Mathematical Logic Quarterly最新文献

We give an extension of the Jónsson-Tarski representation theorem for both normal and non-normal modal algebras so that it preserves countably many infinite meets and joins. In order to extend the Jónsson-Tarski representation to non-normal modal algebras we consider neighborhood frames instead of Kripke frames just as Došen's duality theorem for modal algebras, and to deal with infinite meets and joins, we make use of Q-filters, which were introduced by Rasiowa and Sikorski, instead of prime filters. By means of the extended representation theorem, we show that every predicate modal logic, whether it is normal or non-normal, has a model defined on a neighborhood frame with constant domains, and we give a completeness theorem for some predicate modal logics with respect to classes of neighborhood frames with constant domains. Similarly, we show a model existence theorem and a completeness theorem for infinitary modal logics which allow conjunctions of countably many formulas.

Quantum B-algebras are partially ordered algebras characterizing the residuated structure of a quantale. Examples arise in algebraic logic, non-commutative arithmetic, and quantum theory. A quantum B-algebra with trivial partial order is equivalent to a group. The paper introduces a corresponding analogue of quantale modules. It is proved that every quantum B-module admits an injective envelope which is a quantale module. The injective envelope is constructed explicitly as a completion, a multi-poset version of the completion of Dedekind and MacNeille.

We study notions of genericity in models of $�\mathrm{PA}$, inspired by lines of inquiry initiated by Chatzidakis and Pillay and continued by Dolich, Miller and Steinhorn in general model-theoretic contexts. These papers studied the theories obtained by adding a “random” predicate to a class of structures. Chatzidakis and Pillay axiomatized the theories obtained in this way. In this article, we look at the subsets of models of $�\mathrm{PA}$ which satisfy the axiomatization given by Chatzidakis and Pillay; we refer to these subsets in models of $�\mathrm{PA}$ as CP-generics. We study a more natural property, called strong CP-genericity, which implies CP-genericity. We use an arithmetic version of Cohen forcing to construct (strong) CP-generics with various properties, including ones in which every element of the model is definable in the expansion, and, on the other extreme, ones in which the definable closure relation is unchanged.

We present several examples of hereditary classes of finite structures satisfying the joint embedding property and the weak amalgamation property, but failing the cofinal amalgamation property. These include a continuum-sized family of classes of finite undirected graphs, as well as an example due to Pouzet with countably categorical generic limit.

There is an extensive literature related to the algebraization of first-order logic. But the algebraization of full second-order logic, or Henkin-type second-order logic, has hardly been researched. The question arises: what kind of set algebra is the algebraic version of a Henkin-type model of second-order logic? The question is investigated within the framework of the theory of cylindric algebras. The answer is: a kind of cylindric-relativized diagonal restricted set algebra. And the class of the subdirect products of these set algebras is the algebraization of Henkin-type second-order logic. It is proved that the algebraization of a complete calculus of the Henkin-type second-order logic is a class of a kind of diagonal restricted cylindric algebras. Furthermore, the connection with the non-standard enlargements of standard complete second-order structures is investigated.

We consider forcing axioms for suitable families of μ-complete $�{\mu }^{+}$-c.c. forcing notions. We show that some form of the condition “$��p_1�,�p_2�$ have a $��{\le }_Q�-�\mathrm{lub}�$ in $�Q$” is necessary. We also show some versions are really stronger than others.

I prove that the Boolean Prime Ideal Theorem is equivalent, under some weak set-theoretic assumptions, to what I will call the Cut-for-Formulas to Cut-for-Sets Theorem: for a set F and a binary relation ⊢ on $��P�\left(�F�\right)�$, if ⊢ is finitary, monotonic, and satisfies cut for formulas, then it also satisfies cut for sets. I deduce the CF/CS Theorem from the Ultrafilter Theorem twice; each proof uses a different order-theoretic variant of the Tukey-Teichmüller Lemma. I then discuss relationships between various cut-conditions in the absence of finitariness or of monotonicity.

For (finitary) deductive systems, we formulate a signature-independent abstraction of the weak excluded middle law (WEML), which strengthens the existing general notion of an inconsistency lemma (IL). Of special interest is the case where a quasivariety $�K$ algebraizes a deductive system ⊢. We prove that, in this case, if ⊢ has a WEML (in the general sense) then every relatively subdirectly irreducible member of $�K$ has a greatest proper $�K$-congruence; the converse holds if ⊢ has an inconsistency lemma. The result extends, in a suitable form, to all protoalgebraic logics. A super-intuitionistic logic possesses a WEML iff it extends $�\mathrm{KC}$. We characterize the IL and the WEML for normal modal logics and for relevance logics. A normal extension of $��S�4�$ has a global consequence relation with a WEML iff it extends $��S�4�.�2�$, while every axiomatic extension of $�R^t$ with an IL has a WEML.