ackermann rule we obtain restricted versions of \SSQEMA\ which guarantee canonicity, too. (shrink)

The previously introduced algorithm \sqema\ computes first-order frame equivalents for modal formulae and also proves their canonicity. Here we extend \sqema\ with an additional rule based on a recursive version of Ackermann's lemma, which enables the algorithm to compute local frame equivalents of modal formulae in the extension of first-order logic with monadic least fixed-points \mffo. This computation operates by transforming input formulae into locally frame equivalent ones in the pure fragment of the hybrid mu-calculus. In particular, we prove that (...) the recursive extension of \sqema\ succeeds on the class of `recursive formulae'. We also show that a certain version of this algorithm guarantees the canonicity of the formulae on which it succeeds. (shrink)

We define analogues of modal Sahlqvist formulas for the modal mu-calculus, and prove a correspondence theorem for them.

This chapter presents a new semantics for inductive empirical knowledge. The epistemic agent is represented concretely as a learner who processes new inputs through time and who forms new beliefs from those inputs by means of a concrete, computable learning program. The agent’s belief state is represented hyper-intensionally as a set of time-indexed sentences. Knowledge is interpreted as avoidance of error in the limit and as having converged to true belief from the present time onward. Familiar topics are re-examined within (...) the semantics, such as inductive skepticism, the logic of discovery, Duhem’s problem, the articulation of theories by auxiliary hypotheses, the role of serendipity in scientific knowledge, Fitch’s paradox, deductive closure of knowability, whether one can know inductively that one knows inductively, whether one can know inductively that one does not know inductively, and whether expert instruction can spread common inductive knowledge—as opposed to mere, true belief—through a community of gullible pupils. (shrink)