ABSTRACT
The label “two-dimensional” has been applied to disparate logical systems. The first section of Humberstone 2004 nicely conveys the diversity. For present purposes, we may understand an n-dimensional logic to be one in which sentences acquire truth values relative to n parameters. Thus classical propositional logic, which represents sentences as being true or false simpliciter, is zero-dimensional. Common modal or tense logics, which represent sentences as being true or false relative to a world or a time, are one-dimensional. Logics aiming to represent both tense and modality assign truth values relative to both time and world. (Thomason 1984 provides an overview.) They are two-dimensional. For other linguistic phenomena, other parameters have been employed: for personal pronouns, a speaker, an addressee and a salience order among some class of individuals; for here and there, a pair of locations; for more faithful representations of tense (in a remarkable anticipation of ideas discussed here, dating to Reichenbach 1947), a time of utterance and a time referred to. It may sometimes be appropriate to view some parameters as constructed from others. To represent a notion of historical necessity according to which the past is determined and the future is open, worlds may be seen as constructed out of instants and their temporal order. Proper understanding of actions, events, states, and the like may require that we distinguish among sentences evaluated at instants and intervals and to view one as constructed from the other. For other applications it may be appropriate to view the parameters as primitive and perhaps independent. To represent tense and metaphysical necessity, we may take time and world as independent parameters. Alternatively, we may take time to be something internal to each world, allowing for the possibility that it has different structures in different worlds. All such generalizations of standard modal and tense logics may be regarded as many-dimensional, the dimension corresponding to the number of parameters to which the truth of a formula is relativized.
