We present systems of Natural Deduction based on Strict Implication for the main normal modal logics between K and S5. In this work we consider Strict Implication as the main modal operator, and establish a natural correspondence between Strict Implication and strict subproofs.

We study a hierarchy $\{\mathcal{L}_2^k \}$ of Kalmàr elementary functions on integers based on a classification of LOOP programs of limited complexity, namely those in which the depth of nestings of LOOP commands does not exceed two. It is proved that $n$-place functions in $\mathcal{L}_2^k$ can be enumerated by a single function in $\mathcal{L}_2^{k+2}$, and that the resulting hierarchy of elementary predicates (i.e., functions with 0,1-values) is proper in that there are $\mathcal{L}_2^{k+2}$ predicates that are not in $\mathcal{L}_2^{k}$. Along the way the rudimentary predicates of Smullyan are classified as $\mathcal{L}_2^{2}$.

In this paper we study numeral systems in the $\lambda\beta\eta$-calculus. With one exception, we assume that all numerals have normal form. We study the independence of the conditions of adequacy of numeral systems. We find that, to a great extent, they are mutually independent. We then consider particular examples of numeral systems, some of which display paradoxical properties. One of these systems furnishes a counterexample to a conjecture of Böhm. Next, we turn to the approach of Curry, Hindley, and Seldin. We dwell with the general problem of obtaining their results with the additional requirement of nonconvertibility of numerals. In particular we solve a problem that they left open. Finally, we give the first example of an adequate unsolvable numeral system without a test for zero in the usual sense, thus solving a problem of Barendregt and Barendsen.

If there is a good $\Delta^1_3$ wellordering of the reals, then there is a $\Pi^1_1$ wellfounded relation for which the comparison relation is not projective.

There are eighteen isomorphism types of finite relation algebras with eight or fewer elements, and all of them are representable. We determine all the cardinalities of sets on which these algebras have representations.

This paper studies the topological duality between diagonalizable algebras and bi-topological spaces. In particular, the correspondence between algebraic properties of a diagonalizable algebra and topological properties of its dual space is investigated. Since the main example of a diagonalizable algebra is the Lindenbaum algebra of an r.e. theory extending Peano Arithmetic, endowed with an operator defined by means of the provability predicate of the theory, this duality gives the possibility to study arithmetical properties of theories from a topological point of view. We find topological characterization of $\Sigma_1$-sound theories and of sentences that are $\Sigma_1$-conservative over such a theory.

Let $R$ be a convergent term rewriting system, and let $CR$-equality on (simply typed) combinatory logic terms be the equality induced by $\beta \eta R$-equality on terms of the (simply typed) lambda calculus under any of the standard translations between these two frameworks for higher-order reasoning. We generalize the classical notion of strong reduction to a reduction relation which generates $CR$-equality and whose irreducibles are exactly the translates of long $\beta R$-normal forms. The classical notion of strong normal form in combinatory logic is also generalized, yielding yet another description of these translates. Their resulting tripartite characterization extends to the combined first-order algebraic and higher-order setting the classical combinatory logic descriptions of the translates of long $\beta$-normal forms in the lambda calculus. As a consequence, the translates of long $\beta R$-normal forms are easily seen to serve as canonical representatives for $CR$-equivalence classes of combinatory logic terms for non-empty, as well as for empty, $R$.

The purpose of the paper is to study syntactic refutation systems as a way of characterizing normal modal propositional logics. In particular it is shown that there is a decidable modal logic without the finite model property that has a simple finite refutation system.

Professor Robert Koons provides a welcome guide to our book The Revision Theory of Truth, and he puts forward objections that deserve serious consideration. In this note we shall respond to his principal objection. This objection calls into question our main thesis. As we argue, however, the objection is not successful. We should forewarn the reader that this note is not self-contained. It presupposes familiarity with The Revision Theory of Truth (primarily, Chapter 4) and with the review.