A Formal Sociologic Study of Free Will

We make a formal sociologic study of the concept of free will. By using the language of mathematics and logic, we define what we call everlasting societies. Everlasting societies never age: persons never age, and the goods of the society are indestructible. The infinite history of an everlasting society unfolds by following deterministic and probabilistic laws that do their best to satisfy the free will of all the persons of the society. We define three possible kinds of histories for everlasting societies: primitive histories, good histories, and golden histories. In primitive histories, persons are inherently selfish, and they use their free will to obtain the personal ownerships of all the goods of the society. In good histories, persons are inherently good, and they use their free will to distribute the goods of the society. In good histories, a person is not only able to desire the personal ownership of goods, but is also able to desire that a good be owned by another person. In golden histories, free will is bound by the ethic of reciprocity, which states that "you should wish upon others as you would like others to wish upon yourself". In golden societies, the ethic of reciprocity becomes a law that partially binds free will, and that must be abided at all times. In other words, the verb "should" becomes the verb "must"

Combining Lists with Non-Stably Infinite Theories

http://www.springerlink.comIn program verification one has often to reason about lists over elements of a given nature. Thus, it becomes important to be able to combine the theory of lists with a generic theory $T$ modeling the elements. This combination can be achieved using the Nelson-Oppen method only if $T$ is stably infinite. The goal of this paper is to relax the stable-infiniteness requirement. More specifically, we provide a new method that is able to combine the theory of lists with any theory $T$ of the elements, regardless of whether $T$ is stably infinite or not. The crux of our combination method is to guess an arrangement over a set of variables that is larger than the one considered by Nelson and Oppen. Furthermore, our results entail that it is also possible to combine $T$ with the more general theory of lists with a length function

C-tableaux

The Nelson-Oppen combination method combines decision procedures for first-order theories satisfying certain conditions into a single decision procedure for the union theory. The method is restricted to the combination of stably infinite theories over disjoint signatures. In this report we present C-tableaux, an extension of Smullyan tableaux that generalizes the Nelson-Oppen method to the combination of arbitrary universal theories, not necessarily stably infinite and not necessarily over disjoint signatures. C-tableaux are sound and complete, but not terminating in general. Although C-tableaux do not provide a decidability result in general, in this report we describe two approaches that can be used in order to obtain decidability results using C-tableaux. Using the first approach, we are able to obtain a decidability result when combining theories that share the dense orders. Using the second approach, we are able to obtain a decidability result when combining theories whose union is stably finite

A Decision Procedure for a Fragment of Set Theory Involving Monotone, Additive, and Multiplicative Functions

2LS is a decidable many-sorted set-theoretic language involving one sort for elements and one sort for sets of elements. In this report we extend 2LS with constructs for expressing monotonicity, additivity, and multiplicativity properties of set-to-set functions. We call the resulting language 2LSmf. We prove that 2LSmf is decidable by reducing the problem of determining the satisfiability of its sentences to the problem of determining the satisfiability of sentences of 2LS. Furthermore, we prove that the language 2LSmf is stably infinite with respect to the sort of elements. Therefore, using a many-sorted version of the Nelson-Oppen combination method, 2LSmf can be combined with other languages modeling the sort of elements

C-Tableaux

The Nelson-Oppen combination method combines decision procedures for rstorder theories satisfying certain conditions into a single decision procedure for the union theory. The method is restricted to the combination of stably innite theories over disjoint signatures