Modal Foundations for Predicate Logic

The complexity of any logical modeling reeects both the intrinsic structure of a topic described and the weight of the formal tools. Some of this weight seems inherent in even the most basic logical systems. Notably, standard predicate logic is undecidable. In this paper, we investigatèlighter' versions of this general purpose tool, by modally`deconstructing' the usual semantics, and locating implicit choice points in its set up. The rst part sets out the interest of this program and the modal techniques employed, while the second part provides technical elaborations demonstrating its viability. 1 The modal core of predicate logic The well-known standard semantics for predicate logic has the following key clause: M; a j = 9xx ii for some d 2 jMj : M; x d j = : Tarsk's main innovation here was the use of assignments, which are essential in decomposing quantiied statements, which leave free variables in their matrix. But much less than this is needed to give a compositional semantics for rst-order quantiication. The abstract core pattern which would make the latter work is this: M; a j = 9xx ii for some : R x and M; j = : Here, `assignments' , become abstract states, and the concrete relation = x (which holds between and x d) has become just any binary update relation R x. Evidently, this abstract pattern involves standard poly-modal models, of the form M = (S; fR)xg x2VAR ; I) where S is a set of`states', R x a binary relation for each variable x, and I a `valuation' orìnterpretation function' giving a truth value to each atomic formula Px, Rxy,... in each state. In particular, existential quantiiers 9x become unary existential modalities hxi. This modal state semantics for predicate logic has an independent dynamic appeal: rst-order evaluation is an informational process which changes computational states. The rst-order language then becomes a dynamic logic, with a special choice of atoms and without explicit compound programs. From this modal point of view, conversely, `standard semantics' arises by insisting on three additional mathematical choices, not enforced by the new core semantics. 259 L. 260 Modal Foundations for Predicate Logic (1) States are identiied with variable assignments, (2) `update' must be the speciic relation = x , and (3) all assignments in the function space D VAR must be available to evaluation. The former are issues of implementation, the latter is a strong existence …