In his remarkable paper Formalism 64, Robinson defends his eponymous position concerning the foundations of mathematics, as follows:Any mention of infinite totalities is literally meaningless.We should act as if infinite totalities really existed. Being the originator of Nonstandard Analysis, it stands to reason that Robinson would have often been faced with the opposing position that ‘some infinite totalities are more meaningful than others’, the textbook example being that of infinitesimals. For instance, Bishop and Connes have made such claims regarding infinitesimals, (...) and Nonstandard Analysis in general, going as far as calling the latter respectively a debasement of meaning and virtual, while accepting as meaningful other infinite totalities and the associated mathematical framework. We shall study the critique of Nonstandard Analysis by Bishop and Connes, and observe that these authors equate ‘meaning’ and ‘computational content’, though their interpretations of said content vary. As we will see, Bishop and Connes claim that the presence of ideal objects in Nonstandard Analysis yields the absence of meaning. We will debunk the Bishop–Connes critique by establishing the contrary, namely that the presence of ideal objects in Nonstandard Analysis yields the ubiquitous presence of computational content. In particular, infinitesimals provide an elegant shorthand for expressing computational content. To this end, we introduce a direct translation between a large class of theorems of Nonstandard Analysis and theorems rich in computational content, similar to the ‘reversals’ from the foundational program Reverse Mathematics. The latter also plays an important role in gauging the scope of this translation. (shrink)

When the likelihood ratio approach is employed for evidential reasoning in law, it is often necessary to employ subjective probabilities, which are probabilities derived from the opinions and judgement of a human. At least three concerns arise from the use of subjective probabilities in legal applications. Firstly, human beliefs concerning probabilities can be vague, ambiguous and inaccurate. Secondly, the impact of this vagueness, ambiguity and inaccuracy on the outcome of a probabilistic analysis is not necessarily fully understood. Thirdly, the provenance (...) of subjective probabilities and the associated potential sources of vagueness, ambiguity and inaccuracy tend to be poorly understood, making it difficult for the outcome of probabilistic reasoning to be explained and validated, which is crucial in legal applications. The former two concerns have been addressed by a wide body of research in AI. The latter, however, has received little attention. This paper presents a novel approach to employ argumentation to reason about probability distributions in probabilistic models. It introduces a range of argumentation schemes and corresponding sets of critical questions for the construction and validation of argument models that define sets of probability distributions. By means of an extended example, the paper demonstrates how the approach, argumentation schemes and critical questions can be employed for the development of models and their validation in legal applications of the likelihood ratio approach to evidential reasoning. (shrink)

We present a relational proof system in the style of dual tableaux for a multimodal propositional logic for order of magnitude qualitative reasoning to deal with relations of negligibility, non-closeness, and distance. This logic enables us to introduce the operation of qualitative sum for some classes of numbers. A relational formalization of the modal logic in question is introduced in this paper, i.e., we show how to construct a relational logic associated with the logic for order-of-magnitude reasoning and its dual (...) tableau system which is a validity checker for the modal logic. For that purpose, we define a validity preserving translation of the modal language into relational language. Then we prove that the system is sound and complete with respect to the relational logic defined as well as with respect to the logic for order of magnitude reasoning. Finally, we show that in fact relational dual tableau does more. It can be used for performing the four major reasoning tasks: verification of validity, proving entailment of a formula from a finite set of formulas, model checking, and verification of satisfaction of a formula in a finite model by a given object. (shrink)

Qualitative Reasoning (QR) is an area of research within Artificial Intelligence that automates reasoning and problem solving about the physical world. QR research aims to deal with representation and reasoning about continuous aspects of entities without the kind of precise quantitative information needed by conventional numerical analysis techniques. Order-of-magnitude Reasoning (OMR) is an approach in QR concerned with the analysis of physical systems in terms of relative magnitudes. In this paper we consider the logic OMR_N for order-of-magnitude reasoning with the (...) bidirectional negligibility relation. It is a multi-modal logic given by a Hilbert-style axiomatization that reflects properties and interactions of two basic accessibility relations (strict linear order and bidirectional negligibility). Although the logic was studied in many papers, nothing was known about its decidability. In the paper we prove decidability of OMR N by showing that the logic has the strong finite model property. (shrink)

Combating the identity problem is crucial and urgent as false identity has become a common denominator of many serious crimes, including mafia trafficking and terrorism. Without correct identification, it is very difficult for law enforcement authority to intervene, or even trace terrorists’ activities. Amongst several identity attributes, personal names are commonly, and effortlessly, falsified or aliased by most criminals. Typical approaches to detecting the use of false identity rely on the similarity measure of textual and other content-based characteristics, which are (...) usually not applicable in the case of highly deceptive, erroneous and unknown descriptions. This barrier can be overcome through analysis of link information displayed by the individual in communication behaviours, financial interactions and social networks. In particular, this paper presents a novel link-based approach that improves existing techniques by integrating multiple link properties in the process of similarity evaluation. It is utilised in a hybrid model that proficiently combines both text-based and link-based measures of examined names to refine the justification of their similarity. This approach is experimentally evaluated against other link-based and text-based techniques, over a terrorist-related dataset, with further generalization to a similar problem occurring in publication databases. The empirical study demonstrates the great potential of this work towards developing an effective identity verification system. (shrink)

We introduce a multimodal logic for order of magnitude reasoning which considers a new logic-based alternative to the notion of closeness, we provide an axiom system and prove its soundness and completeness.