# Bath-Bristol Workshop Crosses the Atlantic

24th July 2017

Bristol. Howard House, 4th floor seminar room.

Organised by Can Baskent. Contact Can or Catrin with any questions.

1400: Juliana Bueno-Soler - Popper's Conditional Probability, Negation and Consistency

1445: Walter Carnielli - Consistency, possibilistic, and necessitistic measures

1530: Break

1545: Rohit Parikh - An Epistemic Generalization of Rationalizability

1630: End

### Abstracts

Consistency, possibilistic, and necessitistic measures

Centre for Logic, Epistemology and the History of Science and

Department of Philosophy

State University of Campinas- Unicamp

Possibility and necessity theories rival with probability in representing uncertain knowledge, while offering a more qualitative view of uncertainty. Moreover, necessity and possibility measures constitute, respectively, lower and upper bounds for probability measures, with the advantage of avoiding the complications of the notion of probabilistic independence.

On the other hand, paraconsistent formal systems, especially the Logics of Formal Inconsistency, are capable of quite carefully expressing the circumstances of reasoning with contradictions. In connection to this, a calculus for reasoning with conflicting evidence, LETj, was proposed by (Carnielli & Rodrigues 2016). The aim of this talk is to merge these ideas, by precisely defining new notions of possibility and necessity theories involving the concept of consistency (inspired in (Besnard & Lang 1994)) based on the paraconsistent and paracomplete logic Cie, connecting them to the notion of partial and conclusive evidence. This will also work as an attempt to fill the gap on the formal evidence interpretation for the logic LETj. At the same time, this combination permits a whole treatment of contradictions, both local and global, including a gradual handling of the notion of contradiction, thus obtaining a really useful tool for AI and machine learning, with potential applications in logic programming via appropriate resolution rules.

**Popper's Conditional Probability, Negation and Consistency**

School of Technology

Limeira Campus

State University of Campinas â€“UNICAMP

Popper' s Conditional Probability, Negation and Consistency Juliana Bueno-Soler School of Technology Limeira Campus State University of Campinas â€“UNICAMP The concept of conditional probability is a formidable tool for describing the influence of one event on another and plays a fundamental role in the theory of probability. The usual Kolmogorovian definition based on a ratio is too restrictive and prone to paradoxes, even when dealing with probabilities based on non-classical logics.

Several axiomatizations of probability theory taking into account the concept of conditional probability as a primitive notion have been proposed in an attempt to save the key idea of conditional probability, K. Popper's proposal being the most successful. This paper proposes an extension of Popper's account founded on paraconsistency, by defining a wide notion of conditional probability based on the Logics of Formal Inconsistency. This leads to an interesting new theory of paraconsistent conditional probability which naturally reduces to the standard Popperian axiomatization when events are taken as consistent, and further reduces to ordinary probability theory.

An Epistemic Generalization of Rationalizability

Department of Computer and Information Science

Brooklyn College

CUNY

Rationalizability, originally proposed by Bernheim and Pearce, generalizes the notion of Nash equilibrium. Nash equilibrium requires common knowledge of strategies. Rationalizability only requires common knowledge of rationality. However, their original notion assumes that the payoffs are common knowledge.

We generalize the original notion of rationalizability to consider situations where agents do not know what world they are in, or where some know but others do not know. Agents who know something about the world can take advantage of their superior knowledge. It may also happen that both Ann and Bob know about the world but Ann does not know that Bob knows. How might they act?

We will show how a notion of rationalizability in the context of partial knowledge, represented by a Kripke structure, can be developed.