Let M be a countable recursively saturated model of Th($\mathbb{N}$), and let G$=$Aut(M), considered as a topological group. We examine connections between initial segments of M and subgroups of G. In particular, for each of the following classes of subgroups H$<$G, we give characterizations of the class of terms of the topological group structure of H as a subgroup of G.

(a) $\{ H : H =G_{(K)}$ for some $K\prec_{\rm e} M \}$

(b) $\{ H : H =G_{\{K\}}$ for some $K\prec_{\rm e} M \}$

(c) $\{ H : H =G_{\{M(a)\}}$ for some $ a\in M \}$

(d) $\{ H : H =G_{\{M(a)\}}=G_a$ for some $ a\in M \}$

(Here, M(a) denotes the smallest $I\prec_{\rm e}$M containing a, $G_{\{A\}}=\{g \in G: A=\{gx: x\in A\} \}$, $G_{(A)}=\{g\in G: \forall a\in A\:ga=a\}$, and $G_a=\{g\in G: ga=a\}$.)

Two distinct and apparently "dual" traditions of non-classical logic, three-valued logic and paraconsistent logic, are considered here and a unified presentation of "easy-to-handle" versions of these logics is given, in which full naive set theory, i.e. Frege's comprehension principle + extensionality, is not absurd.

We specify two problems in syllogistic: the lack of functoriality of predicates (although a thief is a person, a good thief may not be a good person) and the change of grammatical role of the middle term, from subject to predicate, in some syllogisms. The standard semantics, the class interpretation, by-passes these difficulties but, we argue, in a manner that is at odds with logical intuition. We propose a semantics that is category theoretic to handle these difficulties. With this semantics we specify when syllogisms are valid and we set limits to the class interpretation. To perform this task we show how to construct the categorical notion of an entity in a system of kinds. We devote two brief sections to an argument that our approach is very much in the spirit of Aristotle.

In this paper we show that the Gupta-Belnap systems $\mathbf{S}^{＃}$ and ${\bf S}^*$ are $\Pi^1_2$. Since Kremer has independently established that they are $\Pi^1_2$-hard, this completely settles the problem of their complexity. The above-mentioned upper bound is established through a reduction to countable revision sequences that is inspired by, and makes use of a construction of McGee.

A formal system is studied having both sentences and diagrams as well-formed representations. Proofs in the system allow inference back and forth between sentences and diagrams, as well as between diagrams and diagrams, and between sentences and sentences. This sort of heterogeneous system is of interest because external representations other than linguistic ones occur commonly in actual reasoning in conjunction with language. Syntax, semantics, and rules of inference for the system are given and it is shown to be sound and complete.

In this paper we study some logics related to the logic of place introduced by von Wright and studied by Segerberg. For every $n \geq 1$ we study the logic of the class of frames whose accessibility relation R satisfies the following condition: if $x \neq y$ then there is $j \leq n$ such that $xR^j y$. For a fixed $n \geq 1$ the logic is the one axiomatized by K $ + [n] \phi \rightarrow [n+1]\phi+ \phi \rightarrow [n]\langle n\rangle\phi$, which we call Kn.4B, where $[n]\phi \hbox{ is } \phi \wedge \square\phi \wedge \dots\wedge \square^n\phi $. We prove that these logics are canonical and hence complete, and that they have the finite model property, being thus decidable. We also characterize their classes of frames. In the way of studying them we also study the logics ${\bf K} + [n]\phi \rightarrow [n+1]\phi $, called Kn.4, and ${\bf K} + \phi \rightarrow [n]\langle n\rangle\phi $, called Kn.B. A translation between these logics and S5 is also presented, and the relation among them all is established.

In 1957, Gödel proved that completeness for intuitionistic predicate logic HPL implies forms of Markov's Principle, MP. The result first appeared, with Kreisel's refinements and elaborations, in Kreisel. Featuring large in the Gödel-Kreisel proofs are applications of the axiom of dependent choice, DC. Also in play is a form of Herbrand's Theorem, one allowing a reduction of HPL derivations for negated prenex formulae to derivations of negations of conjunctions of suitable instances. First, we here show how to deduce Gödel's results by alternative means, ones arguably more elementary than those of Kreisel. We avoid DC and Herbrand's Theorem by marshalling simple facts about negative translations and Markov's Rule. Second, the theorems of Gödel and Kreisel are commonly interpreted as demonstrating the unprovability of completeness for HPL, if means of proof are confined within strictly intuitionistic metamathematics. In the closing section, we assay some doubts about such interpretations.

This essay shows that some recent work by George Weaver can be reformulated in an especially perspicuous way within the theory of closure systems. Closure theoretic generalizations of some theorems of Robert Goldblatt are presented. And, more generally, the relation between closure systems and the deducibility relations of Goldblatt is explored.

Weakening the conditions on the Kripke semantics for propositional intuitionistic logic (J) unearths a family of logics below J. This paper provides a characterization of eleven such logics, using Kripke semantics, proof theory, and algebraic models. Questions about modelling quantification in these logics are also discussed.

This article discusses the nature of numerals and the plausibility of their special semantic and epistemological status as proper names of numbers. Evidence is presented that minimizes the difference between numerals and other devices of direct reference. The availability of intensional contexts within formalised metamathematics is exploited to shed light on the relation between formal numerals and numerals.

Recently Shakurov pioneered the study of subalgebras of diagonalizable algebras of theories of arithmetic. We show that his results extend to weaker theories (namely to theories containing $I\Delta_0 + {\rm exp}$).