We show that there exist 2 ℵ 0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any first-order definable class of relational structures. Using a variant of this construction, we resolve a long-standing question of Fine, by exhibiting a bimodal logic that is valid in its canonical frames, but is not sound and complete for any first-order definable class of Kripke frames (a monomodal example can then be obtained using simulation results of (...) Thomason). The constructions use the result of Erd $\H{o}$ s that there are finite graphs with arbitrarily large chromatic number and girth. (shrink)

I propose a neo-Meinongian theory labelled as “Modal Meinongian Metaphysics” (MMM) to account for the ontology and semantics of fictional discourse. Several competing accounts of fictional objects are originated by the fact that our talking of them mirrors incoherent intuitions: mainstream theories of fiction privilege some such intuitions, but are forced to account for others via complicated paraphrases of the relevant sentences. An ideal theory should resort to as few paraphrases as possible. In section 1, I make the idea explicit (...) via two methodological principles, called the Minimal Revision and the Acceptability Constraint. In section 2, I introduce the standard distinction between internal and external fictional discourse. In sections 3 to 5, I discuss the approaches of traditional Meinongianism, Fictionalism, and Realism – and their main troubles. In section 6 I propose the MMM approach. This is based upon (1) a modal semantics including impossible worlds (sect. 6.1); (2) a Qualified Comprehension Principle for objects (sect. 6.2); (3) a notion of existence-entailment for properties (sect. 6.3). In section 7 I present a formal semantics for MMM based upon a representation operator. And in section 8 I show how MMM solves the problems of the three aforementioned theories. (shrink)