Validity and actuality

The notion of validity for modal languages could be defined in two slightly different ways. The first is the original definition given by S. Kripke, for which a formula Φ of a modal language L is valid if and only if it is true in every actual world of every interpretation of L. The second is the definition that has become standard in most textbook presentations of modal logic, for which a formula Φ of L is valid if and only if it is true in every world in every interpretation of L. For simple modal languages, “Kripkean validity” and “Textbook validity” are extensionally equivalent. According to E. Zalta, however, Textbook validity is an “incorrect” definition of validity, because: (i) it is not in full compliance with Tarski’s notion of truth; (ii) in expressively richer languages, enriched by the actuality operator, some obviously true formulas count as valid only if the Kripkean notion is used. The purpose of this paper is to show that (i) and (ii) are not good reasons to favor Kripkean validity over Textbook validity. On the one hand, I will claim that the difference between the two should rather be seen as the result of two different conceptions on how a modal logic should be built from a non-modal basis; on the other, I will show the advantages, for the question at issue, of seeing the actuality operator as belonging to the family of two-dimensional operators.