In this note we compare propositional logics for closed substitutions and propositional logics for open substitutions in constructive arithmetical theories. We provide a strong example where these logics diverge in an essential way. We prove that for Markov's Arithmetic, that is, Heyting's Arithmetic plus Markov's principle plus Extended Church's Thesis, the logic of closed and the logic of open substitutions are the same.

The purpose of this note is to observe a generalization of the concept "computable in..." to arbitrary partial combinatory algebras. For every partial combinatory algebra (pca) A and every partial endofunction on A, a pca A[f] is constructed such that in A[f], the function f is representable by an element; a universal property of the construction is formulated in terms of Longley's 2-category of pcas and decidable applicative morphisms. It is proved that there is always a geometric inclusion from the realizability topos on A[f] into the one on A and that there is a meaningful preorder on the partial endofunctions on A which generalizes Turing reducibility.

We define an appropriate analog of the Morley rank in a totally transcendental homogeneous model with type diagram D. We show that if RM[p] = α then for some 1 ≤ n < ω the type p has n, but not n + 1, distinct D-extensions of rank α. This is surprising, because the proof of the statement in the first-order case depends heavily on compactness. We also show that types over (D,ℵ₀)-homogeneous models have multiplicity (Morley degree) 1.

In this paper we give a differential lifting principle which provides a general method to geometrically axiomatize the model companion (if it exists) of some theories of differential topological fields. The topological fields we consider here are in fact topological systems in the sense of van den Dries, and the lifting principle we develop is a generalization of the geometric axiomatization of the theory DCF₀ given by Pierce and Pillay. Moreover, it provides a geometric alternative to the axiomatizations obtained by Tressl and Guzy/Point in separate papers where the authors also build general schemes of axioms for some model complete theories of differential fields. We first characterize the existentially closed models of a given theory of differential topological fields and then, under an additional hypothesis of largeness, we show how to modify this characterization to get a general scheme of first-order axioms for the model companion of any large theory of differential topological fields. We conclude with an application of this lifting principle proving that, in existentially closed models of a large theory of differential topological fields, the jet-spaces are dense in their ambient topological space.

We generalize Shelah's analysis of cardinality quantifiers for a superstable theory from Chapter V of Classification Theory and the Number of Nonisomorphic Models. We start with a set of bounds for the cardinality of each formula in some general invariant family of formulas in a superstable theory (in Classification Theory, a uniform family of formulas is considered) and find a set of derived bounds for all formulas. The set of derived bounds is sharp: up to a technical restriction, every model that satisfies the original bounds has a sufficiently saturated elementary extension that satisfies the original bounds and such that for each formula the set of its realizations in the extension has arbitrarily large cardinality below the corresponding derived bound of the formula.

Recall that for any Boolean algebra (BA) A, the cellularity of A is c(A) = sup{|X| : X is a pairwise-disjoint subset of A}. A pseudo-tree is a partially ordered set (T, ≤) such that for every t in T, the set {r ∊ T : r ≤ t} is a linear order. The pseudo-tree algebra on T, denoted Treealg(T), is the subalgebra of ℘(T) generated by the cones {r ∊ T : r ≥ t}, for t in T. We characterize the cellularity of pseudo-tree algebras in terms of cardinal functions on the underlying pseudo-trees. For T a pseudo-tree, c(Treealg(T)) is the maximum of four cardinals c\sbT, ι\sbT, φ\sbT, and μ\sbT : roughly, c\sbT measures the "tallness" of the pseudo-tree T; ι\sbT the "breadth"; φ\sbT the number of "finite branchings"; and μ\sbT the number of places where T "does not branch." We give examples to demonstrate that all four of these cardinals are needed.

We show that in the c.e. weak truth table degrees if b < c then there is an a which contains no hypersimple set and b < a < c. We also show that for every w < c in the c.e. wtt degrees such that w is hypersimple, there is a hypersimple a such that w < a < c. On the other hand, we know that there are intervals which contain no hypersimple set.

Discursive (or discussive) logic, D₂, introduced by Jaśkowski, is widely recognized as a first formal approach to paraconsistency. Jaśkowski applied a quite extraordinary technique at that time to describe his logic. He neither gave a set of the axiom schemata nor presented a direct semantics for D₂ but used a translation function to express his philosophical and logical intuitions. Discursive logic was defined by an interpretation in the language of S₅ of Lewis. The aim of this paper is to present a modified system of the discursive logic that allows some of the weaker versions of Duns Scotus's thesis to be valid. The initial idea is to consider a different characteristic of the connective of negation. We introduce both a direct semantics and an axiomatization of the new system, prove the key metatheorems, and describe labeled tableaux for the system.

We present a new axiomatization of the deontic fragment of Anderson's relevant deontic logic, give an Andersonian reduction of a relevant version of Mally's deontic logic previously discussed in this journal, study the effect of adding propositional quantification to Anderson's system, and discuss the meaning of Anderson's propositional constant in a wide range of Andersonian deontic systems.

Russell and Walker proposed different ways of constructing instants from events. For an explanation of "time as a continuum," Thomason favored Walker's construction. The present article shows that Russell's construction fares as well. To this end, a mathematical characterization problem is solved which corresponds to the characterization problem that Thomason solved with regard to Walker's construction. It is shown how to characterize those event structures (formally, interval orders) which, through Russell's construction of instants, become linear orders isomorphic to a given (or, deriving, to some—nontrivial ordered) real interval. As tools, separate characterizations for each of resulting (i) Dedekind completeness, (ii) separability, (iii) plurality of elements, (iv) existence of certain endpoints are provided. Denseness is characterized to replace Russell's erroneous attempt. Somewhat minimal nonconstructive principles needed are exhibited, and some alternative approaches are surveyed.