# Mathematical Methods in Philosophy (07w5060)

Arriving in Banff, Alberta Sunday, February 18 and departing Friday February 23, 2007

## Organizers

Aldo Antonelli (University of California-Irvine)

Alasdair Urquhart (University of Toronto)

Richard Zach (University of Calgary)

## Objectives

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.

All of these specific applications of mathematical methods in philosophy have proved to yield important results, and each one of them is very much at the forefront of current formal work in philosophy. A workshop which will bring these methods and results into sharper relief and provide an opportunity for researchers to learn about related approaches to similar problems is much-needed and timely, in particular because existing workshops emphasize the areas of application, and not the mathematical methods applied (e.g., someone specializing in modal logic may attend mainly metaphysics conferences, and therefore will not have regular opportunities to interact with researchers applying modal logics to epistemological problems).

One particular aim of the workshop will then be to provide the participants with a sense of what the range of topics, the state of current research, the interconnections, and the important trends are in philosophical logic and related areas. To this end, the organizers will invite selected participants to give survey talks about their area of expertise rather than specialized talks. These surveys will provide an overview of the development of the field in the last 20-50 years, of the current state of the art, of the main open problems, and of anticipated future trends and developments. The emphasis in all this will be on the connection to philosophy, although connections to mathematics, computer science, and other fields will of course be touched upon. The texts of these talks will be collected and published, most likely as a special issue of the Journal of Philosophical Logic.

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.

All of these specific applications of mathematical methods in philosophy have proved to yield important results, and each one of them is very much at the forefront of current formal work in philosophy. A workshop which will bring these methods and results into sharper relief and provide an opportunity for researchers to learn about related approaches to similar problems is much-needed and timely, in particular because existing workshops emphasize the areas of application, and not the mathematical methods applied (e.g., someone specializing in modal logic may attend mainly metaphysics conferences, and therefore will not have regular opportunities to interact with researchers applying modal logics to epistemological problems).

One particular aim of the workshop will then be to provide the participants with a sense of what the range of topics, the state of current research, the interconnections, and the important trends are in philosophical logic and related areas. To this end, the organizers will invite selected participants to give survey talks about their area of expertise rather than specialized talks. These surveys will provide an overview of the development of the field in the last 20-50 years, of the current state of the art, of the main open problems, and of anticipated future trends and developments. The emphasis in all this will be on the connection to philosophy, although connections to mathematics, computer science, and other fields will of course be touched upon. The texts of these talks will be collected and published, most likely as a special issue of the Journal of Philosophical Logic.