A conditional logic for abduction

We propose a logic of abduction that (i) provides an appropriate formalization of the explanatory conditional, and that (ii) captures the defeasible nature of abductive inference. For (i), we argue that explanatory conditionals are non-classical, and rely on Brian Chellas’s work on conditional logics for providing an alternative formalization of the explanatory conditional. For (ii), we make use of the adaptive logics framework for modeling defeasible reasoning. We show how our proposal allows for a more natural reading of explanatory relations, and how it overcomes problems faced by other systems in the literature.     

Completeness and Categoricity,Part II: Twentieth-Century Metalogic to Twenty-first-Century Semantics

This paper is the second in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics so as to shed new light on the relevant strengths and limits of higher-order logic.     

Reasoning About Games

A mixture of propositional dynamic logic and epistemic logic that we call PDL + E is used to give a formalization of Artemov??s knowledge based reasoning approach to game theory, (KBR), [4, 5]. Epistemic states of players are represented explicitly and reasoned about formally. We give a detailed analysis of the Centipede game using both proof theoretic and semantic machinery. This helps make the case that PDL + E can be a useful basis for the logical investigation of game theory.     

Exhibiting interpretational and representational validity

A natural language argument may be valid in at least two nonequivalent senses: it may be interpretationally or representationally valid (Etchemendy in The concept of logical consequence. Harvard University Press, Cambridge, 1990). Interpretational and representational validity can both be formally exhibited by classical first-order logic. However, as these two notions of informal validity differ extensionally and first-order logic fixes one determinate extension for the notion of formal validity (or consequence), some arguments must be formalized by unrelated nonequivalent formalizations in order to formally account for their interpretational or representational validity, respectively. As a consequence, arguments must be formalized subject to different criteria of adequate formalization depending on which variant of informal validity is to be revealed. This paper develops different criteria that formalizations of an argument have to satisfy in order to exhibit the latter’s interpretational or representational validity.     

两水平研究中单维测验信度的估计

在心理、教育和管理等研究领域中,经常会碰到两水平（两层）的数据结构,如学生嵌套在班级中,员工嵌套在企业中。在两水平研究中,被试通常不是独立的,如果直接用单水平信度公式进行估计,会高估测验信度。文献上已有研究讨论如何更准确地估计两水平研究中单维测验的信度。本研究指出了现有的估计公式的不足之处,用两水平验证性因子分析推导出一个新的信度公式,举例演示如何计算,并给出简单的计算程序。     

Probabilistic argumentation systems: A new way to combine logic with probability

Probability is usually closely related to Boolean structures, i.e., Boolean algebras or propositional logic. Here we show, how probability can be combined with non-Boolean structures, and in particular non-Boolean logics. The basic idea is to describe uncertainty by (Boolean) assumptions, which may or may not be valid. The uncertain information depends then on these uncertain assumptions, scenarios or interpretations. We propose to describe information in information systems, as introduced by Scott into domain theory. This captures a wide range of systems of practical importance such as many propositional logics, first order logic, systems of linear equations, inequalities, etc. It covers thus both symbolic as well as numerical systems. Assumption-based reasoning allows then to deduce supporting arguments for hypotheses. A probability structure imposed on the assumptions permits to quantify the reliability of these supporting arguments and thus to introduce degrees of support for hypotheses. Information systems and related information algebras are formally introduced and studied in this paper as the basic structures for assumption-based reasoning. The probability structure is then formally represented by random variables with values in information algebras. Since these are in general non-Boolean structures some care must be exercised in order to introduce these random variables. It is shown that this theory leads to an extension of Dempster–Shafer theory of evidence and that information algebras provide in fact a natural frame for this theory.     

Impact of emotional content on discourse production in patients with unilateral brain damage.

A picture story task was developed to examine expression of emotion via the verbal/lexical channel. The task elicited discourse with either emotional, visual-spatial, or neutral content and was administered to right brain-damaged (RBD), left brain-damaged (LBD), and normal control (NC) right-handed adults. Subjects were matched for gender, age, education, and occupational status. The brain-damaged groups were matched for months post-CVA onset and were similar with respect to intrahemispheric site of lesion. While the number of words produced was equivalent for each of the subject groups, the RBDs and LBDs expressed quantitatively less content than did the NCs. When content differences were examined within each subject group, there were no differences for LBDs and NCs, but the RBDs showed a selective deficit when producing emotional content. This finding suggests a special role for the right hemisphere in the production of emotional content in verbal discourse.     

Kuhn’s notion of scientific progress: “Reduction” between incommensurable theories in a rigid structuralist framework

In the last two sections of Structure, Thomas Kuhn first develops his famous threefold conception of the incommensurability of scientific paradigms and, subsequently, a conception of scientific progress as growth of empirical strength. The latter conception seems to be at odds with the former in that semantic incommensurability appears to imply the existence of situations where scientific progress in Kuhns sense can no longer exist. In contrast to this seeming inconsistency of Kuhns conception, we will try to show in this study that the semantic incommensurability of scientific terms appears to be fully compatible with scientific progress. Our argumentation is based on an improved version of the formalization of Kuhns conception as developed in the 1970s by Joseph Sneed and Wolfgang Stegmüller: In order to be comparable, incommensurable theories need the specification of relations that refer to the concrete ontologies of these theories and involve certain truth claims. The original structuralist account of reduction fails to provide such relations, because (1) it is too structural and (2) it is too wide. Moreover, the original structuralist account also fails to cover important cases of incommensurable theories in being too restrictive for them. In this paper, we develop an improved notion of “reduction” that allows us to avoid these shortcomings by means of a more flexible device for the formalization of (partially reductive) relations between theories. For that purpose, we use a framework of rigid logic, i.e., logic that is based on a fixed collection of objects.     

Łukasiewicz logic and the foundations of measurement

The logic of inexactness, presented in this paper, is a version of the Łukasiewicz logic with predicates valued in [0, ∞). We axiomatize multi-valued models of equality and ordering in this logic guaranteeing their imbeddibility in the real line. Our axioms of equality and ordering, when interpreted as axioms of proximity and dominance, can be applied to the foundations of measurement (especially in the social sciences). In two-valued logic they provide theories of ratio scale measurement. In multivalued logic they enable us to treat formally errors arising in nominal and ordinal measurements.     

Combinatorial Bitstring Semantics for Arbitrary Logical Fragments

Logical geometry systematically studies Aristotelian diagrams, such as the classical square of oppositions and its extensions. These investigations rely heavily on the use of bitstrings, which are compact combinatorial representations of formulas that allow us to quickly determine their Aristotelian relations. However, because of their general nature, bitstrings can be applied to a wide variety of topics in philosophical logic beyond those of logical geometry. Hence, the main aim of this paper is to present a systematic technique for assigning bitstrings to arbitrary finite fragments of formulas in arbitrary logical systems, and to study the logical and combinatorial properties of this technique. It is based on the partition of logical space that is induced by a given fragment, and sheds new light on a number of interesting issues, such as the logic-dependence of the Aristotelian relations and the subtle interplay between the Aristotelian and Boolean structure of logical fragments. Finally, the bitstring technique also allows us to systematically analyze fragments from contemporary logical systems, such as public announcement logic, which could not be done before.     

A logic of comparative obligation

Normal systems of modal logic, interpreted as deontic logics, are unsuitable for a logic of conflicting obligations. By using modal operators based on a more complex semantics, however, we can provide for conflicting obligations, as in [9], which is formally similar to a fragment of the logic of ability later given in [2], Having gone that far, we may find it desirable to be able to express and consider claims about the comparative strengths, or degrees of urgency, of the conflicting obligations under which we stand. This paper, building on the formalism of the logic of ability in [2], provides a complete and decidable system for such a language.     

Error analysis of digital filters using HOL theorem proving

When a digital filter is realized with floating-point or fixed-point arithmetics, errors and constraints due to finite word length are unavoidable. In this paper, we show how these errors can be mechanically analysed using the HOL theorem prover. We first model the ideal real filter specification and the corresponding floating-point and fixed-point implementations as predicates in higher-order logic. We use valuation functions to find the real values of the floating-point and fixed-point filter outputs and define the error as the difference between these values and the corresponding output of the ideal real specification. Fundamental analysis lemmas have been established to derive expressions for the accumulation of roundoff error in parametric Lth-order digital filters, for each of the three canonical forms of realization: direct, parallel, and cascade. The HOL formalization and proofs are found to be in a good agreement with existing theoretical paper-and-pencil counterparts.     

An Operational Logic of Proofs with Positive and Negative Information

The logic of proofs was introduced by Artemov in order to analize the formalization of the concept of proof rather than the concept of provability. In this context, some operations on proofs play a very important role. In this paper, we investigate some very natural operations, paying attention not only to positive information, but also to negative information (i.e. information saying that something cannot be a proof). We give a formalization for a fragment of such a logic of proofs, and we prove that our fragment is complete and decidable.     

Deontic action logic, atomic boolean algebras and fault-tolerance

We introduce a deontic action logic and its axiomatization. This logic has some useful properties (soundness, completeness, compactness and decidability), extending the properties usually associated with such logics. Though the propositional version of the logic is quite expressive, we augment it with temporal operators, and we outline an axiomatic system for this more expressive framework. An important characteristic of this deontic action logic is that we use boolean combinators on actions, and, because of finiteness restrictions, the generated boolean algebra is atomic, which is a crucial point in proving the completeness of the axiomatic system. As our main goal is to use this logic for reasoning about fault-tolerant systems, we provide a complete example of a simple application, with an attempt at formalization of some concepts usually associated with fault-tolerance.     

Synchronization of Logics

Motivated by applications in software engineering, we propose two forms of combination of logics: synchronization on formulae and synchronization on models. We start by reviewing satisfaction systems, consequence systems, one-step derivation systems and theory spaces, as well as their functorial relationships. We define the synchronization on formulae of two consequence systems and provide a categorial characterization of the construction. For illustration we consider the synchronization of linear temporal logic and equational logic. We define the synchronization on models of two satisfaction systems and provide a categorial characterization of the construction. We illustrate the technique in two cases: linear temporal logic versus equational logic; and linear temporal logic versus branching temporal logic. Finally, we lift the synchronization on formulae to the category of logics over consequences systems.     

A Nonmonotonic Modal Formalization of the Logic of Acceptance and Rejection

The problems we deal with concern reasoning about incomplete knowledge. Knowledge is understood as ability of an ideal rational agent to make decisions about pieces of information. The formalisms we are particularly interested in are Moore's autoepistemic logic (AEL) and its variant, the logic of acceptance and rejection (AEL2). It is well-known that AEL may be seen as the nonmonotonic KD45 modal logic. The aim is to give an appropriate modal formalization for AEL2.