Dependent choice, properness, and generic absoluteness

We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to -preserving symmetric submodels of forcing extensions. Hence, not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in and. Our results confirm as a natural foundation for a significant portion of classical mathematics and provide support to the idea of this theory being also a natural foundation for a large part of set theory

Algebraic Principles for Rely-Guarantee Style Concurrency Verification Tools

We provide simple equational principles for deriving rely-guarantee-style inference rules and refinement laws based on idempotent semirings. We link the algebraic layer with concrete models of programs based on languages and execution traces. We have implemented the approach in Isabelle/HOL as a lightweight concurrency verification tool that supports reasoning about the control and data flow of concurrent programs with shared variables at different levels of abstraction. This is illustrated on two simple verification examples

Parametric Weighting Functions

This paper provides behavioral foundations for parametric weighting functions under rankdependent utility. This is achieved by decomposing the independence axiom of expected utility into separate meaningful properties. These conditions allow us to characterize rank-dependent utility with power and exponential weighting functions. Moreover, by restricting the conditions to subsets of the probability interval, foundations of rank-dependent utility with parametric inverse-S shaped weighting functions are obtained. --Comonotonic independence,probability weighting function,preference foundation,rank-dependent utility

Expected utility theory, Jeffrey’s decision theory, and the paradoxes

In Richard Bradley’s book, Decision Theory with a Human Face, we have selected two themes for discussion. The first is the Bolker-Jeffrey theory of decision, which the book uses throughout as a tool to reorganize the whole field of decision theory, and in particular to evaluate the extent to which expected utility theories may be normatively too demanding. The second theme is the redefinition strategy that can be used to defend EU theories against the Allais and Ellsberg paradoxes, a strategy that the book by and large endorses, and even develops in an original way concerning the Ellsberg paradox. We argue that the BJ theory is too specific to fulfil Bradley’s foundational project and that the redefinition strategy fails in both the Allais and Ellsberg cases. Although we share Bradley’s conclusion that EU theories do not state universal rationality requirements, we reach it not by a comparison with BJ theory, but by a comparison with the non-EU theories that the paradoxes have heuristically suggested

On Affine Logic and {\L}ukasiewicz Logic

The multi-valued logic of {\L}ukasiewicz is a substructural logic that has been widely studied and has many interesting properties. It is classical, in the sense that it admits the axiom schema of double negation, [DNE]. However, our understanding of {\L}ukasiewicz logic can be improved by separating its classical and intuitionistic aspects. The intuitionistic aspect of {\L}ukasiewicz logic is captured in an axiom schema, [CWC], which asserts the commutativity of a weak form of conjunction. This is equivalent to a very restricted form of contraction. We show how {\L}ukasiewicz Logic can be viewed both as an extension of classical affine logic with [CWC], or as an extension of what we call \emph{intuitionistic} {\L}ukasiewicz logic with [DNE], intuitionistic {\L}ukasiewicz logic being the extension of intuitionistic affine logic by the schema [CWC]. At first glance, intuitionistic affine logic seems very weak, but, in fact, [CWC] is surprisingly powerful, implying results such as intuitionistic analogues of De Morgan's laws. However the proofs can be very intricate. We present these results using derived connectives to clarify and motivate the proofs and give several applications. We give an analysis of the applicability to these logics of the well-known methods that use negation to translate classical logic into intuitionistic logic. The usual proofs of correctness for these translations make much use of contraction. Nonetheless, we show that all the usual negative translations are already correct for intuitionistic {\L}ukasiewicz logic, where only the limited amount of contraction given by [CWC] is allowed. This is in contrast with affine logic for which we show, by appeal to results on semantics proved in a companion paper, that both the Gentzen and the Glivenko translations fail.Comment: 28 page

The relationship between Mathematical Utility Theory and the Integrability Problem: some arguments in favour

The resort to utility-theoretical issues will permit us to propose a constructive procedure for deriving a homogeneous of degree one, continuous function that gives raise to a primitive demand function under suitably mild conditions. This constitutes the first elementary proof of a necessary and sufficient condition for an integrability problem to have a solution by continuous (subjective utility) functions. Such achievement reinforces the relevance of a technique that was succesfully formalized in Alcantud and Rodríguez-Palmero (2001). The analysis of these two works exposes deep relationships between two apparently separate fields: mathematical utility theory and the revealed preference approach to the integrability problem.Strong Axiom of Homothetic Revelation; revealed preference; continuous homogeneous of degree one utility; integrability of demand.

Bridging the gap between general probabilistic theories and the device-independent framework for nonlocality and contextuality

Characterizing quantum correlations in terms of information-theoretic principles is a popular chapter of quantum foundations. Traditionally, the principles adopted for this scope have been expressed in terms of conditional probability distributions, specifying the probability that a black box produces a certain output upon receiving a certain input. This framework is known as "device-independent". Another major chapter of quantum foundations is the information-theoretic characterization of quantum theory, with its sets of states and measurements, and with its allowed dynamics. The different frameworks adopted for this scope are known under the umbrella term "general probabilistic theories". With only a few exceptions, the two programmes on characterizing quantum correlations and characterizing quantum theory have so far proceeded on separate tracks, each one developing its own methods and its own agenda. This paper aims at bridging the gap, by comparing the two frameworks and illustrating how the two programmes can benefit each other.Comment: 61 pages, no figures, published versio