MCMP – Mathematical Philosophy (Archive 2011/12) MCMP Team

 Society & Culture

Mathematical Philosophy  the application of logical and mathematical methods in philosophy  is about to experience a tremendous boom in various areas of philosophy. At the new Munich Center for Mathematical Philosophy, which is funded mostly by the German Alexander von Humboldt Foundation, philosophical research will be carried out mathematically, that is, by means of methods that are very close to those used by the scientists.
The purpose of doing philosophy in this way is not to reduce philosophy to mathematics or to natural science in any sense; rather mathematics is applied in order to derive philosophical conclusions from philosophical assumptions, just as in physics mathematical methods are used to derive physical predictions from physical laws.
Nor is the idea of mathematical philosophy to dismiss any of the ancient questions of philosophy as irrelevant or senseless: although modern mathematical philosophy owes a lot to the heritage of the Vienna and Berlin Circles of Logical Empiricism, unlike the Logical Empiricists most mathematical philosophers today are driven by the same traditional questions about truth, knowledge, rationality, the nature of objects, morality, and the like, which were driving the classical philosophers, and no area of traditional philosophy is taken to be intrinsically misguided or confused anymore. It is just that some of the traditional questions of philosophy can be made much clearer and much more precise in logicalmathematical terms, for some of these questions answers can be given by means of mathematical proofs or models, and on this basis new and more concrete philosophical questions emerge. This may then lead to philosophical progress, and ultimately that is the goal of the Center.

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Modality and Categories
Steve Awodey (CMU/MCMP) gives a talk at the MCMP Workshop on Modality titled "Modality and Categories".

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Adaptive Logics: Introduction, Applications, Computational Aspects and Recent Developments
Peter Verdée (Ghent) gives a talk at the MCMP Colloquium (8 Feb, 2012) titled "Adaptive Logics: Introduction, Applications, Computational Aspects and Recent Developments". Abstract: Peter Verd ́ee (peter.verdee@ugent.be) Centre for Logic and Philosophy of Science Ghent University, Belgium In this talk I give a thorough introduction to adaptive logics (cf. [1, 2, 3]). Adaptive logics are first devised by Diderik Batens and are now the main research area of the logicians in the Centre for Logic and Philosophy of Science in Ghent. First I explain the main purpose of adaptive logics: formalizing defea sible reasoning in a unified way aiming at a normative account of fallible rationality. I give an informal characterization of what we mean by the notion ‘defeasible reasoning’ and explain why it is useful and interesting to formalize this type of reasoning by means of logics. Then I present the technical machinery of the so called standard format of adaptive logics. The standard format is a general way to define adaptive logics from three basic variables. Most existing adaptive logics can be defined within this format. It immediately provides the logics with a dynamic proof theory, a selection semantics and a number of important metatheoretic properties. I proceed by giving some popular concrete examples of adaptive logics in standard form. I quickly introduce inconsistency adaptive logics, adap tive logics for induction and adaptive logics for reasoning with plausible knowledge/beliefs.
Next I present some computational results on adaptive logics. The adap tive consequence relation are in general rather complex (I proved that there are recursive premise sets such that their adaptive consequence sets are Π1 complex – cf. [4]). However, I argue that this does not harm the naturalistic aims of adaptive logics, given a specific view on the relation between actual reasoning and adaptive logics. Finally, two interesting recent developments are presented: (1) Lexi cographic adaptive logics. They fall outside of the scope of the standard format, but have similar properties and are able to handle prioritized infor mation. (2) Adaptive set theories. Such theories start form the unrestricted comprehension axiom scheme but are strong enough to serve as a foundation for an interesting part of classical mathematics, by treating the paradoxes in a novel, defeasible way. 
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Belief Dynamics under Iterated Revision: Cycles, Fixed Points and Truthtracking
Sonja Smets (University of Groningen) gives a talk at the MCMP Colloquium titled "Belief Dynamics under Iterated Revision: Cycles, Fixed Points and Truthtracking". Abstract: We investigate the longterm behavior of processes of learning by iterated beliefrevision with new truthful information. In the case of higherorder doxastic sentences, the iterated revision can even be induced by repeated learning of the same sentence (which conveys new truths at each stage by referring to the agent's own current beliefs at that stage). For a number of beliefrevision methods (conditioning, lexicographic revision and minimal revision), we investigate the conditions in which iterated belief revision with truthful information stabilizes: while the process of modelchanging by iterated conditioning always leads eventually to a fixed point (and hence all doxastic attitudes, including conditional beliefs, strong beliefs, and any form of "knowledge", eventually stabilize), this is not the case for other beliefrevision methods. We show that infinite revision cycles exist (even when the initial model is finite and even when in the case of repeated revision with one single true sentence), but we also give syntactic and semantic conditions ensuring that beliefs stabilize in the limit. Finally, we look at the issue of convergence to truth, giving both sufficient conditions ensuring that revision stabilizes on true beliefs, and (stronger) conditions ensuring that the process stabilizes on "full truth" (i.e. beliefs that are both true and complete). This talk is based on joint work with A. Baltag.

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Tracking the Truth Requires a Nonwellfounded Prior!
Alexandru Baltag (ILLC Amsterdam) gives a talk at the MCMP Colloquium titled "Tracking the Truth Requires a Nonwellfounded Prior! A Study in the Learning Power (and Limits) of Bayesian (and Qualitative) Update". Abstract: The talk is about tracking "full truth" in the limit by iterated belief updates. Unlike Sonja's talk (which focused on finite models), we now allow the initial model (and thus the initial set of epistemic possibilities) to be infinite. We compare the truthtracking power of various beliefrevision methods, including probabilistic conditioning (also known as Bayesian update) and some of its qualitative, "plausibilistic" analogues (conditioning, lexicographic revision, minimal revision). We focus in particular on the question on whether any of these methods is "universal" (i.e. as good at tracking the truth as any other learning method). We show that this is not the case, as long as we keep the standard probabilistic (or beliefrevision) setting. On the positive side, we show that if we consider appropriate generalizations of conditioning in a nonstandard, nonwellfounded setting, then universality is achieved for some (though not all) of these learning methods. In the qualitative case, this means that we need to allow the prior plausibility relation to be a nonwellfounded (though total) preorder. In the probabilistic case, this means moving to a generalized conditional probability setting, in which the family of "cores" (or "strong beliefs") may be nonwellfounded (when ordered by inclusion or logical entailament). As a consequence, neither the family of classical probability spaces, nor lexicographic probability spaces, and not even the family of all countably additive (conditional) probability spaces, are rich enough to make Bayesian conditioning "universal", from a Learning Theoretic point of view! This talk is based on joint work with Nina Gierasimczuk and Sonja Smets.

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Possible Worlds, The Lewis Principle, and the Myth of a Large Ontology
Ed Zalta (Stanford) gives a talk at the MCMP Workshop on Modality titled "Possible Worlds, The Lewis Principle, and the Myth of a Large Ontology".

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Accuracy, Chance, and the Principal Principle
Richard Pettigrew (University of Bristol) gives a talk at the MCMP Colloquium titled "Accuracy, Chance, and the Principal Principle"