We presume a background theory which allows for indeterminacy of states of affairs involving objects, extending even to indeterminacy of identity between objects. A sentence reporting such an indeterminate state of affairs lacks truth-value. We extend this to a theory of sets, similar to ZFU, in which membership in, and identity between, sets may also be indeterminate.

The aim of this paper is to give a certain algebraic account of truth: we want to define what we mean by De Morgan-valued truth models and show their existence even in the case of semantical closure: that is, languages may contain their own truth predicate if they are interpreted by De Morgan-valued models. Before we can prove this result, we have to repeat some basic facts concerning De Morgan-valued models in general, and we will introduce a notion of truth both on the object- and on the metalanguage level appropriate for such models. The definitions and the existence theorem are extensions of Kripke's, Woodruff's, and Visser's concepts and results concerning three- and four-valued truth models.

The notion of computability is developed through the study of the behavior of a set of languages interpreted over the natural numbers which contain their own fully defined satisfaction predicate and whose only other vocabulary is limited to0, individual variables, the successor function, the identity relation and operators for disjunction, conjunction, and existential quantification.

Hilbert-style axiomatic systems are presented for versions of the modal logics K$ \Sigma$, where $ \Sigma$ $ \subseteq$ {D, 4, 5}, with noncontingency as the sole modal primitive. The classes of frames characterized by the axioms of these systems are shown to be first-order definable, though not equal to the classes of serial, transitive, or euclidean frames. The canonical frame of the noncontingency logic of any logic containing the seriality axiom is proved to be nonserial. It is also shown that any class of frames definable in the noncontingency language contains the class of functional frames, and dually, there exists a greatest consistent normal noncontingency logic.

We show that none of the following statements is provable in Zermelo-Fraenkel set theory (ZF) answering the corresponding open questions from Brunner in ``The axiom of choice in topology'':

(i) For every T₂ topological space (X, T) if X is well-ordered, then X has a well-ordered base,

(ii) For every T₂ topological space (X, T), if X is well-ordered, then there exists a function f : X × W $ \rightarrow$ T such that W is a well-ordered set and f ({x} × W) is a neighborhood base at x for each x $ \in$ X,

(iii) For every T₂ topological space (X, T), if X has a well-ordered dense subset, then there exists a function f : X × W $ \rightarrow$ T such that W is a well-ordered set and {x} = $ \cap$ f ({x} × W) for each x $ \in$ X.

Plausible principles on truth seem to yield contradictory conclusions about paradoxical sentences such as the Strengthened Liar. Those who take the contextualist approach, such as Parsons and Burge, attempt to justify the seemingly contradictory conclusions by arguing that the natural reasoning that leads to them involves some kind of contextual shift that makes them compatible. This paper argues that one cannot take this approach to give a proper treatment of infinite descending chains of semantic attributions. It also examines a related approach taken by Gaifman and argues that it has the same problem.

We work in set theory without the Axiom of Choice ZF. We prove that the Principle of Dependent Choices (DC) implies that the closed unit ball of a uniformly convex Banach space is weakly compact and, in particular, that the closed unit ball of a Hilbert space is weakly compact. These statements are not provable in ZF and the latter statement does not imply DC. Furthermore, DC does not imply that the closed unit ball of a reflexive space is weakly compact.

A theorem due to Shoesmith and Smiley that axiomatizes two-valued multiple-conclusion logics is extended to partial logics.