The workshop on Mathematical Applications in Philosophy took place at the Banff International Research Station, in Banff National Park, Canada, from Sunday, February 18 to Friday, February 23, 2007.

The workshop was organized by Aldo Antonelli (University of California, Irvine), Alasdair Urquhart (University of Toronto), and Richard Zach (University of Calgary).

We are planning to collect selected papers form the workshop in a special issue of the Journal of Philosophical Logic.

• Workshop program
• List of participants

Contributions

• Steve Awodey and Kohei Kishida, Topological semantics for first-order modal logic [extended abstract]
• JC Beall and Michael Glanzberg. Truth and paradox. [slides]
• Kenny Easwaran, Dominance-based decision theory. [slides]
• Harvey Friedman, Concept calculus. [slides]
• Ray E. Jennings and Dorian X. Nicholson. An axiomatization of family resemblance. [paper]
• Hannes Leitgeb, Applications of mathematics in philosophy: four case studies. [slides]
• Eleni Kalyvianaki, Factual content in an algorithmic theory of meaning. [slides]
• Yiannis Moschovakis, Synonymy. [slides]
• Greg Restall, Modal models for Bradwardine’s truth [slides]
• Stewart Shapiro and Gabriel Uzquiano. Ineffability and reflection. [slides]
• Wilfried Sieg, Church without dogma: axioms for computability. [paper] [slides]
• Byeong-Uk Yi, Is logic axiomatizable? [handout]

Goals of the Workshop

The workshop on “Mathematical Methods in Philosophy” will provide a forum for presentation of recent work in logic and probability theory with applications in philosophy and for collaboration and scientific exchange between mathematicians, philosophers, computer scientists, and theoretical linguists working in the intersection of these fields. The meetings of the Association of Symbolic Logic often feature special sessions for selected topics in philosophical logic. Some of the relevant topics are covered in specialized conferences and series of conferences (e.g., the “Formal Epistemology” and the “Advances in Modal Logic” workshops series). These meetings are of course useful and important for specialists in the respective areas, and provide opportunities for interchange between philosophers and logicians working in other fields, e.g., in mathematics or computer science. There is, however, a need for a venue specifically aimed at “theorem-proving philosophers” working on applications of formal and mathematical methods in philosophy. The Banff workshop will help galvanize this group of logicians and philosophers who share a common view on the place of mathematical and formal methods in philosophy.

The specific topics which the workshop will cover are the following:

• Philosophical logics. This includes logical systems such as logics of possibility and necessity, of time, of knowledge and belief, of permission and obligation. This area is unified by its methods (e.g., relational semantics, first introduced by philosophers Saul Kripke and Jaakko Hintikka in the 1950s, algebraic methods, proof theory), but it has diverse applications in philosophy. For instance, logics of possibility and time are mainly useful in metaphysics whereas logics of knowledge and belief are of interest to epistemology. However, the methods employed in the study of these logics is very similar. Other related logics which have important applications in philosophy are many-valued logics, intuitionistic logic, paraconsistent and relevance logics.
• Proof theory. The investigation of the structure of formal proofs has its origins in the philosophy of mathematics, but is more broadly applicable to the philosophy of logic and the philosophy of language. Two important examples of applications of proof theory in these areas are the recent advances in the study of consistency of subsystems of second-order logic and of the proof theory of substructural logics, which in turn is related to the applications of these systems mentioned above.
• Formal theories of truth and paradox. The nature of truth is a central topic in metaphysics and philosophy of logic, and work on truth is closely connected to epistemology and philosophy of language. Significant advances have been achieved over the last 30 years in formal theories of truth, and there are close connections between philosophical work on truth and model theory (especially of arithmetic).
• Formal epistemology. Formal epistemology is an emerging field of research in philosophy, encompassing formal approaches to ampliative inference (including inductive logic), game theory, decision theory, computational learning theory, and the foundations of probability theory.
• Set theory and topology in metaphysics. Set theory has always had a close connection with mereology, the theory of parts and wholes, and topology has also been fruitfully applied in metaphysics.