An Exactification of the Monoid of Primitive Recursive Functions

We study the monoid of primitive recursive functions and investigate a onestep construction of a kind of exact completion, which resembles that of the familiar category of modest sets, except that the partial equivalence relations which serve as objects are recursively enumerable. As usual, these constructions involve the splitting of symmetric idempotents.     

Protoalgebraic Gentzen Systems and the Cut Rule

In this paper we show that, in Gentzen systems, there is a close relation between two of the main characters in algebraic logic and proof theory respectively: protoalgebraicity and the cut rule. We give certain conditions under which a Gentzen system is protoalgebraic if and only if it possesses the cut rule. To obtain this equivalence, we limit our discussion to what we call regular sequent calculi, which are those comprising some of the structural rules and some logical rules, in a sense we make precise. We note that this restricted set of rules includes all the usual rules in the literature. We also stress the difference between the case of two-sided sequents and the case of many-sided sequents, in which more conditions are needed.     

Type Logics and Pregroups

We discuss the logic of pregroups, introduced by Lambek [34], and its connections with other type logics and formal grammars. The paper contains some new ideas and results: the cut-elimination theorem and a normalization theorem for an extended system of this logic, its P-TIME decidability, its interpretation in L1, and a general construction of (preordered) bilinear algebras and pregroups whose universe is an arbitrary monoid. Special Issue Categorial Grammars and Pregroups Edited by Wojciech Buszkowski and Anne Preller     

A Predicate Logical Extension of a Subintuitionistic Propositional Logic

We develop a predicate logical extension of a subintuitionistic propositional logic. Therefore a Hilbert type calculus and a Kripke type model are given. The propositional logic is formulated to axiomatize the idea of strategic weakening of Kripke's semantic for intuitionistic logic: dropping the semantical condition of heredity or persistence leads to a nonmonotonic model. On the syntactic side this leads to a certain restriction imposed on the deduction theorem. By means of a Henkin argument strong completeness is proved making use of predicate logical principles, which are only classically acceptable.     

Causation in a New Old Key

I argue (1) that it is not philosophically significant whether causation is linguistically represented by a predicate or by a sentence connective; (2) that there is no philosophically significant distinction between event- and states-of-affairs-causation; (3) that there is indeed a philosophically significant distinction between agent- and event-causation, and that event-causation must be regarded as an analog of agent-causation. Developing this point, I argue that event-causation's being in the image of agent-causation requires, mainly, (a) that the cause is temporally prior to the effect, (b) that the cause necessitates (is sufficient with necessity) for the effect. Causal necessity is explained as a derivative of nomological necessity, and finally, via a definition of the causal sentence connective, the logic of event-causation is shown to be a part of temporal modal logic.     

Uniform Interpolation and Propositional Quantifiers in Modal Logics

We investigate uniform interpolants in propositional modal logics from the proof-theoretical point of view. Our approach is adopted from Pitts’ proof of uniform interpolationin intuitionistic propositional logic [15]. The method is based on a simulation of certain quantifiers ranging over propositional variables and uses a terminating sequent calculus for which structural rules are admissible. We shall present such a proof of the uniform interpolation theorem for normal modal logics K and T. It provides an explicit algorithm constructing the interpolants. Presented by Heinrich Wansing