Logical Methods in the Humanities Workshop Abstracts Autumn 2005

The logical background of juridical reasoning systems is determined by the tension between
a. arguments should be demonstrably sound
b. decisions have to be achieved within limited time and space.

In this lecture we develop a proof theory of analogical reasoning starting from the breathtaking summation of 1/n^2, an analogical conclusion which made Euler famous.
We demonstrate, how minimalist legal systems (as English Common law)/ maximalist legal systems (as Austrian,German and in general Latin law)make use of analogies to extend/restrict deductions in the sense of mathematical logic. We reduce the well-known specific deduction rules in German law (e.g. E CONTRARIO)to simple forms of analogical reasoning.

It has always been conceded that the 1910 Principia does not deny the existence of classes, but claims only that the theory it advances can be developed so that the apparent commitment to classes is eliminable by the method of contextual analysis. I argue that the work has an illuminating reconstruction within a framework which admits the existence of classes. To preserve many of Principia's most distinctive theses, it is sufficient to maintain that classes are dependent on or presuppose propositional functions in a way I explain. So reconstructed, the work contains a theory of our knowledge of classes whose elegance has not, to my mind, been adequately appreciated.

In the past few decades, logicians, mathematicians, computer scientists, and philosophers have spent much energy developing libraries of formal mathematical texts: they have designed high-level languages that capture various aspects of the mathematical vernacular and implemented computer programs that process these libraries efficiently. In this talk I discuss these libraries and discuss how they are processed, especially with reference to how proof libraries can help one to learn a body of mathematical knowledge. To illustrate this kind of learning I focus on a particular aspect of the recent proof of G��del's completeness theorem carried out within the MIZAR system by Patrick Braselmann and Peter Koepke; this example will illustrate how a proof environment (in this case, the MizarMode environment for the Emacs text editor) can be used to study a formal exposition of a landmark theorem of mathematics. I close by warning against certain tempting conclusions concerning these formal texts, for example, the conclusion that because they have been checked by a computer they are infallible, and that because they are completely formal they provide a universal standard for settling mathematical and philosophical debates.

References:

In this talk I will discuss a number of themes in contemporary set theory that have their origin in the work of Kurt Goedel. Themes will include: the distinction between intrinsic and extrinsic justifications of new axioms; the program for large cardinal axioms; the status of the continuum hypothesis; inner models for large cardinals; and the question of whether there are set-theoretic statements that are in some sense absolutely undecidable.

I will argue that a cluster of results show that there is a good deal of structure and unity beyond ZFC and that there is a precise sense in which every question of complexity strictly ``below'' that of the Continuum Hypothesis is settled by large cardinal axioms. I will then discuss the various ways in which something new emerges at the level of the Continuum Hypothesis and examine various positions to the effect that it does not admit of a solution. I will close by examining recent results of Hugh Woodin and a scenario for axioms extending ZFC that are as complete for the universe of sets as the axioms of PA are for the natural numbers.