Set Of Worlds Research Articles

Modal logics, description logics and arithmetic reasoning

We introduce mathematical programming and atomic decomposition as the basic modal (T-Box) inference techniques for a large class of modal and description logics. The class of description logics suitable for the proposed methods is strong on the arithmetical side. In particular there may be complex arithmetical conditions on sets of accessible worlds (role fillers). The atomic decomposition technique can deal with set constructors for modal parameters (role terms) and parameter (role) hierarchies specified in full propositional logic. Besides the standard modal operators, a number of other constructors can be added in a relatively straightforward way. Examples are graded modalities (qualified number restrictions) and also generalized quantifiers like “most”, “ n%”, “more” and “many”.

Becoming incrementally reactive: on-line learning of an evolving decision tree array for robot navigation

This paper proposes a novel hierarchical multi-layer decision tree for representing reactive robot navigation knowledge. In this representation, the perception space is decomposed into a hierarchical set of worlds reflecting environments which are homogeneous in nature and which vary in complexity in an ordered manner. Each world is used to produce a corresponding decision tree which is trained incrementally. The instantaneous perception of the robot is used to select an appropriate rule from the decision tree and a sequence of rule activations form the complete trajectory. The ability to keep the knowledge complexity manageable and under control is an important aspect of the technique.

Propositions, Numbers, And The Problem Of Arbitrary Identification

Those inclined to believe in the existence of propositions as traditionally conceived might seek to reduce them to some other type of entity. However, parsimonious propositionalists of this type are confronted with a choice of competing candidates – for example, sets of possible worlds, and various neo-Russellian and neo-Fregean constructions. It is argued that this choice is an arbitrary one, and that it closely resembles the type of problematic choice that, as Benacerraf pointed out, bedevils the attempt to reduce numbers to sets – should the number 2 be identified with the set O or with the set O, O? An “argument from arbitrary identification” is formulated with the conclusion that propositions (and perhaps numbers) cannot be reduced away. Various responses to this argument are considered, but ultimately rejected. The paper concludes that the argument is sound: propositions, at least, are sui generis entities.

Mixed-integer column generation algorithms and the probabilistic maximum satisfiability problem

The column generation approach to large-scale linear programming is extended to the mixed-integer case. Two general algorithms, a dual and a primal one, are presented. Both involve finding k -best solutions to combinatorial optimization subproblems. Algorithms for these subproblems must be tailored to each specific application. Their use is illustrated by applying them to a new combinatorial optimization problem with applications in Artificial Intelligence: Probabilistic Maximum Satisfiability. This problem is defined as follows: consider a set of logical sentences together with probabilities that they are true, assume this set of sentences is not satisfiable in the probabilistic sense, i.e., there is no probability distribution on the set of possible worlds (truth assignments to the sentences corresponding to at least one truth assignment to the logical variables they contain) such that for each sentence the sum of probabilities of the possible worlds in which it is true is equal to its probability of being true; determine a minimum set of sentences to be deleted in order to make the remaining set of sentences satisfiable. Computational experience with both algorithms is reported on.

Information and Content

I take it that in his contribution, Stalnaker is arguing for the claim that all there is to is nonconceptual content, and that he reaches his conclusion basically from two premisses: the first premiss says that all there is to is content, and the second says that content' is nonconceptual content. So then, in my comments I will focus on the central notion of informational with the hope that the discussion of them will help me to clarify this key notion in Stalnaker's approach to content. Informational content, we are told, is individuated by truth conditions, propositions as sets of possible worlds or possibilities (in the metaphysical sense of alternative possibilities.) This is, as Stalnaker stresses, the most coarse-grained or minimal sense of content you can get if you endorse the very reasonable assumption that, whatever turns out to be, if something or an state of something has then this something has truth (or, in a somewhat more neutral fashion, conditions of satisfaction) and it has them in virtue of having content. What Stalnaker wants to argue for, then, is that this minimal sense of is all there is to the notion, including mental content. Informational is an abstract object, in particular it abstracts away from the means of representation, whether linguistic

Ways Things Can't Be

Paraconsistent logics are often semantically motivated by considering "impossible worlds." Lewis, in "Logic for equivocators," has shown how we can understand paraconsistent logics by attributing equivocation of meanings to inconsistent believers. In this paper I show that we can understand paraconsistent logics without attributing such equivocation. Impossible worlds are simply sets of possible worlds, and inconsistent believers (inconsistently) believe that things are like each of the worlds in the set. I show that this account gives a sound and complete semantics for Priest's paraconsistent logic LP, which uses materials any modal logician has at hand.

Defining Relative Likelihood in Partially-Ordered Preferential Structures

Starting with a likelihood or preference order on worlds, we extend it to a likelihood ordering on sets of worlds in a natural way, and examine the resulting logic. Lewis earlier considered such a notion of relative likelihood in the context of studying counterfactuals, but he assumed a total preference order on worlds. Complications arise when examining partial orders that are not present for total orders. There are subtleties involving the exact approach to lifting the order on worlds to an order on sets of worlds. In addition, the axiomatization of the logic of relative likelihood in the case of partial orders gives insight into the connection between relative likelihood and default reasoning.

Defining relative likelihood in partially-ordered preferential structures

Starting with a likelihood or preference order on worlds, we extend it to a likelihood ordering on sets of worlds in a natural way, and examine the resulting logic. Lewis earlier considered such a ...

A modal account of similarity-based reasoning

One of the goals of a variety of approximate reasoning models is to cope with inference patterns more flexible than those of classical reasoning. Among them, similarity-based reasoning aims at modeling notions of resemblance or proximity among propositions and consequence relations which make sense in such a setting. One way of proceeding is to equip the set of interpretations or possible worlds with a similarity relation S, that is, a reflexive, symmetric, and t-norm-transitive fuzzy relation. We explore a modal approach to similarity-based reasoning, and we define three multi-modal systems with similarity-based Kripke model semantics. A similarity-based Kripke model is a structure 〈W, S, ‖ ‖〉, in which W is the set of possible worlds, ‖ ‖ represents an assignment of possible worlds to atomic formulas, and S is a similarity function S: W × W → G, where G is a subset of the unit interval [0,1] such that 0,1 ϵ G. We provide soundness and completeness results for these systems with respect to some classes of the above structures.

Kripke models and the (in)equational logic of the second-order λ-calculus

We define a new class of Kripke structures for the second-order λ-calculus, and investigate the soundness and completeness of some proof systems for proving inequalities (rewrite rules) as well as equations. The Kripke structures under consideration are equipped with preorders that correspond to an abstract form of reduction, and they are not necessarily extensional. A novelty of our approach is that we define these structures directly as functors A: W → Preor equipped with certain natural transformations corresponding to application and abstraction (where W is a preorder, the set of worlds, and Preor is the category of preorders). We make use of an explicit construction of the exponential of functors in the Cartesian-closed category Preor W , and we also define a kind of exponential Π Φ ( A s ) s ϵ T to take care of type abstraction. However, we strive for simplicity, and we only use very elementary categorical concepts. Consequently, we believe that the models described in this paper are more palatable than abstract categorical models which require much more sophisticated machinery (and are not models of rewrite rules anyway). We obtain soundness and completeness theorems that generalize some results of Mitchell and Moggi to the second-order λ-calculus, and to sets of inequalities (rewrite rules).

Propositional Quantification in the Topological Semantics for S4

Fine and Kripke extended S5, S4, S4.2 and such to produce propositionally quantified systems ${\bf S5 \pi +}$, ${\bf S4 \pi +}$, ${\bf S4.2 \pi +}$: given a Kripke frame, the quantifiers range over all the sets of possible worlds. ${\bf S5 \pi +}$ is decidable and, as Fine and Kripke showed, many of the other systems are recursively isomorphic to second-order logic. In the present paper I consider the propositionally quantified system that arises from the topological semantics for S4, rather than from the Kripke semantics. The topological system, which I dub ${\bf S4.2 \pi t}$, is strictly weaker than its Kripkean counterpart. I prove here that second-order arithmetic can be recursively embedded in ${\bf S4.2 \pi t}$. In the course of the investigation, I also sketch a proof of Fine's and Kripke's results that the Kripkean system ${\bf S4 \pi +}$ is recursively isomorphic to second-order logic.

Brandom on Representation and Inference

It will take repeated engagements to come to terms with Brandom's huge, cohesive, quirky, and brilliant book. Here I can make only a first pass at one of its architectonic themes. The book is organized around an opposition between two explanatory strategies-representationalism, which Brandom says has been dominant since the Enlightenment (p. xvi), and inferentialism, the complementary approach Brandom adopts. This gives Brandom's story the attractions of bucking a trend. More seriously, he motivates his approach, to a large extent, by its competitor's flaws. The fundamental insight here is that semantics must answer to pragmatics (p. 83). Semantic terms are not intelligible apart from how they pull their weight in enabling us to make sense of what speakers do, in the way we make sense of rational behaviour in general. And inference is a central element in the framework of rational requirements in which we place linguistic behaviour when we make sense of it in that way. Now representationalism, as Brandom conceives it, holds that concepts of the representational directedness of language towards the extra-linguistic world can confer a self-standing intelligibility on the idea of, e.g., the content of an assertion, in advance of our needing to advert to inferential relations between such contents. Consider, e.g., a theorist who undertakes to capture contents of potential assertions by associating sentences with sets of possible worlds, and then to capture inference-warranting relations between such contents in terms of inclusion relations between sets of possible worlds. By using the concept of assertion as he does, this theorist conforms to the letter of the principle that semantics must answer to pragmatics. But suppose he does not mind its seeming an afterthought that the world-directed contents he supposedly brings into view as contents of assertions stand in inferential relations to one another. Since inference is central to the framework in which we make rational sense of linguistic behaviour, such a theorist violates the spirit of the principle. We

A New Semantics for Positive Modal Logic

The paper provides a new semantics for positive modal logic using Kripke frames having a quasi ordering $\leq$ on the set of possible worlds and an accessibility relation $R$ connected to the quasi ordering by the conditions (1) that the composition of $\leq$ with $R$ is included in the composition of $R$ with $\leq$ and (2) the analogous for the inverse of $\leq$ and $R$. This semantics has an advantage over the one used by Dunn in "Positive modal logic," Studia Logica (1995) and works fine for extensions of the minimal system of normal positive modal logic.

Local possibilistic logic

ABSTRACT Possibilistic states of information are fuzzy sets of possible worlds. They constitute a complete lattice, which can be endowed with a monoidal operation (a t-norm) to produce a quantal. An algebraic semantics is presented which links possibilistic formulae with information states, and gives a natural interpretation of logical connectives as operations on fuzzy sets. Due to the quantal structure of information states, we obtain a system which shares several features with (exponential-free) intuitionistic linear logic. Soundness and completeness are proved, parametrically on the choice of the t-norm operation.

Probabilistic conflicts in a search algorithm for estimating posterior probabilities in Bayesian networks

This paper presents a search algorithm for estimating posterior probabilities in discrete Bayesian networks. It shows how conflicts (as used in consistency-based diagnosis) can be adapted to speed up the search. This algorithm is especially suited to the case where there are skewed distributions, although nothing about the algorithm or the definitions depends on skewness of distributions. The general idea is to forward simulate the network, based on the “normal” values for each variable (the value with high probability given its parents). When a predicted value is at odds with the observations, we analyse which variables were responsible for the expectation failure—these form a conflict—and continue forward simulation considering different values for these variables. This results in a set of possible worlds from which posterior probabilities—together with error bounds—can be derived. Empirical results with Bayesian networks having tens of thousands of nodes are presented.

Believing in Language

We propose that the generalizations of linguistic theory serve to ascribe beliefs to humans. Ordinary speakers would explicitly (and sincerely) deny having these rather esoteric beliefs about language—e.g., the belief that an anaphor must be bound in its governing category. Such ascriptions can also seem problematic in light of certain theoretical considerations having to do with concept possession, revisability, and so on. Nonetheless, we argue that ordinary speakers believe the propositions expressed by certain sentences of linguistic theory, and that linguistics can therefore teach us something about belief as well as language. Rather than insisting that ordinary speakers lack the linguistic beliefs in question, philosophers should try to show how these empirically motivated belief ascriptions can be correct. We argue that Stalnaker's (1984) “pragmatic” account—according to which beliefs are dispositions, and propositions are sets of possible worlds—does just this. Moreover, our construal of explanation in linguistics motivates (and helps provide) responses to two difficulties for the pragmatic account of belief: the phenomenon of opacity, and the so-called problem of deduction.

Galileo Gazes at Jupiter and Its Moons

With the public still dazzled by the possibility of life on Mars, space scientists announced new views of another set of alien worlds: Jupiter and its moons. As it swung through the Jovian system, the Galileo spacecraft took new images of Jupiter and another moon, Europa. The data suggest that a watery ocean—one of the criteria for life as we know it—may lie beneath the cracked, icy shell of Europa's surface. And on Jupiter itself, Galileo snapped images of Earth-like thunderclouds, shoring up the idea that thunderstorms sweep across the planet.

The diversity of soil communities, the ?poor man's tropical rainforest?

This paper reviews the various factors that facilitate the high biodiversity of soil communities, concentrating on soil animals. It considers the problems facing soil ecologists in the study of soil communities and identifies the important role such communities play in terrestrial ecosystems. The review also considers diversity and abundance patterns. A range of factors are identified that may contribute to the biodiversity of soil and their role is reviewed. These include diversity of food resources and trophic specialization, habitat favourableness, habitat heterogeneity in space and time, scale and spatial extent of the habitat, niche dynamics and resource partitioning, productivity, disturbance and aggregation. Biodiversity of soil organisms appears high, largely attributable to the nested set of ecological worlds in the soil — the relationship between the range of size groupings of soil organisms relative to the spatial heterogeneity perceived by these various groups — that provide a large ‘area for life” for the micro- and mesofauna. The role of aggregation and how it relates to the spatial scale under consideration and to species interactions amongst soil animals is largely unknown at present. The role of disturbance is equivocal and man's activities more often than not seem to lead to a reduced biodiversity of soil communities. This paper also identifies areas where further work is desirable to improve our understanding of the structure and functioning of soil communities.

Possible Worlds in the Modal Interpretation

An outline for a modal interpretation in terms of possible worlds is presented. The so-called Schmidt histories are taken to correspond to the physically possible worlds. The decoherence function defined in the histories formulation of quantum theory is taken to prescribe a non-classical probability measure over the set of the possible worlds. This is shown to yield dynamics in the form of transition probabilities for occurrent events in each world. The role of the consistency condition is discussed.

Assignment methods for incidence calculus

Incidence calculus is a mechanism for probabilistic reasoning in which sets of possible worlds, called incidences, are associated with axioms, and probabilities are then associated with these sets. Inference rules are used to deduce bounds on the incidence of formulae which are not axioms, and bounds for the probability of such a formula can then be obtained. In practice an assignment of probabilities directly to axioms may be given, and it is then necessary to find an assignment of incidence which will reproduce these probabilities. We show that this task of assigning incidences can be viewed as a tree searching problem, and two techniques for performing this research are discussed. One of these is a new proposal involving a depth first search, while the other incorporates a random element. A Prolog implementation of these methods has been developed. The two approaches are compared for efficiency and the significance of their results are discussed. Finally we discuss a new proposal for applying techniques from linear programming to incidence calculus.