In this paper we carry out a comparative study of $\mathrm{I}\Sigma_1$ and PRA. We will in a sense fully determine what these theories have to say about each other in terms of provability and interpretability. Our study will result in two arithmetically complete modal logics with simple universal models.

We investigate the expressive power of fragments of first-order logic that are defined in terms of prefixes. The main result establishes a strict hierarchy among these fragments over the signature consisting of a single binary relation. It implies that for each prefix p, there is a sentence $\phi_p$ in prenex normal form with prefix p, over a single binary relation, such that for all sentences θ in prenex normal form, if θ is equivalent to $\phi_p$, then p can be embedded in the prefix of θ. This strengthens a theorem of Walkoe.

Extending previous results from work on the complexity of cut elimination for the sequent calculus LK, we discuss the role of quantifier alternations and develop a measure to describe the complexity of cut elimination in terms of quantifier alternations in cut formulas and contractions on such formulas.

The typical rules for truth-trees for first-order logic without functions can fail to generate finite branches for formulas that have finite models–the rule set fails to have the finite tree property. In 1984 Boolos showed that a new rule set proposed by Burgess does have this property. In this paper we address a similar problem with the typical rule set for first-order logic with identity and functions, proposing a new rule set that does have the finite tree property.

Pure Type Systems, PTSs, were introduced as a generalization of the type systems of Barendregt's lambda cube and were designed to provide a foundation for actual proof assistants which will verify proofs. Systems of illative combinatory logic or lambda calculus, ICLs, were introduced by Curry and Church as a foundation for logic and mathematics. In an earlier paper we considered two changes to the rules of the PTSs which made these rules more like ICL rules. This led to four kinds of PTSs. Most importantly PTSs are about statements of the form M:A, where M is a term and A a type. In ICLs there are no explicit types and the statements are terms. In this paper we show that for each of the four forms of PTS there is an equivalent form of ICL, sometimes if certain conditions hold.

Previously, both Soare and Simpson considered sets without subsets of higher $\leq_T$, define the concept of a $\leq_r$-introimmune set. For the most common reducibilities $\leq_r$, a set does not contain subsets of higher $\leq_r$-degree if and only if it is $\leq_r$-introimmune. In this paper we consider $\leq_m$-introimmune and $\leq^P_m$-introimmune sets and examine how structurally easy such sets can be. In other words we ask, What is the smallest class of the Kleene's Hierarchy containing $\leq_r$-introimmune sets for $\leq_r\in\{\leq_m,\leq^P_m\}$? We answer the question by proving the existence of $\leq_m$-introimmune sets in the class $\Pi^0_1$, bi-$\leq_m$-introimmune sets in $\Delta^0_2$, and bi-$\leq^P_m$-introimmune sets in $\Delta^0_1$.

In this paper we investigate nonnormal modal systems in the vicinity of the Lewis system S1. It might be claimed that Lewis's modal systems (S1, S2, S3, S4, and S5) are the starting point of modern modal logics. However, our interests in the Lewis systems and their relatives are not (merely) historical. They possess certain syntactical features and their frames certain structural properties that are of interest to us. Our starting point is not S1, but a weaker logic S1$^0$ (S1 without the schema [T]). We extend it to S1$^0$D, which can be considered as a deontic counterpart of the alethic S1. Soundness and completeness of these systems are then demonstrated within a prenormal idiom. We conclude with some philosophical remarks on the interpretation of our deontic logic.

We construct a novel proof of Cantor's theorem in set theory.