From IF to BI: a tale of dependence and separation

We take a fresh look at the logics of informational dependence and independence of Hintikka and Sandu and Vaananen, and their compositional semantics due to Hodges. We show how Hodges' semantics can be seen as a special case of a general construction, which provides a context for a useful completeness theorem with respect to a wider class of models. We shed some new light on each aspect of the logic. We show that the natural propositional logic carried by the semantics is the logic of Bunched Implications due to Pym and O'Hearn, which combines intuitionistic and multiplicative connectives. This introduces several new connectives not previously considered in logics of informational dependence, but which we show play a very natural role, most notably intuitionistic implication. As regards the quantifiers, we show that their interpretation in the Hodges semantics is forced, in that they are the image under the general construction of the usual Tarski semantics; this implies that they are adjoints to substitution, and hence uniquely determined. As for the dependence predicate, we show that this is definable from a simpler predicate, of constancy or dependence on nothing. This makes essential use of the intuitionistic implication. The Armstrong axioms for functional dependence are then recovered as a standard set of axioms for intuitionistic implication. We also prove a full abstraction result in the style of Hodges, in which the intuitionistic implication plays a very natural r\^ole.Comment: 28 pages, journal versio

Hilbert's Metamathematical Problems and Their Solutions

This dissertation examines several of the problems that Hilbert discovered in the foundations of mathematics, from a metalogical perspective. The problems manifest themselves in four different aspects of Hilbert’s views: (i) Hilbert’s axiomatic approach to the foundations of mathematics; (ii) His response to criticisms of set theory; (iii) His response to intuitionist criticisms of classical mathematics; (iv) Hilbert’s contribution to the specification of the role of logical inference in mathematical reasoning. This dissertation argues that Hilbert’s axiomatic approach was guided primarily by model theoretical concerns. Accordingly, the ultimate aim of his consistency program was to prove the model-theoretical consistency of mathematical theories. It turns out that for the purpose of carrying out such consistency proofs, a suitable modification of the ordinary first-order logic is needed. To effect this modification, independence-friendly logic is needed as the appropriate conceptual framework. It is then shown how the model theoretical consistency of arithmetic can be proved by using IF logic as its basic logic. Hilbert’s other problems, manifesting themselves as aspects (ii), (iii), and (iv)—most notably the problem of the status of the axiom of choice, the problem of the role of the law of excluded middle, and the problem of giving an elementary account of quantification—can likewise be approached by using the resources of IF logic. It is shown that by means of IF logic one can carry out Hilbertian solutions to all these problems. The two major results concerning aspects (ii), (iii) and (iv) are the following: (a) The axiom of choice is a logical principle; (b) The law of excluded middle divides metamathematical methods into elementary and non-elementary ones. It is argued that these results show that IF logic helps to vindicate Hilbert’s nominalist philosophy of mathematics. On the basis of an elementary approach to logic, which enriches the expressive resources of ordinary first-order logic, this dissertation shows how the different problems that Hilbert discovered in the foundations of mathematics can be solved

Empiricism and Philosophy

Though Quine's argument against the analytic-synthetic distinction is widely disputed, one of the major effects of his argument has been to popularise the belief that there is no sharp distinction between science and philosophy. This thesis begins by distinguishing reductive from holistic empiricism, showing why reductive empiricism is false, refuting the major objections to holistic empiricism and stating the limits on human knowledge it implies. Quine's arguments (and some arguments that have been mistakenly attributed to him) from holism against the analytic-synthetic are considered, and while many of them are found wanting one good argument is presented. Holism does not, however, imply that there is no sharp distinction between science and philosophy, and indeed implies that the distinction between scientific and philosophical disputes is perfectly sharp. The grounds upon which philosophical disputes may be resolved are then sought for and deliniated

Empiricism and Philosophy

Though Quine's argument against the analytic-synthetic distinction is widely disputed, one of the major effects of his argument has been to popularise the belief that there is no sharp distinction between science and philosophy. This thesis begins by distinguishing reductive from holistic empiricism, showing why reductive empiricism is false, refuting the major objections to holistic empiricism and stating the limits on human knowledge it implies. Quine's arguments (and some arguments that have been mistakenly attributed to him) from holism against the analytic-synthetic are considered, and while many of them are found wanting one good argument is presented. Holism does not, however, imply that there is no sharp distinction between science and philosophy, and indeed implies that the distinction between scientific and philosophical disputes is perfectly sharp. The grounds upon which philosophical disputes may be resolved are then sought for and deliniated

Saliency Map for Visual Perception

Human and other primates move their eyes to select visual information from the scene, psycho-visual experiments (Constantinidis, 2005) suggest that attention is directed to visually salient locations in the image. This allows human beings to bring the fovea onto the relevant parts of the image, to interpret complex scenes in real time. In visual perception, an important result was the discovery of a limited set of visual properties (called pre attentive), detected in the first 200-300 milliseconds of observation of a scene, by the low-level visual system. In last decades many progresses have been made into research of visual perception by analyzing both bottom up (stimulus driven) and top down (task dependent) processes involved in human attention. Visual Saliency deals with identifying fixation points that a human viewer would focus on the first seconds of the observation of a scene