A probabilistic logic based on the acceptability of gambles

This article presents a probabilistic logic whose sentences can be interpreted as asserting the acceptability of gambles described in terms of an underlying logic. This probabilistic logic has a concrete syntax and a complete inference procedure, and it handles conditional as well as unconditional probabilities. It synthesizes Nilsson’s probabilistic logic and Frisch and Haddawy’s anytime inference procedure with Wilson and Moral’s logic of gambles.Two distinct semantics can be used for our probabilistic logic: (1) the measure–theoretic semantics used by the prior logics already mentioned and also by the more expressive logic of Fagin, Halpern, and Meggido and (2) a behavioral semantics. Under the measure–theoretic semantics, sentences of our probabilistic logic are interpreted as assertions about a probability distribution over interpretations of the underlying logic. Under the behavioral semantics, these sentences are interpreted only as asserting the acceptability of gambles, and this suggests different directions for generalization.     

An epistemic model of an agent who does not reflect on reasoning processes

This paper introduces an epistemic model of a boundedly rational agent under the two assumptions that (i) the agent’s reasoning process is in accordance with the model but (ii) the agent does not reflect on these reasoning processes. For such a concept of bounded rationality a semantic interpretation by the possible world semantics of the Kripke (1963) type is no longer available because the definition of knowledge in these possible world semantics implies that the agent knows all valid statements of the model. The key to my alternative semantic approach is the extension of the method of truth tables, first introduced for the propositional logic by Wittgenstein (1922), to an epistemic logic so that I can determine the truth value of epistemic statements for all relevant truth conditions. In my syntactic approach I define an epistemic logic–consisting of the classical calculus of propositional logic plus two knowledge axioms–that does not include the inference rule of necessitation, which claims that an agent knows all theorems of the logic. As my main formal result I derive a determination theorem linking my semantic with my syntactic approach. The difference between my approach and existing knowledge models is illustrated in a game-theoretic application concerning the epistemic justification of iterative solution concepts.     

Bisimulations and bisimulation games between Verbrugge models

Interpretability logic is a modal formalization of relative interpretability between first-order arithmetical theories. Verbrugge semantics is a generalization of Veltman semantics, the basic semantics for interpretability logic. Bisimulation is the basic equivalence between models for modal logic. We study various notions of bisimulation between Verbrugge models and develop a new one, which we call w-bisimulation. We show that the new notion, while keeping the basic property that bisimilarity implies modal equivalence, is weak enough to allow the converse to hold in the finitary case. To do this, we develop and use an appropriate notion of bisimulation games between Verbrugge models.     

A Routley–Meyer Semantics for Gödel 3-Valued Logic and Its Paraconsistent Counterpart

Routley–Meyer semantics (RM-semantics) is defined for Gödel 3-valued logic G3 and some logics related to it among which a paraconsistent one differing only from G3 in the interpretation of negation is to be remarked. The logics are defined in the Hilbert-style way and also by means of proof-theoretical and semantical consequence relations. The RM-semantics is defined upon the models for Routley and Meyer’s basic positive logic B₊, the weakest positive RM-semantics. In this way, it is to be expected that the models defined can be adapted to other related many-valued logics.     

The Concept of Relevance and the Logic Diagram Tradition

What is logical relevance? Anderson and Belnap say that the “modern classical tradition [,] stemming from Frege and Whitehead-Russell, gave no consideration whatsoever to the classical notion of relevance.” But just what is this classical notion? I argue that the relevance tradition is implicitly most deeply concerned with the containment of truth-grounds, less deeply with the containment of classes, and least of all with variable sharing in the Anderson–Belnap manner. Thus modern classical logicians such as Peirce, Frege, Russell, Wittgenstein, and Quine are implicit relevantists on the deepest level. In showing this, I reunite two fields of logic which, strangely from the traditional point of view, have become basically separated from each other: relevance logic and diagram logic. I argue that there are two main concepts of relevance, intensional and extensional. The first is that of the relevantists, who overlook the presence of the second in modern classical logic. The second is the concept of truth-ground containment as following from in Wittgenstein’s Tractatus. I show that this second concept belongs to the diagram tradition of showing that the premisses contain the conclusion by the fact that the conclusion is diagrammed in the very act of diagramming the premisses. I argue that the extensional concept is primary, with at least five usable modern classical filters or constraints and indefinitely many secondary intensional filters or constraints. For the extensional concept is the genus of deductive relevance, and the filters define species. Also following the Tractatus, deductive relevance, or full truth-ground containment, is the limit of inductive relevance, or partial truth-ground containment. Purely extensional inductive or partial relevance has its filters or species too. Thus extensional relevance is more properly a universal concept of relevance or summum genus with modern classical deductive logic, relevantist deductive logic, and inductive logic as its three main domains.     

Constructive agents

Brouwer’s ideas of construction, proof, and inquiry in mathematics are more widely applicable. On a well-known philosophical view, intuitionistic logic is a general account of meaning and reasoning for natural language and epistemology. In this brief discussion piece, I go one step further, and discuss how intuitionistic semantics fits with information update and belief revision in agency. In the process, I define a number of new logical systems that give rise to several open problems.     

Modeling linear logic with implicit functions

Just as intuitionistic proofs can be modeled by functions, linear logic proofs, being symmetric in the inputs and outputs, can be modeled by relations (for example, cliques in coherence spaces). However generic relations do not establish any functional dependence between the arguments, and therefore it is questionable whether they can be thought as reasonable generalizations of functions. On the other hand, in some situations (typically in differential calculus) one can speak in some precise sense about an implicit functional dependence defined by a relation. It turns out that it is possible to model linear logic with implicit functions rather than general relations, an adequate language for such a semantics being (elementary) differential calculus. This results in a non-degenerate model enjoying quite strong completeness properties.     

Funnel control for fully actuated systems under a fragment of signal temporal logic specifications

Temporal logics have lately proven to be a valuable tool for various control applications by providing a rich specification language. Existing temporal logic-based control strategies discretize the underlying dynamical system in space and/or time. We will not use such an abstraction and consider continuous-time systems under a fragment of signal temporal logic specifications by using the associated robust semantics. In particular, this paper provides computationally-efficient funnel-based feedback control laws for a class of systems that are, in a sense, feedback equivalent to single integrator systems, but where the dynamics are partially unknown for the control design so that some degree of robustness is obtained. We first leverage the transient properties of a funnel-based feedback control strategy to maximize the robust semantics of some atomic temporal logic formulas. We then guarantee the satisfaction for specifications consisting of conjunctions of such atomic temporal logic formulas with overlapping time intervals by a suitable switched control system. The result is a framework that satisfies temporal logic specifications with a user-defined robustness when the specification is satisfiable. When the specification is not satisfiable, a least violating solution can be found. The theoretical findings are demonstrated in simulations of the nonlinear Lotka–Volterra equations for predator–prey models.     

A game semantics for disjunctive logic programming

Denotational semantics of logic programming and its extensions (by allowing negation, disjunctions, or both) have been studied thoroughly for many years. In 1998, a game semantics was given to definite logic programs by Di Cosmo, Loddo, and Nicolet, and a few years later it was extended to deal with negation by Rondogiannis and Wadge. Both approaches were proven equivalent to the traditional semantics. In this paper we define a game semantics for disjunctive logic programs and prove soundness and completeness with respect to the minimal model semantics of Minker. The overall development has been influenced by the games studied for PCF and functional programming in general, in the styles of Abramsky–Jagadeesan–Malacaria and Hyland–Ong–Nickau.     

Soft Constraint Logic Programming and Generalized Shortest Path Problems

In this paper we study the relationship between Constraint Programming (CP) and Shortest Path (SP) problems. In particular, we show that classical, multicriteria, partially ordered, and modality-based SP problems can be naturally modeled and solved within the Soft Constraint Logic Programming (SCLP) framework, where logic programming is coupled with soft constraints. In this way we provide this large class of SP problems with a high-level and declarative linguistic support whose semantics takes care of both finding the cost of the shortest path(s) and also of actually finding the path(s). On the other hand, some efficient algorithms for certain classes of SP problems can be exploited to provide some classes of SCLP programs with an efficient way to compute their semantics.     

Mathematics textbook development for primary grades and its teachers in Mozambique

This is an analysis of the author’s involvement since 1970 in textbook development for primary schools and adult education in Mozambique, focusing on integrating local cultural traditions, covering the period up to 2013. As main example, addition of the type 8?+?5?=?13 is used around the question of whether textbooks (and curricula) advocate counting strategies and/or computation strategies. Different visualisations of these strategies in textbooks and corresponding manipulatives are analysed. During the 1990s, local languages started to be used in adult education, apart from Portuguese, Mozambique’s official language. Unschooled adults showed the importance of verbal computation in Mozambican languages—Bantu languages—most of which use also the auxiliary base five, apart from the base-ten numerals. The 2003 curriculum for primary education introduced the possibility of using local languages, and NGO’s started translating textbooks into 16 Mozambican languages. The paper includes an analysis of some of these textbooks and concludes with the author’s ideas in his teacher’s guides on teaching mathematics in Mozambican languages, showing the opportunities of verbal computation, exploring base-five numerals supported by finger gestures, instead of finger counting.     

A semantic hierarchy for intuitionistic logic

Brouwer’s views on the foundations of mathematics have inspired the study of intuitionistic logic, including the study of the intuitionistic propositional calculus and its extensions. The theory of these systems has become an independent branch of logic with connections to lattice theory, topology, modal logic, and other areas. This paper aims to present a modern account of semantics for intuitionistic propositional systems. The guiding idea is that of a hierarchy of semantics, organized by increasing generality: from the least general Kripke semantics on through Beth semantics, topological semantics, Dragalin semantics, and finally to the most general algebraic semantics. While the Kripke, topological, and algebraic semantics have been extensively studied, the Beth and Dragalin semantics have received less attention. We bring Beth and Dragalin semantics to the fore, relating them to the concept of a nucleus from pointfree topology, which provides a unifying perspective on the semantic hierarchy.     

收益模糊合作对策Shapley值的公理化

研究一类收益模糊的合作对策,这类对策联盟的模糊收益值可以用一个闭区间的形式来表示,本文定义了一个拓展的闭区间空间和一些闭区间线性运算算子,证明了这类对策的Shapley值可以用承载性、可替代性和可加性进行了公理化.     

Quantaloids for Concurrency

S. Abramsky has introduced interaction categories as a new semantics for concurrent computation. We show that interaction categories can be naturally described in the language of quantaloids. More precisely, they are closely related to fix-points of a certain comonad on the category of quantaloids with biproducts.     

Probabilization of Logics: Completeness and Decidability

The probabilization of a logic system consists of enriching the language (the formulas) and the semantics (the models) with probabilistic features. Such an operation is said to be exogenous if the enrichment is done on top, without internal changes to the structure, and is called endogenous otherwise. These two different enrichments can be applied simultaneously to the language and semantics of a same logic. We address the problem of studying the transference of metaproperties, such as completeness and decidability, to the exogenous probabilization of an abstract logic system. First, we setup the necessary framework to handle the probabilization of a satisfaction system by proving transference results within a more general context. In this setup, we define a combination mechanism of logics through morphisms and prove sufficient condition to guarantee completeness and decidability. Then, we demonstrate that probabilization is a special case of this exogenous combination method, and that it fulfills the general conditions to obtain transference of completeness and decidability. Finally, we motivate the applicability of our technique by analyzing the probabilization of the linear temporal logic over Markov chains, which constitutes an endogenous probabilization. The results are obtained first by studying the exogenous semantics, and then by establishing an equivalence with the original probabilization given by Markov chains.     

An epistemic model of an agent who does not reflect on reasoning processes

This paper introduces an epistemic model of a boundedly rational agent under the two assumptions that (i) the agent’s reasoning process is in accordance with the model but (ii) the agent does not reflect on these reasoning processes. For such a concept of bounded rationality a semantic interpretation by the possible world semantics of the Kripke (1963) type is no longer available because the definition of knowledge in these possible world semantics implies that the agent knows all valid statements of the model. The key to my alternative semantic approach is the extension of the method of truth tables, first introduced for the propositional logic by Wittgenstein (1922), to an epistemic logic so that I can determine the truth value of epistemic statements for all relevant truth conditions. In my syntactic approach I define an epistemic logic–consisting of the classical calculus of propositional logic plus two knowledge axioms–that does not include the inference rule of necessitation, which claims that an agent knows all theorems of the logic. As my main formal result I derive a determination theorem linking my semantic with my syntactic approach. The difference between my approach and existing knowledge models is illustrated in a game-theoretic application concerning the epistemic justification of iterative solution concepts.     

Strategies and simulations in a semantic framework

By means of several examples of structural operational semantics for a variety of languages, we justify the importance and interest of using the notions of strategies and simulations in the semantic framework provided by rewriting logic and implemented in the Maude metalanguage. On the one hand, we describe a basic strategy language for Maude and show its application to CCS, the ambient calculus, and the parallel functional language Eden. On the other hand, we show how the concept of stuttering simulation can be used inside Maude to show that a stack machine correctly implements the operational semantics of a simple functional language.