Tutorials in mathematical logic:

**Algebraic modal correspondence - **Alessandra **Palmigiano** (University of Amsterdam)

Sahlqvist correspondence theory is among the most celebrated and useful results of the classical theory of modal logic, and one of the hallmarks of its success. Traditionally developed in a model-theoretic setting (cf. [3], [4]), it provides an algorithmic, syntactic identication of a class of modal formulas whose associated normal modal logics are strongly complete with respect to elementary (i.e. first-order definable) classes of frames.

Sahlqvist's results can equivalently be reformulated algebraically, via the well known duality between frames and complete atomic Boolean algebras with operators (BAO's). This perspective immediately suggests generalizations of Sahlqvist's theorem along algebraic lines, e.g. to the cases of distributive [2] or arbitrary lattices with operators.

We illustrate, by way of examples, the algebraic mechanisms underlying Sahlqvist correspondence for classical modal logic [1], after having discussed the appropriate duality with the relational semantics. We show how these mechanisms work in much greater generality than the classical setting in which Sahlqvist theory was originally developed. Next, we present the newly developed algorithm ALBA [2] which effectively extends the existing most general results on correspondence.

1. W. Conradie, A. Palmigiano, S. Sourabh, Algebraic Sahlqvist Correspondence, in preparation.

2. W. Conradie and A. Palmigiano, Algorithmic Correspondence and Canonicity for Distributive Modal Logic, Annals of Pure and Applied Logic (2011) DOI: 10.1016/j.apal.2011.10.004.

3. H. Sahlqvist, Correspondence and completeness in the first and second-order semantics for modal logic, in Proceedings of the 3rd Scandinavian Logic Symposium, Uppsala 1973, S. Kanger, ed., 1975, pp. 110-143

4. J. van Benthem, Modal logic and classical logic, Bibliopolis, Napoli, 1983.

**Phase transitions for Gödel incompleteness - Andreas Weiermann (Ghent University)**

Kurt Gödel showed that there are assertions A about the natural numbers such that neither A nor its negation follow from the Peano Axioms. We are interested in parameterized versions of such assertions which undergo a phase transition from being provable to being independent from the Peano axioms when the parameter passes a threshold point.

We survey a selection of beautiful results in the area and we will explain the underlying rationale in proving them. The proof technique is based on proof theory, Ramsey theory and real analysis but instead of presenting technical details we will try to explain how to think about such phase transition results. The field is of course open for interested newcomers.

Tutorials in philosophical logic:

**Temporal logics for reasoning about computations in transition systems - Valentin Goranko (Technical University of Denmark)**

This tutorial is about how modal and temporal logics can be used to specify and reason about important properties of discrete transition systems and computations in them. I will introduce and discuss the linear time temporal logic LTL and the branching time temporal logic CTL*. Time permitting, at the end of the tutorial I will also introduce and discuss briefly the alternating time temporal logic ATL for reasoning about multi-player concurrent game models.

**Generalized truth values*** - ***Heinrich Wansing (Ruhr University, Bochum)**

This tutorial will consist of two parts. The first part will introduce generalized truth values understood as subsets of a set of given semantical values, starting from the set of the classical Fregean truth values "The true" and "The false". In particular, the step from the four-element set of truth values of Belnap und Dunn's useful four-valued logic to its sixteen-element powerset is motivated. The resulting algebraic structure is the so-called trilattice SIXTEEN_3. The logic of this trilattice is presented semantically. Interestingly, the logical vocabulary splits up into a truth vocabulary and a falsity vocabulary. Moreover, a truth entailment relation can be distinguished from a falsity entailment relation.

In the second part of the tutorial, the problem of axiomatizing logics related to SIXTEEN_3 is discussed. Moreover, sound and complete sequent calculi for truth entailment and falsity entailment are defined.

The material presented in the tutorial is taken from Y. Shramko and H. Wansing, Truth and Falsehood. An Inquiry into Generalized Logical Values, Trends in Logic Vol. 36, Springer-Verlag, Dordrecht, 2011.