Algebraic Semantics for modal logics

Por Samir Gorsky

In XX century we had a considerable advance on the understanding of the formal meaning of modalities. The Jónsson, McKinsey and Tarski works in fourties enabled the construction of the results of algebraic completeness for the modal systems. In fifties Kripke proposed a interesting semantic for these systems. Such semantics, today known as possible world's semantics, or Kripke's semantics, caused a great impact in the context of analytical philosophy. Articles written by Lemmon in the decade of 60 are supposed to present a synthesis of these two semantics, the algebraic semantic and the possible world's semantic. One interesting result shown in these articles is that the semantic completeness can be inferred from algebraic results through a central theorem. One of the most surprising and interesting results in the paper of Lemmon is the theorem of representation for modal algebras. This theorem of representation for the modal algebra is as a result the connection between the point of view and algebraic point of view of the semantics of possible worlds (or Kripke's semantics). The initial objective of the present work was to extend this same result for algebraic systems of Class Gmnpq proposed by Lemmon and Scott in the ``Lemmon notes''. We argue that the algebraic semantic for modal logic can serve as a basis for answers to the various criticisms directed to the development of modal logic. We'll show, finally, that the algebraic semantics, as a semantics that does not use the concept of possible worlds, may be deemed useful by supporters of modal antirealism.

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