In Writing the Book of the World, Ted Sider argues that David Lewis’s distinction between those predicates which are ‘perfectly natural’ and those which are not can be extended so that it applies to words of all semantic types. Just as there are perfectly natural predicates, there may be perfectly natural connectives, operators, singular terms and so on. According to Sider, one of our goals as metaphysicians should be to identify the perfectly natural words. Sider claims that there is a (...) perfectly natural first-order quantifier. I argue that this claim is not justified. Quine has shown that we can dispense with first-order quantifiers, by using a family of ‘predicate functors’ instead. I argue that we have no reason to think that it is the first-order quantifiers, rather than Quine’s predicate functors, which are perfectly natural. The discussion of quantification is used to provide some motivation for a general scepticism about Sider’s project. Shamik Dasgupta’s ‘generalism’ and Jason Turner’s critique of ‘ontological nihilism’ are also discussed. (shrink)

We review, and in the process unify two techniques , for proving results concerning amalgamation in several classes studied in algebraic logic. The logical counterpart of these results adress interpolation and definability properties in modal and algebraic logic. Presenting them in a functorial context as adjoint situations, we show that both techniques can indeed be seen as instances of the use of the Keisler-Shelah ultrapower Theorem in proving Robinson's Joint Consistency Theorem. Some new results are surveyed. The results of this (...) paper are presented in such a way that further establishes the hitherto existing well developed duality theory between boolean algebras with operators and modal logic. (shrink)

We give a novel application of algebraic logic to first order logic. A new, flexible construction is presented for representable but not completely representable atomic relation and cylindric algebras of dimension n (for finite n > 2) with the additional property that they are one-generated and the set of all n by n atomic matrices forms a cylindric basis. We use this construction to show that the classical Henkin-Orey omitting types theorem fails for the finite variable fragments of first order (...) logic as long as the number of variables available is > 2 and we have a binary relation symbol in our language. We also prove a stronger result to the effect that there is no finite upper bound for the extra variables needed in the witness formulas. This result further emphasizes the ongoing interplay between algebraic logic and first order logic. (shrink)

SC, CA, QA and QEA denote the class of Pinter’s substitution algebras, Tarski’s cylindric algebras, Halmos’ quasi-polyadic and quasi-polyadic equality algebras, respectively. Let . and . We show that the class of n dimensional neat reducts of algebras in K m is not elementary. This solves a problem in [2]. Also our result generalizes results proved in [1] and [2].

We survey various results on the relationship among neat embeddings (a notion special to cylindric algebras), complete representations, omitting types, and amalgamation. A hitherto unpublished application of algebraic logic to omitting types of first-order logic is given.

We generalize two classical results on cylindric algebra to certain expansions of cylindric algebras where the extra operations are defined via first order formulas. The first result is the Neat Embedding Theorem of Henkin and the second is Monk's classical non-finitizability result of the class of representable algebras. As a corollary we obtain known classical results of Johnson and Biro published in the Journal of Symbolic logic.