Leon Horsten

On the one hand, the concept of truth is a major research subject in analytic philosophy. On the other hand, mathematical logicians have developed sophisticated logical theories of truth and the paradoxes. Recent developments in logical theories of the semantical paradoxes are highly relevant for philosophical research on the notion of truth. And conversely, philosophical guidance is necessary for the development of logical theories of truth and the paradoxes. From this perspective, this volume intends to reflect and promote deeper interaction and collaboration between philosophers and logicians investigating the concept of truth than has existed so far.Aside from an extended introductory overview of recent work in the theory of truth, the volume consists of articles by leading philosophers and logicians on subjects and debates that are situated on the interface between logical and philosophical theories of truth. The volume is intended for graduate students in philosophy and in logic who want an introduction to contemporary research in this area, as well as for professional philosophers and logicians

It is often alleged that Cantor’s views about how the set theoretic universe as a whole should be considered are fundamentally unclear. In this article we argue that Cantor’s views on this subject, at least up until around 1896, are relatively clear, coherent, and interesting. We then go on to argue that Cantor’s views about the set theoretic universe as a whole have implications for theology that have hitherto not been sufficiently recognised. However, the theological implications in question, at least as articulated here, would not have satisfied Cantor himself.

We propose an alternative approach to probability theory closely related to the framework of numerosity theory: non-Archimedean probability (NAP). In our approach, unlike in classical probability theory, all subsets of an infinite sample space are measurable and only the empty set gets assigned probability zero (in other words: the probability functions are regular). We use a non-Archimedean field as the range of the probability function. As a result, the property of countable additivity in Kolmogorov’s axiomatization of probability is replaced by a different type of infinite additivity.

In this paper, a general perspective on criteria of identity of kinds of objects is developed. The question of the admissibility of impredicative or circular identity criteria is investigated in the light of the view that is articulated. It is argued that in and of itself impredicativity does not constitute sufficient grounds for rejecting a putative identity criterion. The view that is presented is applied to Davidson’s criterion of identity for events and to the structuralist criterion of identity of places in a structure.

In his Gibbs lecture, Gödel argued for the thesis that either the human mind is not a Turing machine, or there exist absolutely undecidable mathematical propositions. He believed that this disjunction can be deduced with mathematical certainty from certain results in mathematical logic. He thought that his disjunctive thesis is of great philosophical importance. First, Gödel's argument for his disjunctive thesis is discussed. It is argued that thisargument contains an ambiguity. But when it is made precise in a certain way, the argument is seen to be ultimately inescapable. Second, the disjuncts of the thesis are considered in their own right. It is argued that there are today no convincing arguments in favor of the first disjunct, nor are there at present convincing arguments against it. But there do exist today strong reasons to believe the second disjunct to be true, although these reasons do not establish the disjunct with anything like mathematical certainty

This paper investigates the role of pictures in mathematics in the particular case of Cayley graphs—the graphic representations of groups. I shall argue that their principal function in that theory—to provide insight into the abstract structure of groups—is performed employing their visual aspect. I suggest that the application of a visual graph theory in the purely non-visual theory of groups resulted in a new effective approach in which pictures have an essential role. Cayley graphs were initially developed as exact mathematical constructions. Therefore, they are legitimate components of the theory and the pictures of Cayley graphs are a part of practical mathematical procedures.

We investigate and classify the notion of final derivability of two basic inconsistency-adaptive logics. Specifically, the maximal complexity of the set of final consequences of decidable sets of premises formulated in the language of propositional logic is described. Our results show that taking the consequences of a decidable propositional theory is a complicated operation. The set of final consequences according to either the Reliability Calculus or the Minimal Abnormality Calculus of a decidable propositional premise set is in general undecidable, and can be -complete. These classifications are exact. For first order theories even finite sets of premises can generate such consequence sets in either calculus.

This article discusses Peter Galison's views on the structure and evolution of experimental and instrumental cultures in 20th century particle physics, which are unfolded in his recent book Image and Logic. A Material Culture of Microphysics. First a description is given of the uncomfortable predicament in which the Kuhnian tradition finds itself in the past two decades. It is then explained how Galison distinguishes a layered structure in the practice of modern particle physics. Physics as a practice consists of three relatively autonomous levels: the experimental level, the instrumental level and the theoretical level. This paper focuses mainly (but not exclusively) on the instrumental level in particle physics. Subsequently an analysis is given of Galison's metaphor of the wall, which Galison offers as an alternative to the Kuhnian image of paradigms and scientific revolutions. Galison's model is also compared with Laudan's reticulational model of the evolution of scientific research traditions. To conclude, an analysis is made of Galison's attempt to solve the incommensurability problem

Given any finite graph, which transitive graphs approximate it most closely and how fast can we find them? The answer to this question depends on the concept of “closest approximation” involved. In [8,9] a qualitative concept of best approximation is formulated. Roughly, a qualitatively best transitive approximation of a graph is a transitive graph which cannot be “improved” without also going against the original graph. A quantitative concept of best approximation goes back at least to [10]. A quantitatively best transitive approximation is a transitive graph that makes the minimal number of mistakes against the original graph. In other words, the sum of the edges that are removed from and are added to the original graph is minimal.