## Logic and Probability

SSLPS Annual Meeting

Friday, October 28, 2016

## Programme

10:15 - 11:00 | Zoran Ognjanovic | An Introduction to Logics with Probability Operators |

Coffee Break | ||

11:30-12:15 | Ioannis Kokkinis | On the Expected Execution Length of some Randomized Gossip Protocols |

Lunch | ||

14:00 - 14:45 | Sebastiaan Terwijn | Probability Logic |

14:45 - 15:30 | Dragan Doder | Probability in Abstract Argumentation |

Coffee Break | ||

16:00 - 16:45 | Matthias Unterhuber | Convergence Results for Probabilistic and Modal Logics (for Conditionals) with a Connexive Twist |

## Venue

The meeting takes place in Room B5 in the building Exakte Wissenschaften (ExWi) of the University of Bern.Address: Exakte Wissenschaften, Sidlerstrasse 5, 3012 Bern, Switzerland

## Abstracts

#### Zoran Ognjanovic

An Introduction to Logics with Probability Operators

The problems of representing, and working with, uncertain knowledge are ancient problems dating, at least, from Leibnitz. In the last decades there is a growing interest in the field connected with applications to computer science and artificial intelligence. Researchers from those areas have studied uncertain reasoning using different methods. Some of the proposed formalisms for handling uncertain knowledge are based on logics with probability operators. The aim of this presentation is to provide an introduction to such formal systems. The main focus is related to mathematical techniques for infinitary probability logics used to obtain results about proof-theoretical and model-theoretical issues: axiomatizations, completeness, compactness, decidability.

#### Ioannis Kokkinis

On the Expected Execution Length of some Randomized Gossip Protocols

A well-studied phenomenon in network theory is information dissemination by random peer to peer communications. This phenomenon is for example observed in the spreading of epidemics, in censor networks or even in informal chatting between friends. The mathematical objects that are used to model this situation are called gossip or epidemic protocols. In this talk we present some randomized gossip protocols and we show how bounds for their expected execution length can be calculated. This a joint work in progress with Hans van Ditmarsch, Aris Pagourtzis and Tomasz Radzik.

#### Sebastiaan Terwijn

Probability Logic

Probability logic may refer to any kind of combination
of logic and probability, of which there are many, ranging from
philosophy to computer science. In this talk we will survey various
kinds of probability logic, and also discuss some recent work in this
area.

In the context of mathematical logic, the literature may be divided
roughly into two parts. First, there is what may be called the
``probabilities over models'' approach,
where probabilities are imported by considering probability
distributions over classes of models. An example of this are the
various probabilistic logics used in model checking.
Second, there is the ``models with probabilities'' approach, where
the probabilities are internal to the models under consideration.
In this talk we will focus on the second approach, though, as we
will discuss, it turns out that the two approaches are related.
We will discuss various logics falling under the
``models with probabilities'' heading, including logics with
probability quantifiers introduced by Keisler, Valiant, Terwijn,
and Goldbring and Towsner. Though the motivation for studying these
logics is rather different, there are interesting connections,
both technical and conceptual.

#### Dragan Doder

Probability in Abstract Argumentation

Abstract argumentation is nowadays a vivid field within artificial intelligence and has seen different developments recently. The simplest objects used in abstract argumentation are Dung's argumentation frameworks (AFs). They are just directed graphs where vertices represent the arguments and edges indicate a certain conflict between the two connected arguments. The goal is to identify jointly acceptable sets of arguments for which a large selection of different semantics is available. Apart from relations going beyond binary attack, AFs do not handle varying levels of uncertainty This calls for augmenting AFs with probabilities and due to various interpretations of what the probability of an argument or relation is and how it should be used, different methods have been proposed. In this talk, I give an overview of probabilistic argumentation frameworks and present a method of encoding their semantics in probabilistic logic.

#### Matthias Unterhuber

Convergence Results for Probabilistic and Modal Logics (for Conditionals) with a Connexive Twist

In recent decades two systems – Systems P and R – have been found to characterize probabilistic and modal logics for conditionals as well as non-monotonic logics. This paper firstly extends this convergence result by proving that Lewis's System V for counterfactual conditionals and Burgess variant System V* are nothing but System R. Secondly, it is shown that often-ignored connexive principles are at a center stage of the axiomatization of both systems. They introduce a proof-theoretic dependency of two core principles of Systems P and R – Cautious Monotonicity and Rational Monotonicity – even when the connexive principles are formulated as default rules. Thirdly, the impossibility result for connexive principles is strengthened. It is shown that on pain of inconsistency classical conditional logics – conditional logics that allow logic to govern logical connectives other than conditionals (including probabilistic logics) – cannot validate connexive principles unrestrictedly. Implications of the present results are discussed.