MCMP – 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, non-flat, or ‘higher-order’ 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 higher-order rules can be equivalently replaced with flat ones. Building on our joint work with Prof. P. Schroeder-Heister, 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.

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).

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 non-logical terms. The most prominent problem of the interpretational definition is the problem of demarcating the logical from the non-logical 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 non-trivial, 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 non-logical terms, but the more general problem of demarcating the admissible from the inadmissible interpretations.

Torben Braüner (Roskilde) gives a talk at the MCMP Colloquium (17 January, 2013) titled "Hybrid-Logical Proof Theory: With an Application to False-Belief 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 proof-theory for ordinary modal logic. Many modal-logical 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 hybrid-logical proof-theory. 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 proof-theoretical 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 false-belief tasks in cognitive psychology.