Derivation Lengths Classification of G\"odel's T Extending Howard's Assignment

Let T be Goedel's system of primitive recursive functionals of finite type in the lambda formulation. We define by constructive means using recursion on nested multisets a multivalued function I from the set of terms of T into the set of natural numbers such that if a term a reduces to a term b and if a natural number I(a) is assigned to a then a natural number I(b) can be assigned to b such that I(a) is greater than I(b). The construction of I is based on Howard's 1970 ordinal assignment for T and Weiermann's 1996 treatment of T in the combinatory logic version. As a corollary we obtain an optimal derivation length classification for the lambda formulation of T and its fragments. Compared with Weiermann's 1996 exposition this article yields solutions to several non-trivial problems arising from dealing with lambda terms instead of combinatory logic terms. It is expected that the methods developed here can be applied to other higher order rewrite systems resulting in new powerful termination orderings since T is a paradigm for such systems

A New Theory of Content II: Model Theory and Some Alternatives

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Inclusion and Exclusion Dependencies in Team Semantics: On Some Logics of Imperfect Information

We introduce some new logics of imperfect information by adding atomic formulas corresponding to inclusion and exclusion dependencies to the language of first order logic. The properties of these logics and their relationships with other logics of imperfect information are then studied. Furthermore, a game theoretic semantics for these logics is developed. As a corollary of these results, we characterize the expressive power of independence logic, thus answering an open problem posed in (Gr\"adel and V\"a\"an\"anen, 2010)

Introduction to property theory - the fundamental theorems

The market system consists of a price mechanism, built on the foundation of a system of property, and contract. In many developing, and transition economies, the market system functions poorly. In many cases, if not most, the malfunctioning is not simply in the price system (for example, anti-competitive activities), but in the underlying property system (such as contracts being breached, and externalities in the sense of transfers not covered by contracts). Economic theory tends to take the functioning of the system of property, and contract for granted, and focuses on the operation of the price mechanism. Property theory focuses on the underlying system of property, and contract. In this paper, the author inaugurates the mathematical treatment of property theory.In contrast with earlier work in"law and economics", and the"new institutional economics", this approach uses principles drawn from jurisprudence, and does not attempt to reduce"law"to"economics"in the sense of efficiency considerations, such as the minimization of transaction costs. The main results are the two fundamental theorems of property theory that are analogous to the two fundamental theorems of price theory that, in essence, state that: 1) A competitive equilibrium is Pareto optimal. 2) Given a Pareto optimal state, there exists a set of prices such, that a competitive equilibrium at those prices would realize that Pareto optimal state.Environmental Economics&Policies,Labor Policies,Banks&Banking Reform,Municipal Housing and Land,Economic Theory&Research,Environmental Economics&Policies,Banks&Banking Reform,Economic Theory&Research,Municipal Housing and Land,Land and Real Estate Development

Changing a semantics: opportunism or courage?

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201