After a summary of earlier work it is shown that elementary or Kalmar arithmetic can be interpreted within the system of Russell's Principia Mathematica with the axiom of infinity but without the axiom of reducibility.

We establish a completeness theorem for first-order basic predicate logic BQC, a proper subsystem of intuitionistic predicate logic IQC, using Kripke models with transitive underlying frames. We develop the notion of functional well-formed theory as the right notion of theory over BQC for which strong completeness theorems are possible. We also derive the undecidability of basic arithmetic, the basic logic equivalent of intuitionistic Heyting Arithmetic and classical Peano Arithmetic.

We suggest a general formalism of four-valued reasoning, called biconsequence relations, intended to serve as a logical framework for reasoning with incomplete and inconsistent data. The formalism is based on a four-valued semantics suggested by Belnap. As for the classical sequent calculus, any four-valued connective can be defined in biconsequence relations using suitable introduction and elimination rules. In addition, various three-valued and partial logics are shown to be special cases of this formalism obtained by imposing appropriate additional logical rules. We show also that such rules are instances of a single logical principle called coherence. The latter can be considered a general requirement securing that the information we can infer in this framework will be classically coherent.

In this paper, we define a first-order logic CFʹ with strong negation and bounded static quantifiers, which is a variant of Thomason's logic CF. For the logic CFʹ, the usual Kripke formal semantics is defined based on situations, and a sound and complete axiomatic system is established based on the axiomatic systems of constructive logics with strong negation and Thomason's completeness proof techniques. With the use of bounded quantifiers, CFʹ allows the domain of quantification to be empty and allows for nondenoting constants. CFʹ is intended as a fragment of a logic for situation theory. Thus the connection between CFʹ and infon logic is discussed.

Maksimova's principle of variable separation says that if propositional formulas $A_1 \supset A_2$ and $B_1 \supset B_2$ have no propositional variables in common and if a formula $A_1\wedge B_1 \supset A_2\vee B_2$ is provable, then either $A_1 \supset A_2$ or $B_1 \supset B_2$ is provable. Results on Maksimova's principle until now are obtained mostly by using semantical arguments. In the present paper, a proof-theoretic approach to this principle in some substructural logics is given, which analyzes a given cut-free proof of the formula $A_1\wedge B_1 \supset A_2\vee B_2$ and examines how the formula is derived. This analysis will make clear why Maksimova's principle holds for these logics.

The core of the debate between Fregeans and Russellians in the philosophy of language concerns the content of object-dependent propositions, or how we ought to individuate and semantically represent the content of propositions that are about specific individuals. This essay is an investigation of the contemporary status of this debate. My aim is to show how the causal theorists' picture of reference determination entails the need for both Fregean and Russellian conceptions of propositional content in the study of mind and language, and to investigate some of the consequences of this position.

The paper considers extensions of a normal bimodal logic $\textbf{KL}$ in which the two necessity operators are mutual one-sided inverses. A continuum of maximal consistent normal extensions of $\textbf{KL}$ is constructed, each of which has infinitely many quasi-normal Post complete extensions. Some syntactic properties of maximal consistent normal bimodal logics and in particular of such extensions of $\textbf{KL}$ are investigated.

Using methods of abstract logic and the theory of valuation, we prove that there is no paraconsistent negation obeying the law of double negation and such that $\neg(a\wedge\neg a)$ is a theorem which can be algebraized by a technique similar to the Tarski-Lindenbaum technique.

In contrast to Tennenbaum's theorem asserting that there are no nonstandard models of Peano Arithmetic in which either addition or multiplication is recursive, the result here is that there is a nonstandard model of Peano Arithmetic having a recursive divisibility poset and a recursive divisibility lattice.