Em nossa vida cotidiana há duas condições frequentes: 1) há situações nas quais não existem instruções explícitas, mas apenas contingências aos comportamentos e 2) comportamentos que aparentam ser iguais a um observador externo, podem ser originados por raciocínios diversos. Partindo destas premissas, procurou-se neste experimento, desenvolver procedimentos experimentais que levam a determinado comportamento (respostas que refletem transitividade lógica) sem instruções explícitas, averiguando a seguir, quais os raciocínios que realmente estavam atrás das respostas (transitivas ou não) dadas pelos sujeitos. O experimento de Siemann & Delius (1993) utilizado para o estudo da inferência transitiva e que foi replicada por Cruz (20003) em sua dissertação de mestrado, suscitou uma série de temas metodológicos e conceituais abordados para viabilizar a pesquisa do raciocínio transitivo: o conceito de consciência e sua relação com o raciocínio; caracterização do raciocínio por sua explicação; utilização dos conceitos de introspecção, tomada de consciência, metacognição, empatia e teoria da mente para a interpretação de entrevistas. A essência deste experimento é seu método experimental que foi composto de duas fases...

The normal form theorem, proved in R. Laver, On the left distributive law and the freeness of an algebra of elementary embeddings, Advances in Mathematics 91 (1992), 209-231, for the free algebra $\Cal A$ on one generator $x$ satisfying the left distributive law $a(bc) = (ab)(ac)$ is extended by showing that members of $\Cal A$ can be put into a "division form."

The vast quantity of data generated and captured every day has led to a pressing need for tools and processes to organize, analyze and interrelate this data. Automated reasoning and optimization tools with inherent support for data could enable advancements in a variety of contexts, from data-backed decision making to data-intensive scientific research. To this end, we introduce a decidable logic aimed at database analysis. Our logic extends quantifier-free Linear Integer Arithmetic with operators from Relational Algebra, like selection and cross product. We provide a scalable decision procedure that is based on the BC(T) architecture for ILP Modulo Theories. Our decision procedure makes use of database techniques. We also experimentally evaluate our approach, and discuss potential applications.; Comment: FMCAD 2014 final version plus proofs

We consider the quantifier-free languages, Bc and Bc0, obtained by augmenting the signature of Boolean algebras with a unary predicate representing, respectively, the property of being connected, and the property of having a connected interior. These languages are interpreted over the regular closed sets of n-dimensional Euclidean space (n greater than 1) and, additionally, over the regular closed polyhedral sets of n-dimensional Euclidean space. The resulting logics are examples of formalisms that have recently been proposed in the Artificial Intelligence literature under the rubric "Qualitative Spatial Reasoning." We prove that the satisfiability problem for Bc is undecidable over the regular closed polyhedra in all dimensions greater than 1, and that the satisfiability problem for both languages is undecidable over both the regular closed sets and the regular closed polyhedra in the Euclidean plane. However, we also prove that the satisfiability problem for Bc0 is NP-complete over the regular closed sets in all dimensions greater than 2, while the corresponding problem for the regular closed polyhedra is ExpTime-complete. Our results show, in particular, that spatial reasoning over Euclidean spaces is much harder than reasoning over arbitrary topological spaces.

We present Integer Linear Programming (ILP) Modulo Theories (IMT). An IMT instance is an Integer Linear Programming instance, where some symbols have interpretations in background theories. In previous work, the IMT approach has been applied to industrial synthesis and design problems with real-time constraints arising in the development of the Boeing 787. Many other problems ranging from operations research to software verification routinely involve linear constraints and optimization. Thus, a general ILP Modulo Theories framework has the potential to be widely applicable. The logical next step in the development of IMT and the main goal of this paper is to provide theoretical underpinnings. This is accomplished by means of BC(T), the Branch and Cut Modulo T abstract transition system. We show that BC(T) provides a sound and complete optimization procedure for the ILP Modulo T problem, as long as T is a decidable, stably-infinite theory. We compare a prototype of BC(T) against leading SMT solvers.; Comment: CAV 2013 version plus proofs

We consider (fi?nitary, propositional) logics through the original use of Category Theory: the study of the "sociology of mathematical objects", aligning us with a recent, and growing, trend of study logics through its relations with other logics (e.g. process of combinations of logics as ?bring [Gab] and possible translation semantics [Car]). So will be objects of study the classes of logics, i.e. categories whose objects are logical systems (i.e., a signature with a Tarskian consequence relation) and the morphisms are related to (some concept of) translations between these systems. The present work provides the fi?rst steps of a project of considering categories of logical systems satisfying simultaneously certain natural requirements: it seems that in the literature ([AFLM1], [AFLM2], [AFLM3], [BC], [BCC1], [BCC2], [CG], [FC]) this is achieved only partially.; Comment: 10 pages

We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number $\kappa$, let {\sf BC}$_{\kappa}$ denote this generalization. Then ${\sf BC}_{\aleph_0}$ is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, $\neg{\sf BC}_{\aleph_1}$ is equivalent to the existence of a Kurepa tree of height $\aleph_1$. Using the connection of ${\sf BC}_{\kappa}$ with a generalization of Kurepa's Hypothesis, we obtain the following consistency results: (1)If it is consistent that there is a 1-inaccessible cardinal then it is consistent that ${\sf BC}_{\aleph_1}$. (2)If it is consistent that ${\sf BC}_{\aleph_1}$ holds, then it is consistent that there is an inaccessible cardinal. (3)If it is consistent that there is a 1-inaccessible cardinal with $\omega$ inaccessible cardinals above it, then $\neg{\sf BC}_{\aleph_{\omega}} \, +\, (\forall n<\omega){\sf BC}_{\aleph_n}$ is consistent. (4)If it is consistent that there is a 2-huge cardinal, then it is consistent that ${\sf BC}_{\aleph_{\omega}}$. (5)If it is consistent that there is a 3-huge cardinal, then it is consistent that ${\sf BC}_{\kappa}$ holds for a proper class of cardinals $\kappa$ of countable cofinality.; Comment: 15 pages

Problem 2 at the 56th International Mathematical Olympiad (2015) asks for all triples (a,b,c) of positive integers for which ab-c, bc-a, and ca-b are all powers of 2. We show that this problem requires only a primitive form of arithmetic, going back to the Pythagoreans, which is the arithmetic of the even and the odd.

We chart the ways in which closure properties of consequence relations for uncertain inference take on different forms according to whether the relations are generated in a quantitative or a qualitative manner. Among the main themes are: the identification of watershed conditions between probabilistically and qualitatively sound rules; failsafe and classicality transforms of qualitatively sound rules; non-Horn conditions satisfied by probabilistic consequence; representation and completeness problems; and threshold-sensitive conditions such as `preface' and `lottery' rules.

The splitting theorem says that any set of formulae has a finest representation as a family of letter-disjoint sets. Parikh formulated this for classical propositional logic, proved it in the finite case, used it to formulate a criterion for relevance in belief change, and showed that AGMpartial meet revision can fail the criterion. In this paper we make three further contributions. We begin by establishing a new version of the well-known interpolation theorem, which we call parallel interpolation, use it to prove the splitting theorem in the infinite case, and show how AGM belief change operations may be modified, if desired, so as to ensure satisfaction of Parikh’s relevance criterion.

Most areas of logic can be approached either semantically or syntactically. Typically, the approaches are linked through a completeness or representation theorem. The two kinds of theorem serve a similar purpose, yet there also seems to be some residual distinction between them. In what respects do they differ, and how important are the differences? Can we have one without the other? We discuss these questions, with examples from a variety of different logical systems.

We define and examine a notion of logical friendliness, which is a broadening of the familiar notion of classical consequence. The concept is tudied first in its simplest form, and then in a syntax-independent version, which we call sympathy. We also draw attention to the surprising number of familiar notions and operations with which it makes contact, providing a new light in which they may be seen.

This paper considers a puzzling conflict between two positions that are each compelling: (A) it is irrational for an agent to pay to avoid `free' evidence before making a decision, and (B) rational agents may have imprecise beliefs and/or desires. Indeed, we show that Good's theorem (Good, 1967) concerning the invariable choice-worthiness of free evidence does not generalise to the imprecise realm, given the plausible existing decision theories for handling imprecision. A key ingredient in the analysis, and a potential source of controversy, is the general approach taken for resolving sequential decision problems { we make explicit what the key alternatives are and defend our own approach. Furthermore, we endorse a resolution of the aforementioned puzzle { we privilege decision theories that merely permit avoiding free evidence over decision theories for which avoiding free evidence is uniquely admissible. Finally, we situate this particular result about free evidence within the broader `dynamic-coherence' literature.

Chance Neutrality is the thesis that, conditional on some proposition being true (or being false), its chance of being true should be a matter of practical indifference. The aim of this paper is to examine whether Chance Neutrality is a requirement of rationality. We prove that given Chance Neutrality, the Principal Principle entails a thesis called Linearity; the centrepiece of von Neumann and Morgenstern’s expected utility theory. With this in mind, we argue that the Principal Principle is a requirement of practical rationality but that Linearity is not; and hence, that Chance Neutrality is not rationally required.

This paper sheds new light on the subtle relation between probability and logic by (i) providing a logical development of Bruno de Finetti's conception of events and (ii) suggesting that the subjective nature of de Finetti's interpretation of probability emerges in a clearer form against such a logical background. By making explicit the epistemic structure which underlies what we call Choice-based probability we show that whilst all rational degrees of belief must be probabilities, the converse doesn't hold: some probability values don't represent decision-relevant quantifications of uncertainty.

Group decisions must often obey exogenous constraints. While in a preference aggregation problem constraints are modelled by restricting the set of feasible alternatives, this paper discusses the modelling of constraints when aggregating individual yes/no judgments on interconnected propositions. For example, court judgments in breach-of-contract cases should respect the constraint that action and obligation are necessary and sufficient for liability, and judgments on budget items should respect budgetary constraints. In this paper, we make constraints in judgment aggregation explicit by relativizing the rationality conditions of consistency and deductive closure to a constraint set, whose variation yields more or less strong notions of rationality. This approach of modelling constraints explicitly contrasts with that of building constraints as axioms into the logic, which turns compliance with constraints into a matter of logical consistency and thereby conflates requirements of ordinary logical consistency (such as not to affirm both a proposition and its negation) and requirements dictated by the environment (such as budgetary constraints). We present some general impossibility results on constrained judgment aggregation; they are immediate corollaries of known results on (unconstrained) judgment aggregation.

The concept of relevance between classical propositional formulae, defined in terms of letter-sharing, has been around for a long time. But it began to take on a fresh life in the late 1990s when it was reconsidered in the context of the logic of belief change. Two new ideas appeared in independent work of Odinaldo Rodrigues and Rohit Parikh: the relation of relevance was considered modulo the choice of a background belief set, and the belief set was put into a canonical form, called its finest splitting. In the first part of this paper, we recall the ideas of Rodrigues and Parikh, and show that they yield equivalent definitions of what may be called canonical cell/path relevance. The second part presents the main new result of the paper: while the relation of canonical relevance is syntax-independent in the usual sense of the term, it nevertheless remains language-dependent in a deeper sense, as is shown with an example. The final part of the paper turns to questions of application, where we present a new concept of parameter-sensitive relevance that relaxes the Rodrigues/Parikh definition, allowing it to take into account extra-logical sources as well as purely logical ones.

We reflect on lessons that the lottery and preface paradoxes provide for the logic of uncertain inference. One of these lessons is the unreliability of the rule of conjunction of conclusions in such contexts, whether the inferences are probabilistic or qualitative; this leads us to an examination of consequence relations without that rule, the study of other rules that may nevertheless be satisfied in its absence, and a partial rehabilitation of conjunction as a 'lossy' rule. A second lesson is the possibility of rational inconsistent belief; this leads us to formulate criteria for deciding when an inconsistent set of beliefs may reasonably be retained.

In engineering, as in other scientific fields, researchers seek to confirm their models with real-world data. It is common practice to assess models in terms of the distance between the model outputs and the corresponding experimental observations. An important question that arises is whether the model should then be ‘tuned’, in the sense of estimating the values of free parameters to get a better fit with the data, and furthermore whether the tuned model can be confirmed with the same data used to tune it. This dual use of data is often disparagingly referred to as ‘double-counting’. Here, we analyse these issues, with reference to selected research articles in engineering (one mechanical and the other civil). Our example studies illustrate more and less controversial practices of model tuning and double-counting, both of which, we argue, can be shown to be legitimate within a Bayesian framework. The question nonetheless remains as to whether the implied scientific assumptions in each case are apt from the engineering point of view.

The value of actual outcomes or states of affairs often depends on what could have been. Such dependencies create well-known “paradoxes” for decision theory, the best-known perhaps being the so-called Allais Paradox. The primary aim of this PhD thesis is to enrich decision theory such that it includes counterfactual prospects in the domains of desirability (or utility) functions, and show that, as a result, the paradoxes in question disappear. Before discussing the way in which counterfactual propositions influence the desirability of actual outcomes, I discuss the way in which the truth of one factual proposition influences the desirability of another. This examination leads me to reject the Invariance assumption, which states that the desirability of a proposition is independent of whether it is true. The assumption plays an important role in David Lewis’ famous arguments against the so-called Desire-as-Belief thesis (DAB). The unsoundness of Lewis’ argument does of course not make DAB true. In fact, I provide novel arguments against different versions of DAB, without assuming Invariance. To justify the assumptions I make when extending decision theory to counterfactual prospects, I discuss several issues concerning the logic...