LIRa Session: Marta Bílková

Speaker: Marta Bilkova

Date and Time: Thursday, April 8th 2021, 16:30-18:00, Amsterdam time.

Venue: online.

Title: Belief based on inconsistent information.

Abstract. When it comes to information, its potential incompleteness, uncertainty, and contradictoriness needs to be dealt with adequately. Separately, these characteristics have been taken into account by various appropriate logical formalisms and (classical) probability theory. While incompleteness and uncertainty are typically accommodated within one formalism, e.g. within various models of imprecise probability, contradictoriness and uncertainty less so — conflict or contradictoriness of information is rather chosen to be resolved than to be reasoned with. To reason with conflicting information, positive and negative support—evidence in favour and evidence against—a statement are quantified  separately in the semantics. This two-dimensionality gives rise to logics interpreted over twisted-product algebras or bi-lattices, e.g. the well known Belnap-Dunn logic of First Degree Entailment.

In this talk, we introduce many-valued paraconsistent logics for uncertainty which are interpreted over twisted-product algebras based on the [0,1] real interval. They can be seen to account for the two-dimensionality of positive and negative component of (the degree of) belief based on potentially contradictory information. The logics include extensions of Łukasiewicz or Gödel logic with a de-Morgan negation which swaps between the positive and negative component. The extensions of Gödel logic in particular turn out to be extensions of Nelson’s paraconsistent logic N4, or Wansing’s paraconsistent logic I_4C_4, with the prelinearity axiom. The logics inherit completeness and decidability properties of Łukasiewicz or Gödel logic respectively.

They can be applied to reason about belief based on evidence: In [1], a logical framework in which belief is based on potentially contradictory information obtained from multiple, possibly conflicting, sources and is of a probabilistic nature, has been suggested, using a two-layer modal logical framework to account for evidence and belief separately. The logics above are the logics used on the upper level in this framework. The lower level uses Belnap-Dunn logic to model evidence, and its probabilistic extension to give rise to a belief modality.

(Based on joint work with S. Frittella, D. Kozhemiachenko, O. Majer, and S. Nazari.)

[1] M. Bílková, S. Frittella, O. Majer and S. Nazari: Belief based on inconsistent information, DaLi 2020: Dynamic Logic. New Trends and Applications (M.A. Martins and I. Sedlar, editors), LNCS, vol. 12569, Springer, 2020, pp. 68–86.