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Canonical Varieties with No Canonical Axiomatisation
 Trans. Amer. Math. Soc
, 2003
"... We give a simple example of a variety V of modal algebras that is canonical but cannot be axiomatised by canonical equations or firstorder sentences. We then show that the variety RRA of representable relation algebras, although canonical, has no canonical axiomatisation. Indeed, we show that every ..."
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We give a simple example of a variety V of modal algebras that is canonical but cannot be axiomatised by canonical equations or firstorder sentences. We then show that the variety RRA of representable relation algebras, although canonical, has no canonical axiomatisation. Indeed, we show that every axiomatisation of these varieties involves infinitely many noncanonical sentences. Using probabilistic methods...
Erdős Graphs Resolve Fine's Canonicity Problem
 THE BULLETIN OF SYMBOLIC LOGIC
, 2003
"... We show that there exist 2^ℵ0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any firstorder definable class of relational structures. Using a variant of this construction, we resolve a longstanding question of Fine, by exhibiting ..."
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We show that there exist 2^&alefsym;0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any firstorder definable class of relational structures. Using a variant of this construction, we resolve a longstanding question of Fine, by exhibiting a bimodal logic that is valid in its canonical frames, but is not sound and complete for any firstorder definable class of Kripke frames. The constructions use the result of Erdős that there are finite graphs with arbitrarily large chromatic number and girth.
On canonical modal logics that are not elementarily determined. Logique et Analyse
 181:77— 101, 2003. Published October 2004. 20 Robert Goldblatt, Ian Hodkinson, and Yde
, 2004
"... There exist modal logics that are validated by their canonical frames but are not sound and complete for any elementary class of frames. Continuum many such bimodal logics are exhibited, including one of each degree of unsolvability, and all with the finite model property. Monomodal examples are als ..."
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There exist modal logics that are validated by their canonical frames but are not sound and complete for any elementary class of frames. Continuum many such bimodal logics are exhibited, including one of each degree of unsolvability, and all with the finite model property. Monomodal examples are also constructed that extend K4 and are related to the proof of noncanonicity of the McKinsey axiom. 1
Canonical PseudoCorrespondence
 Advances in Modal Logic Volume 2
, 1998
"... Generalizing an example from Fine [1] and inspired by a theorem in J' onsson [4], we prove that any modal formula of the form ß(p q) $ ß(p) ß(q) (with ß(p) a positive formula) is canonical. We also prove that any such formula is strongly sound and complete with respect to an elementary class ..."
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Generalizing an example from Fine [1] and inspired by a theorem in J' onsson [4], we prove that any modal formula of the form ß(p q) $ ß(p) ß(q) (with ß(p) a positive formula) is canonical. We also prove that any such formula is strongly sound and complete with respect to an elementary class of frames, definable by a first order formula which can be read off from ß. 1 Introduction For quite a while now, modal logicians have been interested in the relation between first order logic and canonical modal formulas; recall that the latter are formulas that are valid on the underlying frame of the canonical model. Some very interesting connections have been discovered, but there are also some intriguing open problems. Examples of important results are Fine's Theorem (cf. [1]) that the modal logic of an elementary class of frames is canonical, and Sahlqvist's Theorem (cf. [6]) identifying a class of modal formulas each of which is canonical and corresponds to a first order formula which can...
Algebraic Polymodal Logic: A Survey
 LOGIC JOURNAL OF THE IGPL
, 2000
"... This is a review of those aspects of the theory of varieties of Boolean algebras with operators (BAO's) that emphasise connections with modal logic and structural properties that are related to natural properties of logical systems. It begins with ..."
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This is a review of those aspects of the theory of varieties of Boolean algebras with operators (BAO's) that emphasise connections with modal logic and structural properties that are related to natural properties of logical systems. It begins with
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"... elementarily generated modal logics Abstract. In this paper we consider the normal modal logics of elementary classes defined by firstorder formulas of the form ∀x0∃x1...∃xn∧xiRλxj. We prove that many properties of these logics, such as finite axiomatisability, elementarity, axiomatisability by a s ..."
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elementarily generated modal logics Abstract. In this paper we consider the normal modal logics of elementary classes defined by firstorder formulas of the form ∀x0∃x1...∃xn∧xiRλxj. 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.