Power types in explicit mathematics?

In this note it is shown that in explicit mathematics the strong power type axiom is inconsistent with (uniform) elementary comprehension and discuss some general aspects of power types in explicit mathematic

First order theories for nonmonotone inductive definitions: recursively inaccessible and Mahlo

In this paper first order theories for nonmonotone inductive definitions are introduced, and a proof-theoretic analysis for such theories based on combined operator forms à la Richter with recursively inaccessible and Mahlo closure ordinals is give

A boundedness theorem in ID 1(W)

In this paper we prove a boundedness theorem in the theory ID1(W). This answers a question asked by Feferman, for example in [3]. The background is the following. Let A[X, x] be an X-positive formula arithmetic in X. The theory ID1(PA ) is an extension of Peano arithmetic PA by the following axioms: for arbitrary formulas F; PA is a constant for the least fixed point of A[X, x]. Set-theoretically, PA can be defined by recursion on the ordinals as follows: where is the first nonrecursive ordinal. Now let a ≺ b be the arithmetic relation which expresses that the recursive tree coded by a is a proper subtree of the tree coded by b, and define The least fixed point of Tree[X, x] is the set P Tree of all well-founded recursive trees. We write W or W α for P Tree or , respectively. Since W is complete we have for all α < . If we define for each element a ∈ W its inductive norm ∣a∣ by ∣a∣≔ min{ξ: a ∈ W ξ}, then we have = {∣a∣: a ∈ W} and the elements of W can be used as codes for the ordinals less than . Assume that B[X, x] is an X-positive formula arithmetic in X with the only free variables X and x, and assume that QB is a relation that satisfies If we define then we obviously have PB = I

Annotations on the consistency of the closed world assumption

AbstractThe treatment of negation and negative information in a logic programming environment has turned out to be a major problem. We introduce a relativized version of Reiter's closed world assumption and study it from a logical point of view. In particular, we look at the questions of consistency and conservative extension

Stage levels, states, and the semantics of the copula

The paper investigates the issue whether the stage-level/individual level contrast introduced by Carlson 1977 requires the assumption of two homonymous copulas depending on the categorization of the predicative. We argue that instead of a uniform stage-level/individual level distinction we have to distinguish several similar but independent contrasts, none of which crucially depend on the semantics of the copula. In the second part of the paper, we concentrate on one group of phenomena-the distribution of weak subjects-and propose an explanation in terms of an interaction between topic/comment structure and aspectual properties of the predicate

Towards an explanation of copula effects

This paper deals with a series of semantic contrasts between the copula "be" and the preposition "as", two functional elements that both head elementary predication structures. It will be argued that the meaning of "as" is a type lowering device shifting the meaning of its complement NP from generalized quantifier type to property type (where properties are conceived as relations between individuals and situations), while the copula "be" induces a type coercion from (partial) situations to (total) possible worlds. Paired with van der Sandt's 1992 theory of presupposition accommodation, these assumptions will account for the observed contrasts between "as" and "be"

On the generative capacity of multi-modal Categorial Grammars

In Moortgat 1996 the Lambek Calculus L (Lambek 1958) is extended by a pair of residuation modalities ◊ and □↓. Categorial Grammars based on the resulting logic L◊ are attractive for linguistic purposes since they offer a compromise between the strict constituent structures imposed by context free grammars and related formalisms on the one hand, and the complete absence of hierarchical information in Lambek grammars on the other hand. The paper contains some results on the generative capcity of Categorial Grammars based on L◊. First it is shown that adding residuation modalities does not extend the weak generative capacity. This is proved by extending the proof for the context freeness of L-grammars from Pentus 1993 to L◊. Second the strong generative capacity of L◊-grammars is compared to context free grammars. The results are mainly negative. The set of tree languages generated by L◊-grammars neither contains nor is contained in the class of context free tree languages