### Algebraizable Non-normal Modal Logic.

Author: Carlos C. MartÃnez

December 97

#### Abstract::

This present work is concerned with examining modal logics under an algebraizability viewpoint, a definition introduced by W.J. Blok and D. Pigozzi in their seminal work Algebraizable Logics. Our main result has been to establish the independence of the notions of normal worlds and algebraizability of modal logics by providing an example of an algebraizable non-normal modal logic.

#### Definition:: (Algebraizable Logic)

A class K of algebras is called an algebraic semantics for a logic L if the consequence C can be interpreted, in a natural way, in the equational consequence K^{eq_\models} determined by K on the equational language corresponding to K. If conversely, K^{eq_\models} can be reconstructed on the basis of C, the class K is called an equivalent algebraic semantics. The logic L is algebraizable if it possesses an equivalent algebraic semantics.

#### Background::

A. Tarski in the early 30’s gave a formalization for the study of Logic, through an approach was based on two primitive notions: a formal language and a consequence operation giving an algebraic viewpoint to the subject. In the 70’s H. Rasiowa and R. Sikorski generalized this method to other implicative logics; further generalizations include work of W.J. Blok and D. Pigozzi.

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