MCMP – Logic LudwigMaximiliansUniversität München

 Philosophy

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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HybridLogical Proof Theory: With an Application to FalseBelief Tasks
Torben Braüner (Roskilde) gives a talk at the MCMP Colloquium (17 January, 2013) titled "HybridLogical Proof Theory: With an Application to FalseBelief Tasks". Abstract: Hybrid logic is an extension of ordinary modal logic which allows explicit reference to individual points in a model (where the points represent times, possible worlds, states in a computer, or something else). This additional expressive power is useful for many applications, for example when reasoning about time one often wants to formulate a series of statements about what happens at specific times. There is little consensus about prooftheory for ordinary modal logic. Many modallogical proof systems lack important properties and the relationships between proof systems for different modal logics are often unclear. In my talk I will demonstrate that these deficiencies are remedied by hybridlogical prooftheory. In my talk I first give a brief introduction to hybrid logic and its origin in Arthur Prior's temporal logic. I then describe essential prooftheoretical results for natural deduction formulations of hybrid logic. Finally, I show how a proof system for hybrid logic can be used to formalize what are called falsebelief tasks in cognitive psychology.

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On flattening rules in natural deduction calculus for intuitionistic propositional logic
Grigory K. Olkhovikov (Ural Federal University Yekaterinburg) gives a talk at the MCMP Colloquium (25 April, 2013) titled "On flattening rules in natural deduction calculus for intuitionistic propositional logic". Abstract: Standard versions of natural deduction calculi consist of so called ‘flat’ rules that either discharge some formulas as their assumptions or discharge no assumptions at all. However, nonflat, or ‘higherorder’ rules discharging inferences rather than single formulas arise naturally within the realization of Lorenzen’s inversion principle in the framework of natural deduction. For the connectives which are taken as basic in the standard systems of propositional logic, these higherorder rules can be equivalently replaced with flat ones. Building on our joint work with Prof. P. SchroederHeister, we show that this is not the case with every connective of intuitionistic logic, the connective $c(A,B,C) = (A \to B)\vee C$ being our main counterexample. We also show that the dual question must be answered in the negative, too, that is to say, that existence of a system of flat elimination rules for a connective of intuitionistic logic does not guarantee existence of a system of flat introduction rules.

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Semantic games and hypersequents: a case study in many valued reasoning
Chris Fermüller (Vienna) gives a talk at the MCMP Colloquium (2 May, 2013) titled "Semantic games and hypersequents: a case study in many valued reasoning". Abstract: For a quite a while it had been an open problem whether there is an analytic (cutfree) calculus for infinite valued Lukasiewicz logic, one of threefundamental many valued logics that lie at the centre of interest in contemporary mathematical fuzzy logic. The hypersequent calculus HL presented by Metcalfe, Gabbay, and Olivetti in 2004/5 settled the question positively; but HL did not fit well into the family of sequent and hypersequent systems for related nonclassical logics. In particular it remained unclear in what sense HL provides an analysis of logical reasoning in a many valued context. On the other hand, already in the 1970s Robin Giles had shown that a straightforward dialogue game, combined with a specific way to calculate expected losses associated with bets on the results on `dispersive experiments' leads to a characterisation of Lukasiewizc logic. We illustrate how these seemingly unrelated results fit together: the logical rules of HL naturally emerge from a systematic search for winning strategies in Giles's game. This amounts to a rather tangible interpretation of hypersequents that can be extended to other logics as well.

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Modus Ponens on the Restrictor View
Moritz Schulz (Barcelona) gives a talk at the MCMP Colloquium (7 February, 2013) titled "Modus Ponens on the Restrictor View". Abstract: Recently, Kolodny & MacFarlane (2010) have proposed a new counterexample to modus ponens, which bears interesting relations to the classic counterexample by McGee (1985). By way of resolving the issue, I will see how the potential counterexamples can be analysed on the restrictor view of conditionals. The proposed resolution saves modus ponens by denying that the alleged counterexamples are proper instances of modus ponens. However, this solution raises the question whether there are any genuine instances of modus ponens on this view. To handle this problem, I will focus on the semantics of bare conditionals and develop a framework in which the validity of modus ponens can be addressed (and affirmed).

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From Logic to Behavior
Jakub Szymanik (Amsterdam) gives a talk at the MCMP Colloquium (12 June, 2013) titled "From Logic to Behavior". Abstract: In this talk I will explore the applicability of modern logic and computation theory in cognitive science. I will show how logic can be used to build cognitive models in order to explain and predict human behavior. I will also illustrate the use of logical and computational toolboxes to evaluate (not necessarily logical) cognitive models along the following dimensions: (i) logical relationships, such as essential incompatibility or essential identity; (ii) explanatory power; (iii) computational plausibility. I will argue that logic is a general tool suited for cognitive modeling, and its role in psychology need not be restricted to the psychology of reasoning. Taking Marr's distinctions seriously I will also discuss how logical studies can improve our understanding of cognition by proposing new methodological perspectives in psychology. I will illustrate my general claims with examples of the successful research on the intersection of logic and cognitive science. I will mostly talk about two research projects I have been recently involved in: computational semantics for generalized quantifiers in natural language and logical models for higher order social cognition. The major focus will be computational complexity and its interplay with "difficulty" as experienced by subjects in cognitive science.

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Interpretational Logical Truth: The Problem of Admissible Interpretations
Alexandra Zinke (Konstanz) gives a talk at the MCMP Colloquium (24 January, 2013) titled "Interpretational Logical Truth: The Problem of Admissible Interpretations". Abstract: According to the interpretational definition of logical truth a sentence is logically true iff it is true under all interpretations of the nonlogical terms. The most prominent problem of the interpretational definition is the problem of demarcating the logical from the nonlogical terms. I argue that it does not suffice to only exclude those interpretations from the admissible ones that reinterpret the logical constants. There are further restrictions on admissible interpretations we must impose in order to secure that there are at least some logical truths. Once it is seen that we must impose nontrivial, semantical restrictions on admissible interpretations anyway, the question arises why we should not also accept even further restrictions. I formulate restrictions which would lead to the consequence that all analytical sentences come out as logically true and argue that these restrictions are of the same character as those we already subscribe to. Imposing only some of the restrictions seems arbitrary. The real challenge for proponents of the interpretational definition is thus not just the problem of demarcating the logical from the nonlogical terms, but the more general problem of demarcating the admissible from the inadmissible interpretations.